Liquefaction Potential of Saturated Sand Reinforced by Cement-Grouted Micropiles: An Evolutionary Approach Based on Shaking Table Tests

Cement-grouted injections are increasingly employed as a countermeasure material against liquefaction in active seismic areas; however, there is no methodology to thoroughly and directly evaluate the liquefaction potential of saturated sand materials reinforced by the cement grout-injected micropiles. To this end, first, a series of 1 g shaking table model tests are conducted. Time histories of pore water pressures, excess pore water pressure ratios (ru), and the number of required cycles (Npeak) to liquefy the soil are obtained and modified lower and upper boundaries are suggested for the potential of liquefaction of both pure and grout-reinforced sand. Next, adopting genetic programming and the least square method in the framework of the evolutionary polynomial regression technique, high-accuracy predictive equations are developed for the estimation of rumax. Based on the results of a three-dimensional, graphical, multiple-variable parametric (MVP) analysis, and introducing the concept of the critical, boundary inclination angle, the inclination of micropiles is shown to be more effective in view of liquefaction resistivity for loose sands. Due to a lower critical boundary inclination angle, the applicability range for inclining micropiles is narrower for the medium-dense sands. MVP analyses show that the effects of a decreasing spacing ratio on decreasing rumax are amplified while micropiles are inclined.

A micropile is defined as a drilled and grouted pile with a small diameter (less than 30 cm), which is reinforced typically with a reinforcement bar. The grout is placed or injected into the drilled pile, depending on the construction type of micropile [23]. Micropiles are utilized as foundation systems for new structures and to improve the seismic function of existing structures, which can be used in access-restrictive situations and approximately in all soil types and ground conditions [23]. In seismic areas, micropiles are beneficial in the construction due to their flexibility, ductility, and the ability to resist uplift forces [23][24][25][26]. Therefore, the use of micropiles can be useful regarding their performance in increasing bearing capacity, reducing seismic shear strains, and the liquefaction potential [23][24][25][26][27][28][29][30].

Shaking Table Tests
The study was carried out using a shaking table apparatus at the Engineering Faculty of the University of Guilan, Iran. The shaking table contains an electromotor with a maximum horizontal acceleration of 4.44 (m/s 2 ) and a maximum frequency of 3 (Hz) [16,17].

Rigid Transparent Box
The rigid box used in this study had dimensions of 53 cm in length, 50 cm in width, 45 cm in height, and 1 cm in frame thickness. The apparatus comprised a shaking table placed in a fixed steel frame and a fixed doubled-wall plexiglass box (connected by pins) aloft. The second wall was used to maintain the soil saturated in the box and to keep the overflowing water. A water outlet was installed beneath the plexiglass box to enhance the saturation of the soil from the bottom. A platen with sufficient holes was also placed at the bottom of the box to allow the water to drain freely and uniformly throughout the sample from the base. Sheets of foam covered the inner lateral boundaries of the box (i.e., parallel to the shaking direction). The foam sheets were encased with thin plastic layers to prevent the penetration of soil and water into it. A fine-screen mesh coated the bottom of the container for saturation purposes. Figure 1 shows the testing apparatus along with the instrumentation used in this paper. Two pore pressure transducers (PPT) to monitor the generation and dissipation of excess pore water pressures and a linear variable differential transformer (LVDT) to record the horizontal displacement of the box were utilized.

Instrumentation
Hence, the input acceleration was calculated considering the box displacement; moreover, continuous and precise displacements during the cyclic motion were recorded with an accuracy of 0.01 mm and an amplitude of 37 mm. Three bolts were used to adjust the horizontal location of the sensor, which was vertically fixed on the wall of the box using a sheath. Therefore, the central core of the sensor was able to move simultaneously with the machine during the cyclic motion. Considering the location of the box, and the compression of the central core of the sensor, observed amplitudes were in the range of 0.0 to 37 mm, sending a voltage with a varying magnitude between −4.11 v to 3.12 v to the data logger. Two techniques were implemented to measure and validate the pore water pressure. As the speed of cyclic motion is high, two different pore pressure transducers were located at different depths to measure the generation and dissipation of the excess pore water pressures during the tests, while standpipe piezometers were installed at the same levels to validate the measured pore water pressures. The capacity of the measurement of the used pore water pressure transducers was 0.1 bar (equal to 100 cm of water). Transducers were positioned at vertical distances of 7.5 cm and 17.5 cm from the bottom filter (fine-screen mesh) and in the middle (horizontal dimension) of the box to minimize errors possibly induced by lateral boundaries. Hence, the input acceleration was calculated considering the box dis moreover, continuous and precise displacements during the cyclic motion we with an accuracy of 0.01 mm and an amplitude of 37 mm. Three bolts were use the horizontal location of the sensor, which was vertically fixed on the wall using a sheath. Therefore, the central core of the sensor was able to move simu with the machine during the cyclic motion. Considering the location of the b compression of the central core of the sensor, observed amplitudes were in t 0.0 to 37 mm, sending a voltage with a varying magnitude between −4.11 v to 3 data logger. Two techniques were implemented to measure and validate the pressure. As the speed of cyclic motion is high, two different pore pressure t were located at different depths to measure the generation and dissipation o pore water pressures during the tests, while standpipe piezometers were inst same levels to validate the measured pore water pressures. The capacity of th ment of the used pore water pressure transducers was 0.1 bar (equal to 100 cm Transducers were positioned at vertical distances of 7.5 cm and 17.5 cm from filter (fine-screen mesh) and in the middle (horizontal dimension) of the box t errors possibly induced by lateral boundaries.

Grout-Injection Set-Up
For physical modeling of a grouted micropile, an experimental grouting including an air compressor with 8 bars capacity, an inlet pressure valve, a pr ulating valve, the inlet and outlet grout valves, a grout tank, an injection hose, zle for the injection hose were used. Figure 2 shows the schematic view of the b

Grout-Injection Set-Up
For physical modeling of a grouted micropile, an experimental grouting apparatus including an air compressor with 8 bars capacity, an inlet pressure valve, a pressure regulating valve, the inlet and outlet grout valves, a grout tank, an injection hose, and a nozzle for the injection hose were used. Figure 2 shows the schematic view of the built device. Hence, the input acceleration was calculated considering the box displa moreover, continuous and precise displacements during the cyclic motion were re with an accuracy of 0.01 mm and an amplitude of 37 mm. Three bolts were used t the horizontal location of the sensor, which was vertically fixed on the wall of using a sheath. Therefore, the central core of the sensor was able to move simulta with the machine during the cyclic motion. Considering the location of the box, compression of the central core of the sensor, observed amplitudes were in the r 0.0 to 37 mm, sending a voltage with a varying magnitude between −4.11 v to 3.12 data logger. Two techniques were implemented to measure and validate the por pressure. As the speed of cyclic motion is high, two different pore pressure tran were located at different depths to measure the generation and dissipation of the pore water pressures during the tests, while standpipe piezometers were installe same levels to validate the measured pore water pressures. The capacity of the m ment of the used pore water pressure transducers was 0.1 bar (equal to 100 cm of Transducers were positioned at vertical distances of 7.5 cm and 17.5 cm from the filter (fine-screen mesh) and in the middle (horizontal dimension) of the box to m errors possibly induced by lateral boundaries.

