Interpreting Failure-Related Load Transition in Static Tests of PHC Pipe Piles Using a Work-Based Abrupt Change Method

Abstract

This study proposes a work-based interpretation procedure, hereafter referred to as the IDEA method, for identifying the failure-related transition load in monotonic static load tests of pre-stressed high-strength concrete pipe piles. The method was examined using nine full-scale axial compression tests from a site in the lower reaches of the Yangtze River, China. Cumulative work was reconstructed from the measured load settlement curves, and an incremental work response indicator was fitted with a one-break continuous segmented-regression model. The breakpoint was taken as the IDEA estimate, while bootstrap confidence intervals and delta BIC were used to evaluate numerical stability and model support. For the present nine piles, IDEA showed close agreement with the code-interpreted reference loads and yielded the lowest MAPE among the five Q-s interpretation methods considered, whereas the Davisson method showed slightly lower COV and RMSE. Additional perturbation analyses indicated low sensitivity to moderate settlement noise but clear sensitivity to sparse loading records and missing pre-failure points. A preliminary external application to 10 published pile cases showed generally favorable agreement with reference loads reinterpreted from digitized external Q-s curves using a uniform abrupt-settlement criterion. Because the original settlement–time records of the external cases were unavailable, the external assessment is treated as a curve-based transferability check rather than a strictly code-certified validation.

1. Introduction

Pile foundations are slender structural members capable of safely transferring substantial structural loads from the structure to the foundation soil within a limited area [1]. Among various pile foundations, pre-stressed high-strength concrete (PHC) pipe piles have gained widespread application in construction, highways, bridges, and port terminals due to their numerous advantages, including high load-bearing capacity, efficient construction, strong adaptability to soil conditions, and favorable economic benefits [2]. Beyond these widespread engineering applications, PHC pipe piles have also been the subject of field and analytical studies focusing on installation effects, residual forces, load transfer, and soil–pile interaction. Kou et al. investigated the field performance of open-ended PHC pipe piles jacked into clay and highlighted the influence of installation conditions, soil plugging, and load-transfer characteristics on pile behavior [3]. Kou et al. also reported the development of residual forces in PHC pipe piles, showing that post-installation stress redistribution may affect the interpretation of shaft and end resistances in load tests [4]. Yang et al. examined the field behavior of driven pre-stressed high-strength concrete piles in sandy soils [5]. Li et al. monitored PHC pipe piles under hydraulic jacking using FBG sensing technology and showed that installation-induced strain and locked-in force distributions can be captured directly along the pile shaft [6]. In addition, Huang et al. showed that, when soil–pile interaction is considered, PHC pipe piles may also exhibit a significant structural response under cyclic loading, indicating that PHC pile behavior should be interpreted together with both geotechnical and structural mechanisms [7]. Although these studies have substantially improved understanding of PHC pile installation effects, load transfer, and structural behavior, comparatively less attention has been paid to the objective and repeatable interpretation of full-scale static load-test Q-s curves for identifying ultimate bearing capacity. This provides the motivation for the present study.

Pile bearing capacity is regarded as the most crucial parameter in pile foundation design. For a considerable period, determining the bearing capacity of pile foundations has remained a focal point of numerous theoretical and experimental studies within the geotechnical engineering community. The Standard Penetration Test (SPT) is one of the most widely employed in situ testing methods, used to determine soil density and stability and to assist in calculating pile bearing capacity. Schmertmann proposed a systematic methodology capable of integrating static cone penetration test (CPT) and SPT data for calculating the ultimate bearing capacity (UBC) of pile foundations, thereby enabling a scientific assessment of the ultimate state of pile foundations using multiple in situ test parameters [8]. Meyerhof proposed a formula for estimating the bearing capacity of driven piles using SPT blow counts, based on extensive field data, thereby establishing the theoretical foundation for SPT-related methods [9]. This method was subsequently refined by Decourt, who introduced correction factors for different soil layers and pile types to estimate vertical and lateral resistance [10]. Consequently, the adaptability of the SPT methodology to variations in soil layers was enhanced. In their study of the UBC of PHC pipe piles at varying depths, Wei et al. demonstrated the decisive influence of field survey data accuracy on the reliability of results [11]. Huynh et al. analyzed bearing capacity data from 81 PHC bamboo-jointed piles in Vietnam, confirming that traditional empirical methods may significantly underestimate the bearing capacity of pile foundations with knotted structures [12]. Alielahi investigated the bearing capacity of five bored cast-in-place piles at a port engineering project, discovering that most empirical pile bearing capacity formulas based on SPT or CPT indicators tend to underestimate the actual bearing capacity of pile foundations [13]. Jesswein employed three empirical methods based on the SPT to calculate the UBC of steel piles in glacial till formations, finding that the predicted results from the three approaches exhibited significant dispersion [14]. Jiang et al. demonstrated the significant role of field parameter identification in enhancing the design accuracy and construction quality of pipe piles by modifying the empirical coefficients of the SPT empirical formula [15]. Existing research has found that the applicability of traditional SPT methods is constrained by diverse geological conditions and specific pile foundation types [16]. Because these SPT-based approaches serve mainly as design-stage estimation tools rather than direct interpretations of static load-test response, they are treated in this study as contextual design-stage references rather than equivalent methodological benchmarks.

