Influence of Shaking Sequence on Liquefaction Resistance and Shear Modulus of Sand Through Shaking Table Tests

Abstract

Case histories have shown that the liquefaction behavior of soils can differ depending on the pre-seismic history of sites. Assessing the shear modulus in soils subjected to seismic events is critical for advancing the fundamental understanding of soil behavior and enhancing the accuracy of soil modeling applications. This paper aims to study the effect of small and large pre-shaking sequences on the liquefaction resistance and shear modulus of sand through shaking table tests. The experimental results indicated that small shakings increase liquefaction resistance and shear modulus. Although large shakings leading to liquefaction cause significant densification, they significantly reduce the liquefaction resistance and shear modulus of sand at shallow depths due to the upward water flow during excess pore water pressure dissipation. The high upward flow of water during liquefaction changes the soil structure and increases the horizontal displacement of densified soil in the subsequent shaking. The amplification factor of acceleration was found to be primarily influenced by the excess pore water pressure generated in the soil instead of its relative density at the start of shaking. This paper presents the variations in Ru with shear strain and the relationship between a normalized shear modulus and shear strain considering the pre-shaking history of sand for shallow depths.

1. Introduction

Liquefaction is a catastrophic phenomenon that occurs in sandy deposits during earthquakes. Due to its severe impact on infrastructure and potential loss of human life, extensive research has been conducted in the geotechnical engineering field [1,2,3,4,5]. Liquefaction is caused by a rapid generation of pore water pressure in saturated sandy soils and results in shear strength reduction. The devastating consequence of liquefaction observed during previous earthquakes, including the 1969 Niigata earthquake in Japan, the 1971 San Fernando earthquake, the 2010 Canterbury earthquakes in New Zealand, and the 2018 Sulawesi earthquake in Indonesia, has resulted in extensive investigations to provide insights into the liquefaction mechanism [6]. Traditional approaches employ semi-empirical methods using field data obtained from Standard Penetration Tests (SPTs), Cone Penetration Tests (CPTs), shear wave velocity (Vs), and dilatometer tests (DMTs) to estimate the liquefaction potential of sandy soils [7,8,9,10].

Field observations revealed that the pre-seismic history of sites can influence the liquefaction potential of soil in subsequent earthquakes. Two cases in California with similar geotechnical conditions (in terms of soil type, shear wave velocities, CPT tip resistance, and groundwater level) showed different responses to comparable earthquakes. Both sites were estimated to have undergone liquefaction based on the liquefaction assessment charts [11]. Although Treasure Island F.S. liquefied during the 1989 Loma Prieta earthquake (Mw = 6.9), the 2010 El-Mayor Cucapah earthquake (Mw = 7.2) did not cause liquefaction in Wildlife, indicating higher liquefaction resistance. The main reason behind this discrepancy was the number of previous earthquakes at each site. The Wildlife site had experienced about 70 earthquakes that could cause excess pore pressure, while only two earthquakes had occurred at Treasure Island F.S. [12]. Similarly, sites exposed to a series of pre-seismic events during the 2011 Great East Japan Earthquake showed higher liquefaction resistance [13].

On the other hand, there exists the potential for a recurrence of liquefaction during aftershocks. For instance, Japan experienced re-liquefaction during a 7.1-magnitude earthquake after a devastating 9-magnitude earthquake [14]. Repeated liquefaction at the same sites in Greece within one week during the 2014 Cephalonia Earthquakes was reported [15]. It has been highlighted that a sequence of earthquakes could cause liquefaction at a site during multiple sequential events, as observed in the 2010 Chile and the 2010–2011 Canterbury Earthquakes [14,16].

Numerous experimental studies have been performed to assess the influence of pre-seismic history on the liquefaction potential of fine granular soils. Pre-shakings have typically been categorized into two groups, small and large. In general, small (weak) pre-shaking refers to a seismic event that is insufficient to cause liquefaction, while large (strong) pre-shaking involves applying a higher level of seismic event that either causes liquefaction (Ru = 1) or results in a shear strain of 5% in double amplitude or 3.75% in single amplitude [17]. Several investigations have indicated that small pre-shaking can improve liquefaction resistance, whereas large pre-shaking can significantly reduce the resistance [18,19]. However, further studies revealed that large pre-shaking can improve the liquefaction resistance of dense soils [20]. The number of shakings used as pre-shaking histories in the aforementioned studies were limited.

