Ground Response of Ahmedabad City during Bhuj Earthquake: A Case Study   Jyant Kumar Associate Professor, Civil Engineering Department, Indian Institute of Science, Bangalore jkumar@civil.iisc.ernet.in

ABSTRACT

The seismogram recorded at Ahmedabad city during the occurrence of Bhuj earthquake was analyzed to compute the response of ground mass up to a depth of 10 meters. The horizontal acceleration and shear stress time histories of the ground mass were computed at various depths. It was seen that the amplification of the earthquake base motion was quite predominant in the upper 4 m thickness of the soil mass near the ground surface. Standard penetration conducted at three different locations, around the area where a majority of the building failures were experienced, revealed the presence of loose sand strata in a soil zone over which most of the building foundations were provided. The failure of the building foundations was anticipated due to (i) the placement of the building foundations at very shallow depths over loose alluvial sand deposits, and (ii) the significant amplification of the earthquake motion in the upper 4 m thickness of the soil mass.

Keywords: damping; earthquakes; elasticity; liquefaction; vibrations.

INTRODUCTION

The 2001 Bhuj earthquake (M_w 7.7) was one of the largest seismic events of its kind in India. The epicenter of this earthquake was located at 23.4^oN-70.28^oE, 50 km northeast of the town Bhuj in Gujrat state (Karnath et al., 2001; Thakkar and Goyal 2004). The focus of this earthquake was centered at a depth of 22 km as reported by US Geological Survey. The greatest damage due to this earthquake occurred in the region of Kachchh which is spread over an area of about 45930 km2 and covers roughly about 22% of the area of Gujrat state. Several cities and towns in Kachchh including Bhuj, Bhachau, Rapar, Anjar and Ghadhidham, experienced extensive destruction. A detailed documentation of the liquefaction related features around the epicenter was reported by Tuttle et al. (2002). The earthquake caused serious damage in other parts of the state as well, including the major city Ahmedabad located at a distance of approximately 300 km east of Bhuj. The nearest recorded seismogram for the Bhuj earthquake was available at Ahmedabad. The town of Ahmedabad generally lies on alluvial sandy deposits. The Sabarmati river runs in this city from north to south. A large number of reinforced concrete frame buildings, ranging from three to twelve storied, located in Ahmedabad suffered extensive damage or collapsed completely during the Bhuj earthquake (Humar et al., 2001). Most of these buildings were founded on either isolated or combined footings located generally at depths ranging from 1.5 m to about 3.0 m. It was mentioned by Humar et al. (2001) that the possible reason of the damage to most of these buildings could be attributed (without carrying out any theoretical analysis) due to the amplification of the earthquake motion near the ground surface as most of the buildings were founded on deep loose sediments deposited by the Sabarmati river. Due to very low level of ground water table (20m-50m) in Ahmedabad city, hardly any case of liquefaction was observed in the city of Ahmedabad (Tuttle et al., 2002). No theoretical study seems to have been made so far where the seismogram recorded at Ahmedabad city was utilized to examine the response of ground mass. This was the objective of the current work. In the present investigation, in order to carry out the ground exploration, standard penetration tests were carried out at three different chosen locations upto a maximum depth of 10 meters. By assuming the ground mass to comprise of a visco-linear elastic material, the response of the ground mass was then determined at different depths with the help of the recorded seismogram at ground surface in Ahmedabad. The variation of the acceleration and the shear stress developed at different depths with time was then determined in order to interpret the amplification of the earthquake motion towards the ground surface. A comparison of the obtained results was also made with the studies reported in literature.

SITE INVESTIGATIONS

Three different sites were selected in Ahmedabad town for carrying out the standard penetration tests upto a maximum depth of 10 m. These selected sites were chosen at a maximum distance of about 1.5 km from the west bank of the river Sabarmati. Near the chosen sites 1 and 2, many apartments were either collapsed or subjected to an extensive damage during the occurrence of the Bhuj earthquake; these two sites were located approximately at a distance of 1-1.5 km from the Vasna barrage on the river. The third site was chosen near to Abhay Ghat. Standard penetration tests results, after applying the corrections due to overburden and energy level (70%), are presented in Fig. 1.

