Time-Dependent Bearing Behavior of Uplift Piles in Calcareous Sand

Abstract

As marine engineering advances, the deformation and load-bearing performance of uplift piles in calcareous sand under sustained loading have become increasingly important to the long-term performance of marine foundations. Static uplift tests and sustained-load model tests were conducted in calcareous sand to examine the effects of pile surface roughness and sand gradation. The time-dependent evolution of pile-head displacement, uplift deformation stiffness, pile axial force, and shaft resistance was investigated, and the displacement–time responses were fitted using the Burgers model. The results showed that pile-head creep displacement under sustained uplift loading developed through two stages: primary creep and steady-state creep. Greater pile surface roughness and well-graded calcareous sand effectively limited long-term displacement. Under the maximum sustained loads, uplift deformation stiffness decreased by 27.4–43.0% over 21 d. At the maximum sustained-load level, pile axial force decreased progressively with increasing embedment depth. The time-dependent redistribution of axial force occurred mainly in the upper and middle portions of the pile and gradually stabilized after 3–7 d, with the redistribution becoming more pronounced as pile surface roughness increased. At the same load level, shaft resistance in the peak zone increased with the duration of sustained loading and gradually stabilized. This increase was more pronounced for piles with rougher surfaces, whereas well-graded calcareous sand facilitated the stable mobilization of shaft resistance. The Burgers model accurately captured the creep behavior of the model piles, with coefficients of determination (R²) exceeding 0.967 in all test cases.

1. Introduction

Since the beginning of the 21st century, the continued development of marine resources, island and reef construction, and nearshore infrastructure has increased attention to the time-dependent bearing behavior and long-term safety of pile foundations in reclaimed ground composed of calcareous sand. Calcareous sand, the principal fill material used in coral reef reclamation projects, consists predominantly of biogenic fragments derived from corals, shells, and other marine organisms and is characterized by irregular particle shapes, high intraparticle porosity, and high crushability [1,2,3]. Compared with terrigenous quartz sand, calcareous sand particles exhibit lower strength and greater susceptibility to breakage, resulting in more complex stress-transfer behavior at the pile–soil interface. Consequently, conventional empirical methods for designing piles in siliceous sand cannot be directly applied to calcareous sand [4,5].

In recent years, extensive experimental research has been conducted on the bearing behavior of pile foundations in calcareous sand. Based on direct shear tests on calcareous sand–steel plate interfaces, Kou et al. [6] demonstrated that both steel surface roughness and calcareous sand particle size significantly affected the interfacial shear behavior. Feng et al. [7] investigated the effects of surface roughness and temperature on the shear behavior of calcareous sand–steel interfaces. Wang et al. [8] reported that vertical cyclic loading induced cumulative settlement and load redistribution in calcareous sand, with decreasing shaft resistance and increasing base resistance. Wan et al. [9,10] investigated the bearing behavior of shaft post-grouted piles through model tests under axial compression and lateral loading. They found that shaft post-grouting improved the pile–sand interface conditions and enhanced both the axial and lateral bearing performance of the piles. Wei et al. [11] conducted oblique uplift tests on model piles in calcareous sand and investigated the effects of load inclination and relative density on pile-head deformation, internal forces within the pile, and ultimate uplift capacity. Peng et al. [12] showed that particle interlocking enhanced uplift capacity, whereas corner breakage contributed to abrupt failure in calcareous sand. Wang et al. [13] and Chen et al. [14] found that increasing sand density enhanced the uplift capacity of piles in calcareous sand. Fang et al. [15] investigated the effects of moisture conditions on the uplift behavior of root piles in coral sand. Ouyang et al. [16] identified relative density, embedment depth, and installation method as important factors affecting the vertical bearing behavior of piles in calcareous sand. Collectively, these studies have improved the understanding of the load-bearing mechanisms of piles in calcareous sand. However, most have focused on bearing capacity, deformation, and load-transfer behavior under short-term static loading, whereas the time-dependent bearing behavior of pile foundations in calcareous sand remains insufficiently understood.