The effect of pre-shaking on relative density (Dr), shear wave velocity (Vs), cone tip resistance (qc), and liquefaction resistance has been investigated through shaking table and centrifuge experiments in recent studies. Ha et al. [21] conducted shaking table tests on five different sands to study the role of gradational characteristics on re-liquefaction resistance. It was detected that re-liquefaction resistance is mainly governed by the permeability and compressibility (D₁₀ and Dr) of sands. Shaking table tests conducted by Ecemis et al. [22] demonstrated that both the relative density and the consolidation characteristics (Cv) of sand play a crucial role in re-liquefaction resistance. Using centrifuge tests, El-Sekelly et al. [23] indicated that liquefaction can destroy the pre-seismic history of soils. However, a few small shakings can recover and increase the liquefaction resistance of liquified soils. Darby et al. [24] applied multiple shaking events to loose and dense samples to assess liquefaction and cone penetration resistance in centrifuge tests. They found that the cyclic resistance ratio and cone tip resistance progressively increase due to prior shakings and recurrent liquefaction events. According to Dobry et al. [25], although small pre-shakings significantly improve the tip resistance, the increase in shear wave velocity is not notable. Wang et al. [26] observed that although large pre-shakings increase the relative density of soils, the cone resistance decreases due to disturbance of the soil fabric after the shakings. The existing literature predominantly focuses on the impact of pre-shaking history on liquefaction resistance in fine-grained soils, with an emphasis on relative density (Dr) as the primary variable. However, recent studies highlight the importance of additional factors such as the number of shakings and the sequence of multiple shakings in influencing soil liquefaction behavior [23,25].

The stress–strain behavior of soils plays a key role in the site’s response to earthquakes. Numerical soil models rely on the variations in stress–strain behavior as fundamental input parameters for dynamic analyses. For the seismic design of geotechnical structures, understanding the shear modulus of soils in a strain range of 0.0001–1% is crucial [27]. The shear modulus is defined as the slope of a secant line on a stress–strain loop [28]. Many investigations have employed cyclic triaxial, resonant column, or cyclic simple shear tests to obtain the stress–strain behavior of soils using element tests [28,29,30]. It is possible to evaluate the stress–strain behavior of soil without solely relying on element tests. Acceleration data recorded at various depths within the soil profile can provide valuable insights into its dynamic behavior [31]. However, limited studies used physical model tests to determine the shear modulus in saturated soils, especially in the context of the pre-seismic history of soils.

In conventional liquefaction studies, soils are typically examined in their virgin state, with no consideration of prior loading or seismic history. However, under real-world seismic conditions, sites are often subjected to a sequence of earthquake events, including foreshocks and aftershocks. Within the context of geotechnical engineering, resilience refers to the capacity of soil and soil-structure systems to endure and recover from cyclic or repeated seismic loading. Unlike traditional design methodologies that prioritize ultimate strength or predefined safety margins, resilience-oriented approaches emphasize the sustained functionality and performance of the system following one or more seismic events. In this study, soil resilience is assessed through the lens of pre-shaking history, acknowledging that prior seismic excitations can modify the soil structure, densification, and stiffness characteristics.

This paper seeks to contribute further insights by examining the effect of the shaking sequence on the liquefaction behavior of saturated sand, considering an increase in soil relative density. Moreover, an effort has been made to study how the sequence of small and large shakings influences soil stiffness (the shear modulus, G). To achieve this, a sequence of shaking table tests was conducted on loose sand (Dr = 30%), comprising 19 small and large shakings. Enough time was considered between shakings for excess pore water pressure dissipation. Liquefaction resistance was analyzed based on pore pressure transducer readings while acceleration data was utilized to drive the stress–strain behavior and amplification factor of sand during shaking.

2. Experimental Program

2.1. Apparatus

This study was conducted using a 1 g shaking table facility at Southern Illinois University. The shaking table was equipped with a hydraulic actuator capable of generating unidirectional stroke displacements of up to 25 cm (ranging from −12.5 cm to +12.5 cm), with a maximum load capacity of 150 kN [32]. Sample preparation was carried out in a laminar box measuring 1.22 m in length, 0.51 m in width, and 0.91 m in height. The box consisted of fourteen horizontally arranged rectangular hollow aluminum tubes, which allowed for relative movement through a roller-based system (Figure 1).

