Introduction

Hazard assessment is conducted to estimate quantitatively and define the hazard expected in a particular area due to the strategic location of the place near active faults and lineaments and earthquake events that have occurred in the past and have a probability of taking place in the future. Hazard assessment is generally conducted using a deterministic approach or probabilistic approach. In deterministic seismic hazard analysis (DSHA) the earthquake source causing the highest hazard at a particular location is considered to be the causative source and using attenuation relationships, the hazard level at that site is determined [1]. However in case of probabilistic seismic hazard analysis (PSHA) the uncertainties associated with the size of an earthquake, the location of occurrence and other factors are considered. The critical part of seismic hazard assessment is the determination of maximum horizontal acceleration (MHA) at ground surface (also called peak ground acceleration (PGA)) and spectral acceleration (S a ) of the particular area. The results of such hazard studies are available as seismic hazard maps showing the contours of peak ground acceleration across the area which are essential parameters for earthquake resistant design of structures in seismic prone regions.

It is a challenging task for civil engineers to provide safe and economic design of multi-storied buildings and high rise constructions in urban cities like Kolkata, where the increasing population and accelerating infrastructural growth makes it a necessity. Although Indian standard code on earthquake resistant design of structures IS 1893: Part 1 [2] provides various guidelines for design of foundations (pile foundation, raft foundation, isolated footings) in seismic regions, the influence of various factors like local soil conditions, liquefaction hazard, earthquake parameters and ground response analysis on design of foundations needs considerable attention. Seismic hazard assessment and ground response analyses of various Indian cities have been implemented by several researchers in the past and have observed that the seismic hazard at a particular site is influenced by the topography of the bedrock, local soil conditions existing at the site and duration of shaking of the earthquake motion. These areas for which seismic hazard analysis have been implemented in the past includes Bangalore [3], Chennai [4], Delhi [5], Goa [1], Gujarat [6,7,8], Guwahati [9], Kolkata [10, 11], Kanpur [12], Mumbai [13,14,15] and others. The successful implementation of seismic ground response analysis of soil deposits at a particular location provides a geotechnical engineer with important information on natural period of soil deposits at the site, amplification of acceleration at the ground surface and response spectrum curves which are useful parameters for seismic design of pile foundations.

Various case histories accounting the failure of pile foundation due to liquefaction by treating the pile as beam elements and considering lateral loads to cause bending failure in piles were reported by Hamada [16], Ishihara [17], Tokimatsu et al. [18] and Finn et al. [19]. However piles passing through liquefiable soils are subjected to both axial and lateral loads during an earthquake and bending-buckling interaction causes vulnerable failure of pile foundation due to significant degradation of stiffness [20,21,22,23]. Several researchers have carried out seismic analysis of piles considering both kinematic and inertial loadings using experimental [24,25,26], analytical and numerical techniques [27,28,29,30,31,32,33,34,35,36,37,38,39]. The results obtained from these studies highlighted the considerable need of accurate soil-pile interaction to evaluate the response of piles passing through liquefiable soils in terms of bending moment and deflection under seismic conditions.

However, a thorough procedure on seismic analysis and design of single piles and pile group considering the influence of various parameters like local soil conditions, input ground motions, liquefaction hazards, bending–buckling interaction due to the application of vertical and lateral load on pile foundation for a metropolitan and important city like Kolkata is scarce in literature. Hence, in the present study, empirical correlations between shear wave velocity and field SPT N values have been used to implement soil site characterization of Kolkata city as per NEHRP [40] guidelines and the seismic hazard of Kolkata city is highlighted. Kolkata city is located along the boundary of seismic zone III and IV indicating moderate to high seismic hazard and spread in a north–south direction along the east bank of the Hoogly river with the soil being predominantly soft, thick and alluvial in origin. The alluvial nature of soil located in Kolkata city is prone to soil amplification when subjected to different earthquake motions at the bedrock level, which is obtained from site specific seismic equivalent linear ground response analysis. The results obtained from seismic ground response analysis in the form of response spectrum curves and depth-wise variation of acceleration is utilized in the present study for seismic design of single piles in liquefiable and non-liquefiable soils of Kolkata city, by considering both kinematic and inertial interactions. The presence of vertical loads in addition to input ground motions is considered in the present study and bending–buckling interactions are simulated and the seismic response of pile foundation in terms of pile bending moment and pile deflection are obtained.

