Response of saturated cohesionless soil subjected to irregular seismic excitations

Abstract

Irregular stress waves generated during an earthquake induce irregular shear stresses in the soil, which affect the dynamic behaviour of soil significantly. Laboratory investigations on the response of soils during real earthquake scenario are very limited. This paper reports about the response of saturated sandy soils subjected to irregular cyclic stress histories using stress-controlled cyclic triaxial tests. Tests were conducted at different relative densities (30–90%) and confining stresses (50–150 kPa) representing varying compactness of soil obtained from different confining depths. Cyclic loading was applied as per the three different earthquake motions considered having different peak ground accelerations (PGAs). The responses are presented in terms of stress–strain variations, excess pore-water pressure ratio and shear strain accumulation in the soil specimen. The results indicated that the accumulated shear strains and excess pore-water pressures get significantly affected by the increase in confining stress and simultaneous changes in the relative density. Further, the study emphasized that the strong motions scaled to the same PGA levels produce substantially differing soil responses due to the variation in the associated strong-motion parameters.

1 Introduction

Dynamic behaviour of soils and associated liquefaction aspects are very important considerations in earthquake geotechnical engineering. To study the dynamic behaviour and subsequent evaluation of dynamic soil properties, different laboratory tests were conducted using variety of test specimens and loading conditions (Seed and Idriss 1970; Hardin and Drnevich 1972; Iwasaki et al. 1978; Kokusho et al. 1982; Seed et al. 1986; Chang et al. 1989; Vucetic and Dobry 1991; Ishibashi and Zhang 1993; Stokoe et al. 1995; Sitharam and Govindaraju 2003). Most of the test results, reported in the studies, were evaluated using regular harmonic excitations as cyclic loading. However, it is very well established that the behaviour of soil under real earthquake excitation (irregular) is largely different in comparison with the regular harmonic excitations, as the former is associated with a wide range of frequency content and strong-motion parameters. Shear stresses induced by real-time strong motions are extremely irregular and possess erratic temporal variation of magnitude and frequency. Hence, it is very essential to study the behaviour of soil under real seismic excitations.

The dynamic behaviour of soils is influenced by various parameters, namely stress or strain levels (magnitude of earthquake), soil type, saturation state of soil and in situ stress conditions (Hardin and Drnevich 1972; Kumar et al. 2013). Several researchers have performed cyclic triaxial tests under variety of test conditions, subjected to regular harmonic excitations, to anticipate the effect of the above-mentioned parameters (Seed and Lee 1966; Dobry et al. 1982; Vucetic and Dobry 1988; Ladd et al. 1989; Ishihara 1996). However, instances of laboratory investigation of soil response, using real earthquake motions, are very limited (Ishihara and Yasuda 1972, 1973, 1975; Tsukamoto et al. 2004; Sawada et al. 2006). Due to the versatility in simulating medium-to-large strains in the soil sample, cyclic triaxial tests have extensively been used for investigating the dynamic behaviour of soil. Several researchers have evaluated dynamic properties of soil, considering shear strain up to 1%. However, it is quite common that the dynamic loading conditions, such as earthquakes, result in the generation of high shear strain (> 5%) in the soil (Chang et al. 1989; Sitharam and Govindaraju, 2003; Kirar and Maheshwari, 2013; Kumar et al. 2017). The north-eastern region of India is located in the highest seismically active zone (Zone V as per IS 1892:2002). It is observed that there are no available studies related to the behavioural response of the soils of this region subjected to real seismic motions or irregular excitations. Hence, the findings from this work will provide a significant contribution in the application of geotechnical earthquake engineering for the design of various important structures, namely bridges, dams, tunnels, urbanized habitats and the transportation conveyances. All of these structures built in this region require a thorough understanding of the ground response analysis, which can be successfully carried out with the knowledge base obtained from the reported and similar studies. Therefore, the importance of the present work is to emphasize the response of saturated sand under real earthquake motion and to provide the possible variation of shear strains and pore-water pressures for the consideration in design.

In the present study, response of Brahmaputra river sand (BS), collected from Northeast region of India, subjected to irregular seismic excitations (real earthquake motion) has been investigated through cyclic triaxial tests. The tests have been conducted at different effective confining stresses ($\sigma_{\text{c}}^{\prime } \, = \,50$, 100 and 150 kPa) on the specimens prepared at different relative densities (D_r = 30, 60 and 90%). The specimens were subjected to three different real earthquake excitations, namely 1995 Kobe (PGA = 0.834 g), 2001 Bhuj (PGA = 0.103 g) and 2012 Tezpur (Scaled PGA = 0.36 g) strong motions. The results are reported in terms of stress–strain variations, excess pore-water pressure and shear strain accumulations.

