A Novel Numerical Method for Calculating Vertical Bearing Capacity of Prestressed Pipe Piles

A novel method for calculating the vertical bearing capacity of prestressed pipe piles with the acceptable error was proposed and verified. Soils at the pile side and end were, respectively, simulated by an elastic-plastic model and a new double-line (at soft rock and soil layers) or triple-line model (at hard rock and soil layers); then, a mechanical model was established for simulating vertical bearing capacity of prestressed pipe piles, and the corresponding calculation process was carried out. *e values of pile side resistance, pile end resistance, and pile end elastic displacement were first obtained from the results of high-strain dynamic testing (HSDT) and then were imported into the proposed numerical model for calculating the vertical bearing capacity of prestressed concrete pipe piles. *e static load test was carried out to verify the numerical results. Besides, 20 piles were tested at two typical test sites (soft and hard rock bearing strata), of which 8 piles were arranged at the soft rock bearing stratum site and 12 piles were arranged at the hard rock bearing stratum site. *e numerical results achieved from an empirical formula were also used for making a comparison. *e values obtained by the proposed method were highly close to those achieved from the static load test with an error of within 10%.*e outcomes indicated that the proposed numerical method can be potentially applied to predict the bearing capacity of prestressed pipe piles.


Introduction
In a number of countries, especially in China and Japan, prestressed concrete pipe piles have become one of the most important types of pile. e theoretical solutions employing the bearing capacity to calculate the pile shaft and tip resistance involve setbacks caused by uncertainty factors, such as installation method, stress history, and soil compressibility [1]. e experimental tests, such as standard penetration test (SPT), static cone penetration test (CPT), and static load test (SLT), correlating results of in situ tests with the bearing capacity of pile, are frequently used in evaluating load-bearing capacity of a single pile; note they also involve shortcomings induced by both operator and test procedure [2]. According to the specification, the blow count (N) of SPT (SPT-N), the number of blows to drive a sampler 300 mm in the ground, may significantly vary due to the various experimental and operational conditions in the preliminary investigation of a project site [3], and the SPT-N is widely applied in predicting bearing capacity and assessing the quantificational risk of soil liquefaction [3][4][5]. For CPT, it is taken as one of the most significant field tests into account for the soil characterizations because CPT is a robust, simple, fast, reliable, and economical test, providing continuous soundings of subsurface soil and does not require extensive coring [6]. Generally, CPT is advantageous in the study of geotechnical behavior of a pile due to the resemblance between the cone penetrometer and pile, which consequently has extensive applications in geotechnical engineering [7]. Cone-tip resistance and sleeve friction are measured and recorded simultaneously when the tip penetrates the soil layer [8]. After acquiring the cone penetration data, there are two approaches to apply CPT records for both drilled and driven piles' design [7,8]. One is a direct approach, in which the measured cone-tip resistance and sleeve friction are directly used for calculating the bearing capacity of a pile [9][10][11][12][13][14][15][16], while the other is an indirect approach, in which the measured CPT data are applied first to estimate soil parameters. en, the estimated data are used to obtain the end-bearing capacity, as well as the unit skin friction [17,18]. Moreover, a CPTu-based enhanced unicorn method for pile capacity has been proposed, which could estimate axial pile capacity for a wide variety of types of piles installed in different assortments of geomaterials [19].
High-strain dynamic testing (HSDT) of piles [20], based on one-dimensional wave propagation, is an innovative method in predicting the bearing capacity of buried piles, and the bearing capacity has been proved to be in close agreement with that of SLT [21]. Xu and Li [22] conducted experimental study and theoretical analysis on the vertical bearing capacity of pipe piles. ey combined a large amount of data for a pipe pile foundation project with the measured displacements at the top and bottom of piles under a SLT to determine the mechanical characteristics of pile endbearing layer and revised a formula used for calculating the axial force of pile end. Zhang et al. [23] carried out a series of SLTs to analyze the load-displacement relationship, friction resistance, and pile end resistance. ey used a simple model to describe the degradation of pile side resistance and pile end load-displacement relationship at the pile-soil interface.