Kolkata City: Study Area and Need for Present Study

Kolkata city, located between latitudes 22°18′N–22°50′N and longitudes 88°08′E-88°32′N and covering an area of 185 km2, is one of the largest cities in the world. Located in eastern India, Kolkata is the capital of the Indian state of West Bengal and spread north–south along the east bank of River Hoogly. Kolkata city is a major metro city in India and serves as a gateway to north-east India. Kolkata city is located in the lower delta of River Ganga in eastern India and is also low-lying, having an elevation of 9 m above mean sea level [41]. The soil profile existing in Kolkata city is mostly soft, thick and alluvial in nature and underlain by clay, silt and silty sand sediments, having varying grain size and textures. Moreover a major part of the city, which was initially a wetland, is reclaimed by filling with silty soil, dredged from the River Ganges [10]. Due to immense infrastructural growth, rapid urbanization, increasing population and scarcity of vacant lands, multistoried buildings founded on pile foundations are coming up in Kolkata city. Kolkata city is located at the boundary of seismic zones III and IV as per seismic code of India [2] indicating moderate to high seismic alert. The city is also situated along the Eocene Hinge Zone, a prominent regional basement fault and is surrounded by several other active faults like Jangipur–Gaibandha Fault, Pingla Fault, Rajmahal Fault, Sylhet Fault, Sainthia Bahmani Fault and Dhubri Fault [41, 42]. These are ample proofs of seismic vulnerabilty and hazard associated with Kolkata city and the destructions that might occur if an earthquake occurs in the future.

Kolkata city and its surrounding areas have experienced tremendous vibrations due to various earthquakes in the past. This includes the 1906 Calcutta earthquake and the Calcutta earthquake of 1964 which caused major damages to the city. In the recent years, tremors have been felt across Kolkata city due to various far source earthquakes like 2011 Sikkim, 2012 Indonesia, April 2015 Nepal and 2016 Myanmar earthquakes. This is due to alluvial nature of soil existing in Kolkata city and the significant role played by the local soils in altering the ground motion characteristics of the earthquake. Similar scenarios were also observed in San Francisco bay area during 1989 Loma Prieta earthquake and in Ahmedabad city of Gujarat, India during 2001 Bhuj earthquake. Hence in the present study, Kolkata city is chosen as the study area and using derived V s –SPT N correlations ground response analysis is carried out which is further used for seismic analysis and design of pile foundations in liquefiable soils.

Vs–N Correlations for Kolkata city

The response of soil at a particular site under the influence of dynamic loadings is governed by damping properties, shear wave velocity and shear modulus. Although shear wave velocity can be measured at site using various field tests like seismic reflection test, seismic refraction test, Multichannel Analysis of Surface Wave (MASW) test; however in an urban city like Kolkata in situ measurement of shear wave velocity is not conducted due to lack of required free space. Hence in the absence of sophisticated dynamic field test data, empirical correlations derived between SPT N values and shear wave velocity V s for Kolkata city [41] are used in the present study. Extensive geotechnical data from over 450 boreholes in entire Kolkata city were collected from various soil investigation agencies, government and private institutions and consultants and national labs. The correlations were derived using power model of non-linear regression analysis for various soils existing in Kolkata city of eastern India, i.e., all soils, clay, silt and silty sand and are tabulated in Table 1 [41]. The values of the various statistical parameters like coefficient of correlation (r), coefficient of regression (R 2) and residual mean square error (MS E ) are also tabulated to highlight the accuracy of the results obtained using these empirical correlations. It is observed from Table 1 that the derived correlations have regression coefficients greater than 0.95 and low values of residual mean square error, thereby indicating an accurate prediction of shear wave velocity from SPT N values.

