Fictitious soil pile model for dynamic analysis of pipe piles under high-strain conditions

Abstract

A fictitious soil pile (FSP) model is developed to simulate the behavior of pipe piles with soil plugs undergoing high-strain dynamic impact loading. The developed model simulates the base soil with a fictitious hollow pile fully filled with a soil plug extending at a cone angle from the pile toe to the bedrock. The friction on the outside and inside of the pile walls is distinguished using different shaft models, and the propagation of stress waves in the base soil and soil plug is considered. The motions of the pile−soil system are solved by discretizing them into spring-mass model based on the finite difference method. Comparisons of the predictions of the proposed model and conventional numerical models, as well as measurements for pipe piles in field tests subjected to impact loading, validate the accuracy of the proposed model. A parametric analysis is conducted to illustrate the influence of the model parameters on the pile dynamic response. Finally, the effective length of the FSP is proposed to approximate the affected soil zone below the pipe pile toe, and some guidance is provided for the selection of the model parameters.

Graphical abstract

Keywords

fictitious soil pile / soil plug / pipe piles / high-strain dynamic analysis / one-dimensional wave theory / pile dynamics

1 Introduction

Open-ended pipe piles are increasingly used worldwide as foundations for both land and offshore structures [1,2]; therefore, the characterization of pipe pile capacity and behavior under static and dynamic loading conditions has gained much attention in recent years [3–5]. As an open-ended pipe pile is inserted into the ground, part of the soil surrounding the pile toe enters the pile, creating a soil column known as a soil plug [6–10]. The presence of a soil plug complicates the analysis for evaluating pipe pile response [11]. For example, Gavin and Lehane [12] demonstrated that the shaft resistance of open-ended pipe piles is a complex function of the incremental filling ratio, the relative location of the pile toe in the sand, and the cone penetration test (CPT) q_c value, where the incremental filling ratio is defined as the rate of change in soil plug height with respect to depth advancement [8,13,14]. Wu [15] reported that the wave speed in driven pipe piles decreases as the soil plug height increases. In addition, the soil plug significantly affects the dynamic driving behavior and the time-dependent capacity of pipe piles [6]. Therefore, a better understanding of the interaction mechanism between pipe piles and soil is essential for interpreting the dynamic response and evaluating the bearing capacity of pipe piles.

A dynamic driving analysis of pipe piles conducted during high-strain dynamic tests is typically used to establish the relationship between the observed dynamic resistance and the pile static capacity and to predict pile behavior under static loads. Pile driving analysis is typically conducted numerically using a one-dimensional wave equation [16], and a key factor in solving it is the dynamic interaction model of the pipe pile–soil plug. Existing dynamic interaction models can be classified into four main categories [17]: the equivalent mass model [18], plugging effect model [19], Voigt model [20], and “pile within a pile” model [7,21–23]. The “pile within a pile” model is more rigorous than the other models because it considers both the wave propagation in the soil plug induced by the internal shaft resistance and the soil plug inertia. It simulates the soil resistance on the outside and inside of the pile walls and the soil reactions on the pile annulus and below the soil plug. Moreover, Wu et al. [24,25] proposed an additional mass model to account for the soil plug effect on the dynamic response of pipe piles, but it cannot consider the possible slippage in the pipe pile−soil plug interface or the interaction between soil plug segments.

Smith [16] proposed the first soil resistance model for piling analysis, which uses a dashpot in parallel with a linear spring and a plastic slider in series to simulate soil resistance. Its simplicity prevents it from differentiating soil damping types, and it also have problem of parameter selection due to its empirical natural. Holeyman [26] proposed a shaft model containing a spring and two dashpots in parallel to represent static soil resistance, viscous damping, and radiation damping, respectively. However, the viscous dashpot of this model is always considered active, even before sliding. Another shaft model was proposed by Simons and Randolph [27], which simplifies the soil to an ideal elastoplastic material and only considers radiation damping. This model was improved by Randolph and Simons [22] by adding a viscous dashpot accompanied by a plastic slider to describe the effect of the loading rate. Because this model utilizes physically meaningful input parameters and adheres to the actual mechanics of pile driving, it has gained wide recognition. However, it does not consider soil nonlinearity. Phan [28] introduced soil nonlinearity into the above model based on the study by Chow [29] but did not distinguish between the internal and external soil resistances. Other shaft models [30–34] have also been proposed to refine the details of the soil behavior and provide more rigorous treatment of soil damping, stiffness nonlinearity, and loading rate effects. However, they introduced additional parameters to complicate the analysis.