Given the reliability of the static load test (SLT), the identification and analysis of the load–settlement (Q-s) curve has progressively become the focal point in assessing the UBC of pile foundations. Various mathematical models and graphical methods have been employed to analyze Q-s curves and to investigate the bearing characteristics of pile foundations. Hansen resolved the issue of subjective load capacity determination in engineering when clear inflection points were absent by defining the geometric response logic between load and displacement [17]. Chin hypothesized that when pile foundations approach their ultimate limit state during loading, their Q-S curve takes the form of a hyperbola [18,19]. The Corps of Engineers method proposed by the U.S. Army Corps of Engineers (USACE) is a typical graphical discrimination method, frequently used in engineering practice for rapid interpretation and verification of test results [20]. Davisson proposed a criterion based on the elastic compression of the pile body to determine the UBC of driven precast piles, with the resulting value typically regarded as the safe lower limit of bearing capacity [21]. Adel and Shakir analyzed SLT data from 13 drilled cast-in-place piles, finding that the Chin method and Decourt method generally yielded excessively high critical values; conversely, the Davisson method produced more conservative results [22]. Vural et al. indicated that the deviation in Chin’s method stems from excessive iteration of Q-s curve data, proposing instead a logarithmic coordinate assessment model based on the parabolic region of the Q-s curve [23]. Yang employed a sharp-point mutation model to analyze the Q-s curve of pile foundations, thereby providing a novel applied theory and computational method for determining the UBC of pile foundations [24]. Most existing Q-s interpretation methods still rely on prescribed curve geometry, offset rules, or graphical criteria, and objective breakpoint identification with quantified uncertainty remains limited.

In recent years, there has been a growing recognition that identifying pile foundation damage involves not only observing sudden displacement increases but also detecting changes in the internal energy state of the pile–soil system. These factors are regarded as key elements in assessing structural integrity. Methods such as energy dissipation, entropy increase and damage mechanics theory have been introduced to describe the failure process at the pile–soil interface [25]. Jia et al. investigated the dynamic behavior of soil under complex loading conditions, discovering that the energy distribution during the loading process exhibits distinct phased characteristics, with the energy dissipation rate increasing nonlinearly as damage accumulates [26]. In their study on pile foundations for bridge clusters, Wang et al. quantified the proportional relationship between displacement surge and energy dissipation, indicating that the failure process of pile foundations is accompanied by a dramatic transformation of mechanical energy into plastic dissipation energy [27]. Jiang et al. demonstrated the detrimental effect of material damage within the pile body on the overall stability of pile foundations, emphasizing the necessity of incorporating damage variables in the assessment of pile foundation failure [28]. Xie et al. demonstrated through investigations of structural clay in the Zhan jiang region that the energy dissipation rate can sensitively reflect the damage state of the soil’s micro-structure [29].

Despite substantial progress in the study of pile bearing characteristics, existing work remains focused mainly on empirical estimation methods and geometric interpretation of Q-s curves. SPT-based empirical formulas are useful at the design stage, but they are not equivalent to post-test interpretation methods based on measured load–settlement responses. Meanwhile, most established Q-s interpretation methods still depend on prescribed curve geometry or graphical criteria, and objective breakpoint identification with quantified uncertainty has been less frequently addressed. Accordingly, the present study does not aim to claim a universally superior capacity-prediction model. Instead, it examines whether a work-based breakpoint-identification procedure can provide an objective and reproducible interpretation of failure-related load transition in PHC pile static load tests. Based on nine full-scale PHC piles from one site, four SPT-based empirical methods are treated only as design-stage references, whereas four established Q-s interpretation methods are used as the primary methodological comparators. The proposed IDEA procedure is then examined through segmented regression, bootstrap uncertainty analysis, and sensitivity tests on measurement noise and data density. Finally, a preliminary external check is conducted on a common 10-case subset of published PHC pile Q-s curves. Within this scope, the study focuses on transferability, numerical stability, and comparative performance under the available PHC datasets, while leaving broader generalization to future work.

