Effect of cyclic loading on the mobilization of unit base resistance of model piles jacked in sand

This paper reports the results of a series of monotonic compressive and cyclic load tests performed on a closed-ended jacked model pile installed in a half-cylindrical calibration chamber with image analysis capabilities. The monotonic compressive load tests were carried out before and after performing a displacement-controlled cyclic load test to determine the impact of cycling on unit base resistance. Digital images of the sand and the model pile were taken during cyclic loading and processed using the digital image correlation (DIC) technique to obtain the cumulative displacement and strain fields in the sand domain. The results show that the ultimate unit base resistance can drop significantly after cycling. The magnitude of the drop in ultimate unit base resistance depends on both the magnitude of the cyclic displacement amplitude and the number of cycles. However, the unit base resistance at plunging increases after large-displacement half-amplitude cycling. The increase in unit base resistance at plunging after cycling is linked to the flow of sand particles to the zone below the conical base, the occurrence of sand particle crushing, and the dilative behavior of the sand outside a bulb of crushed particles formed during cyclic loading. The processed DIC data show that for cyclic displacement half-amplitudes ∆wcyclic less than or equal to 0.25 mm (0.0007B), the cumulative radial and vertical displacements in the soil domain normalized by ∆wcyclic are negligible. Values of ∆wcyclic greater than 0.25 mm produce normalized displacements in the soil domain that increase as the cyclic displacement amplitude increases.

Unit base resistance mobilized during loading q b,AC Unit base resistance mobilized in compressive loading performed after cycling q b,BC Unit base resistance mobilized in compressive loading performed before cycling q bL,BC Limit unit base resistance before cycling q bmax Maximum unit base resistance mobilized in each cycle during cyclic loading q bmax0 Maximum unit base resistance mobilized at the first cycle of cyclic loading q bmin Minimum unit base resistance mobilized in each cycle during cyclic loading q bmin0 Maximum unit base resistance mobilized at the first cycle of cyclic loading Tripod and jacket structures are frame structures with three-or four-legs, each supported on an individual monopile [18,29].This type of pile foundation is often selected for offshore wind turbines at depths ranging from 30 to 70 m [1,6,18].Due to the low self-weight of these offshore structures, the piles will likely experience complete load reversals (from tension to compression and back) when subjected to wind and wave loading.When the piles of a jacket/tripod structure are short and rigid, the load cycles can significantly impact pile base resistance, and consequently the overall response of the pile.The boundary-element continuum method (e.g., [26,32]) and the load-transfer method (e.g., [26,13]) are often used in practice for cyclic axial loading analyses of pile foundations.These methods use empirical rules or criteria to simulate pile-soil interaction and predict the effects of cyclic loading on pile response; these effects are degradation of pile capacity and accumulation of permanent displacements.The input parameters in these methods are selected based on engineering judgment, and, in some cases, predictions are validated or calibrated with laboratory-and or field-scale pile load test data [4,12,33,40,41].However, limited experimental data are available to validate predictions considering the effects of cyclic loading on pile base resistance.
Earlier studies using model piles showed that pile response to cyclic loading depends on the number of cycles applied (N), the type of load (one or two-way cyclic loading), and the amplitude of the cyclic load [9].The overall response of the piles has been characterized by different authors [24,35,37,46,51,52] using cyclic interaction diagrams containing three zones (See Fig. 1): (a) a stable zone, where axial displacements stabilize and there is in some cases an improvement in pile capacity; (b) an unstable zone where the axial displacements accumulate in the first 100 or 200 cycles and shaft resistance is reduced drastically; and (c) an intermediate metastable zone where there is an intermediate behavior between (a) and (b) with cyclic failure in a range between 100 and 1000 cycles.Igoe and Gavin [22], using results from field tests and reinterpretation of existing tests at the Dunkirk testing site, updated the boundaries of the proposed interaction diagrams.
The majority of the reported field pile load tests performed on displacement piles in sand to investigate the effects of cyclic loading on their static capacity [23,24] have focused on the tensile shaft capacity.However, results from multiple compressive static and cyclic load tests on bored piles in sand [34] have shown that the total compressive pile capacity can, in some cases, increase after cyclic loading.Puech [34] suggested that the increase of the total pile capacity is a consequence of a substantial increase in pile base capacity that compensates for the loss in shaft capacity; he argued that progressive densification of the sand below the pile base is the cause of this increase.The effects of cyclic loading on the static base resistance of piles have been investigated through model pile experiments in calibration chambers [25,55] and centrifuges [7,28].Le Kouby et al. [25] showed that the static base resistance of jacked and preinstalled model piles (with diameter B equal to 20 mm) decreases after the performance of a displacement-controlled cyclic load test with amplitudes varying from 0.1 to 2.0 mm.Wang et al. [55] using calcareous sand, concluded that cyclic loading reduces the mobilized shaft resistance but increases base resistance due to crushing of particles and densification.In centrifuge tests, Li et al. [28] did not observe any influence of the cyclic displacement on the ultimate base resistance of jacked piles.These results do not address how the magnitude of the cyclic displacement and the number of cycles affect specifically the static unit base resistance mobilization.
Image-based deformation techniques in geotechnical modeling have been used to understand and quantify soil deformation in the boundary-value problems of geomechanics.Study of the cone penetration problem [3,31], pile installation [8,10,56], static pile loading [16,17,45,48], and cyclic loading [15] are some applications of image analysis in this context.This paper presents the results of a series of monotonic compressive and cyclic displacement-controlled load tests performed on a model pile jacked in silica sand in a halfcylindrical calibration chamber with Digital Image Correlation (DIC) capabilities.We consider the effects of cyclic displacement amplitude and number of cycles on the unit base resistance of the model pile.We also present strain and displacement fields in the sand domain around the conical base of the model pile and discuss the primary mechanisms controlling the response of the pile base to cyclic loading.