2 Previous studies

Using dynamic triaxial apparatus, Ishihara and Yasuda (1972) investigated the liquefaction behaviour of saturated cylindrical specimens of Niigata sand subjected to Niigata strong motion and compared the results obtained from stress-controlled tests conducted at a frequency of 1 Hz. It was also reported that the equivalent uniform shear stress about 57% of maximum shear stress was enough for the liquefaction during 20 cycles of load applications. Based on the response of saturated sand, Ishihara and Yasuda (1973) reported that for liquefaction analysis, 47–65% of the maximum shear stress of the irregular stress history can be considered for 20 cycles of harmonic stress cyclic loading. Tsukamoto et al. (2004) have used irregular time histories in dynamic triaxial apparatus to evaluate the settlement of silty sand deposits and established a relation between the factor of safety against liquefaction and the induced maximum shear strain. A relationship between the post-liquefaction volumetric strain and the induced maximum shear strain was also established. Both the relationships were found to be significantly influenced by the soil density and the particle size distribution. Sawada et al. (2006) have also used irregular seismic excitation on anisotropically consolidated unsaturated sandy soil specimens in dynamic triaxial tests to measure the shear strains and volume changes during and after the dynamic excitation. In most of the studies, tests were conducted with different irregular excitations at either a particular relative density or a particular confining pressure. The present study deals with the simultaneous effect of relative density, confining pressure and varying magnitudes of strong motions on the response of soil in terms of accumulated shear strains and excess pore-water pressure.

Kramer (1996) reported that the strain- or stress-time history for a particular strong motion will be highly irregular, and the same would be instrumental in initiating liquefaction in a soil specimen, provided that a threshold strain is reached. Different soils have different limits of threshold strain, and the response to the varying numbers of stress cycles is also unique. Hence, it is imperative to understand the behaviour of different regional soils under irregular excitation for regional ground response analysis. It is a common practice for simplified analysis to consider a regular sinusoidal regular (sinusoidal) time history equivalent to the irregular excitation. In this regard, the effective strain or stress level of the equivalent regular motion is evaluated based on the maximum shear strain or maximum shear stress of an irregular motion. It is assumed that the computed effective strain or stress amplitude will provide the similar response, for specified number of cycles, as that of strong motion of a particular moment magnitude. In conventional cyclic triaxial apparatus, this exercise is carried out to evaluate the dynamic response (dynamic properties and liquefaction behaviour) of soils, considering the equivalent strain or stress cycles generated by a particular strong motion, due to its simplicity. A general practice is to consider the equivalent strain level as 0.65 times the maximum strain level (Kramer 1996). However, it has been shown in previous studies that the scalar multiplier for equivalency varies for different types of soils and is responsible in affecting the dynamic response of regional soils. Therefore, the work presented in this manuscript not only specifically addresses the dynamic response of the regional cohesionless soil under irregular excitation, but would also provide the avenue to use the proper magnitude for the determination of equivalent strain to be used for the assessment of the dynamic properties of the region-specific cohesionless soils.

3 Experimental investigation

3.1 Soil characteristics

Brahmaputra river sand (BS), collected from Guwahati region, Assam (India), has been used for the present study. The particle size distribution of the sand, obtained from dry sieve analysis (as per IS: 2720-IV), is shown in Fig. 1a which reflects that the soil falls within the zone of severely liquefaction-susceptible soils (Tsuchida 1970; Ishihara et al. 1980; Xenaki and Athanasopoulos 2003). The test material has been classified as poorly graded sand (SP) as per ASTM D2487 (2006). Figure 1b shows the field emission scanning electron microscope (FESEM) image of the sand which indicates that the soil particles are sub-angular and possess surficial roughness. The specific gravity of the sand (as per IS: 2720-III) has been found to be 2.7. The minimum and maximum dry unit weights (as per IS: 2720-XIV) were found to be 13.85 and 16.84 kN/m³, respectively. The physical properties of the sand are summarized in Table 1.