The bearing behavior of pile foundations is strongly time-dependent; however, previous studies have focused primarily on piles installed in clay, silt, and conventional sand. Using an effective-stress framework, Li et al. [17] developed a method for predicting the time-dependent capacity of jacked piles in clay that accounts for the installation-induced stress history of the surrounding soil, dissipation of excess pore water pressure, and thixotropic effects at the pile–soil interface. Cui et al. [18] developed a model to predict the long-term load–settlement response and time-dependent capacity of driven piles that accounts for the viscoplasticity and aging of the surrounding clay, as well as softening at the pile–soil interface. Zhou et al. [19] developed a method for predicting the time-dependent bearing capacity of jacked piles that accounts for installation-induced disturbance, thixotropic recovery, and reconsolidation of the surrounding structured clay. Based on uplift tests on model piles following different setup periods, Yang et al. [20] found that thixotropic recovery of the structured clay in the Zhanjiang Formation contributed to increases in shaft resistance and uplift capacity. Jiu et al. [21] reported that soil stress history and pile surface roughness significantly influenced the time-dependent bearing behavior of jacked piles in saturated silt. Multi-site field uplift tests conducted by Carroll et al. [22] showed that pile material, surface roughness, corrosion, and sand properties jointly governed the aging-induced capacity gain of driven piles. Through large-scale model tests, Manthey et al. [23] found that the axial capacity of open-ended steel pipe piles in sand increased markedly with the time elapsed after installation and identified the influence of sand density on pile aging. Although these studies have examined the time-dependent bearing behavior of piles in conventional clay, silt, and sand, corresponding research on uplift piles in calcareous sand remains limited.

To address this research gap, laboratory model tests were conducted to investigate the static uplift behavior of single model piles in calcareous sand and their time-dependent response under different sustained load levels. The effects of pile surface roughness and sand gradation on the evolution of pile-head displacement, uplift deformation stiffness, pile axial force, and shaft resistance were analyzed. The Burgers model was then used to fit the displacement–time curves. The findings may inform the design and construction of pile foundations in calcareous sand.

2. Laboratory Model Tests

2.1. Calcareous Sand

The foundation material used in the tests was collected from a coral reef and comprised a loose, uncemented deposit of coral detritus. Two calcareous sands with different particle-size distributions were used as foundation materials and designated Sand A and Sand B. Coarse particles larger than 2 mm were removed by standard sieving. The basic physical properties of the two sands were then determined and are presented in Figure 1 and Table 1. Sand A was classified as poorly graded medium sand, whereas Sand B was classified as well-graded medium sand.

Figure 1. Particle-size distribution curves of the calcareous sands used in this study.

Table 1. Physical properties of the calcareous sands used in this study.

Sand Type	d50/mm	Cu	Cc	ρd,max/g·cm−3	ρd,min/g·cm−3	ω/%
Sand A	0.46	4.07	2.72	1.81	1.39	0.018
Sand B	0.27	7.75	2.33	1.69	1.27	0.015

Note: d₅₀ is the median particle size, C_u is the coefficient of uniformity, C_c is the coefficient of curvature, ρ_d,max is the maximum dry density, ρ_d,min is the minimum dry density, and ω is the water content.

2.2. Model Pile and Test Chamber

Based on the practical engineering conditions, the prototype uplift pile was a 27 m long steel pipe pile with a diameter of 0.9 m and an elastic modulus of 210 GPa. Based on similitude principles, a geometric similarity ratio of 1:30 was adopted for the model pile. A polyvinyl chloride (PVC) pipe with an elastic modulus of 3 GPa was used as the model pile. The model pile had an embedment depth of 900 mm and a diameter of 30 mm. Starting at the pile toe, strain gauges were bonded symmetrically to opposite sides of the outer pile wall using cyanoacrylate adhesive. The gauge lead wires were routed through predrilled holes into the pile cavity and led out of the pile. A layer of epoxy resin was applied over each strain gauge, and sandpaper was bonded to the outer surface of the pile shaft.

The test chamber had clear internal dimensions of 1000 × 500 × 1800 mm (length × width × height). Its frame was constructed from square steel sections and tempered-glass panels, and one side was fitted with a movable access door. During testing, the model pile was positioned at the center of the chamber. The sand layer beneath the pile toe was 10D thick, and the minimum clearance between the outer pile surface and the inner chamber wall was 7.8D, where D is the model pile diameter. Moreover, the pile diameter exceeded 30 times the median particle size of the sand. Accordingly, boundary and particle-size effects in the model tests were considered negligible [24,25,26]. A schematic of the test chamber and model pile is shown in Figure 2.

Figure 2. Schematic of the model pile and test chamber. (a) Model pile; (b) test chamber.