Table 1 Proposed correlations between shear wave velocity (V s ) and uncorrected SPT N values (modified after Chatterjee and Choudhury [41])
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The proposed correlations are compared with the existing correlations for other Indian cities like Delhi [8], Bangalore [43], Chennai [44] and Mumbai [45] and illustrated in Fig. 1. It is observed from Fig. 1 that the shear wave velocity values computed for Kolkata city using the proposed V s –N correlations are following the similar nature of curves as observed for Bangalore, Chennai, Delhi and Mumbai cities and hence compared reasonably well with the works of previous researchers. Moreover the present curve is observed to coincide with that obtained for Mumbai city due to similar soft soil profiles existing at various locations in both the cities. Further, the results calculated from the present derived correlations for clay, silt and silty sand soils of Kolkata city are compared with the correlations proposed by previous researchers across the world and shown in Fig. 2. It is seen from Fig. 2 that the calculated values of shear wave velocity for clay, silt and silty sand soils of Kolkata city are similar to that calculated using the regression equations given by other researchers. The variation in results may be attributed to the different geotechnical conditions of the study area, variability of SPT equipments, variation in level of ground water table and other similar conditions which may have a profound influence on the results.

Fig. 1
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Comparison of shear wave velocity for soils obtained in the present study with that of different Indian cities

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Fig. 2
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Comparisons between proposed shear wave velocity V s –SPT N correlations for a silt b silty sand and c clay in Kolkata city and other similar relations available worldwide

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The V s –N correlations proposed for all soils have further been used to calculate the magnitudes of V s,30 which is defined as the magnitude of shear wave velocity average over the top 30 m of soil deposits and used for classifying soil sites at the particular location for geotechnical earthquake engineering design problems [40]. The magnitude of V s,30 is calculated according to the following expression:

$$ V_{s,30} = \frac{{\sum\limits_{i = 1}^{n} {d_{i} } }}{{\sum\limits_{i = 1}^{n} {\frac{{d_{i} }}{{V_{s,i} }}} }} = \frac{30}{{\sum\limits_{i = 1}^{n} {\frac{{d_{i} }}{{V_{s,i} }}} }} $$
(1)

where, ∑  n i=1 d i is equal to 30 m, d i denotes the thickness (in metres) and V s,i denotes the shear wave velocity (in m/s) of the ith soil layer respectively, existing in the top 30 m, in a total of n soil layers [41]. The variation of shear wave velocity with depth at various borehole locations is shown in Fig. 3 and the magnitude of V s,30 at each of these locations is tabulated in Table 2. It is seen from Table 2 that the magnitude of V s,30 calculated in these soil sites lies in the range of 165 m/s to 178 m/s, which is less than 180 m/s and thereby classifying these typical Kolkata soil sites in Class E category as per NEHRP [40] guidelines. Hence it can be inferred that several locations in Kolkata city are prone to seismic hazard at different earthquake magnitudes. Moreover the proposed correlations are further utilized for carrying out seismic equivalent linear ground response analysis of various soil sites in Kolkata city and the results obtained in the form of depth-wise variation of maximum horizontal acceleration and response spectrum curves are used for seismic design of pile foundations in Kolkata city.

Fig. 3
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Typical variations of shear wave velocity V s with depth at various locations of Kolkata city

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Table 2 Average shear wave velocity till 30 m depth (V s,30 ) of all soils at particular locations of Kolkata city
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Site-Specific Seismic Equivalent Linear Ground Response Analysis for Kolkata City

The passage of seismic waves from bedrock to the ground surface causes amplification of ground motion over the soil sediments. This happens because the seismic waves gets trapped in the soil sediments, resulting in varying impedance between the underlying bedrock and the soil particles [46]. As a result the amplitude and frequency content of the seismic waves are considerably modified when it travels through the soil deposits and this process where local soil layers modifies the strong motion characteristics of an earthquake is soil amplification [11, 47]. Kolkata city has soft alluvial soil due to its location along the banks of river Hoogly and hence equivalent linear ground response analysis of earthquake motions at the bedrock level is necessary and conducted for Kolkata city.

In the present study seismic equivalent linear ground response analysis is carried out at six different locations of Kolkata city, India as illustrated in Fig. 4. The boreholes are selected in such a manner such that they represent a spatial variation of soil existing at various locations across the city. The soil profiles existing at the six different locations (BH #1–BH #6) of Kolkata city are tabulated in Table 3. As observed from Table 3, the top layers of the typical soil strata BH #1 comprised of loose sand and silty clay underlain by grey silty clay and medium fine sand with grey silty clay at the bottom layers. The soil site (BH #1) is subjected to 5 different input earthquake motions, viz, 1989 Loma Gilroy, 1994 Northridge, 1995 Kobe, 2001 Bhuj and 2011 Sikkim motions, having a wide variation of strong motion parameters like frequency content, bedrock level acceleration and duration of shaking [11, 36] and the analysis is conducted using SHAKE 2000 [48] computer program. The selection of input motions, having different ground motion parameters, are done in such a manner to study its effects at different locations of Kolkata city and the destruction that can occur if any earthquake having such properties strike the city in future. The input motions are applied at the rigid bedrock level and the soil layers were considered horizontal with the ground surface being assumed to be level [11].