Grout-Injection Set-Up
For physical modeling of a grouted micropile, an experimental grouting ap including an air compressor with 8 bars capacity, an inlet pressure valve, a pressu ulating valve, the inlet and outlet grout valves, a grout tank, an injection hose, and zle for the injection hose were used. Figure 2 shows the schematic view of the built

Material Properties
Anzali sand used in this study is composed of fine-grained silica sands. Figure 3 shows the grain size distribution of the studied soil, which is a poorly grained sand (SP) with the curvature coefficient (cc) of 1.23 and the uniformity coefficient (cu) of 1.83. The specific gravity of the soil and the minimum and maximum void ratios are, respectively, 2.65, 0.66, and 0.88. aterials 2023, 16, x FOR PEER REVIEW

Material Properties
Anzali sand used in this study is composed of fine-grained silica sa shows the grain size distribution of the studied soil, which is a poorly grai with the curvature coefficient (cc) of 1.23 and the uniformity coefficient (cu specific gravity of the soil and the minimum and maximum void ratios are 2.65, 0.66, and 0.88.

Boundary Condition
Considering the study conducted by Fishman et al. (1995) [60], the boundary affects the results up to 1.5 to 2 times the height of the model. H sideration of proper treatment with regard to the vertical side boundaries portance. The one-dimensional equation of wave propagation within an el medium is as shown in Equation (1) (Kolsky 1953) [72]: where u = u(x,t) can be the longitudinal displacement in the x-direction du sion waves or transverse displacement perpendicular to the x-direction r shear waves. λ(x) and G(x) are the Lamé constants (G is also called the shea the medium), which are related to Young's modulus and Poisson's ratio. Th sistance to a given particle motion is characterized using its impedance Z = ρ × the mass density and V is the velocity of propagation. In a geotechnical model jected to one-dimensional motion, when a body wave encounters the interfac media having different impedances (i.e., interface soil-wall), the wave energy flected and partially transmitted through the boundary. Wave mode conversio curs, whereby P-waves are converted into S-waves and vice versa. However, i rigid box with absorbing boundaries, as the P-wave propagates from the soil layer, the velocity of propagation slows down due to the low impedance of the [72]. At the interface, the frequency of the propagating wave (f = V⁄λ, where length) must remain constant. Therefore, when the wave propagates from th into the foam, the wavelength must decrease (as shown in Figure 4). This redu length can be associated with energy dissipation. A significant amount of ener

Boundary Condition
Considering the study conducted by Fishman et al. (1995) [60], the rigid lateral boundary affects the results up to 1.5 to 2 times the height of the model. Hence, the consideration of proper treatment with regard to the vertical side boundaries is of great importance. The one-dimensional equation of wave propagation within an elastic isotropic medium is as shown in Equation (1) (Kolsky 1953) [72]: where u = u(x,t) can be the longitudinal displacement in the x-direction due to compression waves or transverse displacement perpendicular to the x-direction resulting from shear waves. λ(x) and G(x) are the Lamé constants (G is also called the shear modulus of the medium), which are related to Young's modulus and Poisson's ratio. The medium resistance to a given particle motion is characterized using its impedance Z = ρ × V, where ρ is the mass density and V is the velocity of propagation. In a geotechnical model container subjected to one-dimensional motion, when a body wave encounters the interface between two media having different impedances (i.e., interface soil-wall), the wave energy is partially reflected and partially transmitted through the boundary. Wave mode conversion therefore occurs, whereby P-waves are converted into S-waves and vice versa. However, in the case of a rigid box with absorbing boundaries, as the P-wave propagates from the soil into the foam layer, the velocity of propagation slows down due to the low impedance of the softer material [72]. At the interface, the frequency of the propagating wave (f = V/λ, where λ is the wavelength) must remain constant. Therefore, when the wave propagates from the soil medium into the foam, the wavelength must decrease (as shown in Figure 4). This reduction in wavelength can be associated with energy dissipation. A significant amount of energy can also be absorbed by the hysteretic damping provided by the foam [64].  Lombardi and Bhattacharya (2012) [73] studied the efficiency of t absorbing boundaries on the dissipation of the wave energy. They de ment to evaluate the energy absorbed by the soft boundary. To conduc the coherence function (as presented in Equation (2)) was calculated co recorded by the accelerometers placed inside and outside of the shaking of rigid and absorbing boundaries for a range of applied frequencies. Ba imental results, suitable performance of soft boundaries was proved: where CSA (f) is the coherence at a given frequency of f, and PSS (f) and PA represent the power spectral density of input and output signals. More cross power spectral density of two signals. To overcome such errors, suggestions of Lombardi and Bhattacharya (2012) [73], and Lombardi e cial soft boundaries (foam sheets) are used as absorbing boundaries. T the applicability of such a method was also proved throughout the sh reported in Hasheminezhad et al. (2022) [22]. Moreover, to increase th experiment results, the minimum distance between the micropiles and l are considered 24 times the diameter of the micropiles.  Lombardi and Bhattacharya (2012) [73] studied the efficiency of the application of absorbing boundaries on the dissipation of the wave energy. They designed an experiment to evaluate the energy absorbed by the soft boundary. To conduct this experiment, the coherence function (as presented in Equation (2)) was calculated comparing the date recorded by the accelerometers placed inside and outside of the shaking box for both cases of rigid and absorbing boundaries for a range of applied frequencies. Based on the experimental results, suitable performance of soft boundaries was proved: where C SA (f) is the coherence at a given frequency of f, and P SS (f) and P AA (f), respectively, represent the power spectral density of input and output signals. Moreover, P SA (f) is the cross power spectral density of two signals. To overcome such errors, and according to suggestions of Lombardi and Bhattacharya (2012) [73], and Lombardi et al. (2015), artificial soft boundaries (foam sheets) are used as absorbing boundaries. The efficiency and the applicability of such a method was also proved throughout the shaking table study reported in Hasheminezhad et al. (2022) [22]. Moreover, to increase the accuracy of the experiment results, the minimum distance between the micropiles and lateral boundaries are considered 24 times the diameter of the micropiles. Furthermore, the distance between the bottom of the micropiles and the horizontal bottom boundary of the box is at least 24 times the diameter of the micropiles, as suggested by Lombardi et al. (2015) [64]. Figure 5 indicates the geometry of the model, the arrangement of micropiles, and distances.