2.2. Sample Preparation and Model Instrumentation

The sand used in this study (Figure 2) is classified as poorly graded sand (SP) based on the Unified Soil Classification System (USCS), with properties of D₁₀ = 0.16 mm, D₅₀ = 0.25 mm, and G_S = 2.66. A sandy specimen with a thickness of 75 cm was prepared at an approximate relative density of 30% using a sand raining technique. Various transducers, as illustrated in Figure 3, were employed in the model test to record acceleration (ACC1–ACC6), pore water pressure (PP1–PP5), settlement (LVDT1), and horizontal displacement (LVDT2–LVDT4) at different depths. To contain the sand sample, a geomembrane bag was employed. The bottom of the laminar box was covered with an 80 mm thick layer of compacted clean gravel to ensure uniform water penetration during saturation. After sample preparation, CO₂ was introduced to replace trapped air in the pores, enhancing the final saturation level of the sample. Water was then slowly released from the bottom of the sample through the gravel layer at a controlled rate, following procedures outlined in previous research [26,33].

2.3. Seismic Events

The experiment involved subjecting the sample to two distinct seismic events, each with a frequency of 2 Hz, applied in a specific sequence. The effect of loading frequency on liquefaction resistance is a controversial topic. Due to discrepancies in research findings regarding frequency effects, this study selected a loading frequency based on previous earthquake records that are widely used in liquefaction studies. A one-dimensional (1D) sinusoidal wave served as the input motion. As the main earthquakes usually include weaker foreshocks and aftershocks, this study applied 3 small shakings before and after the main event. These events were repeated to investigate the effect of all previous shakings and an increase in soil relative density. Event A, as small shakings, had a peak base acceleration (PBA) of about 0.06 g (d = ±3.5 mm) with 5 full cycles, respectively. Event C, as a large shaking, featured a PBA of about 0.17 g (d = ±10 mm) with 15 full cycles. A 5-cycle event (Event A) represented an earthquake with a magnitude of approximately 6, while a 15-cycle event (Event C) corresponded with an earthquake with a magnitude of about 7.5 [8]. Enough silent time intervals were utilized between shakings (S) to allow for the complete dissipation of excess pore water pressure (EPWP) produced during the tests. Regarding the sequence of events, Event A was applied during S1–S3, S5–S7, S9–S11, S13–S15, and S17–S19, while S4, S8, S12, and S16 were designated as Event C (see Figure 4).

2.4. Boundary Conditions

The boundary conditions of the laminar box fixed on a shake table involve the constraints or limitations applied to the edges or surfaces of the box during testing. These conditions are essential in influencing the soil behavior within the box during seismic simulations. To reduce these effects, the rectangular hollow horizontal tubes were designed to move freely on each other via rollers. A flexible geomembrane bag was utilized to contain the soil sample. To monitor the possible effect of boundary conditions on the experimental results, ACC6 and PP5 were located at a 15 cm distance from the laminar box side. Thus, the time history of acceleration (ACC4 versus ACC6) and excess pore water pressure (PP2 versus PP5) at depths of 52.5 and 30 cm from the ground surface were compared during Shaking 4 (S4) in Figure 5 as an example. The other large shakings (i.e., S8 and S12) also showed similar observations. There was a reasonable match between acceleration and the EPWP results recorded at the center and the edge of the laminar box, particularly for the purpose of determining PGA (or the amplification factor) and the maximum EPWP (or Ru). This indicates that the induced boundary effects were limited. The following sections discuss the results of accelerometers and pore water pressure transducers located at the center of the sample to prevent any possible boundary effects. Additionally, an effort was undertaken to mitigate the impact by band-filtering the data (raw acceleration data) to eliminate certain inaccuracies [35].

3. Experimental Results and Discussions

3.1. Liquefaction Analysis

The generation and dissipation of EPWP are critical in evaluating seismic behavior in saturated soils. Pore water pressure transducers were located at depths of 15, 30, 45, and 60 cm below ground level. To assess “liquefaction triggering”, indicating a state where the ratio of pore pressure approaches 1 (Ru ≈ 1), the ratio of EPWP to the initial vertical effective stress (Ru = Δu/ơ′_v) was evaluated. The computation of Ru could potentially be influenced by variances between the intended and actual elevations, a discrepancy that could arise from the movement of embedded pore water pressure transducers during shakings. Despite this potential source of error, this study proceeded to use the prescribed elevations with the absence of a reliable methodology for the exact determination of in situ elevations in line with other literature studies [23,25,36]. Figure 6 demonstrates the Ru measured at four depths in the sample throughout 19 shakings. During Event C at S4, S8, and S12, liquefaction happened at all depths. In subsequent Event A (S5, S9, and S13) after liquefaction, higher Ru values were measured at shallow depths compared with deeper depths. The primary reason for this phenomenon was the upward flow of water through the sample during liquefaction, which disrupted the soil structure and formed localized weak zones at shallow depths, thereby increasing their susceptibility to re-liquefaction during subsequent loading events [37].

