Earthquake Vulnerability of Free Standing Monolithic Statues

European Journal of Scientific Research ISSN 1450-216X Vol.44 No.2 (2010), pp.355-373 © EuroJournals Publishing, Inc. 2010 http://www.eurojournals.com/ejsr.htm

Earthquake Vulnerability of Free Standing Monolithic Statues Dinesh Yadav Earthquake Engineering Research Centre, IIIT Hyderabad Puneet Girdhar Earthquake Engineering Research Centre, IIIT Hyderabad Vinay Kumar Earthquake Engineering Research Centre, IIIT Hyderabad Ramancharla Pradeep Kumar Earthquake Engineering Research Centre, IIIT Hyderabad E-mail: [email protected] Abstract Heritage structures bears a valuable testimony of ancient cultural heritage of any country. Among these structures statues, iconic emblems plays significant role. In the proposed to study the vulnerability of monolithic statues to earthquake ground motions. Initially we used the equations developed by Housner for studying the dynamics of rigid blocks. Parametric study was conducted to study the inuence of size of the block and input parameters such as amplitude and frequency on the behaviour of the block. The study was extended to generate safe surface for the block of given dimensions. This surface gives the information that with any combination of parameters such as amplitude of vibration, frequency of vibration and duration of vibration, if the point lies outside the surface then block overturns otherwise it is safe. To verify the above formulation we selected four statues, viz, Gomateswara statue, Christ the Redeemer, Statue of Liberty, Guinyin Statue for case study. Initially all the statues were idealized like rigid blocks and later safe surfaces were developed for each statue. Since each statue is having different dimensions, surfaces were are also of different nature. Later, eighteen earthquake records were selected and strong ground motion parameters were estimated. Finally every statue was tested for the safety during the earthquake events. It was found that as the height of the statue is increasing its vulnerability is increasing. However, for all the given ground motions, none of the statue overturned. Keywords: Monolithic statues, vulnerability, earthquake ground motion.

356

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar

Introduction When we look at the cultural heritage of any country in broad sense, construction activity of the time plays a significant role. It is because these structures bears a valuable testimony of ancient times and which deserves to be preserved. Structures constructed during the past, ranging from ancient times up to 20th century, can be categorized into two groups, viz historical constructions and monumental constructions. Historical buildings are buildings which have in themselves a cultural value as a whole, while the single building cannot be said as a monument. Where as monumental buildings are constructions having important cultural values. Among monumental structures, towers, statues, iconic emblems are of great importance besides places of religion. In the past, damage and decay of monumental structures have been studied using insitu surveys, measurements of dynamic characteristics using micro-tremors or ambient vibration tests, determination of dimensions and material characteristics using infrared and ultrasonic measurements, engineering analysis by modeling the structures and safety assessments (Sever, 2003). However these structures deserve more detailed studies using advanced computer simulations, together with adequate material and structural characterization (Luranco 2002). In the present study, we are focusing on vulnerability of free standing monolithic statues which are particularly vulnerable to dynamic actions from the point of view of prevailing seismic hazard in the country. In last 2 decades, 7 moderate earthquakes have been witnessed: Bihar-Nepal border (M6.4) in 1988, Uttarkashi (M6.6) in 1991, Killari (M6.3) in 1993, Jabalpur (M6.0) in 1997, Chamoli (M6.8) in 1999, Bhuj (M6.9) in 2001, Sumatra (M9.3) in 2004, and Muzzafarabad (M7.2) in 2005 (Jain 2006). Also, as per seismic zonation map given in the latest revision of IS 1893, 60% of the country may experience moderate to severe earthquake. In this regard, a study is conducted to understand earthquake vulnerability of free standing monolithic blocks to earthquake excitations. For the purpose of the study, we used the well known equations on rocking (Housner 1963, Yim et al. 1980 and Makris and Rousos 2000) on the dynamic response of rigid blocks supported on a base and subjected to horizontal acceleration.