Scaling of Micropiles
In this study, due to the nonlinear behavior of the materials and the geotechnical structure composed of several different materials that interact with each other, it is necessary to use the proper scaling law used in the shaking table (1 g). Table 1 shows used scaling factors. As an example, n L is defined as the ratio of the model length to the prototype length [57,70]. Regarding Wood (2003) [57], the factor of α, as a parameter to correlate the stiffness and the effective stress level in the soil, is considered 0.5 for sandy soil. It should be described considering the fact that the behavior of the saturated soil is governed by the equilibrium of the pore water flow and the mass balance equations [74], and the behavior of the water during earthquake can be approximated by ignoring the viscosity of the water and the wave generated by the motion of the structure [75,76]; the viscosity of the water is not scaled. experiment results, the minimum distance between the micropiles and la are considered 24 times the diameter of the micropiles. Furthermore, the d the bottom of the micropiles and the horizontal bottom boundary of the times the diameter of the micropiles, as suggested by Lombardi et al. (20 indicates the geometry of the model, the arrangement of micropiles, and    The characteristics of the micropiles are shown in Table 2. Regarding the boundary condition, and applying Table 2 where m, and p subscripts, respectively, stand for the model and prototype parameters. Figure 6 shows the cross-section of the modeled micropile schematically. Regarding the boundary condition, and applying Table 2 and Equati factor is obtained 15.7: where m, and p subscripts, respectively, stand for the model and prototy Figure 6 shows the cross-section of the modeled micropile schematically.
Hence, as a link to the real field case, since the scaling factor for the a and ru is a dimensionless parameter, the obtained results can be directly u type with a steel micropile with a casing length of 3.2 m, thickness of 1 diameter of 11 cm.

Model Preparation
In this study, a 33 cm thick sand layer was constructed using the method in 5 uniform sections of 6.6 cm into the transparent box. For each weighed moist sand with a water content of 5% was compacted gently tomed tamper to reach the determined relative density. Samples were grad until full saturation was assured using the pore pressure transducers. T water was conveyed into the sample for 24 h using a valve installed at th container and throughout the mesh screen layer just beneath the soil sam water was kept low enough not to distribute the sample and to affect its r In the grouting process of each micropile, at first, the micropile location w then the perforated rubber casing was placed inside the drilled hole. In th injection operation was carried out under the pressure of 0.1 bar, and im reinforced bar was inserted vertically in the center of the micropile. It shou by performing multiple tests at different injection pressures in a transpa the injection pressure of 0.1 bar was obtained to reach the determined mo (with regards to the affected zone around the grouted micropile) in Figu time of the micropiles was 7 days. Figure 7 illustrates the physical model cropile in the small transparent box used to define the grouting pressure a grouting around the casing. Hence, as a link to the real field case, since the scaling factor for the acceleration is 1, and r u is a dimensionless parameter, the obtained results can be directly used for a prototype with a steel micropile with a casing length of 3.2 m, thickness of 1 cm, and inner diameter of 11 cm.

Model Preparation
In this study, a 33 cm thick sand layer was constructed using the moist tamping method in 5 uniform sections of 6.6 cm into the transparent box. For each section, the preweighed moist sand with a water content of 5% was compacted gently with a flat-bottomed tamper to reach the determined relative density. Samples were gradually saturated until full saturation was assured using the pore pressure transducers. To this end, tap water was conveyed into the sample for 24 h using a valve installed at the bottom of the container and throughout the mesh screen layer just beneath the soil sample. The flow of water was kept low enough not to distribute the sample and to affect its relative density. In the grouting process of each micropile, at first, the micropile location was drilled, and then the perforated rubber casing was placed inside the drilled hole. In the next step, the injection operation was carried out under the pressure of 0.1 bar, and immediately the reinforced bar was inserted vertically in the center of the micropile. It should be noted that by performing multiple tests at different injection pressures in a transparent small box, the injection pressure of 0.1 bar was obtained to reach the determined modeled micropile (with regards to the affected zone around the grouted micropile) in Figure 6. The curing time of the micropiles was 7 days. Figure 7 illustrates the physical model of the built micropile in the small transparent box used to define the grouting pressure and the veins of grouting around the casing. the injection pressure of 0.1 bar was obtained to reach the determined m (with regards to the affected zone around the grouted micropile) in Fig time of the micropiles was 7 days. Figure 7 illustrates the physical mode cropile in the small transparent box used to define the grouting pressure grouting around the casing.

Experimental Results
After constructing the soil sample and performing the grouting operation, the dynamic loading using the shaking table was performed. First, before applying the base excitation, the data acquisition system started to record the data for a duration of 5 s. Next, the transparent box was shaken under the harmonic sine wave with various maximum accelerations according to Table 3.  Figure 8 shows the specification of shaking-induced displacement of the box that illustrates the frequency of shaking, duration of shaking, and the displacement amplitude of the box. As shown, two types of lateral boundaries, two frequencies resulting in two accelerations, two relative densities, three spacing ratios, and two inclination angles are used to study the liquefaction potential of the model. 24 9.5 3 4 7 90 Figure 8 shows the specification of shaking-induced displacement of the box that illustrates the frequency of shaking, duration of shaking, and the displacement amplitude of the box. As shown, two types of lateral boundaries, two frequencies resulting in two accelerations, two relative densities, three spacing ratios, and two inclination angles are used to study the liquefaction potential of the model.  The maximum applied accelerations to the samples are calculated using Equation (4): where r is the displacement amplitude, and f is the frequency of vibration. Figure 9 shows the time histories of recorded excess pore water pressure (EPWP) at the bottom pore pressure transducer (PPT) that indicates the significant impact of lateral boundaries of the box on pore water pressure variations.

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10 of 26 where r is the displacement amplitude, and f is the frequency of vibration. Figure 9 shows the time histories of recorded excess pore water pressure (EPWP) at the bottom pore pressure transducer (PPT) that indicates the significant impact of lateral boundaries of the box on pore water pressure variations. The EPWP ratio (ru) is defined in accordance with Equation (5):