2. Materials and Methods

2.1. Ground Conditions

The in situ test was conducted at a project site in Tongling City, Anhui Province, China. The terrain of the site is largely flat, characterized by a mixed distribution of ponds, farmland and roads. The test piles used in this study are PHC pipe piles with cross-shaped steel tips. The material and structural characteristics of the test piles are summarized in

Table 1. Geometric, material, and structural properties of the PHC piles used in this study.

Test Pile No.	Length (m)	Outer Diameter (mm)	Wall Thickness (mm)	Concrete Grade	Elastic Modulus (GPa)	Compressive Pre-Stress (MPa)	Design Bearing Capacity Ra (kN)
1#	18.25	400	95	C80	38	6	700
2#	19.25	400	95	C80	38	6	700
3#	21.00	500	100	C80	38	6	1220
4#	22.00	500	100	C80	38	6	1220
5#	28.00	500	100	C80	38	6	1220
6#	28.00	500	100	C80	38	6	1220
7#	30.82	500	100	C80	38	6	1400
8#	27.00	500	100	C80	38	6	1200
9#	31.00	500	100	C80	38	6	1400

. As shown in Figure 1, the soil stratification in the test pile area, from top to bottom, are as follows: ① silty clay (locally silty clayey), ② fine sand (slightly dense), ③ fine sand (medium-dense), ④ silty clay, ⑤ rounded gravel, ⑥ strongly weathered muddy sandstone.

Standard penetration tests (SPTs) were conducted in Layer ① (silty clay/mucky silty clay), Layer ② (slightly dense fine sand), and Layer ③ (medium-dense fine sand) in accordance with GB/T 50123-1999 [30]. The UBC of pile foundations may be assessed using empirical formulae based on SPT test results. In accordance with the requirements of the Technical Specification for Testing of Building Foundation Piles (JGJ106-2014) [31], 9 test piles from this project were selected for single-pile vertical compression static load tests (SLT). The slow-load maintenance method was employed, with an incremental loading set at 1/10th of the maximum loading capacity. The first-stage loading was set at twice the incremental loading value. According to JGJ 106-2014, loading shall be terminated when any of the termination conditions specified in the code are met.

The SLT aims to investigate the bearing characteristics of PHC pipe piles at different embedment depths within the bearing layer, providing foundational data for subsequent determination of pile foundation bearing capacity using mathematical modeling. In this study, due to the length of the piles, the piles themselves are only embedded in the first three soil layers (① silty clay/mucky silty clay, ② slightly dense fine sand and ③ medium-dense fine sand). As shown in

Table 2. Basic pile parameters and idealized ground conditions used for the empirical reference calculations.

Test Pile No.	Ground Conditions	Soil Thickness (m)	Na(avg)	qsik	qpk
				(kPa)	(kPa)
1#	Silty clay	0.00–1.60	5.5	55	*
	Fine sand (slightly dense)	1.60–7.80	10.6	26	*
	Fine sand (medium-dense)	7.80–30.20	15.5	48	4400
2#	Silty clay	0.00–1.70	5.5	55	*
	Fine sand (slightly dense)	1.70–6.2	10.6	26	*
	Fine sand (medium-dense)	6.20–31.40	15.5	48	4400
3#	Silty clay	0.00–0.50	5.5	55	*
	Fine sand (slightly dense)	0.50–6.40	10.6	26	*
	Fine sand (medium-dense)	6.40–32.30	15.5	48	4400
4#	Silty clay	0.00–1.90	5.5	55	*
	Fine sand (slightly dense)	1.90–8.50	10.6	26	*
	Fine sand (medium-dense)	8.50–31.70	15.5	48	4400
5#	mucky silty clay	0.00–2.80	1.5	25	*
	Fine sand (slightly dense)	2.80–9.50	10.6	26	*
	Fine sand (medium-dense)	9.50–31.50	15.5	48	4400
6#	Silty clay	0.00–3.80	5.5	55	*
	Fine sand (slightly dense)	3.80–11.50	10.6	26	*
	Fine sand (medium-dense)	11.50–31.40	15.5	48	4400
7#	Silty clay	0.00–3.50	5.5	55	*
	Fine sand (slightly dense)	3.50–11.00	10.6	26	*
	Fine sand (medium-dense)	11.00–31.00	15.5	48	5200
8#	Silty clay	0.00–1.50	5.5	55	*
	Fine sand (slightly dense)	1.50–6.60	10.6	26	*
	Fine sand (medium-dense)	6.60–31.30	15.5	48	4400
9#	Silty clay	0.00–3.40	5.5	55	*
	Fine sand (slightly dense)	3.40–10.60	10.6	26	*
	Fine sand (medium-dense)	10.60–32.00	15.5	48	5200