eir results showed that in the nondestructive load testing, the pile side resistance softening only occurred in shallow soil, and the pile side resistance in deep soil was not fully realized. However, in the destructive load testing, the pile side resistance softening occurred across the whole pile-soil interface. ey further studied the relationship between the pile side resistance and the relative displacement of pile and soil, as well as the relationship between pile end resistance and pile end displacement to establish a load-displacement model [24]. e pile properties are also of considerable significance to better evaluate the bearing capacity of the pile foundation. A new type of offshore oil and gas platform mixed pile was explored with the laboratory test program and the FEM, and the results show that the novel offshore foundation type is suitable for a wide range of sand conditions [25]. e influences of sand piles on improving the bearing capacity of soil foundations as well as controlling the settlement have been studied by partially replacing sand piles with constraints [26]. Moreover, there are still several factors affecting the bearing capacity of the pile foundations, such as external sulfate attack, effective radius, and the elastic modulus of a pile. e degree of soil-plugging should be considered when the bearing capacity of the open-ended pile is studied [27]. e plugging effect of open-ended piles is highly influenced by the pile driving condition, soil condition, and pile geometry [28]. e vertical bearing capacity of prestressed pipe piles generally depends on the soil parameters and soil-pipe interaction. As the soil surrounded a pile is multilayered and heterogeneous, it is complicated to calculate the vertical bearing capacity of the prestressed pipe pile in different layers of soil. It is noteworthy that the calculation involves a series of parameters that include pile side resistance, pile end resistance, pile side elastic displacement, pile end elastic displacement, pile length, pile area, and elasticity modulus of pile.
Considering the high cost of in situ tests, artificial intelligence (AI), combined with mathematics, algorithm, and creativity, has been introduced to establish AI-based predictive models for estimating bearing capacity of piles [29][30][31][32][33][34]. Using the results of CPT, two artificial neural networks (ANNs) and a nonlinear multiple regression model for predicting pile resistance were developed to predict pile resistance. It has been proved that the application of AI and predictive models can be practical, feasible, and they are regarded as fast tools in solving engineering problems [35]. Apart from that, a new on-site camera method based on node displacement with the biological mechanism of phagocytosis to correlate continuous images was developed to calculate the settlement of piles [36]. In addition to in situ tests, numerical simulation methods have also been applied to study the bearing capacity of piles. Chen et al. used a discrete element model (DEM) to assess the pilesand interactions at the microscale [37]. Cai et al. utilized a finite element model (FEM) to calculate the bearing capacity at the tip of the pile for different slope angles [38]. Józefiak et al. also adopted the FEM to obtain the pile bearing capacity and pile settlement of the soil-pile system [39]. For a broad group pile foundation, the field load tests combined with numerical analyses are typically applied to estimate the ultimate performance of the pile foundation [40].
Recently, a number of scholars concentrated on the dynamic testing of piles based on stress wave theory. Liang and Yang [41] developed reliability-based quality control criteria for driven piles based on the framework of acceptance-sampling analysis for both static and dynamic testing methods with the lognormal distribution characteristics. ey also suggested an optimum approach for the number of load tests and the required measured capacities for quality control of driven piles. Salgado et al. [42] proposed an advanced model for dynamic pile driving analysis to develop accurate pile driving formulas. Comparisons of predictions from the proposed formulas with results from static and dynamic load tests showed that reasonably accurate predictions of pile capacity could be achieved. e HSDT has several advantages in pile analysis, while it cannot fully simulate the bearing capacity of end-bearing piles when the bearing capacity of end-bearing piles is tested [43].
In this research, a novel method for calculating the vertical bearing capacity of prestressed pipe piles is proposed via establishing a new mechanical model numerically. e mechanical parameters, such as pile side resistance, pile end resistance, and pile end displacement, are obtained by HSDT and then are imported into a new mechanical model. To verify the accuracy of the numerical method, 20 test piles are arranged in two typical test sites, and their mechanical properties are tested. e novel numerical method can improve the design efficiency and reduce the engineering cost, and it can be potentially applied to design prestressed pipe piles. 2 Advances in Civil Engineering