2.3. Measurement System

Pile-head displacement was measured using a digital dial indicator. Load and strain data were acquired using a GGD-330 measurement controller and a BZ2205C programmable static resistance strain meter, respectively. BX/BF120-5AA strain gauges with a resistance of 120 Ω were used.

2.4. Test Program

2.4.1. Pile Installation Method

A pre-embedding method was adopted for pile installation. For each layer, a fixed mass of calcareous sand (77.0 kg for Sand A and 70.9 kg for Sand B) was weighed using an electronic balance and spread uniformly. The sand was statically compacted using a wooden plate and leveled using a spirit level. The compacted thickness of each layer was strictly controlled at 10 cm to achieve the target relative density (D_r = 42% ± 2%). When the sand bed reached a height of 30 cm, the pile was accurately positioned at the predetermined location, with its axis deviation limited to 0.5% of L, where L is the pile length. Sand placement and compaction then continued until the specified filling height was reached.

2.4.2. Loading System

A pulley–deadweight loading system was used to provide clearly defined load levels and reduce experimental errors. Loading was applied sequentially at 0.3 Q_u → 0.5 Q_u → 0.7 Q_u → 0.9 Q_u, where Q_u is the ultimate uplift capacity of the model pile. Pile-head displacement and pile-shaft strain were recorded immediately after each load level was applied and subsequently at 5 min intervals. When the pile-head displacement rate remained below 0.1 mm/h for 2 consecutive hours, the pile-head displacement was considered stable, and measurements were thereafter recorded once daily. Each load increment was maintained for 21 days, after which the next load increment was applied to the same pile.

2.4.3. Test Matrix

The tests were designed primarily to investigate the effects of pile surface roughness and calcareous sand particle-size distribution on the time-dependent bearing behavior of the model piles. The test matrix is presented in Table 2. The normalized roughness (R_n), listed in Table 2, was calculated using Equation (1).

R_n = R_max/d₅₀

(1)

where R_max is the maximum height difference in the roughness profile on the sandpaper surface, and d₅₀ is the median particle size of the calcareous sand used in the tests.

Table 2. Test matrix.

Test No.	Embedment Depth/mm	Pile Diameter/mm	Sandpaper Grit	Roughness, Rn	Roughness Type
A-Z1	900	30	2000	0.013	Smooth
A-Z2	900	30	320	0.098	Semi-rough
A-Z3	900	30	80	0.565	Rough
B-Z4	900	30	320	0.166	Semi-rough

Note: A and B denote calcareous sand foundations with different particle-size distributions, while Z1, Z2, Z3, and Z4 denote model piles with different surface roughnesses.

The maximum height differences in the roughness profiles on the sandpaper surfaces were measured using a high-precision three-dimensional scanner. The measured R_max values were 0.0065 mm for the 2000-grit sandpaper, 0.045 mm for the 320-grit sandpaper, and 0.260 mm for the 80-grit sandpaper. Following the classification proposed by Paikowsky [27], R_n values of 0.02 and 0.5 were adopted as the boundaries between different interface roughness categories. Accordingly, the model pile surfaces were classified into three roughness categories: smooth, semi-rough, and rough.

3. Results and Discussion

3.1. Determination of the Ultimate Uplift Capacity of the Model Piles

In accordance with the requirements of the relevant standard [28], uplift load tests were conducted on the model piles, and the resulting load–displacement curves are shown in Figure 3. Each curve exhibits a distinct inflection point, at which the pile-head displacement increases abruptly. For this response, the load corresponding to the onset of the abrupt increase in displacement was taken as the ultimate uplift capacity. The resulting values are summarized in Table 3.

Figure 3. Load–displacement curves of the model piles.

Table 3. Ultimate uplift capacity of the model piles.

Test No.	Ultimate Uplift Capacity, Qu/N	Ultimate Pile-Head Displacement, S/mm
A-Z1	700	3.834
A-Z2	1050	4.176
A-Z3	1600	2.975
B-Z4	1100	2.341