Problem Statement Consider a rectangular block with height , breadth and length as shown in figure 1. When block block overturns. When the block is is at rest, the line makes an angle α with vertical and when given initial rotational displacement , the weight of the block will exert a restoring moment , and oscillates about the centers of rotation and . As necessary condition for rocking the block, coeffcient of friction

. So it is assumed that there is sufficient friction

between ground and block so that it enters into rocking without slipping on the ground when subjected to excitation. When the block is rotated, the equation of motion:

Earthquake Vulnerability of Free Standing Monolithic Statues

357

Figure 1: Schematic diagram of the block

where

is mass moment of inertia, is weight of the block and Let us consider a block whose height , breadth and length . Let . From the figure (see Figure 2), we can the block set to oscillate by giving initial rotation notice that the amplitude of vibration and time period of oscillation are reducing after every impact with the ground. This is happening due to loss of energy during impact. If we look at the equation of conservation of momentum about point o0 just before the impact and immediately after the impact: Figure 2: Decay of oscillation due to loss of energy during impact

358

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar

are angular velocities before and after the impact. The ratio of the kinetic energy after where and and before the impact is , which means that the angular velocity after impact is only times the velocity before the impact. From the above, we get

the value of coefficient of restitution given by the above equation is the maximum value of which a block with slenderness will undergo rocking motion.

under

Oscillations Due to Ground Motion The equations that govern the rocking motion under a horizontal ground acceleration

are (4)

(5) The above equations are solved by using finite difference technique and a MATLAB program is written for the purpose of parametric study.

Parametric Study For the purpose of parameteric study following four parameters are selected: Frequency of Input ground Motion Figure 3 describes the variation of ratio versus time for different excitation frequencies. Here we have chosen three different frequencies , and . The amplitude of ground acceleration is taken as . Input ground motion is applied upto and later block is allowed to oscillate block is getting freely. When block is subjected to ground motion, it starts oscillation and till input energy. After every impact block is loosing kinetic energy. As the frequency of input motion is increasing the rate at which the energy is dissipating is also increasing. From the figure we can clearly notice that response due to excitation at higher frequencies is decaying at faster rate that excitation due to lower frequencies. Figure 3: θ/α variation wrt time for different frequencies

Earthquake Vulnerability of Free Standing Monolithic Statues

359

Amplitude of Input Ground Motion Figure 4 shows the variation of ratio

versus time for different excitation amplitude. Here we have

, and . From the figure it is obvious that higher chosen three different amplitudes the amplitude more is the response. From the figure it is quite clear that increasing the amplitude of the ground motion, probability of overturning the block increases. Hence for the line corresponding amplitude shows that the block overturns before [As it is explained in Housners paper, when becomes greater than the value of slenderness of the block then it overturns]. Other block keeps their vibration continuously decreasing as ground acceleration is cut off at . Now it vibrates as free body and after some time it comes to rest. Figure 4: θ/α variation wrt time for different amplitudes

Height of the Block Figure 5 shows the variation of ratio versus time for different block heights. Here we have chosen three different block heights , and . Now looking at the variation of height of the block, its slenderness decreases while increasing the height. Hence due to decrease in slenderness, block is required very small value of , so that it becomes equal to slenderness and will overturn. From , its slenderness becomes . So now it has more chances to overturn. the graph at Therefore at very small ground motion with and , it overturns at . Figure 5: θ/α variation wrt time for different heights