Additionally, several models have been developed to simulate soil base resistance. Lysmer and Richart [35] proposed a base model based on the derivation of a rigid circular footing placed on an elastic half-space consisting of a spring and a dashpot. Nguyen et al. [36] and Warrington [37] improved Lysmer’s model by adding a plastic slider, while Deeks and Randolph [38] used finite element analysis to improve the model further by adding consideration of the inertia effect and radiation damping of soil. However, most of these models consist of simple combinations of rheological elements and do not consider the effect of stress-wave propagation in the base soil. Consequently, they cannot describe the inhomogeneity of the base soil, its stiffness nonlinearity, or the soil depth of the affected zone.

To improve the existing models for the driving analysis of pipe piles with soil plugs, a fictitious soil pile (FSP) model based on the “pile within a pile” approach is proposed. The concept of FSP has been used in low-strain pile integrity tests [39–42], where the soil is regarded as a linear elastic material. In the present study, the FSP method is used to couple the pile−soil system under high-strain conditions by treating the base soil as a nonlinear and elastoplastic material. Considering that the external shaft soil of a pipe pile is radially infinite, whereas the soil plug is limited by the pile wall, the resistance models for external and internal soils are distinguished. Moreover, the nonlinearity and hysteresis of both the shaft and base soils are considered using a nonlinear soil stress−deformation relationship [29]. The FSP model was verified by comparing the calculated pile responses with those obtained from conventional models [16,28]. Furthermore, it was utilized to compute the response of field tests on well-instrumented open-ended pipe piles, and the calculated responses were compared with the measured data. A parametric analysis was performed to evaluate the effects of the discretization scheme, FSP dimensions, soil nonlinearity, and soil plug height. Finally, the effective length of the FSP and soil influence zone are discussed to provide helpful guidance for its application in practical engineering.

2 Model establishment

2.1 Computational conditions and assumptions

A schematic diagram of the pipe pile-FSP coupled model under high-strain conditions is shown in Fig.1, where the base soil is simulated with a fictitious hollow pile fully filled with a fictitious soil plug. To account for the soil distribution around the pile, the pile−soil system is discretized into n pile segments, N′ FSP segments, and N soil plug segments. The inner radius of the pipe pile is denoted by r_{\text{in}}, and the outer radius of the pile is denoted by r_{\text{out}}. In the following analysis, the upper suffixes of “p”, “fp”, “sp”, and “s” are used to distinguish the variables relating to the pipe pile, FSP, soil plug, and external shaft soil, respectively.

The interaction between the pipe pile segments is modeled using linear massless springs, and the spring stiffness k^{\text{p}} is calculated according to Eq. (1).

(1)k^{\text{p}}=\frac{E^{\text{p}}\text{π}\left(r_{\text{out}}^2-r_{\text{in}}^2\right)}{\mathrm{\Delta }{l}^{\text{p}}},

where \mathrm{\Delta }{l}^{\mathrm{p}} and E^{\mathrm{p}} represent the pile segment length and Young’s modulus, respectively.

The nonlinear and elastoplastic behavior of the soil segments are accounted for in the soil model, where the initial compressive stiffness is estimated using Eq. (1), and the development of soil stiffness becomes nonlinear as the deformation increases. The key assumptions used are summarized as follows.

1) The pipe pile is open-ended, elastic, and vertical. When subjected to uniform, high-energy instantaneous pressure loads at the pile head, the pile−soil system undergoes vertical vibrations with significant deformations.

2) Each soil layer around the pile segment (including the FSP segment) is homogenous and isotropic, and the soil stiffness decays nonlinearly with increasing deformation. Only pressure can be transferred between soil segments, and when the interaction force between the pile toe and base soil is zero, a gap is created.

3) The vibrations between the pile and adjacent soil plug segments are out of phase.

4) After the maximal shaft resistance has been attained, the possibility of pile shaft soil softening is not considered.

2.2 Soil model

2.2.1 Base soil model

The reaction from the base soil was simulated using an FSP model. To describe the nonlinear development of the soil stiffness, empirical relationships [29] were used; that is,

(2)k^{\text{fp}}=k_0^{\text{fp}}\left(1-R^{\text{fp}}\frac{q^{\text{fp}}}{q_{max}^{\text{fp}}}\right),

where k^{\text{fp}} represents the vertical stiffness of the FSP; k_0^{\text{fp}} represents the initial stiffness; q^{\text{fp}} represents the stimulated soil resistance; and q_{max}^{\text{fp}} represents the (maximum) ultimate soil resistance, which can be estimated using fundamental soil parameters [43–45]. The nonlinearity coefficients R^{\text{fp}} range from 0 to 1 and can be estimated from laboratory tests.

Equation (2) is transformed into the force−deformation relationship to facilitate subsequent calculations:

(3)q^{\text{fp}}=\frac{q_{max}^{\text{fp}}}{R^{\text{fp}}}\left(1-{\text{e}}^{-\frac{R^{\text{fp}}}{q_{max}^{\text{fp}}}{k}_0^{\text{fp}}{S}^{\text{fp}}}\right),

where S^{\mathrm{f}\mathrm{p}} represents the compression of the FSP.