Effect of Boundary Conditions on Excess Pore Water Pressure Ratio (ru)
where ∆u is the excess pore water pressure and σ0′ is the initial vertical effective stress. As shown in Figure 9, foam sheets as the absorbing boundary decreased the maximum excess pore water pressure ratio (rumax) from 0.88 to 0.82 (7% reduction). Moreover, the pore water pressure was generated faster in the soil embedded in the rigid box than in the box with the absorbing boundary, while the dissipation of pore water pressure started sooner in the box with the absorbing boundary. As discussed, the development of compressive and reflective shaking-induced waves by the rigid walls generated more pressure in the soil sample compared with the absorbing boundary and consequently produced more pore water pressure. In addition, foam sheet (compared with the rigid boundary) decreased the velocity of wave propagation due to its lower impedance, which resulted in a decrease in the pore water pressure. Table 4 presents the rumax values and the The EPWP ratio (r u ) is defined in accordance with Equation (5): where ∆u is the excess pore water pressure and σ 0 is the initial vertical effective stress. As shown in Figure 9, foam sheets as the absorbing boundary decreased the maximum excess pore water pressure ratio (r umax ) from 0.88 to 0.82 (7% reduction). Moreover, the pore water pressure was generated faster in the soil embedded in the rigid box than in the box with the absorbing boundary, while the dissipation of pore water pressure started sooner in the box with the absorbing boundary. As discussed, the development of compressive and reflective shaking-induced waves by the rigid walls generated more pressure in the soil sample compared with the absorbing boundary and consequently produced more pore water pressure. In addition, foam sheet (compared with the rigid boundary) decreased the velocity of wave propagation due to its lower impedance, which resulted in a decrease in the pore water pressure. Table 4 presents the r umax values and the number of cycles required to reach the r umax values (N peak ), illustrating a slight increase in N peak in the box with the absorbing boundary. As a result, for the remaining test plan, foam sheets were used to provide better simulation conditions. To investigate the effect of single micropiles on the liquefaction potential of the saturated sand, pore water pressure was measured at depths of 15.5 cm and 25.5 cm from the sand surface. Time histories of excess pore water pressure in pure and reinforced sand with grouted micropiles are shown in Figure 10.
As illustrated in Figure 10, excess pore water pressure measured at the bottom PPT shows higher values than the upper PPT due to the higher length of the drainage path. In addition, it is observed that the reduction in pore water pressure started from the bottom depth of the sample, and after a few seconds, it followed at the shallower depth. The results show that with increasing the depth of soil, because of the increasing effective stresses, the maximum of the excess pore water pressure ratio (r umax ) decreases. The results presented in Table 4 show that the reinforced sand with nine micropiles has a significant effect to reduce the liquefaction potential.
However, the effect of a single micropile on the reduction in the liquefaction potential is negligible. Such a result (low impact of a single micropile on the liquefaction potential) was also observed in previous numerical or experimental studies [27,33]. The variation of pore water pressure in depth is shown in Figure 11.

Effect of the Number of Micropiles
To investigate the effect of single micropiles on the liquefaction potential of the saturated sand, pore water pressure was measured at depths of 15.5 cm and 25.5 cm from the sand surface. Time histories of excess pore water pressure in pure and reinforced sand with grouted micropiles are shown in Figure 10. As illustrated in Figure 10, excess pore water pressure measured at the bottom PPT shows higher values than the upper PPT due to the higher length of the drainage path. In addition, it is observed that the reduction in pore water pressure started from the bottom depth of the sample, and after a few seconds, it followed at the shallower depth. The results show that with increasing the depth of soil, because of the increasing effective stresses, the maximum of the excess pore water pressure ratio (rumax) decreases. The results presented in Table 4 show that the reinforced sand with nine micropiles has a significant effect to reduce the liquefaction potential.
However, the effect of a single micropile on the reduction in the liquefaction potential is negligible. Such a result (low impact of a single micropile on the liquefaction potential) was also observed in previous numerical or experimental studies [27,33]. The variation of pore water pressure in depth is shown in Figure 11. As shown, with an increasing number of micropiles, the excess pore water pressure significantly decreases. This reduction in the excess pore water pressure can be attributed to the grouted sand zone created around the micropiles. In addition, reducing the lateral displacements of the soil sample after reinforcement by micropiles and the local compres- As shown, with an increasing number of micropiles, the excess pore water pressure significantly decreases. This reduction in the excess pore water pressure can be attributed to the grouted sand zone created around the micropiles. In addition, reducing the lateral displacements of the soil sample after reinforcement by micropiles and the local compression in the soil around the micropiles after drilling and injection steps can be considered as other effective parameters in the reduction in the excess pore water pressure. Figure 12 shows As shown, with an increasing number of micropiles, the excess pore water pressur significantly decreases. This reduction in the excess pore water pressure can be attribute to the grouted sand zone created around the micropiles. In addition, reducing the latera displacements of the soil sample after reinforcement by micropiles and the local compres sion in the soil around the micropiles after drilling and injection steps can be considere as other effective parameters in the reduction in the excess pore water pressure. Figure 12 shows the variations of ru with the number of cycles at the bottom an upper PPTs during shaking.  Figure 12a illustrates that the pure sand needs fewer values of the Npeak (about Npe = 9) than the reinforced sand with two grouted micropiles (about Npeak = 11.6). Such a be havior is also shown in other reinforced samples with a different number of micropiles a presented in Table 4. Similar behavior is also observed at the shallower depths as show in Figure 12b. Increasing the number of cycles required for the liquefaction triggerin (rumax) indicates that with increasing the number of micropiles, the liquefaction resistanc of the reinforced sand is increased. In addition, as presented, values of Npeak obtained a the upper PPT are more than those measured using the bottom PPT. This difference ca be attributed to the proximity of the upper PPT to the grouted micropiles and the groute  Figure 12a illustrates that the pure sand needs fewer values of the N peak (about N peak = 9) than the reinforced sand with two grouted micropiles (about N peak = 11.6). Such a behavior is also shown in other reinforced samples with a different number of micropiles as presented in Table 4. Similar behavior is also observed at the shallower depths as shown in Figure 12b. Increasing the number of cycles required for the liquefaction triggering (r umax ) indicates that with increasing the number of micropiles, the liquefaction resistance of the reinforced sand is increased. In addition, as presented, values of N peak obtained at the upper PPT are more than those measured using the bottom PPT. This difference can be attributed to the proximity of the upper PPT to the grouted micropiles and the grouted sand formed around the micropiles, which confines the excess pore water pressure and reduces the velocity of the pore water pressure generation. Therefore, more cycles are required to reach the maximum pore water pressure in upper PPTs.

Effects of Micropiles Spacing Ratio (S mic /d mic )
In this section, the effects of the micropiles' spacing ratio on the excess pore water pressure generations are determined considering values of 4 and 7 for the micropiles' spacing ratio. Figure 13 shows the variations of r u in the reinforced sand by 2 and 4 vertical micropiles with the spacing ratio of 4 and 7. As illustrated in this figure, a slight reduction (about 3% reduction) in the values of r umax is observed by decreasing s/d from 7 to 4 for the case of using 2 micropiles, while for the case of applying 4 micropiles, more reduction (about 10% reduction) is occurred. In addition, it is important to remark that the N peak and the maximum required time (t peak ) are almost constant with the change of s/d.