Fig. 1

a Particle size distribution and b FESEM image of BS

Table 1 Physical properties of sand

3.2 Testing apparatus

An automated pneumatic controlled cyclic triaxial testing system was used in the present study, as shown in Fig. 2. It consists of a 100-kN capacity loading frame fitted with an actuator (± 15 mm displacement range) operating over a frequency range of 0.01–10 Hz, a triaxial cell having (2000 kPa capacity) and an air compressor having a maximum capacity of 800 kPa. Instrumentations available with the apparatus are: two linear variable differential transducers (LVDTs); one submersible load cell of capacity 25 kN; three pressure transducers of 1000 kPa capacity to measure cell pressure, back pressure and pore-water pressure, and one automatic volume change (AVC) measuring device. One LVDT is attached with the actuator having a measuring range of 0–30 mm, while the other LVDT (i.e. external LVDT as termed in the present manuscript) is placed at the top of triaxial cell having a measuring range of 0–50 mm (marked in Fig. 2). It has been mentioned in the manuscript that the triaxial cell is pneumatic controlled and that the generated compressed air pressure is transferred as applied displacement to the actuator through the air ducts. The LVDT attached to the apparatus is intermittently used to check whether the intended load is properly transferred to the actuator. If the LVDT response is found improper, further calibration of the actuator is carried out. The other LVDT that is attached to the top of the triaxial cell is the primary LVDT used in the present study to measure the displacement response of the soil sample during the test. The testing is controlled by a compact dynamic controller (CDC) unit, which acts as a communicating medium between the software and the testing apparatus.

Fig. 2

Cyclic triaxial set-up and its components

3.3 Testing procedure

Cyclic triaxial apparatus was used for the experimental investigations. The details of instrumentations available with the apparatus are described in Kumar et al. (2017). All the tests were conducted on the remoulded cylindrical soil specimens of dimensions 70 mm diameter and 140 mm height. Dry pluviation technique was adopted to prepare the cylindrical specimens of BS (ASTM D5311 2011). A nominal vacuum pressure of 15–20 kPa has been used to maintain verticality of the specimen (Ishihara et al. 1978). In order to achieve a quick saturation and a substantial replacement of the pore-air, carbon dioxide (CO₂) was flushed through the specimen, for 10–15 min, with a pressure lesser than the applied cell pressure, followed by flushing with de-aired water. To attain the saturation, the cell pressure (CP) and back pressure (BP) were then gradually increased in stages, while maintaining an almost constant differential pressure of 10 kPa and constant monitoring of the pore pressure parameter (B) after each increment of CP. The saturation process was terminated as the back pressure (BP) reached 200 kPa, and the corresponding B-value was obtained to be greater than 0.96 (Kumar et al. 2015). The test specimens were then isotropically consolidated to a targeted effective confining stress and subsequently subjected to irregular seismic loading under undrained condition.

3.4 Irregular seismic excitations

Three different irregular stress-time histories, computed from the acceleration histories of 2001 Bhuj (PGA = 0.103 g), 2012 Tezpur (Scaled PGA = 0.36 g) and 1995 Kobe (PGA = 0.834 g) strong motions, have been used to evaluate the dynamic response of soil. Figure 3 represents the acceleration histories and their corresponding frequency content obtained by the fast Fourier transformations (FFTs) of the strong motions. Frequency-domain representation indicates the variation of energy content over a frequency band. It is observed that the significant energy content of Bhuj and Kobe strong motions is congregated over a frequency band of 0.5–4 Hz, while the same for Tezpur motion is found to be at 2–15 Hz. Apart from these acceleration histories of different PGAs, all the ground motions were scaled to similar PGA levels (0.103 and 0.36 g) for few of the tests. Table 2 presents the ground-motion parameters of the different earthquake motions scaled to PGA of 0.103 g. It can be observed from the table that the three ground motions are different in terms of predominant period (fundamental frequency), duration and energy levels.

Fig. 3

Acceleration-time histories and frequency-domain representation of the input motions

Table 2 Ground-motion parameters of different earthquake motions scaled to PGA of 0.103 g

Cyclic loading has been applied on the test specimen in terms of cyclic stress history, which was evaluated based on the accelerations and the effective stress on the sample. It was considered that the specimens tested at different effective confining stresses ($\sigma_{\text{c}}^{\prime }$) are assumed to be located at different confining depths below the ground level. To evaluate the irregular shear stress (τ) history induced by a real-time earthquake at any depth ‘z’ within a soil deposit, the approach proposed by Seed and Idriss (1971), as exhibited by Eq. (1), has been adopted.

$$ \tau = \frac{{{\text{acc}} . { }(g)}}{g} \times \sigma_{\text{v}} \times r_{\text{d}} $$

(1)

$$ r_{\text{d}} = 1.0 - 0.00765z;\,\,{\text{for}}\,\,z \le 9.15\,{\text{m}} $$

(2a)

$$ r_{\text{d}} \, = \,1.174 - 0.0267z;\,\,{\text{for}}\,\,9.15 \le z \le 23\,{\text{m}} $$