The nonlinear soil stiffness at different loading stages is shown in Fig.2. If q_{max}^{\text{fp}} is reached, the soil remains in a state of complete plasticity, and the resistance is constant with soil deformation. During the unloading or reloading stage, the soil is linearly elastic and its stiffness is the same as the initial stiffness. Furthermore, as the soil cannot transmit tensile stress, the base resistance must be positive; if a tensile force is detected, the resistance is considered to be zero, and a gap formed between the pile toe and base soil. The nonlinear stiffness model can be implemented according to the flowchart shown in Fig.3 and programmed using MATLAB.

2.2.2 Internal shaft model

The internal shaft resistance provided by the soil plug was modeled using a plastic slider connected to a nonlinear spring in parallel with a viscous dashpot, as shown in Fig.4(a). This model is similar to the Smith rheological model [16] by assuming that viscous damping is the dominant damping component of the soil plug. However, to relate the model parameters to the conventional soil parameters, the spring stiffness was estimated using Eq. (4), which is based on the plain strain model according to Novak et al. [46]:

(4)k_{\mathrm{s}}^{\mathrm{s}\mathrm{p}}=\frac{2.75G^{\mathrm{s}\mathrm{p}}}{2\text{π}{r}_{\mathrm{i}\mathrm{n}}},

where k_{\mathrm{s}}^{\mathrm{s}\mathrm{p}} denotes the initial stiffness coefficient, and G^{\mathrm{s}\mathrm{p}} denotes the shear modulus.

Similar to the vertical interaction between FSP segments, the nonlinear behavior of the shaft soil is also considered through the Chow [29] relationship. The spring force in the elastoplastic state is calculated using Eq. (5), and Fig.4(b) shows the deformation states. The shaft resistance {\tau }^{\mathrm{s}\mathrm{p}} is allowed to be negative during the unloading process.

(5){\tau }^{\mathrm{s}\mathrm{p}}=\frac{{\tau }_{max}^{\mathrm{s}\mathrm{p}}}{R^{\mathrm{s}}}\left(1-{\mathrm{e}}^{-\frac{R^{\mathrm{s}}}{{\tau }_{max}^{\mathrm{s}\mathrm{p}}}{k}_{\mathrm{s}}^{\mathrm{s}\mathrm{p}}{u}^{\mathrm{s}\mathrm{p}}}\right),

where u^{\text{sp}} is the displacement of the soil plug; R^{\text{s}} is the shaft nonlinearity coefficient; and {\tau }_{max}^{\text{sp}} is the ultimate soil friction, which can be estimated based on fundamental soil properties [45,47].

The viscous damping caused by the relative motions between the pile segment and adjacent soil plug segment is calculated using Eq. (6) [26]:

(6){\tau }_{\text{visc}}^{\text{sp}}={\tau }_{max}^{\text{sp}}\times c_{\text{s}}^{\text{sp}}\times {\left(v^{\text{p}}-v^{\text{sp}}\right)}^{\alpha },

where {\tau }_{\text{visc}}^{\text{sp}} denotes the viscous damping force; v^{\text{p}} and v^{\text{sp}} denote the velocity of the pile and soil plug, respectively; and c_{\text{s}}^{\text{sp}} and α are input parameters related to the viscous damping of soil. c^{\text{sp}} ranges from 0.1 (for sand) to 1 (for clay soils) and α is close to 0.2 [26].

2.2.3 External shaft model

The Randolph and Simons model [22] was used to model the interaction between the pile and external shaft soil and was enhanced by considering the soil stiffness nonlinearity and hysteresis damping. This model is presented in Fig.5 and consists of two parts.

The first part is formed by a spring related to the displacement (representing static soil resistance) and a dashpot related to the velocity (representing radiation damping) connected in parallel, which describes the outer field soil that has not yet become completely plastic. The spring stiffness is given by Eq. (4), and the nonlinear degradation is based on the relationship described by Chow [29]. The damping coefficient c_{\text{r}}^{\text{s}} is obtained as follows [46]:

(7)c_{\text{r}}^{\text{s}}=\sqrt{{\rho }^{\text{s}}{G}^{\text{s}}},

where G^{\text{s}} and {\rho }^{\text{s}} are the shear modulus and density of soil, respectively.

The second part is a plastic slider (representing the ultimate static resistance) connected in parallel with another dashpot (representing viscous damping), which represents the potential shear band around the pile shaft. As long as the total stresses in the spring and radiation dashpot do not exceed the ultimate shaft resistance, the pile and adjacent soil segments will move together, and the second part will not be activated. Otherwise, when slippage occurs the soil behavior is governed by the slider and viscosity dashpot. Viscous damping due to slippage is also the main source of the soil strength increase owing to the loading rate.