Effects of Micropiles' Inclination
To investigate the effect of inclined micropiles on the liquefaction potential of reinforced sand, the deposits were reinforced by two diagonal and opposed micropiles. Figure 5b shows the arrangement of 2 inclined micropiles with the inclination angle of 12 degrees (with respect to the vertical axis) and the spacing ratios of 4 and 7. Figure 14 illustrates the variations of r u in the reinforced sand by 2 inclined and vertical micropiles with the spacing ratios of 4 and 7. As shown, the arrangement of 2 inclined micropiles with the spacing ratio of 4 has the best performance to reduce the soil liquefaction potential. This result agrees with results of the seismic performance superiority of the inclined micropiles in dry sands compared with the vertical micropiles obtained by other researchers [35,36,39,53].
In this section, the effects of the micropiles' spacing ratio on the excess pore water pressure generations are determined considering values of 4 and 7 for the micropiles' spacing ratio. Figure 13 shows the variations of ru in the reinforced sand by 2 and 4 vertical micropiles with the spacing ratio of 4 and 7. As illustrated in this figure, a slight reduction (about 3% reduction) in the values of rumax is observed by decreasing s/d from 7 to 4 for the case of using 2 micropiles, while for the case of applying 4 micropiles, more reduction (about 10% reduction) is occurred. In addition, it is important to remark that the Npeak and the maximum required time (tpeak) are almost constant with the change of s/d.

Effects of Micropiles' Inclination
To investigate the effect of inclined micropiles on the liquefaction potential of reinforced sand, the deposits were reinforced by two diagonal and opposed micropiles. Figure  5b shows the arrangement of 2 inclined micropiles with the inclination angle of 12 degrees (with respect to the vertical axis) and the spacing ratios of 4 and 7. Figure 14 illustrates the variations of ru in the reinforced sand by 2 inclined and vertical micropiles with the spacing ratios of 4 and 7. As shown, the arrangement of 2 inclined micropiles with the spacing ratio of 4 has the best performance to reduce the soil liquefaction potential. This result agrees with results of the seismic performance superiority of the inclined micropiles in dry sands compared with the vertical micropiles obtained by other researchers [35,36,39,53].

Effects of Micropiles' Inclination
To investigate the effect of inclined micropiles on the liquefaction potential of reinforced sand, the deposits were reinforced by two diagonal and opposed micropiles. Figure  5b shows the arrangement of 2 inclined micropiles with the inclination angle of 12 degrees (with respect to the vertical axis) and the spacing ratios of 4 and 7. Figure 14 illustrates the variations of ru in the reinforced sand by 2 inclined and vertical micropiles with the spacing ratios of 4 and 7. As shown, the arrangement of 2 inclined micropiles with the spacing ratio of 4 has the best performance to reduce the soil liquefaction potential. This result agrees with results of the seismic performance superiority of the inclined micropiles in dry sands compared with the vertical micropiles obtained by other researchers [35,36,39,53].  Figure 15 shows the time history of r u under two maximum accelerations of 0.2 g and 0.32 g. The results for the micropile-reinforced sand indicate that the variation of r u is highly dependent on the applied accelerations so that the maximum pore water pressure generated in pure and reinforced sands increases averagely 20% by changing the maximum acceleration applied from 0.2 g to 0.32 g. The development of soil liquefaction potential by increasing the acceleration in pure sand was also reported in previous studies [77,78]. 0.32 g. The results for the micropile-reinforced sand indicate that the variation of ru highly dependent on the applied accelerations so that the maximum pore water pressu generated in pure and reinforced sands increases averagely 20% by changing the ma mum acceleration applied from 0.2 g to 0.32 g. The development of soil liquefaction p tential by increasing the acceleration in pure sand was also reported in previous stud [77,78]. As shown in Figure 15, the speed of dissipation of the excess pore water pressure 0.32 g acceleration amplitude is faster than 0.2 g acceleration amplitude. It can be d scribed that at the acceleration of 0.32 g, the separation between the equivalent microp and the soil is created, so a path for drainage is generated that reduces the excess wa pressure earlier. Variations of ru versus the normalized number of cycles applied (i N⁄Npeak) are displayed in Figure 16.

Effect of Different Scaled Accelerations on r u Values
As observed in Figure 16 and Table 4, however, reinforced sand samples with 2, and 9 micropiles have a good performance in reducing the liquefaction potential at t acceleration of 0.2 g; at the acceleration of 0.32 g, the reinforced sand with 4 and 9 mic piles showed a significantly reduced liquefaction potential. In addition, increasing t scaled loading acceleration increases the generation rate of the pore water pressure. As shown in Figure 15, the speed of dissipation of the excess pore water pressure at 0.32 g acceleration amplitude is faster than 0.2 g acceleration amplitude. It can be described that at the acceleration of 0.32 g, the separation between the equivalent micropile and the soil is created, so a path for drainage is generated that reduces the excess water pressure earlier. Variations of r u versus the normalized number of cycles applied (i.e., N/N peak ) are displayed in Figure 16.
As observed in Figure 16 and Table 4, however, reinforced sand samples with 2, 4, and 9 micropiles have a good performance in reducing the liquefaction potential at the acceleration of 0.2 g; at the acceleration of 0.32 g, the reinforced sand with 4 and 9 micropiles showed a significantly reduced liquefaction potential. In addition, increasing the scaled loading acceleration increases the generation rate of the pore water pressure.

Effect of the Relative Density of Soil on r u Values
Due to the existence of a broad range of in situ relative densities of Anzali sand (both loose and medium-dense sands form a large area of the coastal shoreline of the Caspian Sea, Iran), 30% and 50% are selected as the index relative densities of the loose and mediumdense sands used in the present study, and the effect of different relative densities on the liquefaction potential of pure and reinforced sands are investigated. Figure 17 shows variations of r u with the relative density.
As shown, the rate of the generation of the excess pore water pressure decreases with the increase in soil density, while the number of required cycles increases. It can be interpreted that this behavior is attributed to the proximity of the void ratio of the loose sand to the critical void ratio of sand compared with the medium sand. Pure sands were also shown to have similar behavior in reducing the volumetric strains and increasing the number of cycles required to liquefy [77]. It can also be inferred that micropiles show a better performance in the loose sands (Dr = 30%) compared with the medium dense sands (Dr = 50%) in reducing the liquefaction potential and increasing the liquefaction resistance. This result will be further discussed in the next sections.

Effect of the Relative Density of Soil on ru Values
Due to the existence of a broad range of in situ relative densities of Anzali sand (both loose and medium-dense sands form a large area of the coastal shoreline of the Caspian Sea, Iran), 30% and 50% are selected as the index relative densities of the loose and medium-dense sands used in the present study, and the effect of different relative densities on the liquefaction potential of pure and reinforced sands are investigated. Figure 17 shows variations of ru with the relative density.
As shown, the rate of the generation of the excess pore water pressure decreases with the increase in soil density, while the number of required cycles increases. It can be interpreted that this behavior is attributed to the proximity of the void ratio of the loose sand to the critical void ratio of sand compared with the medium sand. Pure sands were also shown to have similar behavior in reducing the volumetric strains and increasing the number of cycles required to liquefy [77]. It can also be inferred that micropiles show a better performance in the loose sands (Dr = 30%) compared with the medium dense sands (Dr = 50%) in reducing the liquefaction potential and increasing the liquefaction resistance. This result will be further discussed in the next sections.  Figure 18 shows the upper and lower bounds of variation of ru obtai tests on the pure sand deposits with relative densities of 30% and 50% with  Figure 18 shows the upper and lower bounds of variation of r u obtained from the tests on the pure sand deposits with relative densities of 30% and 50% with the consideration of absorbing boundaries. Figure 17. Effect of the relative density on ru values during shaking. Figure 18 shows the upper and lower bounds of variation of ru obtained from tests on the pure sand deposits with relative densities of 30% and 50% with the cons ation of absorbing boundaries. As shown in Figure 18a, there is a good consistency between the results of the pr study with the previous studies available in the literature [78][79][80]. It should be noted although there are some differences in the applied accelerations and frequencies, tainer dimensions, and types of sands (in view of particle size distribution, phy shapes, etc.), there is still acceptable compatibility between the results of the present s and those available in the literature, which can be used as a prove for the accuracy o modeling and the gained results.

