1. A NUMERICAL PREDICTION OF FUNDAMENTAL FREQUENCY OF 2D BASINS WITH SMALL SHAPE-RATIO(W/H)
Neeraj Kumar and Prof. JP Narayan
Department of Earthquake Engineering

2. OVERVIEW INTRODUCTION 2D DOMINANT FREQUENCY OF BASIN
SITE-CITY INTERACTION EFFECT
DOUBLE RESONANCE
CONCLUSION
New York city: Google

3. INTRODUCTION “EARTHQUAKE DON’T KILL PEOPLE, BUILDINGS DO”
Earthquake is one of the most fearful natural phenomenon which affect the human life.
Michoacan Earthquake (1985). Ms ,000 lives lost and 1,000 construction destroyed at Mexico city, located 400 km, due to very soft soil and matching frequency of site and structures.
Spitka Earthquake (1989), Ms Lekninkan city suffered significant damage (32 km epicentral distance on alluvial basin) compared to Lirovakan city (25 km epicentral distance on rock).
Similarly, earthquakes at San Francisco(1956), Niigata(1964), Mexico(1985,1989), Kobe(1995), Bhuj(2001)and so on….
Site condition largely determine severity of damage.
Maximum effects (economic and human loss) are observed in cities
“EARTHQUAKE DON’T KILL PEOPLE, BUILDINGS DO”

4. Hong Kong city (Google)
The population of world is increasing at a very rapid rate, especially in the metro cities like Delhi, Mumbai etc.
This high demand of real estate sector is met by increasing the size of city and height of buildings.
Many builders are creating sites and constructing the buildings at those seismic vulnerable sites like near river bed, pond filled with loose soil etc
Does City (Group of buildings) influences the seismic ground motion ?
Hong Kong city (Google)

5. Government Flats in Morbi during Bhuj earthquake.
The long and strong ground motion at Mexico city in 1985 earthquake due to interaction with the urban structures has caused considerable damage to the city.
Rapid urbanization growth over the world has made it more critical for safety issue
Government Flats in Morbi during Bhuj earthquake.

6. 2D DOMINANT FREQUENCY OF BASIN
Bard and Bouchon (1985) gave simple relations to determine fundamental frequency of closed rectangular 2 D basin filled with elastic sediment . 2w
It can be observed that for W/H>4, the fundamental frequency of a 2D basin (F02D) varies only slightly and can be approximated by one-dimensional (1D) theory.
Basin models used for simulations
This relation can be extended to any valley shape, taking as equivalent width 2w the length over which the local sediment thickness is greater than half the maximum thickness .
Zhu at el (2016) also predicted the fundamental frequency of trapezoidal basin considering the Bard and Bouchon (1985) proposed equation by taking the equivalent area of trapezoidal basin to that of rectangular basin.

7. Model size and Parameters
The size of model is 6.9 km along horizontal direction and 4.3 km in vertical direction.
The basin is placed at the centre of model.
In the model, thickness of air above the free surface is 120 m.
Horizontally, the numerical model is descritized with grid size of 3.0 m
In vertical direction the grid size is kept 3.0 m up to 300 m and then it is increased to 10.0 m thereafter.
The time step is kept s to make the computation stable.
A receiver is kept at the centre of the basin at the free surface.
Absorbing boundaries are applied at bottom and sides edges up to 200 grids to avoid the edge reflections.

8. The considered rheological parameters for building, basin and rock
S.No.
Material
Density (kg/m3)
Vs (m/s) Qs
Poison’s Ratio
µu (GPa) 1
Basin
1800
350 35
0.25
0.262 2
Rock
2500
180
8.210 3
Building
120 10
0.0064
The role of the shape of the basin on resonant fundamental frequency is studied, using rectangular and elliptical basins.
Elliptical basin -BE1, BE2, BE3, BE4 and BE5 models having depth (H) of 51m and width (2W) as 153 m, 201 m, 249 m, 30 3m and 351 m, respectively are taken.
Rectangular basin -BR1, BR2, BR3, BR4 and BR5 models having depth 51 m and width as 153 m, 201 m, 249 m, 303 m and 351 m respectively are taken.
The parameters of sediment in the basin and bedrock are kept same for all the basin models.

9. A plane horizontal SH-wave front is generated at a depth of 270 m using various point source along a line.
The mathematical form of the Gabor wavelet is given below
Where α=[ ]2 ,fp is predominant frequency, ϒ controls the oscillatory character, ts controls the duration (duration =2 ts) and ϕ is phase shift. Figure shows the generated Gabor wavelet for fp= 2, ϒ= 0.5, ts= 1.5 and ϕ=0 and its spectra. The frequency content in the Gabor wavelet is Hz.

10. SH-wave response of different elliptical basin models (left) and rectangular basin models (right), respectively.
There is tremendous effects of basin shape on the free field ground motion
As the width of basin increases the duration of ground motion is increasing, due back and forth love wave propagation.
The maximum amplitude values are approximately matching with each other.

11. Spectral amplifications at the centre of elliptical and rectangular basins for different shape ratio.
The obtained F01D of basin is 1.71 Hz.
The F02D of elliptical basin is larger than that of the rectangular basin.

