The Liquefaction Behavior of Poorly Graded Sands Reinforced with Fibers

(is study focuses on the performance of fibers, improving the resistance to liquefaction in loose sands, medium sands, and dense sands in Izmir, Turkey. A systematic testing schedule consisting of cyclic triaxial tests was held under stress-controlled and undrained conditions on saturated sand specimens with and without fiber reinforcements.(emajor parameters having effects on the dynamic behavior such as fiber content, fiber length, and relative density on the liquefaction behavior and the excess pore water pressure developments of specimens with and without fibers were investigated. If the fiber content or the fiber length was increased in the specimens, higher number of loading cycles was needed in order to experience the liquefaction of sands. (e reinforcement effect in medium-dense specimens was found to be apparently distinctive compared to loose specimens.(e curves of pore water pressures and shear strains were achieved for the fiber-reinforced sands. (e boundaries of pore water pressure curves presented in the literature on the clean sands were utilized in comparison with the pore water pressure curves of fiberreinforced sands of this study. As a conclusion, the results presented in this study are useful to develop insight into the behavior of clean and fiber-reinforced sands under seismic loading conditions. Based on the test results, it was found that the number of loading cycles had a strong impact on the excess pore pressure generation.


Introduction
e liquefaction phenomenon in a layer of loose sand under dynamic circumstances occurs by the development of excess pore water pressure and decrement of average effective stress which corresponds to a complete loss of shear strength.Liquefaction may cause damages due to bearing capacity loss of strata, large settlements, tilting of structures, and lateral displacements.Condition of soil could be improved by reinforcement to eliminate the liquefaction hazard.Using reinforcement materials such as fibers in soil medium may provide an alternative to reduce the liquefaction potential.Compared to conventional improvement methods using reinforcement, fibers have some advantages like prevention of potential weak planes which mostly form parallel to plane oriented reinforcement and conservation of isotropic shear strength characteristics [1].e reinforced soil behavior of fibers was studied by researchers in the last decades, but focusing the problem only under static conditions [2][3][4][5].It was revealed that using fiber reinforcement in soil increased the shear strength of soil and improved the ductile behavior and reduced the strength loss observed after the peak strength was achieved.Recently, static liquefaction studies explored the possibility of fiber reinforcement to improve the liquefaction resistance of sand.ese studies stated that the occurrence of lateral spreading could be prevented by using fiber reinforcement [6,7].
Wave propagation during the earthquakes originates undrained shear stresses in the soil medium, the particles of soil experience shear strains, and the pore water pressures are generated in the soil medium.Development of excess pore water pressure decreases the stiffness in response to an applied overburden pressure and triggers a vicious circle that causes larger shear strains and higher pore water pressures.
At the nal stage, the excess pore water pressure reaches a level of initial overburden pressure, and liquefaction is initiated.Since 1970s, analyzing and modelling of excess pore water pressures in soils under earthquake excitations gained interest among the researchers of geotechnical earthquake engineering.In this paper, along with the ndings of this study, other literature is also reviewed and compared with the results provided by the experiments.A detailed testing program is followed by conducting experiments on clean sand specimens with varying conditions by using the cyclic triaxial compression testing device.e results of experimental sets are evaluated by stress-based methods, main parameters a ecting the behavior and the uncertainties are considered during analyses, and practical solutions are compared with the existing models.
Over the last years, the e ects of applying reinforcement materials to increase the shear strength of sands and the factors, including reinforcement type and reinforcement material, soil gradation, and reinforcement dispersion, have been studied only under static conditions by monotonic loadings [8][9][10][11][12][13].Studies by Consoli et al. [4] and Consoli et al. [14] focused on consolidated drained triaxial tests to examine the ber reinforcement e ect on the mechanical behavior of sand admixed with varying cement content.Specimens with a relative density of 70% were prepared, and empirical equations to determine peak and residual strength based on cement content, ber content, and con ning pressure were proposed in Consoli et al. [4] and Consoli et al. [14].e resistance to liquefaction in ber-reinforced soils increased the number of cycles required to cause liquefaction under undrained loading conditions [15][16][17][18][19][20].Cyclic triaxial test results have indicated that the shear modulus of reinforced soil is not only under the control of shear strain, but also under the control of many factors such as ber content, loading repetition, and con ning pressure [21].Bhandari and Han [22] worked on the interaction between the geotextile and the soil under a cyclic wheel load applying the discrete element method.e results showed that the geotextile depth had a major e ect on the degree of interaction between the geotextile and the soil.Shuai-dong and Xiang-juan [23] examined the cyclic behavior of reinforced silty sand by performing consolidated undrained cyclic triaxial tests.e dynamic elastic modulus of reinforced soil was reported to increase due to reinforcement, con ning pressure, and consolidation stress ratio, as to the unreinforced soil.e literature is mainly focused on the resistance of soils against liquefaction, but pore water pressure development is a very e ective sign of behavioral change which is mostly left out of the scope [24].
e aim of this study is to identify the liquefaction resistance and the pore water pressure development of berreinforced sand specimens by applying cyclic triaxial tests.e majority of the previous studies have explored the strength and deformation properties of ber-reinforced soil under monotonic loading conditions; this study particularizes the e ectiveness of bers in the liquefaction resistance improvement of poorly graded sand through some series of dynamic testing.
e in uence of bers on the dynamic behavior is investigated in reinforced sand specimens.
e sets of experiments included specimens with 0%, 0.25%, 0.5%, and 1% polypropylene ber contents.Furthermore, the e ect of ber length is investigated by using two di erent bers with lengths of 6 mm and 12 mm.e relative density of the specimens was 30%, 50%, and 70%, representing the di erent sti ness states of the soil.e specimens were consolidated under a con ning pressure of 100 kPa, and a cyclic loading frequency of 0.1 Hz was applied.
e variations of pore water pressure ratios with number of loading cycles, with ber content, and with ber length under constant stress amplitudes are achieved and presented in this study.