3.2. Time-Dependent Analysis of Pile-Head Displacement

Sustained uplift loading tests were conducted on four model piles at different load levels, and the resulting displacement–time curves are shown in Figure 4. As shown in the figure, the pile-head displacement of all four model piles increased with the duration of sustained loading under each constant uplift load, although the rate of displacement increase differed markedly among the load levels. At low load levels, most of the pile-head displacement occurred rapidly during the initial loading period, after which the curves gradually flattened, exhibiting distinct primary creep characteristics. As the load level increased, the displacement increased approximately linearly with time after the primary creep stage, indicating both primary and steady-state creep. For example, under 210 N, the displacement of A-Z1 increased from 0.209 to 0.244 mm, corresponding to an increment of only 0.035 mm. The negligible subsequent increase indicated that the pile remained in the primary creep stage. When the load increased to 490 N, the displacement continued to increase slowly after 3 d, and the displacement–time curve exhibited a slight upward trend with a discernible slope, indicating the occurrence of both primary and steady-state creep. Under 630 N, the displacement–time curve exhibited a pronounced upward trend, and the displacement reached 3.781 mm at 21 d, approaching the ultimate pile-head displacement. This result indicated that shear slip and creep continued to develop at the pile–sand interface under the high load level. A-Z2, A-Z3, and B-Z4 exhibited similar behavior. Because the four load levels were applied sequentially to the same model pile, the response at each higher load level inevitably included the cumulative influence of the preceding sustained-loading stages. Therefore, the differences among the load levels reflect the combined effects of the current load magnitude and the previous loading history, rather than the effect of load level alone.

Figure 4. Pile-head displacement–time curves of the model piles. (a) A-Z1; (b) A-Z2; (c) A-Z3; (d) B-Z4.

Pile surface roughness and the particle-size distribution of the calcareous sand significantly affected the long-term displacement of the model piles. A-Z1, A-Z2, and A-Z3 corresponded to smooth, semi-rough, and rough interfaces, respectively. Under the highest sustained load level, the actual load applied to A-Z3 was 1440 N, considerably greater than the 630 N applied to A-Z1; nevertheless, its pile-head displacement at 21 d remained smaller than that of A-Z1. This result indicated that a rougher interface enhanced the mobilization of pile–sand interface resistance, thereby effectively limiting long-term uplift displacement. A-Z2 and B-Z4 both had semi-rough interfaces but were embedded in calcareous sands with different particle-size distributions. The final displacement of B-Z4 at each load level was generally smaller than that of A-Z2. The smaller displacement of B-Z4 is consistent with previous studies showing that particle-size distribution affects particle packing and the shear behavior of calcareous sand–structure interfaces [6]. However, because A-Z2 and B-Z4 also differed in normalized roughness, this difference cannot be attributed solely to particle-size distribution.

Overall, the pile-head displacement of uplift piles in calcareous sand exhibited pronounced time dependence under different load levels. Higher load levels resulted in greater instantaneous pile-head displacement and long-term creep displacement. Greater pile surface roughness strengthened pile–sand interlocking and suppressed the development of long-term displacement. Well-graded calcareous sand enhanced confinement around the pile through more effective void filling, interparticle interlocking, and densification, thereby reducing the long-term displacement of the model pile.

3.3. Isochronous Load–Displacement Curves and Time-Dependent Uplift Stiffness

For each load level, pile-head displacements recorded at the same elapsed holding times were used to construct the isochronous load–displacement curves of the four model piles (Figure 5). The higher-load segments had lower slopes, indicating larger displacement increments per unit increase in load. For a given pile and load, the curves shifted toward greater displacement with increasing holding time. Curves at different times remained close together at lower loads but separated markedly at higher loads. The spacing between adjacent curves narrowed at later times, indicating slower displacement growth, although displacement continued to accumulate through day 21.

Figure 5. Isochronous load–displacement curves. (a) A-Z1; (b) A-Z2; (c) A-Z3; (d) B-Z4.

For the three piles tested in Sand A, the pile-head displacement increments from 0 to 21 d under their respective maximum sustained loads were 1.625, 1.341, and 0.859 mm for A-Z1, A-Z2, and A-Z3, respectively. Although A-Z3 sustained the highest load, its pile-head displacement at 21 d was smaller than those of A-Z1 and A-Z2. This result is consistent with greater pile surface roughness limiting displacement accumulation under the present test conditions. A-Z2 and B-Z4 both had 320-grit sandpaper surfaces but were tested in poorly graded Sand A and well-graded Sand B, respectively. Under their respective maximum sustained loads, B-Z4 exhibited a smaller displacement increment from 0 to 21 d than A-Z2. This difference may be related to sand gradation, although the two piles also differed in normalized interface roughness.