Analytical Modeling of a Pile.
In order to model a pile analytically, the prestressed concrete pipe piles are theoretically divided into a series of elements with uniform length and diameter ( Figure 1). e axes of each element are assumed to be straight lines, and the angle between the axis and the horizontal plane is denoted by α. For any node located on an element, the static equilibrium equation is formulated as follows [44]: where P is an external force at joints (including force and bending moment), A represents a coefficient matrix, and F is an internal force of the pile at joints (including force and bending moment). X is defined as a displacement matrix of nodes relative to the original axis of a pile under external forces and e as a displacement matrix of the internal force. As a result, the following equation is given: According to previous studies [45,46], B is the transpose of matrix A(B � AT). erefore, equation (2) can be rewritten as e � A T X. (3) We assume that the internal force F in each element has the following relationship with its corresponding internal displacement e: A local coordinate system for each element i is established to construct the coefficient matrix A i and stiffness matrix S i . en, the matrices given by SA T and ASA T are constructed for each element, and the total matrix ASA T in the global coordinate is formed by superposition, i.e., the total stiffness matrix. Based on the total stiffness matrix, the incremental values of internal forces and the displacement of the pile under load are obtained. us, the total internal force and the displacement of each node in the pile elements can be obtained by superposing the results calculated with the initial internal force and displacement at each node of the pile. It is noteworthy that the theoretically developed model can be used to analyze the bearing capacity of vertical or inclined piles under the combined actions of horizontal load, vertical load, and bending moment [47]. e lateral soil resistance was simulated by adding a lateral soil spring at each node of the pile, and the load transfer function of pile side was used to analyze the relationship between the unit side resistance and the pile-soil relative displacement. For the unit side resistance, it is related to the ultimate resistance and elastic displacement of the pile side. e pile end soil resistance was simulated by adding a vertical soil spring at the pile end, and the load transfer function at the pile end was employed to analyze the relationship between the unit end resistance and the pile end displacement. For the unit end resistance (spring coefficient), it is related to the ultimate resistance at the pile end and the elastic limit displacement at the pile end.

Presentation of a Load Transfer Function for Pile Side
Resistance and Parameters of Piles. An elastic-plastic doublefolded line model [48] has been widely used in analysis of pile side resistance because of its simplicity and accuracy in calculation [47,49]. It was thus employed in this paper for establishing a load transfer function at the pile side. With the application of HSDT for test piles, a fully excited pile side resistance was used as the limit value of the pile side resistance.
A load transfer function of the pile side is presented in equation (5), and the load transfer function of the pile side is shown in Figure 2.
where c s is the pile side resistance per unit displacement in elastic soil, q u denotes the design value of pile side resistance achieved according to the results of HSDT, and s u denotes the relative displacement between pile and soil when the pile side resistance reaches the limit value.
In the present research, s u is determined using Table 1, where the empirically determined values of relative displacement corresponding to the maximum lateral resistance values of rock and soil layers are presented when the bearing capacity of prestressed concrete pipe piles is calculated. Expectedly, the relative displacement of rock stratum required to reach the limit value is less than that of the soil layer.

Presentation of a Load Transfer Function for Pile End
Resistance and Parameters of Piles. At present, twofold load transfer functions and threefold load transfer functions are mainly used in formulating load transfer functions for pile end resistance. For instance, Zhang et al. [24] explored the relationship between pile end resistance and its displacement and developed a twofold load transfer function. Luo and Li [50] theoretically analyzed the bearing capacity of piles to develop a unified threefold load transfer function with consideration of the elastic, plastic, and sliding stages of the pile end, and they also formulated an analytical solution. In the abovementioned functions, the displacement at the beginning of pile loading is generally proportional to the load magnitude. However, when the displacement is greater than the elastic limit displacement at the end of the pile, the displacement at the end of the pile may change abruptly after the pile is reloaded. e end resistance is also influenced by the degree of softness and hardness of the rock strata and soil layers at the end of the pile.

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Hence, in this research, a new load transfer function is proposed, and two mechanical models are established according to the soft/hard rock and soil layers at the end of the pile.