Materials.
A clean sand mass was obtained from an excavation site in the city center of Izmir, Turkey.e classi cation of the sand showed that it was poorly graded sand (SP) according to the Uni ed Soil Classi cation System.e e ective size (D 10 ), the diameter corresponding to 30% ner (D 30 ), mean grain size (D 50 ), and the diameter corresponding to 60% ner (D 60 ) of the sand gradation were 0.15 mm, 0.28 mm, 0.53 mm, and 0.70 mm, respectively.e coe cient of uniformity was 4.67, and the coe cient of curvature was 0.75.e maximum void ratio of the sand was 0.84, and the minimum void ratio of the sand was 0.56.e speci c gravity of the sand was 2.67.e grain-size distribution of this sand is shown in Figure 1. e relevant ASTM standards were followed for all index tests (ASTM D6913, ASTM D4253, ASTM D4254, and ASTM D854) [25][26][27][28].
e mono lament polypropylene (PP) ber materials used in this study were also produced in Turkey by a local company.e bers were rectangular in cross section with a speci c density of 0.91.e tensile strength of bers was 400 MPa, and elastic modulus of bers was 1000-2500 MPa.Fiber lengths were 6 and 12 mm (Figure 2).Fiber ratios of 0.25%, 0.5%, and 1% were added to the specimens by dry weight of sand.Fiber-reinforced sand specimens were (d) Cyclic axial strain with the number of cycles for a specimen of ber-reinforced sand.(e) Cyclic deviatoric stress ratio with the number of cycles for a specimen of ber-reinforced sand (D r 30%, ber length 12 mm, ber ratio 0.25%, and σ 0 ′ 100 kPa).
prepared at the same dry density as that of unreinforced sand.e percentage of bers mixed with sand was calculated as a part of the total solids in the void-solid matrix of the sand.e amount of bers added was calculated over the dry mass of sand.

Testing Program.
A testing schedule was planned to investigate the e ect of ber reinforcement in sand specimens.e test cases were combinations of relative density of the sand, ber length, and ber ratio (Table 1).In addition to the planned schedule, some randomly chosen test cases were repeated to check the validation of specimen preparations of same relative density and loading conditions to demonstrate the accuracy of the test results.

Specimen Preparation.
e initial specimen diameter was 50 mm, and specimen height was 100 mm in the experiments.e so-called "undercompaction technique" of Ladd [29] was adopted as the specimen preparation procedure.A porous stone and a circular lter paper were embedded in the base part of the testing device.A membrane made of rubber was placed in the base, and the movement of the membrane was prevented with O-rings.A 4 Advances in Civil Engineering split mold was placed on the lower plate of the triaxial cell; after that, vacuum was applied to the mold.e upper part of the membrane was tted tightly to the mold.A soil composite of dry sand and bers was prepared and transferred to the mold step by step.e soil composite was divided into ten equal parts, and each part was carried into the mold.A wooden rod was used for compaction.In order to achieve su cient bonding, top of every compacted layer was cleared o before locating the following layer.
Another circular lter paper and a porous stone were set above the top of the specimen.e membrane was slipped to the specimen cap from the mold carefully.JGS 0520-2000 [30] was followed for the preparation of the specimens.e dynamic testing procedure of JGS 0541-2000 [31] was followed.