360

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar

Length of the block Figure 6 shows the variation of ratio

versus time for different block lengths. Here we have chosen

three different block heights , and . In the similar way, increasing the length of the block, its slenderness increases So it becomes safer condition for blocks to not get overturn. As already mentioned that overturning of the rocks depend upon the slenderness ( = atan(l/h)). Generally mass does not come into play unless or until if rock is more complex in structure or hollow from inside. It is also clear from the graph that increasing the value of L, stability of the block increases. In the plot, when , it overturns, But when increases its slenderness also increases. Also as slenderness increases the time to come into rest also decreases, thats why when , block comes to rest at . From the above parametric study, we can easily conclude that for a given block size there is a combination of parameters of ground motion which makes the block unstable. And the same can be visualized as the surface which is passing through limiting values of parameters i.e., amplitude, predominant frequency, duration of strong ground motion. The above knowledge is extended to _nd the vulnerability of monolithic statues to earthquake ground motion. The same has been illustrated in the case studies below. Figure 6: θ/α variation wrt time for different lengths

Case Studies Selection of Monolithic Statues For the purpose of studying the affect of vibration on the monolithic statues, following statues were selected: Lord Gomateswara It is a monolithic statue standing at above a hill in a place called Shravanabelagola in the Hassan district of Karnataka state, India. The statue was built by the Ganga minister and Commander Chamundaraya in honour of Lord Bahuabali. It was built in the 10th century AD and is the size Jains believe humans used to be. Hundreds of thousands of pilgrims devotees and tourist from all over the world ock to statue once in 12 years for an event known as Mahamastakabhisheka. (See Figure 7)

Earthquake Vulnerability of Free Standing Monolithic Statues

361

Figure 7: Lord Gomateswara statue

For the case study the above statue is idealized as rectangular block of size length and height and mass .

, width

Christ the Redeemer It is a statue of Jesus Christ in Rio de Janeiro, Brazil. The statue stands 38 meters tall weigh 700 tons and is located at the peak of 700 meters Corcovado Mountain in the Tijuca Forest National park overlooking the city. It is made of reinforced concrete and soapstone. Its construction took nine years from 1922 to 1931. On 7th July, 2007 Christ the Redeemer was named one of the New Seven Wonders of the World in a list compiled by Swiss based The New open World Corporation. (See Figure 8)

362

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar Figure 8: Christ the Redeemer

For the case study the above statue is idealized as rectangular block of size length and height and mass . width

,

Statue of Liberty More formally it is called Liberty Enlightening the World. It was presented to the United states by the people of France in 1886. Standing on Liberty Island in New York Horbor, it welcomes visitors, immigrants, and returning Americans travelling by ship. The copperclad statue, dedicated on October 28, 1886, commemorates the centennial of the signing of the United States Declaration of Independence and was given to the United States to represent the friendship established during the American Revolution. The statue is of a robed woman holding a torch, and is made of a sheeting of pure copper, hung on a framework of steel with the exception of the ame of the torch, which is coated in gold leaf. It stands atop a rectangular stonework pedestal with a foundation in the shape of an irregular eleven-pointed star. The statue is tall, but with the pedestal and foundation it is tall. (See Figure 9)

Earthquake Vulnerability of Free Standing Monolithic Statues

363

Figure 9: Statue of Liberty

For the case study the above statue is idealized as rectangular block of size length and height and mass . width

,

Guanyin Statue The Guanyin Statue of Hainan, also known as Guan Yin of the South Sea of Sanya, is a 108 meter tall statue of the bodhisattva Guan Yin, sited on the south coast of Chinas island province Hainan in the Nanshan Culture Tourism Zone near the Nanshan Temple west of Sanya. The statue has three aspects; one side faces inland and the other side two face the South China Sea, to represent blessing and protection by Guan Yin of China and the whole world. One aspect depicts Guan Yin cradling a sutra in the left hend and gesturing the Vitarka Mudra with the right, the second with her palms crossed, holding a string of prayer beads, and the third holding a lotus. The manta Om mani padme hum is written in Tibetan script around each aspects halo. This is currently the fourth tallest statue in the world. (See Figure 10)

364

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar Figure 10: Guanyin Statue

For the case study the above statue is idealized as rectangular block of size length and height and mass . width

,

Selection of Earthquake Records For the purpose of testing the vulnerability of statues to earthquake ground motion, we selected 16 records. For each record following characteristics were estimated: (figures 11 and 12)

Earthquake Vulnerability of Free Standing Monolithic Statues Figure 11: Earthquake records considered in the study

365

366

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar Figure 12: Earthquake records considered in the study

Amplitude of Ground Motion The most commonly used measure of the amplitude of a ground motion is its peak ground acceleration (PGA) in horizontal direction. PGA for given component of motion is simply the largest (absolute value of horizontal acceleration obtained from the accelerogram of that component. Column 4 of table 2 gives PGA for ground motions.