Test equipment
Model pile tests were performed in a half-cylindrical calibration chamber at Purdue University, USA (see Fig. 2).Details of the chamber and the testing equipment are provided in Table 1.The front wall of the chamber contains three observation windows that allow capturing of digital images of the model pile and the surrounding soil during model pile load testing.The model pile consists of an instrumented half-circular rod with a conical base.Figure 3 shows the brass model pile used in the experiments with the sensor layout.A specially designed tension/compression load cell with a rated capacity of 42 kN was used to measure the load at the pile head.Four electrical-resistance strain gauges installed diametrically opposed to each other in a cylindrical brass rod (load-transfer bar) were used to measure axial deformation of the load-transfer bar and to obtain the load at the base of the model pile.The installation and the monotonic and cyclic loadings of the model pile were performed using a hydraulic actuator mounted on a removable steel frame, as shown in Fig. 2.

Test sand
Ohio Gold Frac sand, a poorly graded silica sand (SiO 2- = 99.7%) with a mean particle size D 50 of 0.62 mm, was used for sample preparation.The index properties and the values of the roundness and sphericity parameters of Ohio Gold Frac sand are summarized in Table 2.

Image analysis
Digital images were taken during cyclic loading using two complementary metal-oxide-semiconductor (CMOS) cameras (see Fig. 3) positioned in front of the top and middle observation windows of the chamber.The images captured were analyzed using the two-dimensional Digital Image Correlation (DIC) technique to obtain the displacement and strain fields in the sand domain surrounding the model pile base during static and cycling loading.The commercial software VIC-2D [14] was used to perform the  3.The fundamentals of the DIC technique are described in [3,16,45,48].

Test procedure and test program
Twelve model pile tests were carried out in the halfcylindrical calibration chamber.The samples were prepared by air pluviation using a large pluviator positioned at the top of the calibration chamber [27].After the sand sample was prepared with the desired density, the loading system and the cameras were carefully positioned (see  Boundary conditions as described in [19,39] b Scale effects on base resistance.B/D 50 should be greater than 20 [20,38] c Complementary Metal-Oxide Semiconductor cameras Fig. 3) in front of the chamber.A surcharge was applied at the top of the sample using a rubber bladder and a reaction steel lid bolted to the chamber.This surcharge produced an initial vertical effective stress r v0 of 33 kPa at the level of the pile base.Next, the model pile was installed at a rate of 1.0 mm/s using jacking strokes of 10 mm to a target base depth of 415 mm (= 10.9B).Once the model pile base reached the desired penetration depth, it was unloaded to simulate the end of installation.Following the installation stage, a compressive load test was performed by pushing the model pile down at a constant rate of 0.1 mm/s for a distance of approximately 12 mm (&0.3B).Next, the pile head load was removed by detaching the loading system from the head of the model pile.Then, the pile was loaded monotonically to a pile base settlement w b of 0.01B, corresponding to a unit base resistance in very dense sand of approximately 28% of the limit unit base resistance q bL,BC measured before cycling.The model pile was subjected to this working load before the cyclic loading stage of the test started.The limit unit base resistance q bL corresponds to the limiting value of the unit base load at which the sand mass surrounding the pile can no longer generate additional resistance, leading to the plunging of the pile [5].The value of q bL,BC was obtained from the compressive load test performed right after the installation of the model pile.The model pile was then subjected to displacement-controlled cycles.The cycles were performed with uniform sinusoidal displacement half-amplitude Dw cyclic ranging from 0.25 mm [Dw cyclic = 0.007B = 0.4D 50 ] to 3.0 mm [Dw cyclic = 0.079B = 4.8D 50 ].The cyclic displacement half-amplitude Dw cyclic was applied at the head of the model pile.The number N c of cycles applied ranged from 100 to 2000 cycles with a frequency f that varied from 0.1 to 1.0 Hz.The ranges of cycles and frequencies selected for these experiments are typical of cyclic loading events to which offshore structures are subjected to [2].Once the cycling stage was completed, any remaining load on the model pile head was removed by disconnecting the loading system from the head of the model pile.In the final stage of the test, the model pile was loaded in compression under displacement-controlled conditions to a depth of at least 1.0 pile diameter (1B = 38.1 mm) at a rate of 0.1 mm/s.Table 4 presents the test conditions of all the tests performed following the procedure described above.The tests were identified by a testing code that gives the information of the cyclic parameters: cyclic displacement half-amplitude Dw cyclic , denoted by CY, and the number N c of cycles, denoted by N. The number that follows the notation letters represents the value of the variable (in millimeters for, Dw cyclic and dimensionless for N c ).All tests were performed in very dense sand samples (relative density D R ranging from 86.9 to 94.3%).The model pile was installed by jacking strokes of 10 mm length.A initial vertical effective stress r 0 ' of 33 kPa at the level of the pile base was applied in all tests that measured in the compressive load test performed before cycling (q b,BC ) (e.g., at w b /B = 0.1, q b,AC- = 0.58q b,BC ).But, the difference between q b,AC , and q b,BC decreases as w b /B increases.In Fig. 4c, q b,AC is less than q b,BC for w b /B \ 0.3.For w b /B [ 0.3, q b,BC remains constant (equal to the limit unit base resistance q bL,BC measured before cycling) while q b,AC increases as the relative settlement increases.At the maximum relative settlement (w b /B = 1), q b,AC is 40% greater than the limit unit base resistance q bL,BC measured before cycling.