In some cases, liquefaction may happen for lower values of Ru due to strong input motions causing extensive strains [38]. Thus, the shear strain caused during shakings at the location of pore water pressure transducers was calculated to confirm the liquefaction occurrence [31]. Figure 7a shows the Ru at the peak EPWP and maximum shear strain calculated in the sample for each shaking. The Ru results indicate that liquefaction occurred during Event C at S4, S8, and S12. This observation was corroborated by the shear strain measurements, which exceeded the threshold of 3.75% single amplitude, confirming the occurrence of liquefaction. Moreover, the variation in the average relative density at the end of each shaking was calculated through surface settlements, as shown in Figure 7b [21,26].

According to Figure 7, an increase in the number of small shakings (Event A) led to Ru value reductions (S1–S3, S5–S7, S9–S11, S13–S15, and S17–S19). Despite a minimal increase in relative density in each of the three subsequent small shakings (A) before large shakings (C), the liquefaction resistance improved as Ru decreased. However, the introduction of large shakings (S4, S8, and S12) led to a decrease in liquefaction resistance. As the number of large shakings increased, the Ru and shear strain caused in the sample decreases. Although liquefaction happened during S4, S8, and S12, considering both Ru and shear strain criteria, Event C at S16 was not able to cause liquefaction due to a considerable increase in the relative density (Dr > 76%) of the sample. Pre-shaking influences soil behavior in two primary ways: it increases soil density (Dr) and induces a reorganization of soil particles. Small shakings improve interlocking among soil particles, while large shakings affect these interlocks because of the higher EPWP within the soil. The liquefaction potential is generally higher in loose soils compared with dense soils, and an increase in relative density is associated with an improvement in liquefaction resistance. As relative density increases, a higher shaking intensity and a greater number of cycles are necessary to trigger liquefaction. Therefore, pre-shaking contributes to increasing soil relative density, allowing it to reach a threshold level that improves resistance to identical seismic shaking. It can be concluded that the sand used, with a relative density greater than 76%, is non-liquefiable under seismic events with magnitudes less than or equal to 7.5 (Event C).

The comparison between Event A (S3 versus S5, S7 versus S9, and S11 versus S13), which happened before and after Event C (liquefaction), clearly shows that large shaking reduces liquefaction resistance. The main reason for the reduced liquefaction resistance is the disturbance of the soil structure due to the upward flow of water during EPWP dissipation. Although soil becomes denser during a large shaking, a higher EPWP destroys the improvement in interlocking among soil particles caused by previous small pre-shakings. These findings provide validation of the hypothesis that liquefaction resets the “aging clock” of sand deposits [39]. On the other hand, an increased relative density during a large shaking can improve soil liquefaction resistance in subsequent shakings in the long term, as the relative density of soil reached the threshold (Dr > 76) during Event C and liquefaction did not occur. Thus, it can be concluded that while a large shaking can reset the improvement of liquefaction resistance induced by prior small shakings, the resulting increase in relative density can increase liquefaction resistance after three liquefaction events.

Figure 8 presents a comparison of the Ru results of this study with a series of large-scale and centrifuge shaking tests [40], strain-controlled cyclic triaxial tests [41], and shaking table tests [36,42] performed on sand. The observed scatter in the results could be attributed to variations in the test methodologies, sample sizes, confining pressure, and initial relative densities of the specimens. As large-scale and centrifuge tests are conducted under a higher confining pressure, the Ru values exhibit a higher rate of increase with shear strain. It could be observed that the Ru values of small shakings (Event A) in this research aligned closely with the boundaries reported by other researchers in shaking table tests [36,42]. However, Ru values generated from large shakings (Event C) were close to the lower boundary of the shaking table test results obtained by Ko et al. [36], which were related to lower overburden pressure (shallow depths). Ko et al. [36] utilized loose- to medium-density sand samples under higher overburden pressures compared with this study. In contrast, the current research was conducted on loose sand with lower overburden pressure, resulting in larger shear strains within the sample during large shakings. The results reveal that as the overburden (confining) pressure decreases, soil shows a higher shear strain for a given Ru. This research presents comprehensive experimental results illustrating the relationship between Ru and shear strain, along with a best-fit line demonstrating acceptable accuracy. The findings are particularly reliable for shallow depths (low confining pressure).