Soft Rock and Soil Layers.
e proportion of load that was borne by pile lateral friction and pile end resistance in soft soil is small due to the shear failure behavior at the end of the pile. Q-S curve ( Figure 3) shows a steep drop, and damages at characteristic points are evident. A new load transfer function is utilized at the end of pile (equation (6)). Figure 4 illustrates utilization of the load transfer function at the end of pile.where where k 1 is the spring coefficient at the end of pile; R c denotes the pile end resistance obtained from the HSDT; s c a L P 1 Nodal soil spring Surface friction resistance Pile element soil spring Several scholars proposed different methods for obtaining s bu . For instance, a number of scholars [51] studied Pile side resistance (kPa) End resistance (kN) Advances in Civil Engineering the bearing capacity using HSDT with sufficient excitation of pile end resistance. It was found that pile end displacement was 2.5% of the pile diameter when the pile end resistance was fully excited. Zhang et al. [23] conducted SLTs to study the relationship between pile end resistance and pile end displacement and attempted to formulate a robust loaddisplacement relationship. ey found that the ultimate displacement in the first stage of a fully excited pile end was 0.5-8.3 mm.

Hard Rock and Soil Layers.
For hard rock and soil layers with high end resistance and those requiring a large displacement to exert end resistance, the Q-S curve ( Figure 5) displays that damages in characteristic points are not evident. A new load transfer function was developed for the end of pile in this research (equation (8)). Figure 6 shows the Q-S curve based on the load transfer function at the end of pile.where where k 1 represents spring coefficient at the end of pile; R c is the pile end resistance obtained from the HSDT; s c denotes the pile end displacement achieved from the HSDT; s bu1 is the first-stage limit displacement at the end of pile; s bu2 indicates the second-stage limit displacement at the end of pile; η 2 represents the second-stage reduction coefficient of spring at the end of pile, generally ranging from 0.4 to 0.8; and η 3 is the third-stage reduction coefficient of spring at the end of pile (range, 0.2-0.4).

Development of Numerical Models
e proposed numerical models were programmed by FORTRAN.
e flowchart of algorithm development is depicted in Figure 7, and the input parameters are summarized in Table 2. e limit value of pile side resistance was measured by HSDT. e spring coefficient at the end of pile could be achieved by calculating the resistance and displacement at the end of pile via testing pile and then fitting a high stress-strain curve. e results of algorithm development could be achieved and then verified by the field testing data.

Experiments
As the softness and hardness of a rock stratum can influence the characteristics of piles in the Q-S curve, both soft rock bearing stratum and hard rock bearing stratum were selected as sites for testing of piles in the present research. A highly weathered mudstone was considered as a soft rock bearing stratum, and a highly weathered argillaceous siltstone was taken as a hard rock bearing stratum into account.    Advances in Civil Engineering rock bearing stratum site, of which 6 were made of ϕ500 (wall thickness, 125 mm) and 2 were made of ϕ400 (wall thickness, 95 mm); the length of the piles was 23-25 m, and the distance between the piles was 7.5 m. Sliding micrometers were preembedded into 4 piles (ϕ500) at this site. In addition, 12 high-strength prestressed concrete pipe piles were arranged on the hard rock bearing stratum site, of which 10 were made of ϕ500 (wall thickness, 125 mm) and 2 were made of ϕ400 (wall thickness, 95 mm); the length of the piles was 22-26 m, and the distance between the piles was 8.0 m. Each test pile consisted of two conjoined pile sections. Besides, 4 piles (ϕ500) at this site contained sliding micrometers.
A geological survey was carried out adjacent to each test pile. e center of the drilling hole was 0.8 m away from the center of the pile hole. e test piles containing the preembedded sliding micrometers were selected to assess the distribution and proportion of friction and the end resistance under various loads, the exertion of friction in different strata, the settlement at the end of pile, and the total compressive deformation of the pile. e field test sites are displayed in Figure 8, and the detailed parameters of each test pile are listed in Tables 3 and 4.