Effect of cyclic displacement half-amplitude
The effect of the cyclic displacement half-amplitude Dw cyclic on the static unit base resistance q b is investigated using the ratio q b,AC /q b,BC of the unit base resistance after cycling to the unit base resistance before cycling, both measured at the same relative base settlement w b /B in the compressive loading tests.Figure 5 shows the curves of q b,AC /q b,BC versus Dw cyclic obtained at four different values of relative settlement w b /B (= 0.1, 0.3, 0.6, and 0.1) and also at plunging.The values of unit base resistance at plunging after cycling were estimated using Chin's method [11].An additional horizontal axis is shown in Fig. 5: Dw cyclic normalized by D 50 .It can be seen that the values of q b,AC /q b,BC are equal to 1.0 for cyclic displacements half-amplitude Dw cyclic less than 0.5 mm, which corresponds to 0.8 times the mean particle size D 50 of the test sand.The results suggest that a value of Dw cyclic equal to 0.5 mm can For a relative pile base settlement w b /B equal to 0.1B (at the ultimate state), Fig. 5 shows that the value of q b,AC / q b,BC decreases from 0.90 to 0.48 when Dw cyclic increases from 0.5 mm (Dw cyclic /D 50 = 0.8) to 1.5 mm (Dw cyclic / D 50 = 2.4).For Dw cyclic [ 1.5 mm, the value of q b,AC / q b,BC tends to increase, at a low rate, with increasing Dw cyclic (q b,AC /q b,BC = 0.63 for Dw cyclic = 3.0 mm).As shown in Fig. 5, the ultimate unit base resistance is significantly affected by uniform cycles of cyclic displacement half-amplitude Dw cyclic in the 0.5 mm to 1.5 mm range.Figure 5 also shows that, for a value of Dw cyclic of 3.0 mm, the ratio q b,AC /q b,BC increases as the relative pile base settlement w b /B increases.The values at plunging show that the limit unit base resistance after cycling is 1.7 times greater than the value of the limit unit base resistance q bL,BC measured before cycling.

Unit base resistance mobilized during cyclic load tests
Figure 6a-d shows the unit base resistance q b and the pile head displacement w (positive for downward pile head movement and negative for upward pile head movement) mobilized during the cyclic stage of tests CY0.25-N100, CY0.5-N100, CY1.0-N100, and CY3.0-N100 as a function of time t.For each cycle, there is a maximum unit base resistance q b,max (i.e., peaks in q b versus t), and a minimum unit base resistance q b,min (i.e., valleys in q b versus t).For a cyclic displacement half-amplitude Dw cyclic of 0.25 mm (test CY0.25-100), the maximum q b,max and minimum q b,min unit base resistances remain approximately constant during cycling for t up to 1000 s.For a cyclic displacement half-amplitude Dw cyclic of 0.5 mm (test CY0.5-100) and 1.0 mm (test CY1.0-100), q b,max decreases as cyclic loading progresses.A different response is observed for the test with Dw cyclic of 3.0 mm (test CY3.0-N100); q b,max decreases during the early stages of cycling (t \ 100 s), but then q b,max increases slightly with time.We also observe that for Dw cyclic [ 0.5 mm, the pile base is fully unloaded during the pull-out phase of each cycle (q b,min always reaches a value of zero).This is an indication of loss of contact between the pile base and the sand.Figure 7 compares the maximum unit base resistance q b,max normalized by the maximum unit base resistance q b,max0 mobilized in the first cycle of cyclic loading (i.e., normalized unit base resistance q b,max /q b,max0 ) for tests CY0. 25 Figure 7 shows that, within the first ten cycles, q b,max /q b,- max0 decreases at a higher rate with increasing Dw cyclic For Dw cyclic = 1.5 mm, q b,max /q b,max0 drops 22% within the first 5 cycles (i.e., q b,max /q b,max0 = 0.78), while for Dw cyclic = 0.25 mm, q b,max /q b,max0 drops only 2% (i.e., q b,max /q b,max0 = 0.98).For 0.25 mm \ Dw cyclic \ 1.0 mm, q b,max /q b,max0 continues decreasing at an approximately constant rate.While for Dw cyclic = 1.0 mm and 1.5 mm, q b,max /q b,max0 tends to stabilize near the end of the test at a value of approximately 0.2.
For reference, Table 5 provides the values of the limit unit base resistance q bL,BC before cycling, the values of the maximum unit base resistance q b,max0 and the minimum unit base resistance q b,min0 mobilized in the first cycle of cyclic loading, and the values of q b,max and q b,max mobilized in the last cycle of cyclic loading (N c = 100 cycles).As expected, q b,max0 increases with increasing cyclic displacement half-amplitude Dw cyclic .For tests CY3.0-N100 (Dw cyclic = 3.0 mm), q bL,BC is equal to q b,max0 ; thus, the base resistance is fully mobilized in the first cycle.For tests CY0.25-N100 (Dw cyclic = 0.25 mm), q b,max0 is approximately equal to 0.37q bL,BC.