2.3 Solution of the proposed model

To activate the ultimate resistance of the soil around the pile, a high-strain dynamic load test (DLT) is typically performed by dropping a hammer to impact the pile top at a weight of approximately 1% to 2% of the designed pile capacity [48]. The impact force–time history can be estimated based on the simplified hammer-cushion-pile model [49]:

for ,

(8)F\left(t\right)=\frac{k_{\text{c}}{v}_{\text{h}}}{w\sqrt{1-{\overline{D}}^2}}{\text{e}}^{-\overline{D}wt}\sin \left(tw\sqrt{1-{\overline{D}}^2}\right),

for \overline{D}>1,

(9)F\left(t\right)=\frac{k_{\text{c}}{v}_{\text{h}}}{w\sqrt{{\overline{D}}^2-1}}{\text{e}}^{-\overline{D}wt}\sinh \left(tw\sqrt{{\overline{D}}^2-1}\right),

for \overline{D}=1,

(10)F\left(t\right)=k_{\text{c}}{v}_{\text{h}}t{\text{e}}^{-wt},

where v_{\text{h}} denotes the hammer impact velocity, k_{\text{c}} denotes the equivalent stiffness of the cushion, and w is the natural frequency estimated by w=\sqrt{k_{\text{c}}/\phantom{k_{\text{c}}{m}_{\text{h}}}{m}_{\text{h}}}.

The energy dissipation of the pile−soil system is represented by the system damping, \overline{D}, which is given by

(11)\overline{D}=\frac{\sqrt{k_{\text{c}}{m}_{\text{h}}}}{2{\rho }^{\text{p}}{A}^{\text{p}}{c}^{\text{p}}},

where A^{\text{p}}, {\rho }^{\text{p}}, and c^{\text{p}} denote the section area, density, and wave speed of the pile, respectively, and m_{\text{h}} denotes the hammer mass.

The dynamic responses of the discrete segments are calculated using a time differentiation scheme. The total time (t) of the loading event is discretized into equal time intervals (n_{\text{t}}\mathrm{\Delta }t). The motions and forces of each segment are assumed to remain constant within each time step. At the start of the analysis, the displacement (u), velocity (v), acceleration (a), spring force (P), and soil resistance (R) of each segment are set to zero, that is,

(12)\text{when}j=0:u\left(i,j\right)=v\left(i,j\right)=a\left(i,j\right)=P\left(i,j\right)=R\left(i,j\right)=0,

where i represents the segment number and j represents the time step.

Once the impact force is applied to the pile head, the acceleration of the pile head segment is used as the input to the system.

1) Calculate the displacement and velocity of each segment

(13)v\left(i,j\right)=v\left(i,j-1\right)+a\left(i,j-1\right)\mathrm{\Delta }t,

(14)u\left(i,j\right)=u\left(i,j-1\right)+v\left(i,j-1\right)\mathrm{\Delta }t.

2) Calculate the deformation of each spring

(15)s\left(i,j\right)=u\left(i,j\right)-u\left(i+1,j\right),

where s denotes the spring deformation.

The forces in the pile springs were calculated using Eq. (1). The spring force of the soil segments can be calculated using Eq. (2) by considering the stiffness nonlinearity.

3) Calculate the shaft resistances based on the shaft model described in Subsection 2.2.

4) Calculate the acceleration of each segment according to the dynamic equilibrium equations.

For a typical pile segment,

(16)a^{\text{p}}\left(i,j\right)=\left(P^{\text{p}}\left(i-1,j\right)-P^{\text{p}}\left(i,j\right)-R^{\text{s}}\left(i,j\right)-R^{\text{sp}}\left(i,j\right)\right)/\phantom{\left(P^{\text{p}}\left(i-1,j\right)-P^{\text{p}}\left(i,j\right)-R^{\text{s}}\left(i,j\right)-R^{\text{sp}}\left(i,j\right)\right)m^{\text{p}}\left(i\right)}{m}^{\text{p}}\left(i\right),

where P^{\text{p}} denotes the axial force provided by the pile spring; R^{\text{s}} and R^{\text{sp}} represent the outer-shaft resistance and inner-shaft resistance, respectively; and m^{\text{p}} denotes the segment mass.

For a typical soil plug segment,

(17)a^{\text{sp}}\left(i,j\right)=\left(R^{\text{sp}}\left(i,j\right)+P^{\text{sp}}\left(i-1,j\right)-P^{\text{sp}}\left(i,j\right)\right)/\phantom{\left(R^{\text{sp}}\left(i,j\right)+P^{\text{sp}}\left(i-1,j\right)-P^{\text{sp}}\left(i,j\right)\right)m^{\text{sp}}\left(i\right)}{m}^{\text{sp}}\left(i\right),

where P^{\text{sp}} represents the axial force provided by the soil plug spring and R^{\text{sp}} represents the inner shaft resistance.