3.2. Acceleration Response

As indicated in Figure 3 and Figure 5, accelerometers were used to record the acceleration time history induced during each shaking at depths of 7.5, 22.5, 37.5, 52.5, and 67.5 cm, and another one (ACC: Input) was located on the shake table to measure the input motion. The amplification factor was calculated as the ratio of the recorded acceleration at various depths to the input motion. Figure 9 displays the amplification factor at various depths in the sample during 19 shakings. The results for all events indicate a progressive growth of acceleration amplitude from the bottom to the surface of the sample. This trend aligns with historical earthquake reports and experimental research, where seismic waves tend to amplify as they approach the ground surface [24,43]. It was observed that acceleration amplified in the sample during large shakings until liquefaction happened. After liquefaction, the sample behaved like liquid and the acceleration amplitude decreased (see Figure 5a). The amplification factor before liquefaction occurrence for large shakings (Event C) was considerably higher compared with small shakings (Event A). Similar observations have been reported that as shaking intensity increases, the amplification factor increases [24,44]. Castiglia et al. [44] revealed that as the acceleration amplitude increases, the instantaneous frequencies of some layers approach the natural frequency of the soil, which leads to higher amplification. This phenomenon, referred to as soil resonance, is linked to the production of significant pore water pressure. The findings of this study confirm that larger shakings, which induce higher EPWP, are associated with greater amplification factors.

Figure 10 indicates the peak ground acceleration (PGA) recorded at shallow depths throughout 19 shakings. The measured PGAs during small shakings (Event A) were between 0.08 and 0.12 g, while this value was about 0.36, 0.54, 0.7, and 0.31 g for Event C at S4, S8, S12, and S16, respectively. It can be observed that the PGA trend changes as the sample becomes denser (Dr > 76%) and Ru decreases. The wave propagation is higher at S12 compared with S4 and S8 because the number of cycles required for the liquefaction of densified samples increases. In contrast, further increases in relative density in S16 result in a lower Ru and reduced wave propagation. Previous studies concluded that the PGA increases as relative density increases [45]. However, the findings of this study indicate that the PGA is influenced by the EPWP generated, rather than by relative density. Similarly, a comparison among small shakings (Event A) revealed that the PGA recorded did not directly change with the relative density. As the Ru generated during S5 and S9 was higher, the PGA recorded in the sample was higher compared with other small shakings (Event A).

3.3. Horizontal Displacement of Soil Profile

The relative horizontal displacement of soil refers to the horizontal movement of the soil profile during each shaking. Figure 11 illustrates the maximum relative horizontal displacement recorded during small shakings (Event A) (Figure 11a) and large shakings (Event C) (Figure 11b) at depths of 8, 20, and 39 cm from the ground surface. It can be observed that the horizontal displacement increases from the bottom to the top of the sample due to a higher amplification factor at shallow depths. According to Figure 11a, as the number of small shakings (Events A) increases, the horizontal displacement decreases. Despite a less than 1% increase in relative density, the average reduction in horizontal displacement from S1 to S3 was about 40%. A similar reduction trend in horizontal displacement was noted from S5 to S7, S9 to S11, S13 to S15, and S17 to S19. Small shakings cause a slight increase in relative density, which mainly contributes to the improvement in interlocking among soil particles. In contrast, large shakings increase the horizontal displacement of soil in subsequent shakings despite a noticeable increase in soil relative density. Although considerable densification happened during liquefaction (S4, S8, and S12), subsequent Event A (S5, S9, and S13) measurements showed higher horizontal displacement compared with previous Event A measurements. This tendency is mainly linked to soil-structure disturbance due to the upward flow of water during large shakings. Thus, interlocking among soil particles plays a more significant role in horizontal displacement behavior compared with densification. According to Figure 11b, a corresponding decrease in horizontal displacement is observed as the number of large shakings (Event C) increases. Although the relative density of the sample increased from 31% to 69% throughout S4 to S12 (including three liquefaction events), the average horizontal displacement reduction was about 49%.