Particle crushing effects
Figure 8 shows the digital images obtained at the end of the cycling stage of tests CY0.5-N100, CY1.0-N100, CY1.5-N100, and CY3.0-N100.For tests CY0.5-N100 [Fig.8a] and CY1.0-N100 [Fig.8b], the zone of crushed particles near the pile base seems to be smaller than those for tests CY1.5-N100 [Fig.8c] and CY3.0-N100 [Fig.8d].We estimated the amount of crushing around the conical base at the end of 100 loading cycles by measuring the area where we detected crushed particles (crushed particles are lighter than uncrushed particles).The estimated crushing area A c was normalized by the projected area A b of the conical base.The results for the tests in Fig. 8 show that the area with crushed particles, developed after 100 cycles, Table 5 Limit unit base resistance q bL,BC before cycling, maximum unit base resistance q b,max0 and minimum unit base resistance q b,min0 mobilized in the first cycle, and maximum unit base resistance q b,max and minimum unit base resistance q b,min mobilized in the last cycle for cyclic tests with 100 displacement cycles Test code (a)  D R (%) q bL,BC (MPa) q b,max0 (MPa) q b,max at cycle 100 (MPa) q b,min0 (MPa) q b,min at cycle 100 (MPa)

Image analysis results
Figures 9 and 10 show the color map and contours lines of cumulative radial displacement u (positive when soil elements move away from the centerline of the model pile and negative when they move toward it) and cumulative vertical displacement v (positive when soil elements move upward and negative when they move downward) at the end of the cycling stage of tests CY0.25-N100, CY0.5-N100, CY1.0-N100, CY1.5-N100, and CY3.0-N100.The cumulative radial and vertical displacement u and v mobilized during cyclic loading are both normalized by the value of the cyclic displacement half-amplitude Dw cyclic of the corresponding test and plotted at the initial locations of the soil elements at the beginning of the cyclic loading stage.The x-axis of plots shown in Figs. 9 and 10 corresponds to the horizontal distance r from the centerline of the model pile to the soil element, normalized by the model pile radius r p ; the y-axis corresponds to the vertical distance h of the soil element with respect to the pile base (h = 0 at the pile base, positive above it, and negative below it), also normalized by the model pile radius r p .Figures 9 and 10 also show the region where DIC analysis results are not available.For test CY3.0-N100, this region is significantly larger than for tests CY0.25-N100, CY0.5-N100, CY1.0-N100, and CY1.5-N100, particularly next to the conical shoulder, as shown in Figs.9e and 10e.Because of the large displacements and rotations of the soil particles located next to the shoulder of the conical base during cycling, the DIC algorithm fails to track these soil elements.Considering the DIC data available for test CY3.0-N100, the results in Fig. 9e show that soil elements with u/Dw cyclic C 0.1 extend to a radial position r/r p equal to 4 and a vertical distance h/r p of -3.0.For the case of cumulative vertical displacements [see Fig. 10e], soil elements located initially between h/r p of -3.0 and 0 move down more than 20% the value of the cyclic displacement (Dw cyclic = 3.0 mm). Figure 11d and e show that, immediately next to the cone shoulder, a zone of dilation is formed.This dilative zone increases considerably when Dw cyclic is 3.0 mm.An additional area of dilation is observed in test CY3.0-N100[Fig.11e] underneath the conical base between h/r p- = -1 and h/r p = -4 (-2B \ h \ -0.5B).These results support the fact that q b,AC values for w b / B [ 0.3B are greater than the limit unit base resistance q bL,BC before cycling.
Figure 12 shows the heat map of volumetric strain E vol , plotted at the deformed location of the soil elements, mobilized in the cyclic loading stage of test CY3.0-N100.Figure 12a, c, e, and g show the contours of E vol at the maximum downward movement of the pile (push-in) in cycles 10, 20, 50, and 100, respectively.Figure 12b, d, f,  and h show the contours of E vol at the maximum upward movement of the pile (pull-out) in cycles 10, 20, 50, and 100, respectively.As shown in Fig. 12b, d, f, and h, a gap between the sand and the inclined surface of the conical This process, which repeats after each cycle, leads to the addition of sand particles into the zone where the gap forms near the conical base.As shown in Fig. 7, q b,max reaches values always greater than 0.4q b,max0 = 5.8 MPa, which are values of q b sufficiently large to produce crushing of silica sand particles [48,50].Therefore, the sand particles that flow inside this gap end up being crushed in the downward movement of the pile during cycling, as evidenced by the growing bulb of crushed particles formed around the conical base.The digital images shown in Fig. 12 confirmed that the load at the pile base is fully removed during each pullout, as observed in Fig. 6d.
Figure 12 also shows that, after 20 cycles, soil elements around the conical base contract (E vol \ 0).For N c [ 20 cycles, as cycling progress, some soil elements surrounding the bulb of crushed particles start dilating (e.g., point B in Fig. 12).Although DIC results inside the bulb of crushed particles are not available, the lower unit base resistance measured in the compressive load test after cycling for  4c] suggests that the crushed material in the bulb the conical base is less dense than that before cycling.As the conical pile base passes the bulb of crushed sand particles, it goes through densified sand with dilative tendency, resulting in higher unit base resistance.