Predictive Model for the Potential of Liquefaction
A comparison with the results of available stress-controlled cyclic triaxial tests i conducted in Figure 18b. As shown, the upper bound and lower bound results pres in the present study still cover the results of cyclic triaxial tests of Lee and Albaisa ( and De Alba et al. (1975) [67,68]. This agreement is more considerable for the N⁄Npea ues less than 0.5. Differences observed for N⁄Npeak values between 0.5 to 1 can be d the high loading acceleration induced by the shaking table. As shown in Figure 1   As shown in Figure 18a, there is a good consistency between the results of the present study with the previous studies available in the literature [78][79][80]. It should be noted that although there are some differences in the applied accelerations and frequencies, container dimensions, and types of sands (in view of particle size distribution, physical shapes, etc.), there is still acceptable compatibility between the results of the present study and those available in the literature, which can be used as a prove for the accuracy of the modeling and the gained results.
A comparison with the results of available stress-controlled cyclic triaxial tests is also conducted in Figure 18b. As shown, the upper bound and lower bound results presented in the present study still cover the results of cyclic triaxial tests of Lee and Albaisa (1974) and De Alba et al. (1975) [67,68]. This agreement is more considerable for the N/N peak values less than 0.5. Differences observed for N/N peak values between 0.5 to 1 can be due to the high loading acceleration induced by the shaking table. As shown in Figure 18, the results of samples reinforced by micropiles are not included in the proposed ranges for the pure sand. The results of sand samples reinforced with micropiles are separately prepared and compared with the available literature on the cemented sand [71]. Therefore, new upper and lower bounds are proposed for the micropile-reinforced sand as shown in Figure 19 and are compared with the reported results of the cemented sand in cyclic triaxial tests conducted by Porcino et al. (2015) [71].
Due to the lack of a model for predicting r u values in the sand reinforced by cement-groutinjected micropiles, the proposed ranges can be a considerable help to engineers to assess the safety factor of foundations improved by micropiles against the liquefaction phenomenon.

Evolutionary Polynomial Regression Modeling (EPR)
The evolutionary polynomial regression (EPR) modeling approach known as a symbolic grey box technique can make structured model expressions for a given dataset. It is a hybrid regression method, which employs the genetic programming symbolic regression technique and merges it with the best features of conventional numerical regressions. As the main solution strategy, an evolutionary computing scheme is applied to search for a model of the under-modeling system, and to find the most accurate constants based on the least squares in a parameter estimation framework [81]. Assigning a fixed maximum number of terms, exponents of different polynomial functions are searched in this technique, while genetic programming deals with a general evolutionary search during the evolutionary procedure. That is why the development of mathematical expressions in EPR is not a lengthy-in-time procedure. As another superiority, optimum term numbers can be introduced to the model for each execution. The general form of the expression used to develop an EPR model is shown in Equation (6) [81]: where y is the estimated output vector, a j represents constants, F is a function constructed during the process, X shows the input variables' matrix, f is a function defined by the user, and m stands for the maximum term numbers for the desired output. The flow diagram used as the modeling procedure in EPR is illustrated using Figure 20 [82].  Due to the lack of a model for predicting ru values in the sand reinforced by cementgrout-injected micropiles, the proposed ranges can be a considerable help to engineers to assess the safety factor of foundations improved by micropiles against the liquefaction phenomenon.

Evolutionary Polynomial Regression Modeling (EPR)
The evolutionary polynomial regression (EPR) modeling approach known as a symbolic grey box technique can make structured model expressions for a given dataset. It is a hybrid regression method, which employs the genetic programming symbolic regression technique and merges it with the best features of conventional numerical regressions As the main solution strategy, an evolutionary computing scheme is applied to search for a model of the under-modeling system, and to find the most accurate constants based on the least squares in a parameter estimation framework [81]. Assigning a fixed maximum number of terms, exponents of different polynomial functions are searched in this technique, while genetic programming deals with a general evolutionary search during the evolutionary procedure. That is why the development of mathematical expressions in EPR is not a lengthy-in-time procedure. As another superiority, optimum term numbers can It is widely applied to a variety of geotechnical engineering problems to find the relationship between a complex function and its affecting parameters [47,[83][84][85][86][87][88]. In this paper, to predict the excess pore water pressure ratio and the required cycles to achieve the peak EPWP ratio values, an evolutionary polynomial regression (EPR) model is devel- It is widely applied to a variety of geotechnical engineering problems to find the relationship between a complex function and its affecting parameters [47,[83][84][85][86][87][88]. In this paper, to predict the excess pore water pressure ratio and the required cycles to achieve the peak EPWP ratio values, an evolutionary polynomial regression (EPR) model is developed, and the effectiveness of each parameter on the output models are obtained using sensitivity analyses. The parameters used in the EPR models are acceleration ratio (a/g), number of micropiles (N), the angle of micropile to the horizontal axis (θ) expressed in radian, the relative density of soil (Dr), and the spacing ratio of micropiles (s/d). The EPR model developed for peak values of the EPWP ratio at the bottom PPT is shown in Equation (7).
The EPR model developed for peak values of the EPWP ratio at the upper PPT is shown in Equation (8).  It is widely applied to a variety of geotechnical engineering problems to find the lationship between a complex function and its affecting parameters [47,[83][84][85][86][87][88]. In this p per, to predict the excess pore water pressure ratio and the required cycles to achieve t peak EPWP ratio values, an evolutionary polynomial regression (EPR) model is dev oped, and the effectiveness of each parameter on the output models are obtained usi sensitivity analyses. The parameters used in the EPR models are acceleration ratio (a⁄ number of micropiles (N), the angle of micropile to the horizontal axis (θ) expressed radian, the relative density of soil (Dr), and the spacing ratio of micropiles (s⁄d). The EP model developed for peak values of the EPWP ratio at the bottom PPT is shown in Equ tion (7).   As described, since the nature of the problem is a multiple-variable parameter target function (r u ), interactive effects of concerning parameters (N, a/g, Dr, s/d, and θ) are more important compared with the effect of each individual parameter. Hence, the following section investigates simultaneous effects of loading, geometrical, soil, and configuration parameters on the potential of liquefaction.