3.4. Stress–Strain Behavior of Soil

The soil sample in the shaking table test was assumed to experience 1D shear loading by Zeghal et al. [46]; therefore, a shear beam equation was utilized to obtain shear stress ($\tau$) at a specific depth ($z$) using Equation (1) as follows:

$$\tau (z)={\displaystyle \underset{0}{\overset{z}{\int }}\rho (z)a(z)dz}$$

(1)

where $\rho$ and $a$ refer to the mass density of soil and acceleration at a specific depth ($z$), respectively.

As it is almost impossible to record reliable surface acceleration time histories in model tests due to the lack of good contact between the accelerometer and soil, it has been suggested by Brennan et al. [31] to determine surface acceleration using the linear extrapolation of acceleration data from the top pair to the surface. Similarly, a linear interpolation can be used to obtain acceleration data from a pair of accelerometers at different depths [31,46].

To determine shear strain, displacements ($u$) can be obtained from a double integration of the acceleration data. Then, shear strain induced in soil can be computed at the midpoint of the pair of accelerometers ($z$), which is where the pore pressure transducers are located (Equation (2)).

$${\gamma }_i(t)={\displaystyle \frac{u_i-u_{i-1}}{z_i-z_{i-1}}}$$

(2)

The method described above is frequently employed to extract shear-stress–strain loops from downhole accelerometer array data and physical model tests [31,36,46,47].

Figure 12 shows typical stress–strain loops at locations PP1 and PP4 at depths of 15 and 60 cm during small (S1) and large (S4) shakings as an example. In general, at small shakings, the stress–strain loops tend to be relatively vertical, and the shear strain induced is extremely limited. However, the stress–strain loops display a progressively flattening trend during large shakings. As shear strain increases during seismic events, soil stiffness reduces due to significant EPWP and liquefaction.

After plotting stress–strain loops, the shear modulus (G), the slope of each loop, can be determined from Equation (3).

$$G={\displaystyle \frac{{\tau }_{\max }-{\tau }_{\min }}{{\gamma }_{\max }-{\gamma }_{\min }}}$$

(3)

To investigate the impact of small shakings on soil dynamic behavior, the shear modulus for Event A was analyzed. Figure 13 presents a comparison of the shear modulus between S5 and S7 as well as S9 and S11 as representative examples. It indicates that small shakings improve soil stiffness and decrease the shear strain caused in the soil. The finding highlights that small shakings can mitigate the liquefaction potential by enhancing the shear modulus in susceptible soils.

Figure 14 shows the shear modulus variations before and after Event C (S4 and S16) during an identical seismic event (Event A). Liquefaction (S4, S8, and S12) reduces the shear modulus and leads to higher shear strains in the subsequent small shakings. Due to higher densification caused by previous shakings, Event C at S16 could not cause higher excess pore water pressure. As a result, it led to an improvement in soil stiffness. Thus, it can be noted that shakings generating a higher Ru and shear strain can reduce soil stiffness due to disturbance of the soil structure. However, a shaking inducing a lower Ru and shear strain can improve soil stiffness.

Figure 15 compares the soil stiffness during large shakings (Event C) in the soil. In general, soil stiffness progressively increases with an increase in the number of previous shakings at deeper depths (PP3–PP4). This phenomenon can be attributed to the higher densification of deeper depths during shakings due to the higher effective stress. However, soil stiffness remains almost unchanged at shallow depths (PP1–PP2) due to the development of weak zones caused by the upward flow of water during large shakings [37]. Due to the greater effective stress at deeper depths (more than 30 cm), these layers undergo higher densification, resulting in higher stiffness. In contrast, shallow depths remain vulnerable to strain-induced stiffness loss.

The shear modulus curve is typically represented by plotting the shear modulus (G) against shear strain (γ) based on laboratory cyclic loading test results in soil dynamic studies. Consequently, the shear modulus is typically normalized by the maximum shear modulus (G₀) at γ = 0.0001%. However, as shaking table tests produce higher shear strains, the shear modulus is normalized by the shear modulus (G) at γ = 0.01% [36]. Thus, the results of this study are presented in the form of $G/G_{\gamma =0.01\%}$. As the shear strains induced in small shakings were close to 0.01%, the shear modulus ($G_{\gamma =0.01\%}$) at each depth was determined using the best-fit trendline (Figure 16).