Unit base resistance mobilized during static load tests
Figure 13 compares the unit base resistance q b,AC measured in the compressive load test after cycling normalized by the limit unit base resistance q bL,BC measured before cycling for tests with cyclic loading stage performed with the same cyclic displacement half amplitude Dw cyclic , but with a different number of cycles.Figure 13a shows that for Dw cyclic = 0.25 mm, the q b,AC /q bL,BC versus w b / B curves for the tests with cyclic loading stage ended at 100 (test CY0.25-N100), 1000 (test CY0.25-N1000), and 2000 (test CY0.25-N2000) cycles are comparable, indicating that for this cyclic displacement half amplitude, the number of cycles has a minimal effect on the q b,AC /q bL,BC ratio.
For Dw cyclic = 0.5 mm [see Fig. 13b], increasing the number of cycles from 100 cycles (test CY0.5-N100) to 1000 cycles (test CY0.5-N1000) results in lower values of q b,AC /q bL,BC at w b /B = 0.1 (q b,AC /q bL,BC = 0.93 and 0.78 for 100 and 1000 cycles, respectively), but the values of q b,AC /q bL,BC at w b /B = 1 are the same (q b,AC /q bL,BC = 1.0 for 100 and 1000 cycles).
For Dw cyclic = 1.0 mm, [see Fig. 13c], the values of q b,AC /q bL,BC at w b /B = 0.1 for tests with cyclic loading stages ended at 100, 200, and 1000 cycles are comparable (i.e., q b,AC /q bL,BC = 0.57, 0.47, 0.56 for 100, 200, and 1000 cycles, respectively).However, for w/B [ 0.12, q b,AC / q bL,BC increases about 9% when the number of cycles increases from 200 cycles to 1,000 cycles.These results suggest that, for a given cyclic displacement half-amplitude Dw cyclic , there is a threshold number of cycles below which the lowest ultimate unit base resistance is reached.Table 6 provides the ratios of q b,AC /q bL,BC for values of relative pile base settlement w b /B of 0.1, 0.3, 0.6 and 1.0 for the tests shown in Fig. 13.
Fig. 13 Effect of number N c of cycles on the unit base resistance q b,AC measured in the static load test performed after cycling normalized by the limit unit base resistance q bL,BC measured before cycling for tests with cyclic displacement half amplitudes Dw cyclic of a 0.25 mm, b 0.5 mm, and c 1.0 mm

Unit base resistance mobilized during cyclic load tests
Figure 14 shows q b,max /q versus number N c of cycles for cyclic load tests ended at 1,000 cycles or more (i.e., tests CY0.25-N2000, CY0.5-N1000, and CY1.0-N1000) and q b,max /q b,max0 versus N c for test CY3.0-N100.Figure 14 shows that, for test CY0.25-N2000, the value of q b,max /q b,max0 remains approximately constant throughout the 2,000 cycles.For test CY0.5-N1000, the value of q b,- max /q b,max0 decreases continuously as the number of cycles increases up to 1,000 cycles.However, for test CY1.0-N1000, the q b,max /q b,max0 versus N c curve has a minimum value at N c = 133 and, for N c [ 133 cycles, the value of q b,max /q b,max0 increases slightly as N c increases.This result suggests that (i) there is a number of cycles that leads to a minimum value of q b,max /q b,max0 , and (ii) the minimum value of q b,max /q b,max0 is reached with a smaller number of cycles as Dw cyclic increases.For example, for q bL,BC is the limit unit base resistance measured in the compressive load test performed before cyclic loading q b,AC is the unit base resistance measured in the compressive load test performed after cyclic loading Fig. 14 Maximum unit base resistance q b,max normalized by the maximum unit base resistance q b,max0 measured in the first cycle of cyclic loading versus number of cycles for tests CY3.0-N100, CY0.25-N2000, CY0.5-N1000, and CY1.0-N1000 Dw cyclic = 3.0 mm the minimum value of q b,max /q b,max0 occurs at c = 9 cycles, as shown in Figs.7 and 14, while for Dw cyclic = 1.0 mm, the minimum value of q b,max /q b,max0 occurs much later, at N c = 133.15e and plotted as a function of the number of cycles.Figure 15e also shows the evolution of q b,max / q b,max0 with cyclic loading.The plots of A c /A b versus N c and q b,max /q b,max0 versus N c indicate that, during the first 20 cycles, q b,max /q b,max0 decreases from 1 to 0.45 at a high rate, while A c /A b increases from 10 to 23%.Both curves tend to stabilize at around 100 cycles.From N c = 133 cycles to 1,000 cycles, q b,max increases slightly, from 0.18q b,max0 (i.e., q b,max- = 2.6 MPa) to 0.27q b,max0 (i.e., q b,max = 3.9 MPa) and A c / A b grows, from 35% in cycle 133 to 148% in cycle 1,000.