Note: N_a(avg) denotes the average SPT blow count for each idealized stratum, calculated from all boreholes within that same stratum across the site. It does not represent an average taken across different soil types or distinct geological sub-layers. “*” indicates that the corresponding soil layer is not the pile-tip bearing layer; therefore, qpk is not used in the calculation. In this study, the pile tip is located in the third soil layer, and only the qpk value of this layer is used in the calculation.

, for the empirical reference calculations, the subsurface profile has been simplified into three major idealized strata, representing the primary soil categories influencing pile response. Each N_a(average) value denotes the mean standard penetration test (SPT) blow count obtained from all boreholes within the same idealized stratum, under identical soil types and engineering characteristics.

2.2. Methods

2.2.1. SPT-Based Empirical Methods as Design-Stage References

• JGJ94-2008 method [32]. This method calculates the ultimate bearing capacity Q_uk of pile foundations based on their geometric dimensions and soil properties, using the following formula:

  $$Q_{\mathit{uk}}=Q_{\mathit{sk}}+Q_{\mathit{pk}}=\mathit{u}{\displaystyle \sum _{i=1}^n{q}_{\mathit{sik}}{l}_i+q_{\mathit{pk}}{A}_p}$$

  (1)

  where Q_sk denotes the ultimate lateral friction resistance (kN); Q_pk denotes the ultimate pile tip resistance (kN); q_sik denotes the standard value of ultimate lateral resistance for each soil layer (kPa); q_pk denotes the standard value of ultimate tip resistance (kPa); u denotes the pile circumference (m); A_p denotes the pile tip area (m²); l_i denotes the length of contact between the pile foundation and soil layer i (m).
• Meyerhof method [9]. Meyerhof proposed an empirical correlation between SPT-N and pile bearing capacity, which effectively estimates the ultimate bearing capacity Q_uk of pile foundations. The specific formula is as follows:

  $$Q_{\mathit{uk}}=q_{\mathit{pk}}{A}_p+u{\displaystyle \sum _{i=1}^n{q}_{\mathit{sik}}{l}_i}$$

  (2)

  $$q_{\mathit{pk}}=a{N}_b$$

  (3)

  $$q_{\mathit{sik}}=b{N}_{\mathit{si}}$$

  (4)

  where q_pk denotes the unit ultimate resistance at the pile tip (MPa); q_sik denotes the unit ultimate lateral resistance of the soil in the i-th layer (MPa); u denotes the pile circumference (m); l_i denotes the length of contact between the pile and the soil in the i-th layer (m); A_p denotes the pile tip area (m²); N_b represents the average SPT value measured within the range of 10 D above the pile tip to 5 D below; N_si is the average N value for the i-th soil layer where the pile foundation is located. The bearing layer at the pile tip in this study is sandy soil. Calculations are performed as for driven piles, with parameters set as a = 0.4 and b = 0.002.
• Decourt method [10]. Decourt estimates the end resistance and lateral resistance of pile foundations by introducing specific empirical coefficients, adjusting their values according to different soil types and pile configurations. This enables the calculation of the ultimate bearing capacity Q_uk, with the specific formula as follows:

  $$Q_{\mathit{uk}}=q_{\mathit{pk}}{A}_p+u{\displaystyle \sum _{i=1}^n{q}_{\mathit{sik}}{l}_i}$$