Test Procedure.
e stress-controlled cyclic triaxial tests were performed.e triaxial testing system includes a vertical pressure loading unit with air and water panel, a triaxial cell, a pneumatic sine loader, an electric measurement unit, including, pressure, displacement, and volume change transducers, strain ampli ers, and a dynamic data acquisition system.e specimens were initially ooded with carbon dioxide; after this step, the specimen was ooded with deaired water, and back pressure was applied to saturate the specimens.Skempton (B) parameter de ning the saturation was assured to vary between 0.96 and 1.00.e specimens were isotropically consolidated under 100 kPa of e ective stress, and undrained cyclic loading was subsequently applied in a stress-controlled manner.In the liquefaction tests, the loading sequence applies a certain number of cycles necessary to reach a speci ed level of cyclic stress under a frequency of 0.1 Hz until the specimen develops a double-amplitude (DA) axial strain of 5%.During cyclic loading, continuous digital data were recorded for the following parameters: cyclic axial strain (ε c ), excess pore water pressure (u), and the cyclic deviator stress ratio applied to the specimen.JGS 0541-2000 considers two criteria to de ne liquefaction.If the amplitude of cyclic axial load is relatively large, the number of cycles needed to cause liquefaction is accepted as the number of cycles needed to reach a maximum value of excess pore water pressure equal to 95% of the effective con ning stress; otherwise, it is recognized as the number of cycles needed to reach a double amplitude of 5% of the axial displacement of the specimen.e experiments progress until all specimens reach 10% of the axial displacement.
Figure 3 shows the cyclic trixial test results of sand specimen prepared at a relative density of 30%.An e ective con ning pressure of 100 kPa was applied to the specimen.Figure 3(a) illustrates the development of the stress path of the specimen.During cyclic testing, a continuous sequence of compression and extension loadings having the same intensity was regularly applied to simulate dynamic conditions, which was seen from the steady change of q/σ′ 0 between +0.20 and −0.20. Figure 3(b) illustrates the variation of the stress path with the cyclic axial strain.e initial stage of the test started with a small cyclic axial strain, but as the applied cycles progressed, the strains became more dominant on the compression side to a level of 8% strain.Pore water pressure development with the number of loading cycles is given in Figure 3(c).A steady trend was observed in the progress of the pore water pressure ratio with the number of cycles.After 3 cycles, the liquefaction criterion was satis ed, and the cyclic axial strains varied over a wider range.Figure 3(d) shows the state when the pore water pressure ratio equals the e ective pressure.Strain levels of +8% and −5% were achieved in the Advances in Civil Engineering compression and extension sides, respectively, resulting in a total strain of 13%. Figure 3(e) shows the cyclic deviator stress ratio with the number of cycles as an example of a measured record during undrained cyclic triaxial testing.

Results and Discussion
e main parameters of this study, namely, relative density, ber length, and ber ratio on the liquefaction resistance, are presented and discussed.It should be noted that all the  6 Advances in Civil Engineering outcomes of this study belongs to the experiments which were performed under 100 kPa e ective con ning pressure.e liquefaction criterion of the tests was to achieve the number of cycles when the specimen developed a doubleamplitude (DA) axial strain of 5%.