Particle crushing effects
We observed that the minimum value of q b,max /q b,max0 coincides with the onset of local flow of particles from the region above the shoulder of the conical base into the gap left during the pull-out stage of each cycle.Videos generated from the digital images taken during the cyclic loading stage of test CY1.0-N1000showed that, between cycles 100 and 300, the majority of the particles dropping are crushed particles coming from the crushed particle band next to the pile shaft (formed during installation [48,50]).After about 300 cycles, uncrushed and crushed particles from above the shoulder of the pile base flow into the gap.As shown in Fig. 15b, some particles reach positions below the tip of the cone, and others accumulate below the cone shoulder.For N c [ 300 cycles, it can be seen that some sand particles are crushed during the downward movement of the model pile.

Image analysis results
Figure 16 shows the trajectory of soil elements located near the conical base (elements A, B, C, and D) during the cycling stage of test CY1.0-N1000.Table 7 shows the position of these soil elements at the beginning of the cycling stage.Element A was chosen to represent soil Figure 16a shows that, for N c \ 100 cycles, the average trajectory of soil element A forms an angle h with the horizontal (r/r p axis) of approximately 135 degrees (measured counter-clockwise from the r/r p axis on the right side and clockwise on the left side of the pile).For N c [ 100 cycles, the angle h changes slightly to 120 degrees.) has experienced mainly vertical displacement in the first 100 cycles of the cyclic loading stage (i.e., after 100 cycles, u = -0.04mm and v = 0.12 mm). Figure 17 shows that, between N c equal to 100 cycles and 180 cycles, the cumulative radial and vertical displacement of elements A, B, and C remain approximately constant.However, for N c [ 200 cycles, there is a reactivation of the radial and vertical movement of elements A and B and the vertical movement of element C; this agrees well with the increase in the area of crushed particles surrounding the conical base and the small increase in q b,max /q b,max0 , as can be seen in Fig. 15e.  Figure 18 shows the evolution of the cumulative radial strain E rr , vertical strain E zz , shear strain E rz , and volumetric strain E vol during the cycling stage of test CY1.0-N1000 for the same four elements listed in Table 7. Figure 18a

Summary and conclusions
In this paper, we reported the results of a series of monotonic and cyclic load tests performed on a closed-ended jacked model pile with a conical base.The model pile load tests were performed in dense silica sand samples prepared in a half-cylindrical calibration chamber that allows image collection.The digital images collected cyclic loading were processed using the DIC technique to obtain the displacement and strain fields in the sand domain near the conical base.
Cycling performed with a cyclic displacement halfamplitude Dw cyclic equal or less than 0.25 mm (= 0.0065B = 0.4D 50 ) had a minimal effect on the unit base resistance regardless of the number of cycles.The results from DIC showed that soil elements near the conical base are minimally disturbed during cyclic loading.In contrast, after 100 cycles with a cyclic displacement half-amplitude Dw cyclic in the 0.5 mm (= 0.8D 50 ) to 1.5 mm (= 2.4D 50 ) range, the ultimate unit base resistance (q b at w b /B = 0.1) decreased considerably with increasing Dw cyclic and the In current practice, the effects of cyclic loading on base resistance are not considered in the most well-known pile design equations.Although more research is needed before the results of this research can be adopted by the industry, this research shows that reducing the calculated base capacity of piles due to cyclic loading would be a good design practice.The results of this research show that the unit base resistance decreases up to 50% for a range of cyclic displacement amplitudes (0.5 to 1.5 mm) that piles typically experience in onshore and offshore applications.
Funding Open Access funding provided by Colombia Consortium.
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r p Radius of model pile r Radial distance from soil element to centerline of model pile r 0 ' Initial vertical effective stress (before jacking) at the final installation depth u Cumulative radial displacement of soil element v Cumulative vertical displacement of soil element Dw cyclic Cyclic displacement half-amplitude w Pile head displacement w b Pile base settlement w b /B Relative settlement at the pile base 1 Introduction