Figure 6 shows the variation in uplift deformation stiffness with time for the four model piles under their respective maximum load levels. Over 21 d, the stiffness of A-Z1, A-Z2, A-Z3, and B-Z4 decreased from 293, 350, 684, and 625 N/mm to 167, 232, 488, and 454 N/mm, respectively. The corresponding reductions were 43.0%, 33.8%, 28.7%, and 27.4%. For all four piles, stiffness decreased rapidly during the initial holding period and more slowly thereafter. The curves were still declining gradually at 21 d.

Figure 6. Uplift deformation stiffness–time curves.

For A-Z1, A-Z2, and A-Z3, all tested in Sand A, stiffness at 21 d increased and the relative reduction decreased as pile surface roughness increased. Under the same relative load level in this test series, the rougher pile–sand systems therefore showed greater resistance to displacement accumulation. B-Z4 had a higher stiffness and a smaller relative reduction at 21 d than A-Z2. However, the two piles differed in both sand gradation and normalized interface roughness, so this difference cannot be attributed to gradation alone. Overall, the decline in stiffness is consistent with the continued increase in pile-head displacement described above and reflects the deformation of the pile–sand system under sustained loading.

3.4. Internal Force Distribution in the Model Piles

3.4.1. Time-Dependent Behavior of Axial Force in the Model Piles at the Maximum Load Level

At the maximum load level, strain variations in the model piles were monitored at different times, and the pile axial force was calculated using Equation (2):

N_i = E_pA_iε_i

(2)

where N_i is the axial force at section i of the model pile, E_p is the elastic modulus of the pile material, A_i is the cross-sectional area at section i, and ε_i is the axial strain of the pile shaft at section i.

Figure 7 presents the variation in pile axial force with embedment depth at different times. At the maximum load level, the axial force in all four model piles decreased progressively from the pile head toward the pile toe, consistent with the typical load-transfer behavior of uplift piles. From 0 to 21 d, the overall shapes of the axial-force profiles remained essentially unchanged, indicating that sustained loading did not alter the fundamental pattern of load transfer from the pile head toward the pile toe. However, the axial forces in the upper and middle portions of the piles changed appreciably with sustained-loading duration, whereas only minor changes occurred near the pile toe. For example, at an embedment depth of 5 cm, the axial forces in A-Z1 and A-Z3 increased from initial values of 625 and 1436 N to 652 and 1609 N at 21 d, corresponding to increases of 4.32% and 12.04%, respectively. Most of the time-dependent changes in the axial-force profiles occurred during the early stage of sustained loading, with the most pronounced changes occurring during the first 3–7 d. After 7 d, the profiles gradually converged, indicating that the pile–sand system evolved from rapid interfacial adjustment toward a relatively stable load-transfer state. Previous studies have demonstrated that creep of the soil surrounding the pile, particle crushing, and localized changes in soil fabric can modify the stress state around the pile and consequently redistribute axial force along the pile shaft [12,29].

Figure 7. Variation in axial force with depth in the model piles at different times. (a) A-Z1; (b) A-Z2; (c) A-Z3; (d) B-Z4.

Pile surface roughness had a pronounced effect on the time-dependent redistribution of pile axial force. A-Z1, A-Z2, and A-Z3 were embedded in Sand A and had normalized roughness values, R_n, of 0.013, 0.098, and 0.565, respectively. As pile surface roughness increased, the separation among the axial-force profiles obtained at different times generally became more pronounced in the upper and middle portions of the piles. This indicated that pile–sand load transfer at rougher interfaces was more sensitive to sustained-loading duration. This behavior can be attributed to the enhanced mechanical interlocking and particle wedging between the calcareous sand particles and the pile surface with greater surface roughness, which promotes the progressive mobilization of the shear zone surrounding the pile during continued relative displacement. This behavior is consistent with previous interface shear studies showing that surface roughness significantly affects the shear resistance of calcareous sand–structure interfaces [6,7].