(2b)

$$ \sigma_{\text{d}} = 2 \times \tau = 2 \times \frac{{{\text{acc}} . { }(g)}}{g} \times \sigma_{\text{v}} \times r_{\text{d}} $$

(3)

where τ is shear stress, acc. (g) is acceleration-time history, σ_v is total overburden vertical stress and r_d is the stress reduction factor (Eq. 2; Youd et al. 2001) accounting for the deformable characteristics of the soil specimen. The deviatoric stress (σ_d) history at confining depths of 2.5, 5.0 and 10 m, to be applied during the experiment, was evaluated from the strong-motion stress histories as per Eq. (3). Cyclic stress ratio (CSR) of the cyclic loading can be evaluated as $t/\sigma_{\text{c}}^{\prime }$. Figure 4a shows the typical applied σ_d time histories of different ground motions with different PGAs for a test specimen at effective confining stress of 100 kPa, i.e. at an approximated confining depth of 5.0 m. Figure 4b presents the measured σ_d time histories corresponding to the input σ_d as presented in Fig. 4a. It is observed that the measured σ_d is relatively lesser (nearly 5–10%) than the input σ_d, which is attributed to the energy loss caused by the soil deformation.

Fig. 4

Typical variations for specimens tested at $\sigma_{\text{c}}^{\prime }$ = 100 kPa and subjected to different earthquake excitations: a input deviatoric stress and b measured deviatoric stress

4 Results and discussion

In addition to the nature of the seismic excitation, relative compactness and confinement of soil also govern the dynamic soil behaviour. Three strong-motion excitations have been chosen (Bhuj, Tezpur and Kobe motions as described earlier) to study the behaviour of BS specimens under irregular seismic excitations at different relative densities and confining depths. Stress-controlled cyclic triaxial tests (undrained condition) conducted on BS specimens are summarized in Table 3. Test specimens were prepared at different relative densities (30, 60 and 90%) and were subjected to irregular excitations at different effective confining stresses, i.e. 50, 100 and 150 kPa. Cyclic loading has been applied on the soil specimens in the form of irregular excitations as explained earlier. Test results were presented in terms of developed shear strains and excess pore-water pressures. Excess pore-water pressures (u_e) are represented as excess pore-water pressure ratio ($r_{\text{u}} \, = \,u_{\text{e}} /\sigma_{\text{c}}^{\prime }$).

Table 3 Investigation parameters of irregular excitations

4.1 Effect of relative density

Relative density, representing the compactness of soil specimen and indicative of the degree of inter-particle interaction, plays a major role in defining the dynamic behaviour of cohesionless soils. At different D_r, Fig. 5 presents the variations of shear stress and shear strain obtained during cyclic triaxial tests subjected to Bhuj (Fig. 5a), Tezpur (Fig. 5b) and Kobe strong motions (Fig. 5c). Figure 5a shows that for BS specimens subjected to the Bhuj motion, the shear modulus (G₁ = initial secant modulus) increases with increase in D_r. Similar responses were observed when BS specimens were subjected to the Tezpur and Kobe motions, as shown in Fig. 5b, c, respectively.

Fig. 5

Stress–strain relationship obtained during irregular loading of BS specimen at different D_r: a Bhuj, b Tezpur and c Kobe strong motions

Figure 6 presents the effect of relative density on the onset of liquefaction of the BS specimens at a confining stress of 100 kPa and subjected to different stress histories. From the responses of specimens subjected to Bhuj motion (Fig. 6a), it can be observed that r_u decreases with the increase in D_r; maximum r_u values of 0.13, 0.09 and 0.08 are obtained for test specimens at D_r of 30, 60 and 90%, respectively. Owing to the higher ratio of solid particles in a fixed volume representing a denser state, the quantity of induced pore-water pressure decreases, and hence, a reduced excess pore-water pressure ratio is observed during the dynamic shaking. As a consequence, tests conducted at higher relative densities revealed lesser shear strain accumulation (< 0.04%). Similar observations are illustrated in Fig. 6b, c, when the BS specimens were subjected to Tezpur and Kobe motions, respectively, although they are depicting larger strains and excess pore-water pressure ratios. When specimens are subjected to Bhuj motion, no liquefaction was observed (r_u ≪ 1), however, due to the increase in PGA, for both Tezpur and Kobe motions, BS specimens exhibited liquefaction (r_u = 1). Further, it can be observed that the accumulation of excess pore-water pressure is of sudden onset when subjected to Tezpur and Kobe motions, and such behaviour results in sudden and quick liquefaction phenomena. This feature is attributed to the impulsive nature of Tezpur and Kobe strong motions, where the PGA is reached suddenly, unlike the Bhuj motion which shows a gradual attainment of PGA (Fig. 3 ).