Fig. 1
Fig. 1 Cyclic interaction diagram for full-scale piles [51] (Numbers indicate number of cycles developed in each test)

Fig. 2
Fig. 2 Experimental setup at the Bowen Laboratory-Purdue University, USA

Fig. 3
Fig.3Schematic of the model pile used in the experiments[50]

3. 1 . 1 Figure 4
Figure4shows the measured unit base resistance q b versus relative settlement w b /B at the pile base for the compressive load tests performed before and after the cyclic loading stage of tests CY0.25-N100, CY1.0-N100, and CY3.0-N100.These tests differ only by the cyclic displacement half-amplitude Dw cyclic (= 0.25 mm, 0.5 mm, 1.0 mm, and 1.5 mm).For Dw cyclic = 0.25 mm [see Fig.4a], the q b versus w b /B curves from the pre-cyclic and the post-cyclic compressive load tests are comparable.For Dw cyclic- = 1.0 mm [see Fig.4b], at small pile base settlements, the unit base resistance measured in the compressive load test performed after cycling (q b,AC ) is significantly smaller than

Frequencya
Test code CY'#' = cyclic displacement half amplitude Dw cyclic followed by its value in mm, N'#' = number of cycles b

Fig. 4 Fig. 5
Fig. 4 Effect of cyclic displacement half-amplitude Dw cyclic on the unit base resistance q b mobilized in the compressive load tests performed before and after cycling versus relative settlement w b /B at the pile base for tests: a CY0.25-N100, b CY1.0-N100, and c CY3.0-N100

Fig. 7
Fig.7Effect of cyclic displacement half amplitude Dw cyclic on the cyclic unit base resistance q b,max mobilized during the cyclic loading stage of tests with 100 displacement cycles: maximum unit base resistance q b,max normalized by the maximum unit base resistance q b,max0 mobilized in the first cycle of cyclic loading versus number N c of cycles

Fig. 9
Fig. 9 Contours of normalized radial displacement u/Dw cyclic (positive when soil moves away from the model pile centerline) near the conical base after 100 cycles with cyclic displacement half amplitude Dw cyclic of a 0.25 mm, b 0.5 mm, c 1.0 mm, d 1.5 mm, and e 3.0 mm

Figure 11
Figure 11 shows the heat map and the cumulative volumetric strain E vol mobilized during the cyclic loading stage for the same tests shown in Fig. 8.The solid mechanics sign convention is followed in this figure: positive values of E vol indicate dilation.The volumetric strains are calculated using the expressions presented in [44].Negligible cumulative volumetric strains (|E vol | \ 0.1%) were measured after 100 cycles for Dw cyclic = 0.25 mm.However, for Dw cyclic C 0.5 mm, the magnitude of the volumetric strains and the zone of soil undergoing volumetric deformation increase as Dw cyclic increases.Except for test CY3.0-N100(Dw cyclic = 3.0 mm), the soil elements next to the inclined surface of the conical base undergo contraction (negative E vol ).Figure11dand e show that, immediately next to the cone shoulder, a zone of dilation is formed.This dilative zone increases considerably when Dw cyclic is 3.0 mm.An additional area of dilation is observed in test CY3.0-N100[Fig.11e]underneath the conical base between h/r p- = -1 and h/r p = -4 (-2B \ h \ -0.5B).These results support the fact that q b,AC values for w b / B [ 0.3B are greater than the limit unit base resistance q bL,BC before cycling.Figure12shows the heat map of volumetric strain E vol , plotted at the deformed location of the soil elements, mobilized in the cyclic loading stage of test CY3.0-N100.Figure12a, c, e, and gshow the contours of E vol at the maximum downward movement of the pile (push-in) in cycles 10, 20, 50, and 100, respectively.Figure12b, d, f, and h show the contours of E vol at the maximum upward movement of the pile (pull-out) in cycles 10, 20, 50, and 100, respectively.As shown in Fig.12b, d, f, and h, a gap between the sand and the inclined surface of the conical

Fig. 11
Fig. 11 Contours of volumetric strain E vol (positive values indicating dilation) near the conical base after 100 cycles with cyclic displacement half amplitude Dw cyclic of a 0.25 mm, b 0.5 mm, c 1.0 mm, d 1.5 mm, and c 3.0 mm

Fig. 12
Fig. 12 Heat map of volumetric strain E vol (positive values indicating dilation) near the conical base at the maximum downward movement of the pile (push-in) in cycles a 10, c 20, e 50, and g 100, and at the maximum upward movement of the pile (pull-out) in cycles b 10, d 20, f 50, and h 100 for test CY3.0-N100

Figure 15a -
Figure15a-d shows the digital images at the end of cycles 100, 200, 400, and 1,000 for the cyclic loading stage of test CY1.0-N1000.For each digital image, the area with crushed particles A c is estimated and normalized by the projected area A b of the conical base.The values of A c /A b are shown in Fig.15eand plotted as a function of the number of cycles.Figure15ealso shows the evolution of q b,max / q b,max0 with cyclic loading.The plots of A c /A b versus N c and q b,max /q b,max0 versus N c indicate that, during the first 20 cycles, q b,max /q b,max0 decreases from 1 to 0.45 at a high rate, while A c /A b increases from 10 to 23%.Both curves tend to stabilize at around 100 cycles.From N c = 133 cycles to 1,000 cycles, q b,max increases slightly, from 0.18q b,max0 (i.e., q b,max- = 2.6 MPa) to 0.27q b,max0 (i.e., q b,max = 3.9 MPa) and A c / A b grows, from 35% in cycle 133 to 148% in cycle 1,000.