For a typical FSP segment,

(18)a^{\text{fp}}\left(i,j\right)=\left(P^{\text{fp}}\left(i-1,j\right)-P^{\text{fp}}\left(i,j\right)-R^{\text{s}}\left(i,j\right)-R^{\text{sp}}\left(i,j\right)\right)/\phantom{\left(P^{\text{fp}}\left(i-1,j\right)-P^{\text{fp}}\left(i,j\right)-R^{\text{s}}\left(i,j\right)-R^{\text{sp}}\left(i,j\right)\right)m^{\text{fp}}\left(i\right)}{m}^{\text{fp}}\left(i\right),

where P^{\text{fp}} represents the axial force provided by FSP spring.

The acceleration calculated at the current time step (j) is treated as a new input to the pile−soil system, and a new cycle of calculations from Eq. (13) to (18) begins at the next time step. The procedure is repeated until a predetermined moment after the end of the pile head dynamic loading. The process was coded using MATLAB and then used to calculate the pile and soil responses at all positions.

3 Model validation

3.1 Validation using existing soil models

Using the developed model, the vertical vibration responses of a pipe pile under high-strain conditions were analyzed, and a comparison was made between the results from the FSP model and those calculated by the existing soil models proposed by Smith [16] and Phan [28]. Because soil plug movement cannot be considered in the Smith model, it is treated as a part of the pipe pile (i.e., equivalent mass model). The Phan model uses an external shaft model (see Fig.5) similar to that of the proposed model, but it does not distinguish between the internal and external shaft soils. And the reactions below the pipe pile and soil plug are based on the Deeks and Randolph [38] model.

The dimensions of the analyzed pipe pile are shown in Fig.6(a). Two values of soil plug length H (i.e., 0.5L and L) were considered to simulate different filling degrees of the soil plug. Tab.1 lists the soil parameters used in the numerical models.

The impact load can be simulated using a one-pulse load in the form of a half-sine if (see Eq. (8)) [49]. Various loading durations of the half-sine function were employed, as shown in Fig.6(b). The loading durations were divided into three groups according to the range of “wave number, N_c” [50], which is defined by N_{\text{c}}=T/\phantom{T\left(L/\phantom{L{c}^{\text{p}}}{c}^{\text{p}}\right)}\left(L/\phantom{L{c}^{\text{p}}}{c}^{\text{p}}\right). Because the impact load disappears before the reflected wave reaches the pile top in the short loading duration group (), the reflected wave from the pile toe cannot be superposed on the incident wave; thus, the wave effect in the pile is relatively obvious. For the intermediate group (), although the wave effect is weakened, it should still be considered in the pile load capacity interpretation. For longer loading durations (N_{\text{c}}>10), the test is considered a rapid load test, and the stress wave effect can be ignored [51].

The displacement responses of the pile head calculated using different models were compared for all loading conditions with soil plugs either partially filled (H = L/2) or fully filled (H = L), as shown in Fig.7.

For H = L/2, the displacement curves obtained from the FSP model are consistent with those obtained from the Phan model, as shown in Fig.7(a) and Fig.7(b), but they are slightly larger for longer loading durations, as shown in Fig.7(c). However, the results calculated using the Smith model are significantly different. Under short loading durations, the peak displacement from the Smith model is similar to that of the other models, but the displacement amplitude decays more rapidly and the calculated residual displacement deviates more significantly. As T increases, the Smith model increasingly underestimates both the peak and permanent displacements, which demonstrates that ignoring the effect of the soil plug could lead to large errors in the dynamic analysis of pipe piles subjected to impact loading for long durations.

For H = L, the pile displacements are smaller than those for H = L/2 owing to the increase in the inner soil resistance, as shown in Fig.7(d), Fig.7(e), and Fig.7(f). The calculated displacements from the FSP model still agree well with those of the Phan model, whereas the results of the Smith model are significantly different even for short loading durations. These results further confirm the significant effect of the soil plug on the dynamic response of pipe piles and that the soil plug must be considered in the analysis, particularly for completely filled piles.

3.2 Validation using field measured results

The proposed model was used to simulate the dynamic behavior of pipe piles during the field high-strain DLTs, and the pile responses were analyzed to further confirm the validity of the model. Two open-ended tested piles, one tested spun concrete pile (TSC) and one tested steel pipe pile (TSP) used for the berth foundation of the Thi Vai International Port [28,52], were analyzed. The soil profile at the analyzed pile, shown in Fig.8(a), consisted of soft clay, clayey sand, hard silty clay, etc., and the soil strength parameters {\tau }_{max} and q_{max} of each soil layer were estimated and adjusted from the standard penetration test (SPT) N-value according to Matsumoto and Hoang [52], as shown in Fig.8(b). Other basic soil parameters for each soil layer were assumed to be those summarized by Matsumoto and Hoang [52] and Phan [28].