Earthquake Vulnerability of Free Standing Monolithic Statues

367

Duration of Ground Motion It is important because the extent of cumulative damage incurred by the structures (in case of elastic limit being exceeded) increases with the number of loading cycles. Duration of strong ground motion proposed by Trifunac and Brady (1975) was adopted. According to them it is the interval in which a signi_cant contribution takes place to the integral. (6)  where

is acceleration. column 5 of table 2 shows duration of strong ground motion.

Frequency content For finding out predominant frequency, time history of ground motion is converted into frequency domain using FFT. Later the frequency to which amplitude is largest is selected as predominant frequency. Column 6 of table gives predominant frequency of ground motions.

Results and Discussion Vulnerability Assesment of Gomatswara statue Figures 13 shows the variation of ratio, versus time for different frequencies . Statue is oscillating at all the three frequencies, however, at frequencies near

,

and its

amplitude of vibration is higher than at other frequencies. Figures 14 shows the variation of ratio, versus dirrerent amplitudes of excitation, , and . It can be seen from the figure that the statue is becoming unstable under excitation amplitude of 0:66g just after 4sec of excitation. We can use the above information and evaluate the extreme points at which statue becomes unstable. Later an approximate surface can be fitted passing through these points. This surface shows the safety of the statue for given vibration characteristics (see Figure 15). Figure 13: θ/α variation wrt time for different frequencies

368

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar Figure 14: θ/α variation wrt time for different amplitudes

Figure 15: Surface showing the safety of statue during vibration

Table 3 column 1 describes that none of the selected records can shake the statue upto collapse. It means that all the points are lying well within the surface generated. Vulnerability Assesment of Christ the Redeemer Figure 16 shows the variation of ratio,

versus time for different frequencies

and

amplitude of vibration is choosen as A = 0:95g. As already explained earlier that increase in the angular frequency, decreases the time period hence time for the block to move from one center of rotation to other is less. This does not provide su_cient amount of energy to get overturn. From the , it gets overturn. Similarly figure 17 shows the variation of figure 16 we can see that when ratio, versus time for different amplitudes A = 0:22g; 0:23gand0:24g. As acceleration increase, the magnitude of displacement increases for a given statue. From the Figure we can observe that at amplitude A = 0:24g statue gets overturn at t = 2:79sec. Following the procedure described as in figure 15 we can generate surface describing the safety of statue. Figure 18 describes the surface for Christ the

Earthquake Vulnerability of Free Standing Monolithic Statues

369

redeemer. Compared to Gomateswara statue it overturns at lower values of amplitude and frequency level of vibration. The reason being the height, which is almost equal to twice of that of statue of Gomateswara. Figure 16: θ/α variation wrt time for different frequencies

Figure 17: θ/α variation wrt time for different amplitude

370

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar Figure 18: Surface showing the safety of statue during vibration