Fig. 4
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Map of Kolkata city showing the location of the six boreholes considered in the present study

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Table 3 Soil profiles of BH #1–BH #6 considered in the present study
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The variation of acceleration with time at the ground surface due to 1995 Kobe and 2001 Bhuj earthquake motions is illustrated in Fig. 5. It is observed that the peak ground acceleration at the ground surface is 1.57 g when subjected to 1995 Kobe motion and 0.24 g when subjected to 2001 Bhuj motion. This indicates that the input acceleration is amplified by 1.88 and 2.28 times when subjected to 1995 Kobe and 2001 Bhuj seismic motions, respectively. In Table 4 the dynamic soil properties at different layers of the soil column in BH #1 is calculated. The average shear wave velocity is computed to be 187.1 m/s while the fundamental time period is determined to be 0.81 s. The acceleration response spectrum at the ground surface for BH #1 due to the input ground motions, considering 5% damping ratio, and its comparison with IS 1893: Part 1 [49] is illustrated in Fig. 6. It is observed that the maximum spectral acceleration recorded is 1.94 g at a time period of 0.86 s, which is close to the fundamental time period of the soil column, i.e., 0.81 s, when subjected to 1995 Kobe motion. The peak spectral acceleration for 1989 Loma Gilroy, 1994 Northridge, 2001 Bhuj and 2011 Sikkim earthquake motions are 0.69 g at 0.42 s, 1.14 g at 0.3 s, 0.46 g at 0.26 s and 0.75 g at 0.21 s, respectively. This clearly indicates that while 1995 Kobe motion is having a detrimental influence over long, flexible structures resting on soft soils and having a long fundamental period, 2011 Sikkim motion is vulnerable for small structures having a short fundamental time period and resting on soft soils. Figure 7 shows the variations of Fourier amplification ratio with frequency for BH #1. It is observed that 2001 Bhuj and 2001 Sikkim motions produced the maximum amplifications of 4.26 and 5.34 at a frequency of 1.125 Hz and 1.25 Hz, respectively. The minimum amplification recorded is 3.82 at a frequency of 1 Hz due to 1995 Kobe motion. This is due to the higher duration and frequency content of 2001 Bhuj and 2011 Sikkim seismic motions resulting in higher amplification as compared to 1995 Kobe motion which, in spite of having a higher acceleration at bedrock level (0.834 g), do not amplify significantly due to its lower duration. The depth-wise variation of maximum horizontal acceleration (MHA) and the amplification factor is illustrated in Fig. 8 and tabulated in Table 5. It is observed for 1995 Kobe motion the MHA at the ground surface (which is also called the peak ground acceleration (PGA)) is 1.57 g against the bedrock level acceleration (a max ) of 0.834 g; thereby indicating an amplification factor of 1.88. Similarly for 1989 Loma Gilroy, 1994 Northridge, 2001 Bhuj and 2011 Sikkim motions the corresponding PGA are observed to be 0.63 g, 1.08 g, 0.24 g and 0.51 g, respectively. The corresponding amplification factors for the same sequence of motions are calculated to be 1.69, 1.90, 2.28 and 2.52, respectively. Thus it can be inferred that the amplification of acceleration is significantly affected by the frequency content and duration of the input seismic motions and the bedrock level acceleration has little influence over the same. Moreover, due to soft alluvial and clay soil layers existing in Kolkata city, high values of amplification of bedrock motion, varying between 1.6 and 2.5, are obtained in the present study. As a result soft soil sites, which undergoes large amplification, should be given proper attention for seismic analysis and design of pile foundations in Kolkata city.