The particle-size distribution of calcareous sand also affected the time-dependent axial load-transfer behavior. A-Z2 and B-Z4 both had surfaces covered with 320-grit sandpaper and were embedded in poorly graded Sand A and well-graded Sand B, respectively. As shown in Figure 7b,d, the axial-force profiles of B-Z4 at different times were generally more closely clustered during the later stage of sustained loading. This indicated that, under the test conditions investigated, well-graded calcareous sand helped maintain a more stable pile–sand load-transfer state. This behavior can be attributed primarily to the more effective void filling and interparticle interlocking among particles of different sizes, which form a more densely packed granular skeleton around the pile, enhance radial confinement, and restrict continued slip at the pile–sand interface. It should be noted that although A-Z2 and B-Z4 used the same 320-grit sandpaper, the d₅₀ values of Sand A and Sand B were 0.46 and 0.27 mm, respectively. According to Equation (1), the corresponding normalized roughness values were 0.098 and 0.166. Therefore, strictly speaking, the differences between the two sets of profiles reflected the coupled effects of particle-size distribution and normalized interface roughness and should not be attributed solely to particle-size distribution [5].

Overall, pile axial force in uplift piles embedded in calcareous sand exhibited pronounced attenuation with depth and time-dependent redistribution. Long-term sustained loading primarily redistributed the axial force in the upper and middle portions of the piles, with most changes occurring during the early loading stage and gradually stabilizing after 3–7 d. Greater pile surface roughness strengthened interfacial shearing and increased the sensitivity of load transfer to sustained-loading duration, whereas a well-graded particle-size distribution promoted the formation of a more stable granular skeleton and load-transfer path in the sand surrounding the pile.

3.4.2. Time-Dependent Behavior of Shaft Resistance in the Model Piles at the Maximum Load Level

At the maximum load level, the shaft resistance mobilized along each pile segment was determined from the difference in pile axial force between adjacent measurement sections. The shaft resistance was calculated using the following equation:

$${{\displaystyle q}}_i={\displaystyle \frac{{{\displaystyle N}}_i-{{\displaystyle N}}_{i+1}}{\mathit{\pi D}\Delta L}}$$

(3)

where q_i is the average unit shaft resistance along pile segment i, N_i and N_i+1 are the axial forces at sections i and i + 1, respectively, D is the model pile diameter, and ΔL is the distance between the two sections.

Figure 8 presents the shaft-resistance profiles with depth at different times. At the maximum load level, the shaft resistance of all four model piles exhibited a markedly nonuniform distribution with depth. It was relatively low near the pile head, increased progressively with depth, reached a peak at an embedment depth of 45–55 cm, and then decreased toward the pile toe. From 0 to 21 d, the fundamental distribution pattern remained unchanged. However, shaft resistance in the peak zone increased moderately with sustained-loading duration and gradually stabilized after 3–7 d. Accordingly, the time-dependent evolution of shaft resistance under sustained uplift loading can be divided into three stages: early redistribution, progressive mobilization in the peak zone, and subsequent stabilization. This evolution was closely associated with the time-dependent changes in pile-head displacement and pile axial force. During the initial loading stage, relative pile–sand displacement developed rapidly, initiating the mobilization of interfacial shear resistance. As sustained loading continued, the pile–sand load-transfer state gradually adjusted, which allowed further mobilization of shaft resistance in the peak zone.

Figure 8. Variation in shaft resistance with depth in the model piles at different times. (a) A-Z1; (b) A-Z2; (c) A-Z3; (d) B-Z4.

Pile surface roughness significantly affected the time-dependent mobilization of shaft resistance. Comparisons among A-Z1, A-Z2, and A-Z3 showed that as R_n increased from 0.013 to 0.565, shaft resistance generally increased, and its time-dependent changes became more pronounced in the peak zone along the middle and lower portions of the piles. This result is consistent with previous interface shear studies showing that increasing surface roughness enhances the shear resistance of calcareous sand–structure interfaces [6,7]. Particle rearrangement and particle crushing were not directly investigated in the present tests. Therefore, their possible roles in pile–calcareous sand interaction are discussed only with reference to previous studies [12].

Particle-size distribution also affected the measured time-dependent shaft-resistance response. Compared with A-Z2, B-Z4 exhibited a relatively more stable load-transfer response during the later stage of sustained loading. Interface dilation and particle breakage were not directly measured in the present tests; therefore, their evolution could not be quantitatively determined. Previous large-scale interface shear tests have shown that particle-size distribution significantly affects the shear strength and particle breakage of calcareous sand–structure interfaces. Particle rearrangement and interlocking contribute to the initial dilative response, whereas progressive particle breakage can reduce dilation and promote contraction [30]. These findings provide a possible particle-scale explanation for the present observations. However, because A-Z2 and B-Z4 also differed in normalized interface roughness, the measured differences should be regarded as the coupled effects of particle-size distribution and interface roughness.