Fig.9 shows the measured axial force at the pile head for the TSC and TSP piles, which were calculated based on measurements from accelerometers and strain gauges attached 2.7 m (3.86D) and 3.5 m (3.5D) below the pile top of the TSC and TSP, respectively. The measured forces were then used as the input forces in the FSP model. According to Paik et al. [53], if no measurements are available, the soil plug length can be assumed to be between 70% and 80% of the pile embedment length. However, considering that the shaft resistance of the shallow soil layer (see Fig.8(b)) was almost negligible, the soil plug length was assumed to be equal to the pile embedment length to facilitate the numerical calculation. Tab.2 lists other relevant parameters used in the calculations.

The velocity and displacement time histories from the FSP and Phan models were compared with those measured at the TSC pile head, as shown in Fig.10. It shows that the computed velocities and displacements from both the FSP and Phan models are comparable and agree well with the measured values. The FSP model predictions were closer to the measured results, especially for the initial part of the records and at the velocity and displacement peaks, whereas the Phan model overestimated the results.

Fig.11 shows a comparison between the static load−displacement curves from the DLT and the static load test (SLT) results for two loading cycles of the TSC pile. Note that the static resistance response in the SLT is slightly higher than that derived from the DLT, which is attributed to the set-up phenomena (i.e., the increase in pile capacity over time) [54]. In this specific case, the DLT was conducted immediately at the end of pile driving, whereas the SLT was performed 17 d after the DLT.

The calculated velocities and displacements of the pile and surrounding soil at different positions are shown in Fig.12, where Z and Z^{\prime } represent the distance from the pile head (or soil plug head) and pile toe to the calculated point, respectively. Note that both the pile and soil vibrations propagate in the form of a one-dimensional wave and decay along the pile (or soil depth). Furthermore, as shown in Fig.12(b), the velocity and displacement of the base soil decrease sharply owing to the sudden reduction in the stiffness at the pile toe-soil interface, and the soil motion almost stops beyond H^{\prime } = 3 m below the pile toe. Fig.12(c) shows that the soil plug displacement is much smaller than that of the pile segments, with the maximum displacement being less than 10% of the pile head maximum displacement, which implies that the soil plug motion can be neglected.

Fig.13 shows a comparison of the calculated and measured TSP responses. Most of the measured pile head velocities and displacement time histories matched the computed ones, although the residual displacement was slightly overestimated. However, the Phan model considerably underestimated the maximum displacement and overestimated the residual displacement, as shown in Fig.13(b).

Fig.14 shows a comparison between the derived static load curve and the static load-settlement curve of the TSP. The derived soil resistance is also lower than that of the response in the SLT, but the difference is not significant compared to the case of the TSC, although the TSP had a longer testing rest period (48 d). The different set-up amounts of the TSC and TSP might be explained by the differences in the pile configurations. The TSC had a much thicker wall thickness (100 mm) than the TSP (12 mm); therefore, the surrounding soil of the TSC was compressed more densely and caused a greater excess pore-water pressure.

The vibration time histories of the TSP pile−soil system at different depths are displayed in Fig.15, which indicates propagation and decay patterns similar to those of the TSC. However, the base soil influence zone is deeper (up to 8 m).

4 Parametric analysis

4.1 Discretization of pile−soil system

The effect of the pile segment length (ΔL) is shown in Fig.16, where the soil plug length H = L and loading durations were 25 and 60 ms, respectively. It shows that the calculated displacements of the pile and soil plug tend to increase and gradually plateau as ΔL decreases from 4.0 to 0.25 m for both impact conditions. The ΔL influences the initial displacement curve more strongly than the residual displacement curve for the pile. However, both the maximum and residual displacements of the soil plug were affected by ΔL. Moreover, the calculated displacement curves at the pile head and soil plug head tend to stabilize when ΔL ≤ 0.5 m.

Fig.17 presents the calculated results combined with the field tests. The pile head responses vary in a similar manner; that is, ΔL affects the pile head peak displacement and the initial part of the curve, but the calculation accuracy is sufficiently high when ΔL < 1.0 m.

Fig.18 shows the effect of the FSP segment length (ΔH′), which indicates that the displacements of both the pile head and soil plug head increase as ΔH′ decreases. It also shows that the FSP should be discretized into smaller segments for the results to plateau. ΔH′ affects the peak and residual displacements, but has almost no effect in the initial ascending region. These conclusions can be drawn from the calculated results combined with field tests, as shown in Fig.19. This indicates that ΔH′ has a significant influence on the final part of the displacement curves. Moreover, ΔH′ must be smaller (0.25−0.5 m) than the pile segment length (ΔL = 0.5−1.0 m) to obtain accurate results, which is attributed to the difference in the elastic modulus of the pile and soil.