Fig. 5
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Acceleration–time history at the ground surface for BH #1 when subjected to a 1995 Kobe and b 2001 Bhuj earthquake motions

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Table 4 Typical dynamic soil properties calculated in the present study for BH #1
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Fig. 6
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Response spectrum curves at ground surface at 5% damping at BH #1 obtained in the present study

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Fig. 7
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Fourier amplification ratio curves at the ground surface at BH #1 obtained in the present study

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Fig. 8
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Variation of a maximum horizontal acceleration (MHA(g)) and b amplification of acceleration (MHA(g)/a max (g)) along depth for BH #1

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Table 5 Amplification factor of acceleration at the ground surface due to different earthquake motions considered in the present study
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Seismic Analysis of Piles in Liquefiable Soil

The analysis of piles passing through liquefiable and non-liquefiable soils and subjected to earthquake motions is an interesting area of research and that too for a densely populated city like Kolkata where most of the buildings are constructed on pile foundations. The soil site characterization as per NEHRP [40] and amplification of bedrock level acceleration at the ground surface is an indication of the seismic risk the city is exposed to. Although most of the existing methods of seismic analysis of pile foundations consider the effect of bending failure due to lateral loads and buckling failure due to vertical loads separately; however an accurate design procedure should consider bending-buckling interaction mechanism of pile failure in liquefiable soil. In the present study a vertical concrete pile having length l, diameter d and flexural stiffness E p I p is embedded in a two-layered soil medium comprising of a liquefiable sand layer of thickness L liq underlain by a non-liquefiable stiff clay layer of thickness L nliq , as shown in Fig. 9. The pile is subjected to a vertical compressive load having magnitude P applied at the centre of the pile section on the ground surface. In addition to it, the pile tip is also subjected to a dynamic loading generated due to an input ground motion and horizontal lateral loads at various nodes along the pile depth which is calculated according to Chatterjee et al. [36]. The pile will experience lateral deflection dependent upon the magnitude and direction of the applied loading. The lateral resistance developed by the pile when subjected to combined loading is analyzed using the modulus of subgrade reaction approach, which is considered to be dependent on depth below the ground surface. Hence the governing differential equation for determining the horizontal deflection of the pile (y) along the depth (z) when subjected to the above mentioned combined loadings is given as [50, 51]:

$$ E_{p} I_{p} \frac{{d^{4} y}}{{dz^{4} }} + P\frac{{d^{2} y}}{{dz^{2} }} + \eta_{h} d(y - y_{g} ) = 0 $$
(2)
$$ y_{g} = g_{x} \cos \frac{{\pi \left[ {z - L_{liq} } \right]}}{{2L_{nliq} }} $$
(3)

where, η h is the modulus of subgrade reaction in kN/m3 in liquefiable soil, y g is the permanent ground displacement profile varying with depth along the pile length and g x is the soil displacement which is maximum at the ground surface and varies with depth z [51, 52]. The degradation of subgrade modulus is observed for liquefiable soil with increasing displacement and is given as [18, 52, 53]:

$$ \eta_{h} = \eta_{hn} \times s_{f} $$
(4)

where, s f is stiffness degradation factor in a liquefiable soil and considered as 0.01 in the present study [54]. η hn is the modulus of subgrade reaction for non-liquefiable soil in MN/m3 [53, 55] and expressed as:

$$ \eta_{hn} = 80E_{o} d^{ - 0.75} $$
(5)
$$ E_{o} = 0.7N $$
(6)

where, E o is the modulus of deformation in MN/m2, N denotes the SPT value at a particular depth and d is the diameter of the pile in cm. The governing differential equation is solved using the stiffness method based on finite element approach by considering a single pile element having two degrees of freedom (translation and rotation) at each node. A pile element is chosen and unit displacement is applied at a node keeping the rotation as zero while in the other node both displacement and rotation are kept zero. In a similar manner by applying appropriate boundary conditions and equilibrium equations, the element stiffness matrix for each pile elements are formulated which are assembled together to obtain the global stiffness matrix, thereby relating the forces with the displacement at each pile node [36]. After calculating the displacement and rotation at various nodes, finally the force and moment of the corresponding pile elements are calculated using the load–displacement relationship. The entire procedure is accomplished by writing a code using the mathematical tool MATLAB [56] to calculate the deflection and bending moment at various nodes along the pile depth when it is subjected to the combined loadings.