After determining the shear modulus at γ = 0.01%, the shear modulus calculated for each cycle during Event A and C throughout 19 shakings was normalized. Figure 17 compares the normalized shear modulus curves from this research with those from the shaking table tests conducted on loose- to medium-density sand [36]. It also compares the results with the model suggested by Seed and Idriss [48], which is derived from small-scale dynamic experiments on virgin clean sand under higher confining pressure and has become a widely referenced model. There is a reasonable match between the results of this study and those obtained by Ko et al. [36]. As shaking table tests are conducted under low overburden pressure, the shear strain increase with EPWP generation is higher compared with other dynamic tests. As a result, the shear modulus degradation curves derived from the shaking table experiments exhibit lower values compared with those presented by Seed and Idriss [48]. The upper and lower boundaries of the results of Ko et al. [36] for virgin samples were generally obtained from higher and lower overburden pressures (depths), respectively. Given that the present study was conducted under even lower overburden conditions, the results for Event A, which influenced the soil within the elastic range, align more closely with the lower boundary of Ko et al. [36]. However, pre-shaking improved soil stiffness during Event C, especially at deeper depths. As a result, the observed results for large shakings exceeded the upper boundary of Ko et al. [36], which was based on tests conducted on virgin sand.

Although shear modulus values are influenced by variations in effective overburden stress (i.e., depth) and pre-shaking history, the normalized curves obtained from the tests demonstrate that the combined effects of soil nonlinearity and excess pore water pressure (EPWP) generation on soil degradation were consistently captured across both small and large shaking events during the shaking table experiments. Therefore, the derived normalized curves offer valuable insights and are suitable for integration into engineering practice.

4. Conclusions

The primary aim of this investigation was to investigate how the sequence of small and large pre-shakings influenced the liquefaction resistance and stiffness of soil via shaking table tests. To this end, a saturated loose sand sample was subjected to 19 small and large shakings. The results provide insights into how pre-shaking history influences soil behavior under subsequent events. The key findings and outcomes of this experimental research are summarized as follows:

• Small shakings lead to an improvement in liquefaction resistance as the Ru generated decreases, even with minimal changes in relative density. However, liquefaction initially decreases liquefaction resistance due to higher EPWP. As the number of large shakings leading to significant densification (Dr > 76%) increases, the ability of subsequent large shaking (Event C) to trigger liquefaction is decreased. Liquefaction resistance increase during both small and large shakings at deeper depths is higher compared with shallow depths. This is related to the development of weak zones at shallow depths resulting from the upward flow of water during EPWP dissipation.
• The peak ground acceleration (PGA) increases with soil densification during large shakings but begins to decline beyond a relative density of 76%. Conversely, for small shakings, the PGA remains irrelevant to changes in relative density. The amplification factor was found to rely on the EPWP produced during shakings instead of relative density. A higher Ru leads to a higher amplification factor.
• Small shakings improve shear modulus values whereas large shakings significantly reduce the shear modulus in cases of liquefaction occurrence. Pre-shakings lead to higher stiffness at deeper depths of soil. As the normalized shear modulus versus the shear strain is almost constant for small and large shakings, the results can apply to engineering practice.
• Although the relative density increased by successive small shakings is minimal, the reduction in horizontal displacement is significant. Large shakings (liquefaction) cause significantly higher densification whereas the horizontal displacement surprisingly increases. Interlocking among soil particles resulting from small shakings plays a crucial role in controlling soil horizontal displacement compared with the significant densification induced by large shakings.
• In general, pre-shaking has a twofold influence on soil behavior: firstly, it increases soil density, and secondly, it induces a reorganization of soil particles. Small shaking events primarily contribute to enhancing the interlocking between soil particles as the increase in relative density remains minimal. On the other hand, large shakings cause a major increase in relative density while the soil structure is disturbed by the upward flow of water during liquefaction. The disturbance in the soil structure caused by large shakings (liquefaction) can be mitigated by subsequent small shakings and an increase in relative density has a lasting impact on liquefaction resistance increase.
• The analyses and outcomes presented in this study, although primarily applicable to low effective stresses, offer significant contributions to the understanding of underlying mechanisms and the calibration of numerical models used for liquefaction purposes.