The effect of the calculation time interval on the analysis results is evaluated by considering different values, 0.5t_{\text{cri}}, 0.8t_{\text{cri}}, 0.9t_{\text{cri}} and t_{\text{cri}}, where t_{\text{cri}}=\mathrm{\Delta }L/\phantom{\mathrm{\Delta }L{c}^{\text{p}}}{c}^{\text{p}}. The results obtained using \mathrm{\Delta }t=0.8t_{\text{cri}} were almost equal to those obtained using \mathrm{\Delta }t=0.\text{5}{t}_{\text{cri}}, but no solution could be obtained for \mathrm{\Delta }t⩾0.\text{9}{t}_{\text{cri}}. Therefore, \mathrm{\Delta }t=0.8t_{\text{cri}} is suggested to achieve acceptable accuracy and efficiency.

4.2 FSP dimensions

Fig.20 shows the influence of FSP length H′ combined with the field test case. To ensure that ΔH′ remained constant (0.25 m), the segment number was set to be proportional to H′. It can be observed that H′ affects the peak and residual values of the displacement curves, despite its small effect on the first ascending region. Additionally, the displacements increase with an increase in H′ and tend to stabilize. The change in distance between the pile toe and bedrock could explain this result. When the distance is small, the base soil movement is more restricted by the fixed boundary, and the base soil deformation is therefore smaller. Moreover, pile vibration has an influence depth in the base soil, which is equivalent to the FSP effective length. Beyond the effective length, the value of H′ no longer affects the pile response.

Fig.21(a) and Fig.21(b) illustrate cone angle θ has a significant effect on the final region of the displacement curve. For the TSC, both the maximum and residual displacements decrease with an increase in θ, but only the residual displacement decreases for the TSP. When θ = 15° and 12° for the TSC and TSP, respectively, the calculated results were much closer to the measured results.

4.3 Soil nonlinearity coefficient

By assuming that the vertical soil stiffness has the same nonlinearity coefficient as the shaft soil, the influence of soil nonlinearity (R) was investigated. A higher R-value indicates a higher degree of soil stiffness. If R = 0, the spring is linearly elastic; if R converges to 1, the soil reaches its maximum degree of nonlinearity. As the R-value increases, so does the u_{max} required to fully stimulate the ultimate resistance (see Fig.2). For a constant ultimate resistance, Fig.22 shows the calculated displacements of the piles as R varies from 0.01 to 0.99. It shows that as R increases, the pile displacement increases while that of soil plug decreases.

The displacement curves in Fig.23 of the field test piles (TSC and TSP) show that the displacements are almost unaffected by the value of R. This can be explained by the excessive displacement of the test piles. According to the nonlinear stiffness models, when the displacement of the pile (or soil) is much greater than u_{max}, the soil is in the fully plastic state, and the soil resistance are controlled by the ultimate resistance. In this case, the effect of R-value is not significant. However, if the displacement amplitude is close to u_{max}, the responses of the pile and soil are more influenced by R-value. This reveals that conventional soil resistance models are more applicable only when the tested pile displacement during the high-strain DLT is much greater than u_{max}. This, however, necessitates a large impact energy on the pile head, which raises the testing cost as well as the risk of destroying the pile head.

4.4 Length of soil plug

Fig.24 shows the influence of soil plug length (H) on the calculated responses. In Fig.24, H = L, 0 < H < L, and H = 0 indicate a pipe pile with a full plug, partial plug, and no plug, respectively. It shows that the pile head displacement amplitude decreases as H increases, owing to the increase in inner shaft resistance. However, as H increases, the impact force transferred to the soil plug also increases as the top surface of the plug approaches the pile head. Therefore, the soil plug displacement gradually increases.

The pile head displacement results for the field test piles, assuming different H values, were compared with the measured results, as shown in Fig.25. Fig.25(a) and Fig.25(b) show that although H changed significantly, the calculated results varied only slightly, indicating that H had a negligible effect in these analyses. One of the reasons could be that, as shown in Fig.8(b), the soil plug was assumed to be much weaker than the outer shaft soil owing to the excess pore water pressure generated in the soil around the pile during the piling process, which is difficult to dissipate after a brief resting period [52].

5 Discussion

The effective length (H_e) of the FSP is determined by the depth of the base soil-affected zone. When the pile toe is close to the bedrock layer, the length of the FSP can be set as the vertical distance between the bedrock and the pile toe. However, if the bedrock layer is far away from the pile toe, a reasonable FSP length should be considered to ensure accuracy and efficiency. Once the FSP length is determined, the FSP cone angle can be back calculated by wave matching analysis.