  (5)

  $$q_{\mathit{pk}}=k_b{N}_b$$

  (6)

  $$q_{\mathit{si}}=\alpha \left[2.8N_{\mathit{si}}+10\right]$$

  (7)

  where q_pk denotes the unit ultimate bearing capacity at the pile tip (MPa); q_sik denotes the unit ultimate lateral bearing capacity of the soil layer at position i (MPa); u denotes the pile circumference (m); l_i denotes the length of soil contact between the pile and layer i (m); A_p denotes the pile tip area (m²); N_b and N_si represent the average SPT values (N) measured at the pile tip and the i-th soil layer, respectively; for fine-grained soils, α is taken as 1.0, with k_b set to 0.10 for driven piles and 0.08 for cast-in-place piles; and for coarse-grained soils, α = 0.5–0.6 (with an average value of 0.55 adopted in this study) and k_b = 0.325.
• Schmertmann method [8]. Schmertmann employs the N-value to determine the ultimate end bearing capacity and ultimate axial friction, as shown in

  Table 3. Schmertmann method.

  Soil Type	qc/N	Friction Ratio (%)	Ultimate End Bearing Capacity (kPa)	Ultimate Shaft Friction Capacity (kPa)
  Clean sand with various densities	0.3745	0.60	342.4 × Nb	2.03 × Ni(avg)
  Mixed sand with clay and silt Plastic clay	0.2140	2.00	171.2 × Nb	4.28 × Ni(avg)
  Plastic clay	0.1070	5.00	74.9 × Nb	5.35 × Ni(avg)
  Shell-containing sand and soft argillaceous limestone	0.4280	0.25	385.2 × Nb	1.07 × Ni(avg)

  , where qc/N represents the ratio of tip resistance to standard penetration value; N_b denotes the average N value within the range extending 3 D above the pile tip to 3 D below.

2.2.2. Established Q-S Curve Interpretation Methods

• Chin method [18,19]. Chin hypothesized that the failure stage of pile foundation Q-s curves exhibits hyperbolic characteristics. The graph is redrawn with the pile head settlement value s as the horizontal axis and the ratio of the settlement value to the corresponding load as the vertical axis. We perform a linear regression on the last three loading points, defining the reciprocal of the slope of the fitted line as the UBC of the pile foundation, denoted as Q_uk. The specific mathematical expression is as follows:

  $${\displaystyle \frac{s_i}{P_i}}=m{s}_i+c$$

  (8)

  $$Q_{\mathit{uk}}={\displaystyle \frac{1}{m}}$$

  (9)

  where P_i denotes the i-th level of vertical load (kN) applied to the pile cap; s_i represents the settlement (mm) corresponding to the i-th level of vertical load; m is the slope of the fitted straight line; c is the intercept of the fitted line on the vertical axis.
• Corps of Engineers method [20]. In this method, the Q-s curve is drawn first, and then uses three distinct criteria—displacement control, slope control, and inflection point control—to obtain three representative bearing capacity values. The first load (Q₁) is the load corresponding to the 6.4 mm settlement level. We draw tangents to the initial segment and final segment of the curve, respectively. The load corresponding to the intersection point of these two tangents is the second one (Q₂). The third one (Q₃) is the load corresponding to the point where the line makes an angle of 0.025 mm/kN intersecting with the Q-s curve. The average of these three load values is considered as the ultimate bearing capacity Q_uk of the pile.

  $$Q_{\mathit{uk}}={\displaystyle \frac{Q_1+Q_2+Q_3}{3}}$$

  (10)
• Davisson method [21]. In this method, an offset curve is plotted in the Q-s coordinate system, and the corresponding load at the intersection point of the Q-s curve and offset curve is defined as the ultimate bearing capacity Q_uk The formula for determining the offset curve S is as follows:

  $$S={\displaystyle \frac{\mathit{QL}}{\mathit{AE}}}+\left(3.8+{\displaystyle \frac{D}{120}}\right)$$

  (11)

  where Q denotes the vertical load applied to the pile cap (kN); A denotes the cross-sectional area of the pile body (m²); E denotes the elastic modulus of the pile foundation material (kPa); L denotes the pile length (m); D denotes the diameter of the pile foundation (mm).
• Hansen method [17]. The Hansen method transforms each measured load Q_i and its corresponding settlement s_i into transformed quantities y_i. Subsequently, using s_i as the independent variable and y_i as the dependent variable, a linear regression is performed on the final two measured data points via the least squares method. The slope a and intercept b of the fitted straight line are extracted, and the regression coefficients are utilized to determine the ultimate bearing capacity Q_uk of the pile foundation. The specific mathematical expression is as follows:

  $$y_i={\displaystyle \frac{\sqrt{s_i}}{Q_i}}$$

  (12)

  $$y_i=\mathit{as}+b$$

  (13)

  $$Q_{\mathit{uk}}={\displaystyle \frac{1}{\sqrt{\mathit{ab}}}}$$

  (14)