Results of Testing of Piles.
e results of testing of 6 prestressed pipe piles made of ϕ500 (piles no. 1, 2, 3, 4, 5, and 6) at the soft rock bearing stratum site and those of 9 prestressed pipe piles made of ϕ500 (piles no. 1, 2, 3, 4, 7, 8, 10, 11, and 12) at the hard rock bearing stratum site were statistically analyzed. It should be noted that pile #5 failed in the field test, and, thus, its data were not presented. e pile side resistance was obtained based on the results of HSDT and measurements of sliding micrometers. e average pile side resistance for each soil layer is shown in Tables 5 and 6. e resistance and elastic limit displacement of soil at the end of the pile were obtained from the results of HSDT. e resistance, elastic limit displacement, and spring coefficient of soil at the end of each test pile are presented in Tables 7 and 8. e results of the test piles (ϕ500) at the soft (hard) rock bearing stratum site showed that the elastic limit displacement at the end of the pile was 6.9-11.5 mm (7.4-14.4 mm), and the spring coefficient at the end of the pile was 83-282 kN/mm (200-420 kN/mm). e lateral resistance, end resistance, and distribution ratio of four piles (ϕ500) tested with the preembedded sliding micrometers are summarized in Tables 9 and 10. ese results are obtained from the measurements and fitting results of HSDT of piles on the soft rock bearing stratum site and hard one, as shown in Figures 9(a)-10(d), respectively. e curves of the sliding micrometer related to piles at the soft rock bearing stratum site and hard one are illustrated in Figures 11(a)-12(d), respectively. To explain the principle of HSDT that was employed in this research and the results that are presented in Figures 9(a)-10(d), the analysis of pile #1 in Figure 9(a) is taken as an example. e satisfactory agreement between solid line and dashed line (top-left corner subfigure in Figure 9(a)) indicates acceptable results of field testing. e solid line represents the results of field testing, and the dashed line shows the results of numerical simulation. e top-right corner subfigure in Figure 9(a) illustrates the excitation profiles at the end of pile. e bottom-left corner subfigure in Figure 9(a) displays the pile top displacements.
e calculated values of pile side and pile end resistance could be obtained from the bottom-right corner subfigure in Figure 9(a).

Analysis of Test Piles.
Four test piles (ϕ500) that were buried with a preembedded sliding micrometer at both soft and hard rock bearing stratum sites were mechanically simulated, and a numerical algorithm was developed. Four test piles at each site were modeled using the following parameters. e Sato elastic-plastic twofold line model [45] was used for transferring load into the pile side. e ultimate resistance of the pile side was measured by HSDT (Tables 5  and 6). e ultimate displacement s u of the pile side was determined according to the results of systematic testing. e values of the ultimate displacement s u of the pile side in each soil layer are presented in Table 1.
Equations (6) and (8) were used for load transfer at the pile end at the soft rock bearing stratum site and hard one, respectively. e corresponding average values of the spring coefficient k 1 were 166 and 290 kN/mm, respectively, on the basis of the results of HSDT (Tables 7 and 8).
e reduction coefficient η of the spring at the end of the pile on the soft rock bearing stratum site was 0.25, while on the hard rock bearing stratum site, the values of the reduction coefficient of the spring at the pile end were 0.80 (in the second stage) and 0.25 (in the third stage), respectively. Tables 7 and 8 show that the elastic limit displacement of the pile end ranges from 6.9 to 11.5 mm with an average value of 8.7 mm (at soft rock bearing stratum site) and from 7.4 to 14.4 mm with an average value of 10.6 mm (at hard rock bearing stratum site).
In the present study, for piles at the soft rock bearing stratum site, the ultimate displacement s bu of the pile end was the elastic limit displacement of the pile end given by the results of HSDT, and the model predicted the elastic limit displacement of the pile end to be equal to 8.5 mm. For the hard rock bearing stratum site, the limit displacement s bu2 of the pile end for the second stage corresponding to the elastic limit displacement of the pile end could be achieved by the results of HSDT (10.5 mm). e ultimate displacement s bu1 of the pile end for the first stage was within 1.5-2.0 mm that was smaller than that for the second stage. e predicted ultimate displacement s bu1 for the first stage at the pile end was 9.0 mm. e results of numerical simulation and SLTs of four test piles at each site are depicted in Figures 13(a)-14(d). Tables 11 and 12 show the ultimate vertical bearing capacity and the differences between the four methods at both sites. To compare the differences between different methods with the SLTs, the corresponding values were normalized to values of the SLTs (Figure 15).  Advances in Civil Engineering −12.4% (7.9%), respectively. However, the differences between the results of numerical simulation and the SLTs were 9.1% (2.8%), 0% (9.1%), 0% (8.3%), and −7.1% (0%), respectively. It is noteworthy that the calculation of vertical bearing capacity by a standard empirical formula is very conservative and the maximum error reaches −26.9% and −20.0% for piles at the soft rock bearing stratum site and hard one, respectively. Note. e embedment depth of pile #1 into rock is for fully weathered argillaceous siltstone, while that of other test piles is for highly weathered argillaceous siltstone.