Table 3 column 2 describes that none of the selected records can shake the statue upto collapse. It means that all the points are lying well within the surface generated. Vulnerability Assesment of Statue of Liberty Figure 19 shows the variation of ratio, versus time for di_erent frequencies and0:43 . Amplitude of vibration is choosen as A = 0:60g. Here also we can observe that increase in the angular frequency, decreases the time period hence time for the block to move from one center of rotation to other is less. From the _gure 19 we can see that when ! = 0:14_, it gets overturn. Similarly figure 20 shows the variation of ratio, versus time for different amplitudes A = 0:40g; 0:50gand0:60g. As acceleration increase, the magnitude of displacement increases for a given statue. From the figure we can observe that at amplitude A = 0:60g statue gets overturn at t = 2:9sec. Following the procedure described as in figure 15 we can generate surface describing the safety of statue. Figure 21 describes the surface for statue of Liberty. Compared to safety surface of Gomateswara statue and also with the safety surface of Christ the Redeemer, it overturns at lower values of amplitude but higher values of frequency level of vibration. The reason being the height, which is more compared to both the statues. Figure 19: θ/α variation wrt time for different frequencies

Earthquake Vulnerability of Free Standing Monolithic Statues

371

Figure 20: θ/α variation wrt time for different amplitudes

Figure 21: Surface showing the safety of statue during vibration

Table 3 column 3 describes that none of the selected records can shake the statue upto collapse. It means that all the points are lying well within the surface generated. Vulnerability Assesment of Guanyin Statue 0:50 Figure 22 shows the variation of ratio , versus time for di_erent frequencies and0:60 . Amplitude of vibration is choosen as A = 0:95g. Here again we can observe that increase in the angular frequency, decreases the time period hence time for the block to move from one center of rotation to other is less. From the _gure 22 we can see that when ! = 0:40_, it gets overturn. Similarly figure 23 shows the variation of ratio, versus time for different amplitudes A = 0:95g; 1:05gand1:15g. As acceleration increase, the magnitude of displacement increases for a given statue. From the _gure we can observe that at amplitude A = 0:95g statue gets overturn at t = 2:2sec. Following the procedure described as in figure 15 we generated surface describing the safety of statue. Figure 24 describes the surface for statue of Guanyin. Compared to safety surface of earlier three statues, it overturns at much

372

Dinesh Yadav, Puneet Girdhar, Vinay Kumar and Ramancharla Pradeep Kumar

lower values of amplitude and frequency level of vibration. The reason being the height, which is significantly more compared to all the three statues. Figure 22: θ/α variation wrt time for different frequencies

Figure 23: θ/α variation wrt time for different amplitudes

Earthquake Vulnerability of Free Standing Monolithic Statues

373

Figure 24: Surface showing the safety of statue during vibration

Table 3 column 4 describes that none of the selected records can shake the statue upto collapse. It means that all the points are lying well within the surface generated.

Conclusions A study has been conducted for finding the vulnerability of long standing monolithic statues for earthquake ground shaking. For the purpose of study first we studied the inuence of parameters of the ground motion on the behaviour of the rigid blocks. Later four statues were selected and were idealized like rigid blocks. Surfaces were generated for each statue. A set of earthquakes were identified and strong ground parameters were calculated for each earthquake. Finally safety of each statue for the given earthquake was estimated. It was found the as height of the statue is increasing its vulnerability was increasing however, none of the statue was beyond the safety limits.

References [1] [2] [3] [4] [5] [6] [7]

Luranco, P. (2002): Computations of historical masonry constructions, Progress in Structural Engineering and Materials, 4(3), 301-319, 2002. Sever, G (2003): Use of advanced techniques to asist heritage conservation decisions in seismic zones, Fifth national conference on earthquake engineering, 26-30 May, Istanbul, Turkey. Jain (2006): Housner, G.W. (1963): The behaviour of inverted pendulum structures during earthquake, BSSA, 53(2). Yim, S.C.S, Chopra, A.K., Penzien, J. (1980): Rocking response of rigid blocks to earthquakes, Earthquake Engineering and Structural Dynamics, 8(6). Makris, N. and Roussos, Y. (2000): Rocking response of rigid blocks under nearsource ground motions, Geotechnique 50(3). Trifunac, M.D. and Brady, A.G. (1975): A study on the duration of strong earthquake ground motion, BSSA, 65, 581-626.