Figure 1. Variation of N value with depth for three different chosen sites in Ahmedabad

Disturbed soil samples collected at different locations were also examined for finding the percentage of fines passing through sieve size of 75 mm. The variation of the percentage of fines with depth has also been provided in Fig. 2. It can be noted that for soil deposits at a depth ranging from 2m to 6m, the values of N were found to be generally smaller than 14 and the corresponding percentage of fines were found to be generally less than about 18%. These field tests, therefore, indicate that the ground mass at these sites comprises of mostly loose sandy material at a depth ranging from about 2m to 6m from the ground surface.

Figure 2. Percentage of fines with depth for three different chosen sites in Ahmedabad

RESPONSE OF GROUND FOR THE SEISMOGRAM RECORDED AT AHMEDABAD

The study was carried out by extending the previous analysis carried out by the author (Kumar 2005) for determining the response of ground mass subjected to an harmonic excitation; the analysis takes into account the variation of the shear modulus, damping ratio and density of the material with depth. The analysis deals with the determination of the response associated with the vertical propagation of shear wave through visco-elastic medium. For the derivations of various expressions, the papers of Schnabel et al. (1972) and Kumar (2005) can be referred. The previous formulation of the author was extended in this paper to deal with the earthquake random vibration by using Fourier transformation. A digitized seismogram with n equidistant acceleration values a_j at time interval Dt was represented by a finite sum of harmonic motions having different frequencies. The formulation of the Fourier transformation is presented below.

(1)

where w_s, s = 0, 1, ......, n/2, are equidistant frequencies; i = ; and a_s and b_s are the complex Fourier coefficients as defined by the expressions presented below.

(2)

(3)

The representation of a discrete motion with its Fourier transform gives an exact representation of the motion at discrete points t =jDt, j =0,1, n-1. Cyclic repetition of the motion with the period T = nDt was implied in the solution.

MATERIAL INPUT PARAMETERS

The average specific gravity of the different collected soil samples was found to be 2.66. The minimum and maximum dry unit weights of various soil samples were found to vary in between 14.9 and 17.5 kN/m3. It provides the corresponding maximum and minimum void ratio of the soil samples to vary in between 0.75 and 0.49, respectively. The analysis was carried out by choosing two different values of void ratio, namely 0.65 and 0.75, which corresponds to relative density of approximately 38.5% and 0%, respectively. However, it was assumed that the values of the void ratio do not vary with the depth. From the electron microscope pictures of the collected soil samples, the shapes of the different grains were found to vary in between rounded to sub-angular. The following relationship between the maximum shear modulus G, mean effective principal stress sm and void ratio e, as recommended by Hardin and Black (1968) and Hardin and Drnevich (1972) for round-grained sands, where the void ratio is smaller than 0.80, was assumed to be applicable:

(4)

The magnitude of s_m at a depth z is defined below.

(5)

g^/ is the effective unit weight of the soil mass. The value of the earth pressure coefficient (K_o) was taken equal to 0.5 in the present study; this value approximately corresponds to a loose normally consolidated sand with friction angle equal to 30o. For carrying out the analysis, the average unit weight (g) of the soil mass was assumed to be equal to 16.0 kN/m³ without considering its variation with depth. It should be mentioned that even though the magnitude of the unit weight was assumed to be constant with respect to depth, however, the value of the shear modulus increases continuously with increase in depth. It is known that the magnitudes of D and G vary with the level of shearing strain (Richart et al. 1970; Hardin and Drnevich 1972). However, due to lack of any available information about the variation of D and G with the shearing strain, no such variation could be taken into account in the present study. The non-linearity of the ground can be incorporated either via performing an equivalent linear analysis or with the use of non-linear finite element method (Kausel and Roesset 1984; Li et al. 1992; Pyke 1992; Borja et al. 1999; Borja et al. 2002). Since the damping value (material) associated with the loose sand strata is usually quite significant (Richart et al., 1970), the analysis was carried out for two different values of material damping ratio (D) namely 5% and 10%.