Overall, the shaft resistance of uplift piles in calcareous sand exhibited pronounced depth- and time-dependent behavior. Its peak occurred mainly along the middle and lower portions of the pile, and its evolution under sustained loading was characterized by early redistribution, enhancement in the peak zone, and subsequent stabilization. Greater pile surface roughness promoted the mobilization of interfacial resistance and made its time-dependent redistribution more pronounced, whereas a well-graded particle-size distribution facilitated the formation of a more stable granular skeleton around the pile. Therefore, evaluations of the long-term bearing performance of uplift piles should not assume that shaft resistance is uniformly distributed along the pile and independent of time.

3.5. Creep Model for the Model Piles

The test results indicate that the creep response of the model piles can be represented by the Burgers model, which consists of a Maxwell element and a Kelvin–Voigt element connected in series, as illustrated in Figure 9.

Figure 9. Burgers model.

Let σ_M and ε_M denote the stress and strain of the Maxwell element, respectively; σ_K and ε_K denote the stress and strain of the Kelvin–Voigt element, respectively; and σ and ε denote the total stress and strain of the Burgers model, respectively. The corresponding relationships are given by Equation (4).

$$\left\{\begin{array}{l}\sigma ={\sigma }_M={\sigma }_K\\ \epsilon ={\epsilon }_M+{\epsilon }_K\end{array}\right.$$

(4)

The Maxwell model consists of a Newtonian dashpot (N element) and a Hookean spring (H element) connected in series. Its creep equation is given by Equation (5).

$${\epsilon }_M={\displaystyle \frac{{\sigma }_M}{E_M}}+{\displaystyle \frac{{\sigma }_M}{{\eta }_M}}t$$

(5)

The Kelvin–Voigt model consists of a Newtonian dashpot (N element) and a Hookean spring (H element) connected in parallel. Its creep equation is given by Equation (6).

$${\epsilon }_K={\displaystyle \frac{{\sigma }_K}{E_K}}(1-{\mathrm{e}}^{-{\displaystyle \frac{E_K}{{\eta }_K}}t})$$

(6)

Substituting Equations (5) and (6) into Equation (4) yields the creep equation of the Burgers model, as expressed in Equation (7).

$$\epsilon ={\displaystyle \frac{\sigma }{E_M}}+{\displaystyle \frac{\sigma }{{\eta }_M}}t+{\displaystyle \frac{\sigma }{E_K}}(1-{\mathrm{e}}^{-{\displaystyle \frac{E_K}{{\eta }_K}}t})$$

(7)

where E_M and E_K are the elastic moduli of the springs in the Maxwell and Kelvin–Voigt elements, respectively; η_M and η_K are the viscosity coefficients of the dashpot elements; and t is the elapsed loading time.

In the present tests, the model pile was subjected to a constant load Q, and its creep displacement is denoted by S(t). The creep equation can therefore be rewritten as Equation (8). Defining L = 1/E_M, P = 1/η_M, U = 1/E_K, and V = E_K/η_K, Equation (8) can be simplified to Equation (9).

$$S\left(t\right)={\displaystyle \frac{Q}{E_M}}+{\displaystyle \frac{Q}{{\eta }_M}}t+{\displaystyle \frac{Q}{E_K}}(1-{\mathrm{e}}^{-{\displaystyle \frac{E_K}{{\eta }_K}}t})$$

(8)

S(t) = Q[L + Pt + U(1 − e^−Vt)]

(9)

Equation (9) was used to fit the displacement–time curves of the model piles at different load levels, as shown in Figure 10. The curves fitted using the Burgers model showed good overall agreement with the experimental measurements. At all load levels, the model effectively captured the complete time-dependent evolution of pile-head displacement, including the rapid initial increase in displacement and the subsequent progressive reduction in displacement rate as the creep response approached a relatively stable state.

Figure 10. Fitted displacement–time curves of the model piles. (a) A-Z1; (b) A-Z2; (c) A-Z3; (d) B-Z4.

The Burgers-model parameters obtained from the fitting are presented in Table 4, Table 5, Table 6 and Table 7. The coefficient of determination, R², exceeded 0.967 for every test group, and the fitted mechanical parameters generally decreased with increasing uplift load. Because the same model pile was used throughout the loading sequence, these variations reflect the combined effects of the increasing load level and the accumulated loading history from the preceding stages.