Multiple-Variable Parametric (MVP) Study on the Potential of Liquefaction
As described, there are different parameters affecting the liquefaction potential of the studied sandy soil. Some of them are related to the loading conditions (e.g., a/g ratio), some others represent soil conditions (e.g., relative density), and other ones describe the micopiles' configuration and geometrical properties (e.g., number of micropiles, spacing ratio, and the inclination angle of the micropiles). Among these factors, increasing the a/g ratio and the spacing ratio increase the liquefaction potential, while there is a reverse relationship with the number of micropiles and the relative density of soil with the r u . The inclination angle of the micropiles as another affecting parameter does not have a known direct/reverse relationship with the liquefaction susceptibility. Such behaviors make it a complicated task to assess the liquefaction potential of micropile-reinforced sand with simultaneous changes of described, affecting parameters. Hence, the presentation of a multiple-purpose, parametric study based on the developed model for the liquefaction potential can be effective.
7.1. Simultaneous Effects of a/g, Inclination Angle, and Relative Density Figure 22 shows the analysis of the liquefaction behavior of the studied soils with different relative densities and under different loading accelerations. To take the effect of the inclination of micropiles into consideration, the evolutionary relationship (Equation (7)) developed for the liquefaction potential is employed. some others represent soil conditions (e.g., relative density), and other ones desc micopiles' configuration and geometrical properties (e.g., number of micropiles, spa tio, and the inclination angle of the micropiles). Among these factors, increasing the and the spacing ratio increase the liquefaction potential, while there is a reverse rela with the number of micropiles and the relative density of soil with the ru. The inc angle of the micropiles as another affecting parameter does not have a known direct relationship with the liquefaction susceptibility. Such behaviors make it a complica to assess the liquefaction potential of micropile-reinforced sand with simultaneous of described, affecting parameters. Hence, the presentation of a multiple-purpose, p ric study based on the developed model for the liquefaction potential can be effecti 7.1. Simultaneous Effects of a⁄g, Inclination Angle, and Relative Density Figure 22 shows the analysis of the liquefaction behavior of the studied so different relative densities and under different loading accelerations. To take the the inclination of micropiles into consideration, the evolutionary relationship (E (7)) developed for the liquefaction potential is employed.  A 3-dimensional representation of the relationship with the consideration of s/d = 7 and N = 2 better illustrates the simultaneous effects of the inclination angle of the micropiles and the scaled loading acceleration on the liquefaction potential of the model. It should be noted that, in this figure, the axis presenting the "angle" is in radian and also represents the angle of micropiles with respect to the ground surface (the angle complementary to the micropiles' inclination angle). As shown, there is a threshold inclination angle where the liquefaction behavior changes. With increasing the angle between the micropiles and the ground surface, first, the liquefaction potential decreases, and after reaching a critical boundary inclination angle, the behavior changes and the soil tends to liquefy. This shows that inclining micropiles is not always efficient and there is an optimum inclination angle. The value of the boundary micropiles' angle (with respect to the ground surface) in the loose sand is lower than that for dense sands. As another point of view, instead of the angle between the micropiles and the ground surface, the problem can be reconsidered with special attention to the inclination angle of the micropiles. From this viewpoint, it can be concluded that the inclination of the micropiles (with respect to the vertical micropiles) can decrease the potential of liquefaction for both loose and dense sands. However, after reaching the critical, boundary inclination value, the behavior changes. The value of this boundary inclination angle (angle from the vertical direction) is higher for the loose sand (the value of the boundary critical angle between the micropiles and the ground surface is lower for the loose sand), emphasizing that in loose sands, micropiles can be constructed in a wider range of inclination angles still mitigating the liquefaction susceptibility. In other words, the more inclining the micropiles are, the higher the liquefaction resistivity until reaching the critical boundary inclination value where the reserve behavior is observed after inclinations higher than it. On the other hand, although inclining micropiles (for the values of inclination angles lower than the critical boundary inclination angle) still have a positive effect on mitigating the liquefaction potential of medium-dense sands, the applicable range of inclination of micropiles is less than that for loose sands as the value of the critical boundary inclination angle in dense sands is lower than loose sands (in other words, the value of the critical boundary angle between micropiles and the ground surface is higher for the medium-dense sand). This is because such inclinations in dense sands followed by the grout injection can disturb their compacted grain formation leading to instabilities, which will result in increasing the liquefaction susceptibility. In the present study, for the loose sand (Dr = 30%), the critical angle between micropiles and the ground is shown as 0.4 rad (23 • ) (which corresponds to the inclination angle of 67 • ), and the corresponding value for the medium dense sand (Dr = 50%) is 1 rad (57 • ) (which corresponds to the inclination angle of 33 • ). Furthermore, as depicted in Figure 22, the constant rate of variation of r u with the inclination in different acceleration levels shows that input ground acceleration does not have a significant effect on the variation of r u with the inclination (the whole r u -angle curve shifts as the acceleration varies). The same behavior is also observed for the effect of the inclination of micropiles on the variation of r u with the scaled input acceleration.  can be reconsidered with special attention to the inclination angle of the micropile this viewpoint, it can be concluded that the inclination of the micropiles (with re the vertical micropiles) can decrease the potential of liquefaction for both loose an sands. However, after reaching the critical, boundary inclination value, the b changes. The value of this boundary inclination angle (angle from the vertical di is higher for the loose sand (the value of the boundary critical angle between the piles and the ground surface is lower for the loose sand), emphasizing that in loos micropiles can be constructed in a wider range of inclination angles still mitiga liquefaction susceptibility. In other words, the more inclining the micropiles higher the liquefaction resistivity until reaching the critical boundary inclinatio where the reserve behavior is observed after inclinations higher than it. On the othe although inclining micropiles (for the values of inclination angles lower than the boundary inclination angle) still have a positive effect on mitigating the liquefac tential of medium-dense sands, the applicable range of inclination of micropile than that for loose sands as the value of the critical boundary inclination angle i sands is lower than loose sands (in other words, the value of the critical boundar between micropiles and the ground surface is higher for the medium-dense sand) because such inclinations in dense sands followed by the grout injection can distu compacted grain formation leading to instabilities, which will result in increasing uefaction susceptibility. In the present study, for the loose sand (Dr = 30%), the angle between micropiles and the ground is shown as 0.4 rad (23°) (which corresp the inclination angle of 67°), and the corresponding value for the medium dense s = 50%) is 1 rad (57°) (which corresponds to the inclination angle of 33°). Furtherm depicted in Figure 22, the constant rate of variation of ru with the inclination in d acceleration levels shows that input ground acceleration does not have a significa on the variation of ru with the inclination (the whole ru-angle curve shifts as the a tion varies). The same behavior is also observed for the effect of the inclination o piles on the variation of ru with the scaled input acceleration.   Mutual effects of the scaled acceleration and the number of micropiles for different soil's relative densities are also investigated using Figure 23.