Fig. 6

Strain accumulation and excess PWP ratio in BS specimens of different D_r, confined at 100 kPa, and subjected to a Bhuj, b Tezpur and c Kobe strong motions

Maximum shear strains were observed to be in the range of 0.5–0.7% for the specimens subjected to Tezpur motion (Fig. 6b), while the range is 5–15% for Kobe motion (Fig. 6c), which is due to an enhancement in PGA value. It has also been noticed from Fig. 6 that the strain accumulation and rise of pore pressure reduced with the increase in D_r, for a particular $\sigma_{\text{c}}^{\prime }$. Thus, it is evident that relative density plays a significant role in governing the onset of liquefaction of cohesionless specimens subjected to seismic shaking. Table 4 enumerates the findings of the experimental investigations conducted on BS specimens prepared at different relative densities. It can be noticed that for a given earthquake motion, variation in the relative density of the specimen showed a significant difference in maximum shear strain values; the difference is being larger (3 times) for higher PGA motion (Kobe). For BS specimens subjected to Tezpur motion, it can be observed that the specimen prepared at D_r = 90% attained a near-liquefaction state with maximum r_u, value (r_u,max) of 0.9, whereas the specimen prepared at D_r = 60% showed a distinct onset of liquefaction, although the difference in the maximum shear strains (γ_max) is marginal, 0.5 and 0.52, respectively. From this observation, it may be stated that the threshold shear strain manifesting the onset of liquefaction for these sets of tests may be about 0.5%, and the corresponding maximum cyclic stress ratio (CSR_max) is being 0.32. Table 4 clearly indicates that higher relative density (compactness) of the sample results in the reduction in excess pore-water pressure (PWP) ratio, and the effect is being more pronounced at lower PGA level. At higher PGA levels, though there is a significant reduction in the resulting shear strains, liquefaction condition becomes inevitable since the excess pore-water pressure ratio reaches 1.0. The table also enlists the shear modulus (G) of different specimens, evaluated as a ratio of the maximum shear stress (applied) to the maximum shear strains (observed) for a given test, and as obvious, the shear modulus is found to increase with the increase in the relative density.

Table 4 Summary of investigations on BS specimens prepared at different D_r

4.2 Effect of confining stress

Effect of confining stress illustrates the resulting stress ratio (${\text{CSR}}\, = \,\sigma_{\text{d}} /2\sigma_{\text{c}}^{\prime }$) due to a given earthquake motion on the soil specimens corresponding to different confining depths. In this attempt, the test specimens prepared at D_r = 30% are considered for different confining depths. Figure 7 illustrates the variations of shear stress and shear strain obtained during cyclic triaxial tests subjected to Bhuj (Fig. 7a), Tezpur (Fig. 7b) and Kobe motions (Fig. 7c), at different $\sigma_{\text{c}}^{\prime }$ and D_r = 30%. It is seen that the shear modulus (G₁ = initial secant modulus) increases with the increase in $\sigma_{\text{c}}^{\prime }$, when BS specimens are subjected to the Bhuj motion (Fig. 7a), whereas when BS specimens are subjected to the Tezpur motion, it showed decreased value of G₁ with increasing $\sigma_{\text{c}}^{\prime }$ (Fig. 7b). This is attributed to the high shear strain mobilized at $\sigma_{\text{c}}^{\prime }$ = 100 and 150 kPa in comparison with that mobilized at $\sigma_{\text{c}}^{\prime }$ = 50 kPa, as shown in Fig. 7b. Figure 7c represents that the responses are similar to the Tezpur motion. Moreover, Fig. 7c illustrates that shear strain in the soil specimen, at similar numbers of cycles, increases with the increase in $\sigma_{\text{c}}^{\prime }$.