DEFINITIONS OF THE ACCELERATION REDUCTION FACTOR AND THE STRESS REDUCTION COEFFICIENT

The peak magnitude of the horizontal acceleration at a given depth z can be expressed in terms of peak horizontal acceleration at the free surface (z = 0) and the acceleration reduction factor (x_d) as defined by the equation given below.

(6)

The magnitude of the peak shear stress (t) on a horizontal plane developed at a depth z from the ground surface (refer Fig. 3) was calculated by using the following expression,

(7)

where C_d is the stress reduction coefficient at a given depth z, a_g is the peak value of the acceleration (a_gp) at the ground surface, and u is the magnitude of horizontal displacement at any given time t and depth z.

Figure 3. Geometry of the problem and the boundary conditions

RESULTS

Predominant Frequency

For the Bhuj earthquake, three different recorded seismograms were available at the ground floor building of the passport office in Ahmedabad. These seismograms comprise of the acceleration time histories in a vertical and two orthogonal horizontal directions. Each recorded seismogram comprises of 26706 data points equally spaced at time interval of 0.005 seconds. In the present study the seismogram providing the horizontal acceleration time history, with the greater acceleration peak, was chosen; this seismogram is shown in Fig. 4a. The peak magnitude of the acceleration was noted to be 0.106g; where g is the acceleration due to gravity. By using Fourier transformation, the recorded seismogram was expressed as a superposition of a number of harmonic motions having 26706 different frequencies.

Figure 4. (a) Recorded seismogram (b) Seismogram generated using Fourier transformation 

Since a very large number of harmonics were chosen, the obtained seismogram using Fourier transformation was found to be invariably the same; the seismogram generated on the basis of the Fourier transform is illustrated in Fig. 4b. By using Eq.(1), the acceleration amplitudes of the harmonic motions associated with different chosen frequencies were obtained. A variation of the acceleration amplitudes with change in frequency is shown in Fig. 5.

Figure 5. A variation of acceleration amplitude with frequency in Fourier transform

It was noticed the peak acceleration occurs at a predominant frequency corresponding to 1.198 Hz. It is known that in order to find the liquefaction potential of the site, the cyclic triaxial shear tests are often conducted at a frequency in between 1 and 2 Hz. The seismogram recorded at Bhuj also provides the predominant frequency close to that often used in the determination of the liquefaction potential of the soil samples using the cyclic shear tests.

Ground Response at Different Depths

In order to generate acceleration and shear stress time histories of the ground mass at different depths, a computer program was written in FORTRAN. The program incorporates all the complex variables. To account for the variation of the increasing shear modulus with depth, the soil mass was discretized into 100 layers having equal thickness (0.1m). For the seismogram generated with the help of Fourier transform, as shown in Figure 4b, a time interval of 0.05 seconds was used to obtain the horizontal acceleration and shear stress time histories at different depths; however, in order to reduce the size of the manuscript, only the results providing the acceleration time histories at different depths were provided in this paper. Starting from the ground surface, the values of the acceleration (on a horizontal plane) were obtained at a depth interval of 2 m; the results were obtained upto a maximum depth of 10 m. The obtained variation of the acceleration time histories, for depths equal to 2m, 6m and 10m, is shown in Figs. 6-9. The magnitude of the acceleration was expressed in non-dimensional manner with respect to peak amplitude of the acceleration at the ground surface. It was noticed that the peak magnitude of the acceleration decreases with the increase in depth. It was also observed that the peak value of the ratio, t/(a_z a_gp) decreases continuously with increase in the depth; where t is shear stress at any depth z, agp is the peak magnitude of the acceleration at the ground surface and r is the density of the soil mass. By identifying the peak values of the acceleration and shear stress for a given time history, the values of x_d and C_d were obtained at different depths. The obtained variation of x_d and C_d with depth, for values of void ratio 0.65 and 0.75, and damping ratio 5.0 % and 10.0 %, is shown in Tables 1 and 2.