Discussion
e existing methods for calculating the vertical bearing capacity of prestressed pipe piles include theoretical analysis, empirical formulas, and in situ testing [22,[52][53][54]. However, developing a theoretical analysis has a problem that is related to difficulty in determining parameters required for calculation; besides, the use of a standard empirical formula has a problem that is related to oversimplification, resulting in conservative outcomes; in situ HSDT cannot fully simulate the vertical bearing capacity of end-bearing pile, while SLT is costly.
Compared with empirical formulas, the numerical parameters used in the proposed method are based on in situ testing data, while the numerical parameters used in empirical formulas are on the basis of empirical values or data retrieved from geotechnical tests; thus, the numerical results seem to be more reasonable in the current study. It is noteworthy that previously reported data related to geotechnical tests cannot truly reflect actual conditions of a site. In this research, the data related to pile side resistance, pile end resistance, and pile end displacement could be collected from HSDT. ese data were then imported into the novel numerical method presented in this research.  Advances in Civil Engineering e proposed numerical method uses the results of HSDT to indicate the pile side ultimate resistance and the pile end spring coefficient, and then, the novel model calculates the vertical ultimate bearing capacity of prestressed concrete pipe piles based on the ultimate displacement of the pile end. is process could avoid errors caused by inadequate excitation of the pile end-bearing capacity in HSDT. In addition, multiple pile observational points were used in this study to achieve more realistic results than HSDT.
us, the developed method is more accurate than the original theoretical method, the empirical formula, and HSDT in calculating the vertical bearing capacity of prestressed pipe piles. e results given by this method provide the same Q-S curves as the SLTs showed the same pile side resistance, pile end resistance, and pile end displacement values as HSDT.
us, the proposed method could combine the advantages of SLT and HSDT and could provide the R b -S curve of pile end resistance and pile end displacement. Overall, there is a great potential to apply the proposed method for prediction of the bearing capacity of prestressed concrete pipe piles and analysis of bearing mechanism of prestressed concrete pipe piles in a rock bearing stratum. e novel numerical method was applied to a typical highly weathered mudstone and a highly weathered  16 Advances in Civil Engineering   argillaceous siltstone bearing stratum. Additional tests need to be carried out to determine parameters required for other typical rock bearing strata or to extract the data from previous high-strain tests, aiming to eventually establish a database. Additional research will also be required to better classify and calculate the mechanical parameters of various soil layers that surround piles.

Conclusions
In this study, a novel numerical method was developed for calculating the vertical bearing capacity of prestressed concrete pipe piles. is model was used to assess the geotechnical behavior of 20 test piles in two typical pile endbearing strata, and the numerical results were compared with those achieved by in situ testing. us, the following conclusions can be drawn: (1) e results of novel numerical method were very similar to those achieved by SLT with error of within 10% compared with HSDTand empirical formulas at both soft rock bearing stratum site and hard one. e high-strain curve fitting data showed that the values of mechanical parameters, such as pile side resistance, pile end resistance, and pile end displacement were more reliable than those provided by the geotechnical testing methods, and therefore, the values of those parameter are of great practical value. (2) High-strain curve fitting can be used to determine the parameters of pile side resistance, pile end resistance, and pile end displacement from testing of prestressed pipe piles. e standard mechanical parameters corresponding to various types of soil layers can be obtained for engineering investigation.
(3) e novel numerical method can be applied with little cost of computational resources, and thus, it can be potentially used for estimating vertical bearing capacity of prestressed pipe piles.
Data Availability e data that support the findings of this study are available upon request from the corresponding author.  Figure 15: Comparison the results between different calculation methods and SLTs (normalized values) at (a) the soft rock bearing stratum site and at (b) the hard rock bearing stratum site. 18 Advances in Civil Engineering