2.2.3. Segmented-Regression-Based Idea Method

Theoretical Basis

As illustrated conceptually in Figure 2, the response of the pile–soil system during loading may be interpreted from a work-based perspective. Under monotonic static loading, the external load Q acting through the corresponding settlement increment produces external work W, which represents the total mechanical work input to the pile–soil system along the load–displacement path. This quantity reflects the overall deformation and resistance mobilization process of the system.

It should be emphasized that the proposed IDEA indicator is a work-based response indicator derived from the measured monotonic load–settlement path. Because no unloading branch or independent strain-energy measurement is available in the present static load tests, the cumulative work cannot be rigorously decomposed into recoverable elastic energy and irreversible plastic dissipation. Therefore, the identified breakpoint should not be interpreted as a strict mechanical threshold or as a complete damage variable. Instead, it represents a reproducible response transition point at which the measured Q-s behavior changes from relatively stable deformation to a more pronounced nonlinear deformation regime.

This interpretation also implies that the method does not fully capture path-dependent soil behavior, cyclic degradation, or highly nonlinear pile–soil interaction mechanisms that cannot be inferred from a monotonic Q-s curve alone. The IDEA method is therefore used here as a physically motivated interpretation criterion for failure-related transition loads, rather than as a universal mechanistic capacity model.

From this perspective, the evolution of the incremental work per unit load response can still provide useful information on the transition of the pile–soil system from relatively stable deformation to pronounced nonlinear deformation. At low load levels, the incremental response to a unit load increase remains relatively small, indicating that the system behavior is mainly governed by pile compression and small-strain soil deformation. As the load increases, progressive mobilization of shaft resistance and local inelastic development in the surrounding soil lead to a gradual increase in the response indicator. When the load approaches the failure-related stage, the response increases more abruptly, indicating the onset of rapid nonlinear deformation and more intense pile–soil interaction.

In this sense, the IDEA method is used as a physically motivated, work-based interpretation criterion for identifying failure-related transition loads, rather than as a strict decomposition of elastic and plastic energy components. In the present study, the work-based indicator is introduced as a response-sensitive scalar derived from the measured load–settlement path. It reflects the rate at which external work accumulates with increasing load and is used to detect changes in the deformation regime of the tested pile system. This interpretation is physically motivated because abrupt changes in the measured Q-s response are expected to be accompanied by abrupt changes in work accumulation. However, the indicator should not be interpreted as a strict decomposition of elastic strain energy and irreversible dissipation, nor as a complete damage variable. Its role in the present paper is more limited: it serves as a reproducible transition indicator for identifying failure-related changes in monotonic static tests.

Computational Procedure

First, based on Q-s curve data, the cumulative external work W during loading is calculated using the trapezoidal area integration method [33]. Let s_i denote the total settlement corresponding to the i-th load level Q_i. The calculated W_i represents the total work done by external forces on the pile–soil system up to the i-th load level. The specific formula is as follows:

$$W_i={\displaystyle \sum _{i=1}^n\frac{Q_i+Q_{i-1}}{2}}\left(s_i-s_{i-1}\right)$$

(15)

Second, a work-normalized incremental response indicator is calculated from the cumulative work curve. For consistency with the previous notation, this indicator is denoted as IDER in the present study, but its physical meaning is clarified here as the incremental work accumulated per unit load increment. Because ΔW_i is the increment of external work and ΔQ_i is the corresponding load increment, IDER_i = ΔW_i/ΔQ_i has the dimension of displacement when W_i is expressed in kN·mm and Q_i is expressed in kN. Therefore, IDER should not be interpreted as an energy quantity, a direct stiffness-degradation variable, or a strict measure of plastic dissipation. Instead, it is used as a scalar response indicator that amplifies changes in the measured load–settlement path and helps to identify abrupt changes in the deformation regime of the tested pile system. The specific formula is as follows:

$${IDER}_i={\displaystyle \frac{\Delta W_i}{\Delta Q_i}}$$

(16)

where ΔW_i denotes the incremental external work between two adjacent loading stages, and ΔQ_i denotes the corresponding load increment. The IDER value therefore represents the incremental work per unit load increment, with the dimension of displacement.