Fig. 9
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Schematic sketch of a single pile passing through liquefiable soil layer and underlain by a non-liquefiable soil layer as considered in the present study

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A free headed single pile of M30 grade concrete with a floating tip having length 10 m, diameter 600 mm and flexural stiffness 174.3 kN m2 is inserted into a layered soil (site BH #1), the properties of which are tabulated in Table 6. The allowable load carrying capacity of the pile is calculated to be 420 kN according to IS 2911: Part 1/Section 1 [57] and is assumed to be the safe vertical load acting on the pile top. The peak ground acceleration obtained from seismic equivalent linear ground response analysis for BH #1 site in Kolkata city are 0.63 g for 1989 Loma Gilroy motion, 1.57 g for 1995 Kobe motion, 0.24 g for 2001 Bhuj motion and 0.51 g for 2011 Sikkim motion. The lateral load (H) acting at the pile head is calculated according to Chatterjee et al. [36]. The analysis is initially conducted for inertial loading (when the pile is subjected to the input seismic motion only) and then for combined loadings. The effect of inertial loading is obtained by subtracting the deflection or bending moment observed due to kinematic loading from the corresponding deflection or bending moment observed due to combined loading [35, 58].

Table 6 Ground parameters at BH #1 for conducting soil-pile interaction analysis under liquefiable soil conditions
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The influence of depth of liquefiable soil layer on pile response when subjected to combined loading is analyzed in the present study. The depth of liquefiable soil layer (L liq ) is varied in terms of total length of the pile (l) and 5 different combinations, i.e., L liq /l = 0.2, 0.4, 0.6, 0.8 and 1.0, is considered. The variation of pile bending moment and pile deflection along pile depth for combined loading conditions and when subjected to 1989 Loma Gilroy and 2001 Bhuj motions for different L liq /l ratio is illustrated in Fig. 10 and Fig. 11, respectively. It is observed for 1989 Loma Gilroy motion that maximum deflection is 34.6 cm when L liq /l is 0.2 and it increases to 74 cm when L liq /l increases to 0.6. However, when L liq /l = 1.0, i.e., the entire soil is liquefiable the deflection at the pile head is 52.6 cm while at the pile tip is -14.8 cm. This is because the soil has lost its shear strength due to liquefaction and the pile cannot stand in such a liquefiable soil medium. In a similar manner for 2001 Bhuj earthquake motion, when the L liq /l ratio increases from 0.2 to 0.6, the pile head deflection increases from 16.5 cm to 42 cm. When L liq /l ratio rises to 1.0, the deflection at the pile head is 30.9 cm while at the pile tip it is -9.1 cm. The deflection is observed to be more for 1989 Loma Gilroy motion as compared to 2001 Bhuj motion due to the higher magnitude of peak ground acceleration generated from ground response analysis and greater magnitude lateral loads acting at the pile head. The maximum bending moment is observed to occur at the interface of the liquefiable and non-liquefiable soil layers and the maximum bending moment occurs when L liq /l ratio is 0.6, i.e., the depth of the liquefiable soil layer is around 60% of the total pile length. Any further increase in depth of liquefiable soil layer will cause a reduction in the magnitude of bending moment. The maximum bending moment due to 1989 Loma Gilroy motion is observed to be 702 kN m when L liq /l ratio is 0.6 and it reduces to 280 kN m when L liq /l ratio increases to 1.0. Similarly, for 2001 Bhuj motion, the maximum bending moment reduces from 400 to 134 kN m when L liq /l ratio increase from 0.6 to 1.0. This is observed because the soil has already failed and lost its shear strength due to liquefaction before pile failure and the stresses developed in the soil under such circumstances are more than the corresponding shear strength of the soil.