Fig. 7

Stress–strain relationship obtained during irregular loading of BS specimen at different $\sigma_{\text{c}}^{\prime }$: a Bhuj, b Tezpur and c Kobe strong motions

Figure 8 shows the results of such specimens subjected to different irregular excitations. Figure 8a illustrates the accumulation of shear strain (γ) and development of r_u in the BS specimen subjected to Bhuj motion (PGA = 0.103 g). It is observed that the γ_max is nearly 0.01, 0.03 and 0.03% at $\sigma_{\text{c}}^{\prime }$ of 50, 100 and 150 kPa, respectively. An increase in the confining depth implies that the sample has been subjected to higher shear stress, which resulted in increased shear strain. Maximum excess pore pressures observed are very low (r_u,max = 0.1 ≪ 1), which is due to the low CSR values (0.097, 0.092 and 0.078 at $\sigma_{\text{c}}^{\prime }$ = 50, 100 and 150 kPa due to 0.103 g PGA). It is also observed that the specimens at different $\sigma_{\text{c}}^{\prime }$ = 100 and 150 kPa showed identical response in terms of shear strain and pore pressure, which is again attributed to the low PGA level.

Fig. 8

Strain accumulation and excess PWP ratio histories of BS specimens at D_r = 30% and different $\sigma_{\text{c}}^{\prime }$ for a Bhuj, b Tezpur and c Kobe strong motions

BS specimens at different depths subjected to scaled Tezpur motion (PGA = 0.36 g) with CSR ranging between 0.28 and 0.35 exhibited higher peak shear strains in the range of 0.06–1.8% (Fig. 8b). Specimens subjected to $\sigma_{\text{c}}^{\prime }$ of 100 and 150 kPa exhibited a clear onset of liquefaction as r_u reaches nearly 1, while it is significantly lesser (r_u,max = 0.25 ≪ 1) for 50 kPa confining pressure. This behaviour indicates that a homogeneous BS stratum in the field located at a particular depth, corresponding to the above-stated range of confining stresses (100–150 kPa), is likely to liquefy. It is also observed that as the specimen liquefies, a significant residual shear strain is manifested indicating the strength reduction in soil. Similar response of larger residual shear strain (> 6%) has also been reported from ground response analysis studies using SHAKE and DEEPSOIL (Suetomi and Yoshida 1998; Kumar et al. 2014a, b; Singhai et al. 2016). As indicated in Fig. 8c, when subjected to Kobe motion (PGA = 0.834 g with CSR range of 0.65–0.80), BS specimens at any of the confining pressures exhibited r_u = 1 and substantial residual shear strain (> 5%), thus clearly exhibiting the occurrence of liquefaction in the specimen. From the above illustration, it can be stated that the behaviour of BS specimens at different confining pressures is indicative of their supposed behaviour at different depths in the field subjected to strong motions.

Table 5 summarizes the results demonstrating the effect of confining pressures (depth) and PGA levels of chosen strong motions. It can be stated that the BS specimens will manifest the onset of liquefaction behaviour beyond a PGA value of 0.36 g. It has been noticed that the developed maximum shear strain (γ_max) exceeds 0.5% for soils exhibiting liquefaction. Furthermore, at particular D_r and confining depth, the input motion of higher PGA reflects higher strain accumulation. Table 5 also illustrates that for a particular D_r, generation of excess PWP or liquefaction susceptibility of soil increases with the increase in confining depth for all input motions.

Table 5 Summary of investigations on BS specimen subjected to different $\sigma_{\text{c}}^{\prime }$

Based on the individual effects of relative density and confining pressure as portrayed in earlier figures (Figs. 6, 8), Fig. 9 depicts the response of BS specimens, when confining pressure and relative density increased simultaneously such that higher relative density of the test specimen was considered at a larger depth (for example, 150 kPa and 90% at 10 m depth). Although the situation described herein may not be necessarily universal, such occurrences of shallower sandy stratum of lesser relative density overlying the denser and deeper sandy stratum are quite common for sedimentary beds. Adoption of such parameters might be realistic to the field scenario, and it can be noted that, in this case as well, there is an increment in the γ and r_u values with the increase in confining depth, although the relative increment is lesser compared to the homogeneous soil consideration (Fig. 8a). Thus, from the above-illustrated results, it can be concluded that in comparison with the relative density, confining stress has a more pronounced effect on the enhancement of the liquefaction susceptibility of soil. It is to be understood that soils at higher confining depth will require more deviatoric stress to achieve the onset of liquefaction (as indicated by laboratory investigations). However, such high deviatoric stress at higher depths may not be available in the field when subjected to strong motion, and the zone of liquefaction is thus restricted more at shallower depths. The confining depths considered in the present study, as such, pertain to shallower depth (≤ 15 m) and hence do not violate the field experience.