Last, an automatic breakpoint-identification procedure was introduced in this study. For each test pile, the IDER-Q data were fitted using a one-break continuous segmented linear regression model:

$$IDER(Q)={\beta }_0+{\beta }_{1}Q+{\beta }_2{(Q-\psi )}_{+}$$

(17)

where (Q − ψ)₊ = max(Q − ψ, 0), β₀ is the intercept, β₁ is the pre-breakpoint slope, β₂ represents the slope increment after the breakpoint, and ψ is the unknown breakpoint. The slope before the breakpoint is β₁, whereas the slope after the breakpoint is β₁ + β₂. The breakpoint ψ was taken as the IDEA-based failure-related transition load.

The breakpoint ψ was estimated by minimizing the residual sum of squares of the segmented-regression model under minimum-point constraints on both sides of the breakpoint. In this study, at least three data points were required before the breakpoint and at least two data points after the breakpoint. This formulation eliminates manual selection of linear segments and allows the breakpoint to be identified in a reproducible and programmable manner.

To quantify the uncertainty of the breakpoint estimate, a residual bootstrap procedure with 1000 resamples was performed for each pile. The breakpoint was then re-estimated, and the 95% confidence interval (CI) of Ψ was obtained from the 2.5th and 97.5th percentiles of the bootstrap distribution. To evaluate whether the one-break segmented model was statistically more appropriate than a single linear model, the Bayesian information criterion (BIC) was calculated for both models. A larger positive value of ΔBIC = BIClinear − BICone-break indicates stronger support for the segmented model.

In the present study, all 9 PHC test piles were loaded to structural failure during the static loading tests. Therefore, the abrupt transition identified from the IDER-Q relationship is interpreted as a failure-related breakpoint in the measured load–settlement response, and for the piles considered here, this breakpoint corresponds to the structural failure stage of the tested pile system. Accordingly, the segmented-regression-based IDEA method is used to identify the failure-related transition point in the static load-test response, rather than to claim a purely geotechnical ultimate capacity of the surrounding soil alone.

The present method is an interpretation tool for measured monotonic Q-s curves rather than a design formula independent of test data. Therefore, the reference loads used in the present comparison should be understood as benchmark interpretations from the same static load tests according to JGJ 106-2014, rather than as external ground truth in a strict predictive sense. In this regard, close agreement with the reference interpretation indicates methodological consistency and practical competitiveness, but it does not by itself prove universal superiority. The method is expected to be most reliable when the loading record contains sufficiently dense stages around the transition region and extends into the failure-related response branch. Its applicability to incomplete tests terminated before a clear transition, to highly sparse loading records, or to pile types and soil conditions beyond the presently examined PHC cases remains to be established. The code implementing the IDEA workflow is available from the corresponding author upon reasonable request.

Representative Example

Taking Test Pile No. 1^# as an example,

Table 4. Variations in IDER values for Test Pile 1^# calculated using the proposed method.

Loading Steps	Load (kN)	Settlement (mm)	∆Q (kN)	$\frac{({\mathbf{Q}}_{\mathbf{i}}+{\mathbf{Q}}_{\mathbf{i}-1})}{2\times ({\mathbf{s}}_{\mathbf{i}}-{\mathbf{s}}_{\mathbf{i}-1})}$= ∆W (kN·mm)	ΣWi (kN·mm)	IDER = ΔW/ΔQ (mm)
-	0	0	-	-	-	0
1	296	0.82	296	121.36	121.36	0.41
2	444	1.34	148	192.4	313.76	1.3
3	592	2.02	148	352.24	544.64	2.38
4	740	2.8	148	519.48	871.72	3.51
5	888	3.62	148	667.48	1186.96	4.51
6	1036	4.46	148	808.08	1475.56	5.46
7	1184	5.98	148	1687.2	2495.28	11.4
8	1332	8.9	148	3673.36	5360.56	24.82
9	1480	42.64	148	47,438.44	51,111.8	320.53

lists the ΔW and IDER values between each loading stage. Based on these, the IDER response for each stage of this pile was calculated. As shown in Table 4, when the load increased from 296 kN to 1036 kN, the IDER value gradually rose from 0.41 to 5.46. Upon reaching 1184 kN, the IDER value increased from 5.46 to 11.4, subsequently rising to 24.82 at 1332 kN, before surging sharply to 320.53 at 1480 kN. The aforementioned process may be summarized as follows: As the load increases, the IDER value first changes gradually, indicating a relatively stable response of the tested pile system. When the load approaches the failure-related stage, the IDER value rises much more rapidly, reflecting an abrupt change in the measured load–settlement response. Following preliminary inspection of the IDER evolution, the failure-related transition interval for Test Pile No. 1^# was judged to lie approximately between 1184 and 1480 kN. The final transition load was then identified objectively using the segmented-regression procedure.