Table 4. Fitted Burgers-model parameters for A-Z1.

Load/N	EM/N·mm−1	EK/N·mm−1	ηM/N·d·mm−1	ηK/N·d·mm−1	R2	RMSE/mm
210	1004.327	7839.677	513,795.619	14,704.022	0.987	0.0009
350	568.714	2287.546	500,966.648	12,297.546	0.973	0.0062
490	435.804	1760.176	22,266.652	1243.637	0.986	0.0203
630	293.296	1063.837	12,879.646	898.795	0.995	0.0263

Table 5. Fitted Burgers-model parameters for A-Z2.

Load/N	EM/N·mm−1	EK/N·mm−1	ηM/N·d·mm−1	ηK/N·d·mm−1	R2	RMSE/mm
315	615.185	8120.515	1,826,017.174	5716.367	0.997	0.0005
525	513.190	1587.083	130,954.748	2256.057	0.977	0.0143
735	398.515	1578.276	7,762,194.206	3112.472	0.997	0.0061
945	349.964	1533.592	26,009.658	1491.530	0.997	0.0078

Table 6. Fitted Burgers-model parameters for A-Z3.

Load/N	EM/N·mm−1	EK/N·mm−1	ηM/N·d·mm−1	ηK/N·d·mm−1	R2	RMSE/mm
480	1300.481	4851.844	1,858,403.539	2274.453	0.991	0.0004
800	1064.332	5700.956	525,716.657	3416.371	0.967	0.0064
1120	860.678	3938.248	105,198.877	4879.731	0.999	0.0035
1440	683.654	3914.942	63,126.557	3773.513	0.998	0.0072

Table 7. Fitted Burgers-model parameters for B-Z4.

Load/N	EM/N·mm−1	EK/N·mm−1	ηM/N·d·mm−1	ηK/N·d·mm−1	R2	RMSE/mm
330	1552.604	17,710.594	3,170,343.821	22,285.142	0.973	0.0008
550	1005.022	4080.163	4,810,061.261	4318.245	0.999	0.0009
770	781.680	2954.388	1,759,020.485	2775.162	0.997	0.0038
990	624.871	2893.727	81,684.384	3752.548	0.998	0.0065

4. Conclusions

(1)

Uplift piles in calcareous sand exhibited pronounced time-dependent displacement under sustained uplift loading, with pile-head displacement progressing through primary and steady-state creep stages. Increasing pile surface roughness and using a well-graded particle-size distribution both effectively reduced long-term displacement.

(2)

The slopes of the isochronous load–displacement curves decreased as the load increased. At a given load, the curves shifted toward larger displacements as the holding time increased, while the spacing between successive curves narrowed at later times. Under their respective maximum sustained loads, the uplift deformation stiffness decreased by 27.4–43.0% over 21 d, with the reduction being smaller for rougher piles in Sand A.

(3)

At the maximum load level, pile axial force decreased progressively with embedment depth. Sustained loading primarily redistributed the axial force in the upper and middle portions of the pile, with the distribution stabilizing after 3–7 d. Greater pile surface roughness enhanced the time-dependent redistribution of load, whereas a well-graded particle-size distribution helped maintain a stable load-transfer path.

(4)

At the maximum load level, shaft resistance increased with embedment depth to a peak at approximately 45–55 cm and then decreased toward the pile toe. Within this peak zone, shaft resistance generally increased with sustained-loading duration and gradually stabilized after 3–7 d. This time-dependent increase was more pronounced for piles with rougher surfaces.

(5)

The Burgers model effectively described the displacement–time relationship of uplift piles in calcareous sand. As the pile-head load increased, the elastic moduli and viscosity coefficients in the Burgers model generally decreased, indicating that the pile–sand system became less resistant to long-term deformation.

5. Limitations and Future Work

(1)

Each load stage lasted 21 d. Model-scale and boundary effects and the use of dry, medium-dense sand limit the extrapolation of the observed 3–7 d stabilization to field piles. Future work will include longer-duration and field-scale tests.

(2)

Sequential loading on the same pile, without separate specimens for each load level or replicate tests, confounded loading-history effects and prevented assessment of repeatability. Future tests will use independent, replicated piles.

(3)

The A-Z2/B-Z4 comparison partly confounded particle-size distribution with normalized interface roughness, while interface dilation, particle rearrangement, and breakage were not directly measured. Future tests will match normalized roughness and include post-test particle analyses.