Fig. 9

Strain accumulation and excess PWP ratio histories of BS specimens at different D_r and $\sigma_{\text{c}}^{\prime }$, subjected to Bhuj motion of PGA 0.103 g

4.3 Effect of similarly scaled strong motions

This section reports the results of the investigations conducted on the BS specimens with different strong motions scaled to specific PGA (0.103 g and 0.36 g). Figure 10 depicts the effect of the three earthquake motions (with same PGA) on the BS specimens prepared at D_r = 60% and subjected to $\sigma_{\text{c}}^{\prime }$ = 100 kPa. It was observed that none of the ground motions, when scaled to PGA = 0.103 g, could initiate liquefaction in the BS specimen, while liquefaction was observed in the specimens for any of ground motions scaled to 0.36 g. Though the PGA is same, the specimens exhibited different shear strain levels under different excitations. The overall behaviour of the specimens can clearly be observed from the summary of the results presented in Table 6. It can be noted that BS specimens, prepared at particular D_r and $\sigma_{\text{c}}^{\prime }$, subjected to the similarly scaled strong motions result in different magnitudes of maximum shear strain due to the variation in the associated strong-motion parameters. From the table, it can be stated that Tezpur motion shows lowest values of γ_max and r_u,max in comparison with the Bhuj and Kobe motions, the highest magnitudes being manifested by the Bhuj motion. The reason for such behaviour is attributed to the varying strong-motion parameters (Table 2) such as Arias intensity, specific energy density, which exhibited similar trend of variation as that observed from the response of test results. Although other strong-motion parameters such as predominant period, mean period, bracketed duration and significant duration (Table 2) might have a direct or indirect effect on the observed responses; however, the study of their individual effects is outside the scope of the present attempt.

Fig. 10

Strain accumulation and excess PWP ratio in BS specimens prepared at D_r = 60% and $\sigma_{\text{c}}^{\prime }$ = 100 kPa, subjected to scaled earthquake motions of PGA 0.103 and 0.36 g

Table 6 Summary of results subjected to ground motions with same PGA

4.4 Stress cycles and liquefaction initiation

Since the actual acceleration-time history of an earthquake motion is in irregular form, the shear stresses induced during an earthquake in the ground vary randomly both in magnitude and in frequency (Ishihara and Yasuda 1972; 1975). For accounting the effects of irregular stress history, the laboratory experimental investigations are carried out considering an average uniform shear stress in the regular harmonic cycles which is equivalent to 65% of the maximum shear stress of strong ground motions (Seed and Idriss 1971). The regular sinusoidal deviatoric stress (σ_d), applied on the soil specimens during cyclic load, is derived from Eq. (4).

$$ \tau_{\text{cyc}} = 0.65 \times \frac{{a_{\hbox{max} } }}{g} \times \sigma_{\text{v}} \times \, r_{\text{d}} \Rightarrow \tau_{\text{cyc}} = 0.65 \times \tau_{\hbox{max} } $$

(4)

In this equation, τ_cyc is the average cyclic shear stress, and a_max is the peak acceleration of the applied acceleration-time history. To obtain the CSR values, the average cyclic shear stress was normalized with effective confining stress, i.e. ${\text{CSR}} = \tau_{\text{cyc}} \sigma_{\text{c}}^{\prime } \, = \,\sigma_{\text{d}} /2\sigma_{\text{c}}^{\prime }$. In the present study, based on the PGA of irregular excitation and Eq. (4), the CSR values for regular stress cycle were found to be in the range of 0.05–0.5.

Figure 11a presents the variations in r_u for BS specimens (D_r = 30%; $\sigma_{\text{c}}^{\prime }$ = 50, 100 and 200 kPa) at different CSR ($\sigma_{\text{d}} /2\sigma_{\text{c}}^{\prime }$) values ranging from 0.05 to 0.3. It was observed that the specimens with CSR = 0.2 exhibit initiation of liquefaction 10, 4 and 3 numbers of stress cycles (N) when tested at $\sigma_{\text{c}}^{\prime }$ = 50, 100 and 200 kPa, respectively, whereas for CSR = 0.3, the same is reflected at N = 2 and 1.5 for $\sigma_{\text{c}}^{\prime }$ = 50 and 100 kPa, respectively. Figure 11a also illustrates that the liquefaction did not initiate for CSR = 0.05 and 0.1 till 100 cycles at $\sigma_{\text{c}}^{\prime }$ = 50 kPa. The specimens showed liquefaction at N = 122 and 62 for $\sigma_{\text{c}}^{\prime }$ = 100 and 200 kPa, respectively, when subjected to CSR = 0.1. Thus, it can be stated that the liquefaction in BS specimens is more pronounced with the increase in $\sigma_{\text{c}}^{\prime }$. This is attributed to the simultaneous increase in γ and σ_d on the soil specimens with the increase in $\sigma_{\text{c}}^{\prime }$. Similar observations have been reported by Simatupang and Okamura (2017).