E ect of Relative Density.
e cyclic stress ratio is calculated as the cyclic strength normalized by the e ective stress.In Figures 4-6, variations of the cyclic stress ratio (CSR) with the number of cycles of ber-reinforced soils with relative densities of 30%, 50%, and 70% are shown, respectively.
e control group consisting of specimens without bers was also prepared in aforementioned relative densities and tested.eir results could also be viewed in Figures 4-6.An increase in ber length increases the number of cycles that would trigger liquefaction for the specimens having a relative density of 30%.For specimens having a relative density of 50% and ber length of 6 mm, lowest liquefaction resistance was obtained from specimens having a ber ratio of 0.25%.Figure 4 shows the specimens constituted by using 6 mm bers with sand and compacted to a relative density of 30% resulted a CSR value of 0.361 and the specimens prepared by using 12 mm bers with sand and compacted to a relative density of 30% resulted a CSR value of 0.384.If the relative density was increased to 50%, a CSR value of 0.424 was found for the specimens with 6 mm bers and a CSR value of 0.438 was found for the specimens with 12 mm bers (Figure 5).When the relative density was attained as 70%, the CSR values corresponding to 6 mm and 12 mm were 0.434 and 0.547, respectively (Figure 6).In order to compare the variation of number of cycles with CSR values, the upper boundary of CSR was taken as 0.6.If Figures 4-6 were interpreted amongst themselves, the relative density was found the dominating factor among other variables such as ber length and ber ratio.It was seen that the maximum CSR values were obtained for specimens with 12 mm bers and compacted to a relative density of 70%, resulting in the maximum resistance to liquefaction cycles [19,20].
e specimens with a ber ratio of 0.5% and the ones without bers showed a similar liquefaction resistance.e highest resistance was achieved in specimens that contain 1% of bers.For specimens having a relative density of 50% and ber length of 12 mm, liquefaction resistance of specimens that contain no ber, 0.25%, and 0.5% showed  Advances in Civil Engineering a liquefaction resistance varying in a narrow band.e most remarkable improvement against liquefaction was obtained in specimens with 1% ber content.For medium dense specimens, the e ective length of ber required to develop shear strength increased with the increase in ber length at a constant ber ratio.In this condition, the slippage taking place between individual bers was reduced with ber length increment, resulting in improved performance of bers in soil.

E ect of Fiber Ratio.
e ber ratios of specimens were chosen as 0.25%, 0.5%, and 1.0%.For all test cases, the liquefaction resistance was highest for specimens with D r 70% and FL 12 mm, and it was lowest for specimens with D r 30% and FL 6 mm (Figure 7). is nding is valid for all ber ratios used in this study.Besides, medium dense specimens (D r 50%) with a ber length (FL) of 6 mm showed the second best performance against liquefaction.
is trend is followed in the same manner by loose specimens.is nding is associated with the fact that the voids in the soil matrix are covered by the ber addition, inducing an additional densi cation of the solid matrix.is nding is in congruence with the results of Ibraim et al. [6].

E ect of Fiber Length.
e cyclic stress ratios corresponding to 20 loading cycles at 5% double-amplitude axial strain are given in Figure 8.When the e ect of ber length is observed for ber-reinforced specimens, CSR values are greater for longer bers (FL 12 mm).As an example, CSR is 0.264 for medium dense specimens having a ber ratio of 0.25 and ber length of 6 mm (Figure 8(a)), and CSR is 0.335 for medium dense specimens (D r 50%) having a ber ratio  ).e increase in CSR is 27%.erefore, it may be said that the ber length increment also improves the liquefaction resistance of the poorly graded sand.If a comparison is made between loose specimens with ber ratios of 0% and 1%, improvement in liquefaction resistance is 52% for loose specimens that contain 6 mm long bers, and improvement in liquefaction resistance is 59% for loose specimens that contain 12 mm long bers.A notable development in liquefaction resistance is observed when the poorly graded sand is reinforced with bers.