Fig. 10
figure 10

Variation of pile bending moment (combined loading) with pile depth for a 1989 Loma Gilroy and b 2001 Bhuj earthquake motion for different combinations of L liq /l ratio in liquefiable soil

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Fig. 11
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Variation of pile deflection (combined loading) with pile depth for a 1989 Loma Gilroy and b 2001 Bhuj earthquake motion for different combinations of L liq /l ratio in liquefiable soil

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The influence of 4 different seismic motions on dynamic analyses of piles having L liq /l ratio 0.6 and subjected to combined loadings is analyzed in the present study and illustrated in Fig. 12. The maximum pile bending moment is observed to be 1620 kN m when subjected to 1995 Kobe motion. Similarly for 1989 Loma Gilroy, 2001 Bhuj and 2011 Sikkim earthquake motions, the bending moment generated are 702, 400 and 490 kN m respectively. The pile head deflections generated are 74, 154, 42 and 60 cm due to 1989 Loma Gilroy, 1995 Kobe, 2001 Bhuj and 2011 Sikkim earthquake motions, respectively and when subjected to combined loadings. However, for kinematic loading only and when L liq /l ratio is 0.6, the pile head deflection is measured to be 30.3, 50, 14 and 19.6 cm while the maximum bending moment is observed to be 248, 464, 108 and 158 kN m when subjected 1989 Loma Gilroy, 1995 Kobe, 2001 Bhuj and 2011 Sikkim earthquake motions respectively, as illustrated in Fig. 13. The influence of the inertial load is obtained after subtracting the kinematic response from the combined response [58]. The bending moment and deflections obtained due to inertial component of loading is tabulated in Table 7. The inertial component of loading of 1989 Loma Gilroy motion is observed to have 59.1 and 64.7% influence on pile head deflection and maximum pile bending moment of the total loading, while for 2001 Sikkim motions the corresponding percentage influences are 67.3 and 67.8%. The variation in pile response due to inertial loading for the various seismic motions may be attributed to the different magnitudes of depth-wise variation of soil displacement and lateral load acting on the pile. It is also observed that although kinematic loading affects the initial pile head deflection and pile bending moment, the contribution due to inertial loading on pile response (deflection and bending moment) is significant and hence should be considered for evaluating seismic design of piles in liquefiable soil.

Fig. 12
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Variation of a pile bending moment and b pile deflection with pile depth due to combined loading when L liq /l ratio = 0.6

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Fig. 13
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Variation of a pile bending moment and b pile deflection with pile depth due to kinematic loading when L liq /l ratio = 0.6

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Table 7 Magnitudes of pile head deflection and pile bending moment obtained for different component of loadings in the present study
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Conclusions

The following conclusions are obtained from the present study:

  • The magnitude of shear wave velocity till 30 m depth (V s,30 ) at various soil sites in Kolkata city is observed to lie in the range of 165 to 178 m/s and thereby classifying these typical Kolkata soil sites in Class E category as per NEHRP [40] guidelines. It is an indication of the seismic risk associated with these areas.

  • The seismic ground response analysis is significantly affected by the bracketed duration and frequency content of the input ground motions and the bedrock level acceleration (a max ) has little influence over the same. Thus 2001 Bhuj and 2011 Sikkim earthquake motions caused higher ground amplifications at various soil sites of Kolkata city as compared to 1995 Kobe motion due to the higher duration and frequency content of the former two motions.

  • The amplification factor of bedrock level acceleration at the ground surface lies within 1.6–2.5 at various soil sites of Kolkata city. This is due to the soft alluvial and clayey soil layers located in Kolkata city.

  • The response spectrum curves obtained in the present study for the different input earthquake motions provides valuable information like spectral acceleration which will be beneficial for geotechnical engineers for earthquake resistant design of structures like foundations, retaining walls, embankments at various locations of Kolkata city.

  • The period of spectral acceleration of 1995 Kobe motion is observed to coincide with natural period of the soil column and hence is an indication of the destruction that might occur if an earthquake of such high magnitude occurs in Kolkata city in future.

  • The magnitude of pile bending moment and pile head deflection is considerably increased in liquefiable soil layers due to the degradation in stiffness of the soil which reduces the shear strength of the soil significantly.

  • The depth of liquefiable soil layer has a profound impact on pile head deflection and pile bending moment. Maximum bending moment is observed to occur at the interface of the liquefiable and non-liquefiable soil layers. Further, bending moment is maximum when depth of the liquefiable soil layer is around 60% of the total pile length, i.e., L liq /l ratio is 0.6.

  • In addition to kinematic loadings, the influence of inertial loads should be considered for seismic analysis and design of piles passing through liquefiable soil.

Thus the present study is important for geotechnical engineers since it considers the influence of local soil sites for seismic analysis of pile foundations in Kolkata city and proper care should be taken for design of piles passing through liquefiable soil layers.