Simultaneous Effects of Number of Micropiles, Spacing Ratio, and a/g
As shown, compared with the effects of spacing and a/g ratios, the effect of the number of micropiles on the liquefaction potential is more considerable. As depicted in Figures 23 and 24, if at least the minimum number of required micropiles, as the critical required number (this range is identified to be 5 in the presented figures), is applied, the effects of the spacing and a/g ratio will be negligible, and there will not be a meaningful difference between s/d amounts of 1 or 7, and also a/g values of 0.2 or 0.4 (in view of the liquefaction potential). In general, this boundary can be treated as the solution for an optimization problem with regard to some other pre-defined variables for minimizing the cost of micropiling projects (considering the desired value of the potential of liquefaction, configurations, and the material state).
As shown, compared with the effects of spacing and a/g ratios, the effect of th ber of micropiles on the liquefaction potential is more considerable. As depicted in 23 and 24, if at least the minimum number of required micropiles, as the critical r number (this range is identified to be 5 in the presented figures), is applied, the e the spacing and a/g ratio will be negligible, and there will not be a meaningful di between s/d amounts of 1 or 7, and also a/g values of 0.2 or 0.4 (in view of the liqu potential). In general, this boundary can be treated as the solution for an optim problem with regard to some other pre-defined variables for minimizing the cos cropiling projects (considering the desired value of the potential of liquefaction, rations, and the material state).  Figure 25 investigates the effect of simultaneous changes in the inclination an spacing ratio of micropiles, and also the relative density of the studied soil on th faction potential. For this purpose, the case of a/g = 0.2 and N = 2 is used for the m variable, parametric study.   Figure 25 investigates the effect of simultaneous changes in the inclination angle, the spacing ratio of micropiles, and also the relative density of the studied soil on the liquefaction potential. For this purpose, the case of a/g = 0.2 and N = 2 is used for the multiple-variable, parametric study. number (this range is identified to be 5 in the presented figures), is applied, the the spacing and a/g ratio will be negligible, and there will not be a meaningful d between s/d amounts of 1 or 7, and also a/g values of 0.2 or 0.4 (in view of the liq potential). In general, this boundary can be treated as the solution for an opt problem with regard to some other pre-defined variables for minimizing the co cropiling projects (considering the desired value of the potential of liquefaction rations, and the material state).  Figure 25 investigates the effect of simultaneous changes in the inclination a spacing ratio of micropiles, and also the relative density of the studied soil on t faction potential. For this purpose, the case of a/g = 0.2 and N = 2 is used for the variable, parametric study.  As shown in Figure 25, higher liquefaction resistivity is gained for the lower spacing ratios and the range of (3-4) can be considered as the optimum range for the spacing ratio (in view of both material consumption and enhancing the soil's resistivity against the liquefaction). Decreasing the liquefaction potential with a decreased spacing ratio is even more considerable while micropiles are inclined. Hence, it can be concluded that inclining micropiles amplifies the effect of spacing ratio. This effect also increases as the inclination angle increases (corresponds to the case that the angle between micropiles and the ground surface decreases). Furthermore, as stated earlier, the effect of inclination on reducing the liquefaction potential of loose sands is more significant compared with the medium dense sand.

Conclusions
In this study, a series of small-scale 1 g shaking table tests are performed to study the liquefaction resistance of Anzali sand (commonly found in northern Iran), with and without grouted micropiles considering the soil-micropile interaction. To investigate the efficiency of grout veins around the micropiles on the generation and dissipation of pore water pressure, a specifically designed grouting apparatus is used. The effects of variations of different parameters, such as scaled acceleration, boundary condition, relative density of soil, micropiles spacing ratio, number of micropiles, and their inclination on the liquefaction susceptibility are studied. Based on the gained test results, upper and lower bound curves of the liquefaction potential versus the normalized number of cycles are presented and compared with the available literature. Evolutionary polynomial regression models, for both the bottom and upper PPTs, are developed to predict r umax using the obtained experimental data, and parametric studies were performed to estimate the effects of each parameter. Three-dimensional, multiple-variable parametric studies on the developed EPR model are carried out to investigate the effects of simultaneous changes in input parameters on the liquefaction potential of the micropile-reinforced sand in the shaking table. Based on the results, the following points can be concluded:

•
A more accurate response for the excess pore water pressure is obtained using foam sheets as the artificial lateral boundaries by a reduction in reflected and generated waves, in which its effect is well illustrated by a 7% decrease in r umax .

•
The application of only one micropile has a negligible effect on the liquefaction potential of the soil at different seismic excitations. On the other hand, 2, 4 and 9 micropiles reduce r umax values averagely by 27%, 46%, and 66%, respectively. Therefore, the results clearly show that micropile reinforcement is an effective technique to decrease the liquefaction potential of sands, especially in samples with nine micropiles.

•
The spacing ratio of micropiles has a small effect on r umax and N peak values in the reinforced sand by two vertical micropiles, while its effect is more considerable in specimens reinforced by four vertical micropiles.

•
The reinforced sand by two inclined micropiles exhibits a greater resistance to liquefaction compared with vertical micropiles.

•
The results indicate that specimens of reinforced sand in all micropile arrangements have more liquefaction resistance in comparison with pure sand, due to the increase in the required number of cycles (N peak ) to liquefy. • With increasing the scaled input acceleration, the liquefaction potential of pure and reinforced sand increases. Moreover, the dissipation of pore water pressure occurred faster with an increase in the applied excitations due to the separation between soil and micropiles.

•
The increase in relative density of the sand significantly reduces the liquefaction potential. In addition, this positive effect has a better efficiency in loose sand compared with the medium sand.

•
Modified upper and lower bounds are proposed to predict the values of r u in pure sand at 2 accelerations of 0.2 g and 0.32 g, which improves the previous ranges suggested by Lee and Albisa (1974) and De Alba et al. (1975) [67,68].
• New upper and lower bounds are suggested for the prediction of the liquefaction potential of micropile-induced sand, which can be an efficient controlling tool for design engineers. • High-accuracy EPR models are proposed for the prediction of r u of the sand reinforced with micropiles.
MVP studies on the proposed EPR models illustrated that: • The impact of N, Dr, and a/g on r u is more significant compared with other affecting parameters. • Inclined micropiles have a better performance in the mitigation of liquefaction potential for loose sands compared with the medium-dense sand.

•
The applicable range of inclination of micropiles in medium-dense sands is less than its applicable range for loose sands. • Critical, boundary inclination angle in dense sands (33 • ) is lower than loose sands (67 • ).

•
The range of (3-4) is introduced as the optimum range for the spacing ratio of micropiles (in view of both material consumption and enhancing the soil's resistivity against the liquefaction).

•
With inclining micropiles, the effect of the spacing ratio on the liquefaction potential is amplified.

•
The number of micropiles plays a more important role in the liquefaction potential compared with the spacing ratio and the scaled input acceleration. With the application of at least five micropiles, the effects of s/d and a/g are shown to be negligible.