In the present study, H_e-value was determined by defining the critical depth (Z_cr) of the influence zone corresponding to u_j^{max}/u_1^{max}⩽1\mathrm{\%} in the base soil, where u_j^{max} and u_1^{max} represent the maximum displacement of the jth segment and first segment, respectively. For instance, according to the displacement curves along the FSP depth (see Fig.26(a)) and the maximum displacement attenuation through a dimensionless presentation (see Fig.26(b)), the FSP effective length is 3.5 m (T = 0.06 s; F = 2.0 MN).

There is an obvious correlation between H_e and θ of the FSP because θ affects the spring stiffness of the FSP. Fig.27 presents the relationship between H_e and θ for various pile diameters under the four impact conditions at the pile head. Note that D represents the outer diameter of the pile and the pile wall thickness remains constant (0.01 m). The analyzed pile−soil system has the same properties as those used in Subsection 3.1. Note that H_e increases as θ increases from 0° to 45° for all cases and tends to be stable when θ is close to 45° for longer load durations (T = 0.06 s). In addition, H_e also increases with an increase in D from 0.4 to 1.5 m, which reveals the rationality of using the FSP model for large-diameter pipe piles. As shown in Fig.28, similar results can be drawn from the calculated results of the field test piles, in which the H_e values stabilize (maximum) when θ is close to 45° and are then equal to 3.0 and 9.1 m for the TSC and TSP, respectively. This indicates that the greater the vertical stiffness of the base soil, the deeper the vibration is transferred downward in the soil and the larger the influence soil zone. Moreover, a comparison of Fig.27(a) and Fig.27(b) (or Fig.27(c) and Fig.27(d)) reveals that the loading duration significantly affects H_e; it increases as T increases And the effect of the impact load amplitude is less remarkable.

For a particular DLT, the test pile dimensions and properties and the impact force are usually known (measurable), whereas soil parameters are not always measured and are typically estimated (e.g., correlations with SPT-N values). Therefore, they are typically iteratively matched according to the measured pile responses. When using the FSP model to simulate the DLT results, the maximum and minimum values of H_e (i.e., corresponding to θ = 45° and 0°) can be established (see Fig.27) and then, combined with the assumed soil parameters, utilized to determine H_e and θ within this range.

Fig.29 shows the effects of the base soil parameters on the maximum (H_{\text{e}}^{max}) and minimum (H_{\text{e}}^{min}) effective length to provide some guidance for adjusting the FSP parameters. The H_{\text{e}}^{max} and H_{\text{e}}^{min} correspond to θ = 45° and 0°, respectively. The analyzed parameters included the soil Young’s modulus (E), ultimate shear stress ({\tau }_{max}), and relative height of the soil plug (H′/L). Fig.29(a) shows that both H_{\text{e}}^{max} and H_{\text{e}}^{min} increase as E increases from 1 to 90 MPa, but the rate of increase gradually decreases. Meanwhile, Fig.29(b) shows that H_{\text{e}}^{max} and H_{\text{e}}^{min} initially increase as {\tau }_{max} increases from 10 to 40 kPa and then remain constant. Finally, Fig.29(c) shows that both H_{\text{e}}^{max} and H_{\text{e}}^{min} increase slightly as H′/L increases.

6 Conclusions

1) To simulate the dynamic behavior of an open-ended pipe pile−soil system under high-strain dynamic conditions, a novel FSP model is proposed. Unlike conventional soil models, the FSP model distinguishes the soil resistance models for the inside and outside shafts of pipe piles and accounts for soil nonlinearity and wave propagations of the soil plug and pile base soil.

2) The accuracy of the FSP model predictions was validated by comparing the results with those from conventional numerical model calculations and field test measurements of full-scale pipe piles.

3) The discretization parameters, that is, the calculation time interval (Δt), pipe pile segment length (ΔL), FSP segment length (ΔH′), FSP length (H′), cone angle (θ), soil nonlinearity coefficient (R), and soil plug length (H), were examined, and the following recommendations are provided for their selection: ΔL between 0.5–1.0 m; ΔH between 0.25 and 0.5 m; and Δt ≤ 0.8t_cri.

4) The pile head displacement varied with the model parameters; it increased and gradually plateaued as H^{\prime } increased and decreased as θ decreased. The pile displacement increased as R increased, whereas the soil plug displacement tended to decrease. In addition, the pile head displacement decreased, whereas the soil plug displacement increased as H increased.

5) The effective length increases and then plateaus as θ increases from 0° to 45° and increases as the pile diameter increases. In addition, the soil’s Young’s modulus, shear strength, and soil plug height also affect the effective length to some extent.

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