12. Rectangular (BR) Basin
The obtained F02D of rectangular basin models very much corroborates with the relationship given by Bard and Bouchon (1985).
The F02D of elliptical basin models have only minor deviation (<5%) as compared with the same computed using the concept of equal area (Zhu and Thambiratnam, 2016) and using the Bard and Bouchon (1985) empirical relation for the rectangular basin.
This may be due to use of different numerical method and discrete frequency in FFT.
Basin size (m)
Basin shape ratio
Lowest F01D (Hz)
Amplat F01D
Rectangular (BR) Basin
Elliptical (BE) Basin
Depth
Width
Model
F02D (BB)
F02D Simu.
Amp. (F02D) 51
153
0.67
1.71
6.15
BR1
2.06
1.98
12.76
BE1
2.35
2.21
13.46
201
0.51
BR2
1.92
1.85
11.54
BE2
2.12
2.08
12.41
249
0.41
BR3
1.78
10.38
BE3
2.00
1.95
12.14
303
0.34
BR4
1.81
1.73
9.28
BE4
1.91
1.88
11.53
351
0.29
BR5
8.60
BE5
1.86
11.12

13. The obtained spectral amplification at F02D is larger in the elliptical basin as compared to that in case of rectangular basin.
This may be due to larger amplitude of the Love wave generated in elliptical basin than that generated in the rectangular basin.
An empirical relation has been developed between the shape ratio (H/W), F01D and F02D for the elliptical basin, as shown in figure and corresponding equation is given below.
Variation of ratio of 2D to 1D resonance frequency with shape-ratio of elliptical basin models.

14. SITE-CITY-INTERACTION EFFECTS
Site–city interaction (SCI) includes the combined effects of basin, kinematic soil–structure interaction and inertial structure–soil interaction on a global scale.
Kinematic Interaction
The SSI effect which is associated with the stiffness of the structure is termed as kinematic interaction. The deformation caused by kinematic interaction alone can be computed by assuming that the structure and foundation has stiffness but no mass.
Inertial Interaction
The mass of structure and foundation causes them to respond dynamically. The SSI effect which is associated with the mass of the structure is termed as inertial interaction.
Kinematic interaction analysis (Kramer,1996)

15. The first experiment was carried out by Jennings (1970) to estimate the ground motion imported in the soil by the building vibrations.
A nine-story Millikan Library Building at the Caltech campus, San Francisco was excited into resonance by two vibration generators on the 9th floor and ground motion was recorded at distances up to a few kilometers from the building.
Later on, Kanamori et al. (1991) studied the effects of high-rise buildings in Los Angeles, excited by atmospheric shock waves generated by the space shuttle Columbia on its return.
Inertial interaction analysis
Gueguen et al. (2000) based on Volvi experiment reported the radiated motion as 20% and 5% of the building base motion at distances of two and ten times of the characteristics length of the structure, respectively.
Guéguen (2008) reported the ground motion developed by the World Trade Centre (WTC) during the terrorism attack based on an analytical modeling and several records done by the Lamont network.

16. In order to study the SCI effects on the building response ten storey (B10) buildings situated in a basin was considered having plan of 63 m x 63 m and height is 30 m.
The effective density of the building was obtained as 350 kg/m3
The time period of building is calculated for the framed system using the following formula (IS:1893) with building height H m.
A 1D sediment-filled basin of depth 87m is considered for the study of SCI effects. The basin sediment and bedrock are homogenous and viscoelastic in nature.
The total length of the city is 963 m and number of B10 buildings in the city is 13. The distance between two buildings is 12. So, the effective city-density is 0.85 (Sahar et al., 2015).
The obtained F02D of the building block is 0.97 Hz and the F01D of the 1D basin is 1.0 Hz. Both the building and the 1D basin are under resonance condition.

17. DOUBLE RESONANCE
(a)
(c)
(b)
Single building in basin (left) and city in basin (right);
SH-wave response of single building in basin (left) and city in basin (right);
Spectral amplification at the top of single building in basin (left) and at the top of a building situated at the centre of city in basin (right).

18. The SCI effects has caused a considerable decrease of both the amplitude and duration of ground motion at the top of building as compared to the top of a single building situated in basin.
The spectral amplification at the top of single B10 building has reduced to 22.5 from 45.0 when the same building is at the centre of the city.
Means, SCI has caused a reduction of spectral amplification of the order of 50% at the top of the building situated at the centre of the city corresponding to the double resonance frequency.

19. CONCLUSION
There is an excellent correlation between the computed F02D of the rectangular basin using the FD response and the same computed using empirical relationship given by Bard and Bouchon (1985) .
The empirical relationship given by Bard and Bouchon (1985) for elliptical basin is not so effective for predicting F02D.
The numerically computed F02D of the elliptical basins very much matches with the same computed using the concept of equal area proposed by Zhu and Thambiratnam (2016).
The simulated results have been used to develop a new empirical relationship based on the regression analysis to predict the F02D of the elliptical basin.
 It is alarming to know that the spectral amplification at F02D of basin is more than twice to that of F01D of the same basin computed assuming it as 1D basin.
The computed SCI effects on the building response revealed a reduction of building response up to 50% under double resonance condition.
This finding calls for the study of SCI effects in a 2D and 3D basins where both the buildings and the basin are in 2D and 3D resonance condition.

21. THANK YOU...