Figure 6. Acceleration time history with e = 0.75 and D = 5% for (a) z = 2m (b) z = 6 m, and (c) z =10 m

Figure 7. Acceleration time history with e = 0.75 and D = 10% for (a) z = 2m, (b) z = 6 m, and (c) z =10 m

Figure 8. Acceleration time history with e = 0.65 and D =5% for (a) z = 2m, (b) z = 6 m, and (c) z =10 m

Figure 9. Acceleration time history with e = 0.65 and D =10% for
(a) z = 2m, (b) z = 6 m, and (c) z =10 m

Also shown in Table 1 are the corresponding lower and upper bound values of C_d as recommended by Seed and Idriss (1971). It can be seen that the values of x_d and C_d increase with (i) decrease in the void ratio of the material (which leads to increase in the shear modulus), and (ii) increase in the damping ratio of the material. With increase in the void ratio of the material, the value of x_d at a given depth becomes smaller; it implies the amplification of the earthquake base motion towards the ground surface will become more pronounced for a loose sand. For a given value of e and D, up to a depth of 4m, it can be noted that C_d > 0.93 and x_d > 0.85; it implies that the amplification of the ground motion remains very significant in the upper 4m of the ground mass. Due to an increase in the value of the shear modulus with the depth, the magnitude of C_d for a given value of z remains always greater than that of x_d.

 Table 1. Stress Reduction Factor (C_d)

Table 2. Acceleration Reduction Factor (x_d)

COMPARISONS

It is known that the response of ground mass often gets amplified towards the ground surface (Idriss and Seed 1968; Kramer 1996). Therefore, the observed decrease in the values of x_d with depth is in accordance with the information available in literature. Due to deformability nature (not rigid) of the soil mass, the magnitude of x_d and C_d decreases continuously with increase in the depth (Idriss and Seed 1968). The increase of C_d with increase in shear modulus (decrease in void ratio) is quite obvious. The increase of x_d and C_d with increase in the value of damping ratio also matches with the observation made by Harrop-Williams (1988) and Kumar (2005) for harmonic vibration. The lower bound values of C_d given by Seed and Idriss (1971) compare reasonably well with the present results for the Bhuj earthquake.

REMARKS

It should be mentioned that most of the building foundations in Ahmedabad were provided at a depth varying in between 1.5 m and 4.0 m. The size (B) of the most of the footings was generally smaller than 1.0m (Humar et al., 2002). Considering the square footing, the effective stressed zone below the footing level will extend to a maximum depth of about 2B (Bowles 1996), that is, 2m. Therefore, in most of the cases, the superstructure load will transfer to the soil mass up to a maximum depth of about 6m (considering the footing to be located at a maximum depth of 4m). The soil mass upto a depth of 6 m was found to comprise mostly of loose alluvial sand deposits. The amplification of the ground motion was noted to be very significant (x_d > 0.85) in the upper 4m thickness of the soil mass near the ground surface. Since the depth of the water table in this region was quite deep and the liquefaction does not seem to have occurred around these selected sites in Ahmedabad during the occurrence of the earthquake, the foundation failures of the different apartment buildings were expected due to the occurrence of the differential settlements of the footings placed over the loose sand strata (2m-6m) and also on account of the amplification of the ground motion in the upper ground strata (0-4m).

CONCLUSIONS

By incorporating the increase of shear modulus with depth, the seismogram recorded at Ahmedabad was utilized to obtain the response of ground mass upto a depth of 10 m. The amplification of the ground base motion was seen to be very significant in the upper 4 m depth of the material near the ground surface. Standard penetration tests conducted near the area, where many of the buildings failures were reported during the Bhuj earthquake, revealed the presence of loose sand alluvial deposits especially in the upper 2m-6m depth. The predominant frequency of the recorded seismogram was found to be 1.2 Hz. The computed values of x_d and C_d were found to be greater for lower values of void ratio and for higher values of damping ratio. The values of C_d obtained in this study were found to compare reasonably well with the lower bound values of Seed and Idriss.

ACKNOWLEDGMENTS

The financial assistance provided by the Department of Science and Technology, New Delhi in carrying out this work under the DST project ”Liquefaction assessment for the recent Bhuj earthquake” is gratefully acknowledged.

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