Fig. 15
Fig. 15 Digital images at the end of a 100 cycles, b 200 cycles, c 400 cycles, and (d) 1,000 cycles; and e maximum unit base resistance q b,max normalized by the maximum unit base resistance q b,max0 in the first cycle of cyclic loading (on the left vertical axis) and normalized area of crushed particles A c /A b (on the right vertical axis) versus number N c of cycles for test CY1.0-N1000 Figure 16b shows that element B follows approximately the same trajectory (h = 110 degrees) during the first 100 cycles.Then, for N c [ 100 cycles, element B moves following an angle h equal to approximately 35 degrees with the horizontal.Figure 16c shows that element C moves slightly upward during the first 100 cycles; then, it moves downward for N c [ 100 cycles.From Fig. 16d, the average trajectory of element D (the element closest to the inclined faces of the conical base) varies from h & -80 degrees (sub-vertical), for N c \ 100 cycles, to h & 10 degrees (nearly horizontal), for 300 cycles \ N c \ 500 cycles.Element D could not be tracked using DIC after 500 cycles.

Fig. 16
Fig. 16 Trajectory of soil elements (A, B, C, D) near the conical base of the pile during the cycling stage of test CY1.0-N1000:normalized radial position versus normalized vertical position of a soil element A [r/r p = 2.5, h/r p = 1.7], b soil element B [r/r p = 2.5, h/r p = 0], c soil element C [r/r p = 0, h/r p = -1.6],and d soil element D [r/r p = 1.0, h/r p = 0.8].The selected N c values are indicated in the red boxes Figure 17a also shows that, for N c [ 200 cycles, the radial displacement u of element A decreases with increasing N c , indicating that soil element A continues moving radially toward the pile axis, as observed for N c- \ 200 cycles.In contrast, the radial displacement u of element B increases with increasing N c , indicating that the direction of the radial displacement changed after 200 cycles (i.e., element B moves radially away from the pile axis for N c [ 200).The change in the direction of the radial displacement of element B results from the flow of sand particles to the region below the conical base when the pile moves upward during the cycling stage, pushing element B radially away from the conical base.As shown in Fig. 17b, element C experiences an increase in downward movement after 200 cycles (i.e., from 200 to 500 cycles, the permanent vertical displacement v changes from 0.02 mm to -0.64 mm); this movement is affected by the increase in the deposition of sand particles below the conical base, which pushed down element C.
Figure 17a and b show that, of elements A through D, element D has the widest range of displacements around the mean paths of displacement versus the number N c of cycles in the first 100 cycles.A and B experience the least oscillations.
and b show that the deformation mechanism in element A is the opposite of that in element B. While element A stretches radially and compresses vertically, element B compresses radially and stretches vertically.During the first 180 cycles, the cumulative shear strain [see Fig. 18c] in elements A and B is less than 0.1%; therefore, for N c \ 180 cycles, the radial and vertical strains of elements A and B are approximately equal to the principal strains.For N c [ 180 cycles, the shear strain of elements A and B become negative and positive, respectively, indicating that soil element A distorts in the opposite direction of element B. In terms of volumetric strains [see Fig. 18d], element A, B, and C exhibit volumetric contraction during the first 300 cycles.Then, for N c [ 300 cycles, these elements tend to dilate.

Fig. 17
Fig. 17Displacements of soil elements (A, B, C, and D) near the conical base during the cycling stage of test CY1.0-N1000(f = 1.0 Hz): a radial displacement u (positive when soil elements move away from the centerline of the model pile and negative when they move toward it) and b vertical displacement v (positive when soil elements move upward and negative when they move downward) Fig. 17Displacements of soil elements (A, B, C, and D) near the conical base during the cycling stage of test CY1.0-N1000(f = 1.0 Hz): a radial displacement u (positive when soil elements move away from the centerline of the model pile and negative when they move toward it) and b vertical displacement v (positive when soil elements move upward and negative when they move downward)

Fig. 18
Fig. 18 Cumulative strains of soil elements (A, B, C, and D) around the conical base during the cycling stage of test CY1.0-N1000(f = 1.0 Hz): a radial strain E rr (positive when soil elements stretch radially); b vertical strain E zz (positive when soil elements stretch vertically); c shear strain E rz ; d volumetric strain E vol

Table 4
Testing program

Table 6
Values of q b,AC /q bL,BC obtained at four different relative pile base settlements w b /B for tests with cyclic displacement half amplitude Dw cyclic of 0.25 mm, 0.5 mm, and 1.0 mm