Pore Pressure Development.
In liquefaction tests, the pore water pressure develops continuously and reaches the initially applied con ning stress after a certain amount of loading cycles.Pore water pressure generation depends on the relative density of the soil and the existing cyclic stress ratio.In addition, shear strain of the soil is a dominating property that relates to the e ect of number of loading cycles on the level of pore water pressure.Since four decades, an interest has been raised to evaluate the generation of excess pore water pressure of sands considering the above mentioned parameters, and some numerical models were proposed in literature.However, the behavior of reinforced soils is still not clear and requires further research.In this section, the main parameters of this study, namely, relative density, ber length, and ber ratio on the pore water pressure generation curves, are presented and discussed.It should be noted that all the outcomes of this study belong to the experiments which were performed under 100 kPa e ective con ning pressure.e liquefaction criterion of the tests was to achieve the number of cycles when the specimen developed 5% double-amplitude axial strain.
In order to model the pore water pressure generation in ber-reinforced specimens, two di erent stress-based models were considered, and α coe cients of these models are achieved by calculating the smallest mean square error in each model.e model proposed by Seed et al. [32] can be stated in a closed-form solution in where r u is the pore water pressure ratio, N is the number of equivalent uniform loading cycles, and N liq is the number of cycles required to produce initial liquefaction (r u 1.0).α is a function of the soil properties and test conditions.In this study, the model o ered by Seed et al. [32] is used to evaluate the pore-water pressure ratio data.α coe cients are calculated with a con dence interval of 95% by using the bootstrapping, which allows assigning measures of accuracy to specimen estimates of ber-reinforced sand specimens.α coe cients are presented as individual points along with all specimen data.α coe cients versus test numbers are given in Figure 9. α coe cient calculated with a con dence interval of 95% was found as 4.920 in Figure 9 [24].
e relationship between the pore-water pressure ratio and number of cycles required to initiate liquefaction for the specimens having 30%, 50%, and 70% relative densities which were consolidated under 100 kPa of overburden pressure are given in Figure 10.Advances in Civil Engineering the curves obtained by calculating the smallest mean square error for α coe cient and using (1).α coe cient is 4.92 on the average, which is higher than the upper limit value o ered in the work of Seed et al. [32].In Figure 10(a), it is seen that when the pore-water pressure ratio is 30%, the number of cycle ratio is 10%.However, if (1) is used to calculate α coe cients, then the pore-water ratio becomes 50% from the same point where the number of cycles ratio is 10% (Figure 10(b)).As α coe cients are highly dependent on soil conditions, ber-reinforced medium resulted in much higher α coe cients, which is depicted as curve arcs with steep slopes, than the curves observed for the clean sands.Booker et al. [33] proposed another equation ( 2) similar to the pore-water pressure model proposed by Seed et al. [32].
Model parameters such as r u , N, N liq , and α have the same de nitions with (1).α coe cients are calculated with a con dence interval of 95% by using the bootstrapping methodology to provide comparison with the results of Seed's equation.α coe cients are presented as individual points along with lower limit, average, and upper limit values of all specimen data in Figure 11.
e same procedure explained in methodology of Seed et al. [32] is also followed for the model suggested by Booker et al. [33] to derive the curves of pore-water pressure rationumber of cycles ratio.α coe cient is derived as 3.66.Figure 12(a) shows the curves where pore-water ratio values are normalized with the number of cycles ratio values, while For the initial 10-20% of number of cycles ratio, the pore water pressure development is relatively quick, and pore pressure ratio is in the range of 50-60%.A statistical evaluation of methodology of Seed et al. [32] was performed by Polito et al. [34].According to Polito et al. [34], it was concluded that the model coe cient (α) should be estimated as a function which includes the variables of cyclic stress ratio (CSR), nes content of the soil (FC), and relative density of the soil (D r ): ( In this study, the ner passing through number 200 mesh was around 0.14%, therefore α 1 could be neglected.Using the bootstrap algorithm, α 2 , α 3 , and α 4 values were calculated as 2.230, 1.757, and 3.222, respectively, and their dispersion among with the tests are given in Figure 13. Pore-water pressure ratio (PWP) versus number of cycles ratio (N/N L ) obtained from the results of the experiments are shown in Figure 14(a).Figure 14(b) shows pore-water pressure ratio (PWP) versus number of cycles ratio (N/N L ) obtained by using α coe cients in (3) by the bootstrap method of the model proposed by Polito et al. [34].
e number of cycles ratio of 0.2 corresponds to a PWP of 50%.By using (3), minimizing all mean square errors was possible, and for every α coe cient, pore water pressure versus number of cycles ratio could be obtained (Figure 14(b)).High values of α coe cients were calculated; this means high pore water pressure values were produced in the beginning number of cycles ratio in conformity with Figure 14(b).e critical point was 50% of pore pressure where the number of cycles ratio was 0.1.

E ect of Double Amplitude (DA) of Axial Strain.
e e ect of number of loading cycles on the pore pressure level is basically a shearing strain function [35].e change of double-amplitude axial strain with the number of cycles in each testing condition with corresponding cyclic stress ratio (CSR) is given in Figure 15.
Liquefaction occurred instantly in reinforced sand specimens with lower relative densities; because of this, double amplitude of axial strain follows a perpendicular path (Figure 15(a)).e values of CSR bigger than 0.230 required number of cycles less than 20 to observe liquefaction.As the CSR decreased, the number of cycles to liquefaction increased.In Figure 15(b), at a constant CSR value of 0.286, the number of cycles to liquefaction was 14 for specimens with 0.25% of bers, 16 for specimens with 0.50% of bers, and 33 for specimens with 1% of bers.Increment of ber ratio dramatically increased the number of cycles at a constant value of CSR. is nding proves the e ciency of ber ratio in sand specimens.Fiber ratio has a direct e ect on the number of cycles resulting in liquefaction in reinforced specimens.