Fig. 11

Comparative variation of a r_u and b γ in BS specimens prepared at D_r = 30% and tested under different $\sigma_{\text{c}}^{\prime }$ and CSR

Figure 11 illustrates the accumulation of γ during cyclic loading at D_r = 30% and different $\sigma_{\text{c}}^{\prime }$, subjected to different CSRs ranging from 0.05 to 0.3. It reflects that for CSR = 0.2, the accumulated γ was 0.45, 2.5 and 6.8% when the initiation of liquefaction occurred at $\sigma_{\text{c}}^{\prime }$ = 50, 100 and 200 kPa, respectively, whereas for CSR = 0.3 the accumulated γ was greater than 1% for $\sigma_{\text{c}}^{\prime }$ = 50 and 100 kPa. It also shows that the liquefaction was not observed at CSR = 0.05 and 0.1 for $\sigma_{\text{c}}^{\prime }$ = 50 kPa; the accumulated γ was nearly equal to 0.02% till 500 cycles.

Figure 12 presents the variation in CSR with number of stress cycles required to liquefy (N_L) the BS specimens prepared at D_r = 30–90% and subjected to $\sigma_{c}^{\prime } \, = \,100$ kPa. It shows that, for any constant CSR (for example, CSR = 0.2), the liquefaction resistance increases with the increase in D_r (i.e. from D_r = 20–90%). It is also seen that, for any constant D_r, the liquefaction resistance decreases with the increase in CSR (i.e. from CSR = 0.15–0.3). Thus, it can be stated that the BS soil in loose state is highly vulnerable to liquefaction even at low magnitudes of shaking (based on the CSR value), whereas it is relatively resilient at higher densities.

Fig. 12

Variation of CSR with N_L for BS specimens subjected to $\sigma_{\text{c}}^{\prime }$ = 100 kPa

5 Conclusions

The present study illustrates the effect of irregular seismic excitations (Bhuj, Tezpur and Kobe motions) on the dynamic response of Brahmaputra sand. The generation of excess pore-water pressure, accumulation of shear strain due to cyclic loading and the onset of liquefaction was observed to be significantly affected by the state of the specimen manifested in terms of relative density and confining depth. Based on the present study, the following conclusions can be drawn:

• 1. Due to higher inter-particle interaction and degree of compactness, an increase in the relative density leads to the decrement of the accumulated shear strain and the excess pore-water pressure ratio. The effect on accumulated shear strain is more prominent for higher PGA of the input motions, while lower PGA induces prominent effect on the developed excess pore-water pressure.

• 2. BS specimens subjected to any of the strong motions exhibited onset of liquefaction when the maximum shear strain exceeded 0.5%, and hence, this magnitude of shear strain can be stated as the threshold shear strain for liquefaction.

• 3. The time taken for the onset of liquefaction is governed by the nature of the applied strong motion. An impulsive strong motion, e.g. Tezpur/Kobe motion, will exhibit a quick liquefaction phenomenon, while Bhuj motion will exhibit a gradual attainment of the onset due to its gradual rise towards PGA.

• 4. An increase in the confining depth leads to the enhancement in the accumulated shear strain and excess pore-water pressure ratio and hence increases the liquefaction susceptibility of the sample. Such effect gets more pronounced with higher PGA of the input motion.

• 5. Simultaneous increment of confining depth and relative density of the sample exhibited that the confining depth has more dominant effect on the enhancement of liquefaction susceptibility of BS specimens in comparison with the effect of relative density.

• 6. Based on the results of BS specimens subjected to similarly scaled strong motions, it can be stated that specimens at any relative density will liquefy under the following optimum conditions: PGA > 0.36 g, CSR > 0.3 and γ_max > 0.5%.

• 7. Cohesionless soil specimens subjected to similarly scaled strong motions exhibits varying accumulated shear strains and excess pore-water pressure ratios due to the variation of the other associated strong-motion parameters such as arias intensity, specific energy density, predominant period, mean period, bracketed duration and significant duration. Individual and simultaneous effect of the strong-motion parameters is bound to affect the dynamic response of cohesionless soil and needs to be studied in rigorous and minute detail.

• 8. Liquefaction in BS specimens is more pronounced with the increase in $\sigma_{\text{c}}^{\prime }$. This is due to the substantial increase in σ_d and γ, with the increase in $\sigma_{\text{c}}^{\prime }$; consequently, the number of cycles required for the onset of liquefaction decreased. It is also seen that the liquefaction resistance increases with the increase in D_r.