Conclusions
A series of dynamic experiments was carried out by performing cyclic triaxial tests on poorly graded sand specimens.e sand was obtained in bulk form in an excavation site in Izmir-Turkey.e liquefaction and stress-strain behavior of the sands were investigated in laboratory triaxial tests performed on reconstituted specimens.e con ning pressures were 100 kPa re ecting the actual overburden pressure in situ conditions.e frequency of testing was held at 0.1 Hz. ree di erent relative density values of the sand were considered: loose (D r 30%), medium dense (D r 50%), and dense (D r 70%).e parameters a ecting  the liquefaction behavior of soil were considered as ber content (0.25%, 0.50%, and 1%), ber length (6 mm and 12 mm), and relative density.Specimens without bers were also prepared and tested as a control group in the relative densities of 30%, 50%, and 70%.Liquefaction behavior was observed and explained through cyclic stress curves, and the excess pore water pressure developments of specimens with and without bers were modeled with the relevant equations in the literature, and model parameters considering the e ect of bers were proposed.e following conclusions could be summarized in this study: (1) e ber existence causes a major change in the liquefaction behavior and reduces the susceptibility of soil to liquefy.If the ber ratio was increased, the number of cycles triggering liquefaction was also increased.Maximum improvement in resistance to liquefaction was for sand specimens reinforced with 1% bers at D r 50%, and FL 12 mm.e results apparently imply that ber reinforcement could be an e ective solution improving the liquefaction resistance of the poorly graded sand.(2) CSR values increased with the increment of ber length.is result is attributed to the development of a better mesh structure in the soil matrix as more grains can interrelate with a longer ber.(3) e liquefaction resistance of the poorly graded sand increases with an increase in relative density.In medium dense specimens (D r 50%), the reinforcement e ect was found to be distinctive compared to the loose specimens (D r 30%).( 4) In this study model parameter (α), mean square error of α coe cients are obtained for the pore water pressure models o ered by Seed et al. [32] and Booker et al. [33].Mean square errors of both models were assured to be less than 0.02.(5) Seed et al. [32] and Booker et al. [33] were preferred due to the similar nature and to provide comparison to the ndings of this study.e accuracy of the models of Seed et al. [32] and Booker et al. [33] are checked by mean square errors.ese values were found very close to each other.However, the experimental results of ber-reinforced sand specimens showed that α coe cient is a ected from the relative density of the reinforced medium, cyclic shear strength ratio, and the properties of bers such as length and ratio.Obtaining higher coe cients for the reinforced soil should be interpreted in this manner.12 Advances in Civil Engineering (6) Statistical calculations depending on the ber length, ber ratio, and CSR values are compared with the statistical model o ered by Polito et al. [34].In this method, nes content is taken as 0%.e statistical parameters showed that ber length, ber ratio, and relative density of the medium highly a ect and change the development of pore water pressure.(7) Fiber reinforcement could be an alternative in lowering or eliminating the lateral movement of the sands caused by liquefaction.Further research is planned to focus on lateral spreading of berreinforced poorly graded sand.

Figure 1 :
Figure 1: Grain-size distribution of the sand.

Figure 3 :
Figure 3: (a) Development of stress path.(b) Variation of stress path with cyclic axial strain.(c) Pore water pressure ratio with the number of cycles.(d)Cyclic axial strain with the number of cycles for a specimen of ber-reinforced sand.(e) Cyclic deviatoric stress ratio with the number of cycles for a specimen of ber-reinforced sand (D r 30%, ber length 12 mm, ber ratio 0.25%, and σ 0 ′ 100 kPa).

Figure 4 :Figure 5 :
Figure 4: Variation of CSR with the number of cycles considering the e ect of ber length (a) FL 6 mm and (b) FL 12 mm (WO: without bers, FR: ber ratio, D r 30%, and σ 0 ′ 100 kPa).

Figure 6 :
Figure 6: Variation of CSR with the number of cycles considering the e ect of ber length (a) FL 6 mm and (b) FL 12 mm (WO: without bers, FR: ber ratio, D r 70%, and σ 0 ′ 100 kPa).

Figure 10 :
Figure 10: (a) Pore-water pressure ratio (PWP) versus cycle ratio number (N/N L ) obtained from the results of experiments.(b) Porewater pressure ratio (PWP) versus number of cycles ratio (N/N L ) obtained by using α coe cients from the model proposed by Seed et al. [32].

Figure 12 :
Figure 12: Pore-water pressure ratio (PWP) versus cycle ratio number (N/N L ) (a) obtained from the results of experiments and (b) obtained by using α coe cients from the model proposed by Booker et al. [33].

Table 1 :
Test cases conducted in this study.