To determine the IDEA-based failure-related transition load, the complete IDER-Q dataset was subsequently fitted using the one-break continuous segmented-regression model described in the Computational Procedure section. As shown in Figure 3, the estimated breakpoint was ψ = 1324.72 kN, which falls within the preliminarily identified interval and was therefore taken as the IDEA-based failure-related transition load of Test Pile 1^#. The corresponding 95% confidence interval was 1322.99–1326.28 kN. In addition, the pre-breakpoint and post-breakpoint slopes (k₁ and k₂) were 0.0105 and 1.9980, respectively, indicating an abrupt increase in the work-based response indicator after the breakpoint. The large ΔBIC value of 71.21 further indicates that the segmented model is strongly preferred over a single-line model.

2.2.4. Statistical Metrics for Method Comparison

For consistent comparison among the Q-s curve interpretation methods, statistical indicators including the mean ratio (MR), coefficient of variation (COV), mean absolute percentage error (MAPE), root mean square error (RMSE), and mean bias (MB) were adopted.

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{Q_{pred,i}-Q_{r,i}}{Q_{m,i}}\right|}\times 100\%$$

(18)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(Q_{pred,i}-Q_{r,i}\right)}^2}}$$

(19)

$$r_i={\displaystyle \frac{Q_{pred,i}}{Q_{r,i}}},$$

(20)

$$\mathrm{COV}={\displaystyle \frac{\mathrm{SD}(r_i)}{{\overline{r}}_i}}\times 100\%$$

(21)

$$\mathrm{MR}={\displaystyle \frac{1}{n}}{\displaystyle \sum \frac{Q_{pred}}{Q_m}}$$

(22)

$$\mathrm{MB}={\displaystyle \frac{1}{n}}{\displaystyle \sum (Q_{pred}-Q_m)}$$

(23)

where n is the total number of test piles, Q_pred,i denotes the predicted ultimate bearing capacity of the i-th test pile derived from the target method, Q_m,i was interpreted from the SLT according to JGJ106-2014, and SD(r_i) and $\overline{r}$_i are the standard deviation and arithmetic mean of the ratio series {r_i}.

3. Results and Discussion

3.1. Results of Vertical Static Load Tests for Single Piles

Figure 4 displays the Q-s curves obtained from the static load tests on the test piles. The reference capacities Q_m interpreted from the same static load tests according to JGJ 106-2014 for each pile are marked on the graph [31]. In the present study, Q_m denotes the reference load interpreted from the in-house static load-test records according to the single-pile vertical compressive bearing-capacity determination principle in JGJ 106-2014 [31]. Specifically, when the pile-head settlement under a given loading stage was greater than twice the settlement under the previous loading stage and the settlement did not reach the relative stability criterion within 24 h, the previous loading stage was adopted as the reference bearing capacity. This abrupt-settlement criterion was used to determine the Q_m values marked in Figure 4. Therefore, Q_m is used in this study as a code-interpreted reference load from the same static load-test record, rather than as an independently measured ground-truth capacity.

It should be emphasized that all nine test piles were loaded to structural failure during the static load tests. Therefore, the abrupt settlement increases observed in Figure 4 are interpreted in this study as failure-related responses associated with the structural limit state of the PHC pile sections, rather than purely geotechnical failure of the surrounding soil. In this sense, the ultimate loads discussed in this study correspond to the failure-related load levels recorded in the static load tests of the tested PHC pile system. Accordingly, the subsequent interpretation methods, including the IDEA procedure, are used here to identify the load level associated with the observed failure transition in the measured Q-s response.

3.2. Results of SPT-Based Empirical Methods (Engineering Reference Comparison)

It should be noted that the SPT-based empirical methods considered in this section are not used as independent validation benchmarks for the proposed IDEA method. These methods rely mainly on soil-index parameters and empirical design assumptions, whereas IDEA is an interpretation procedure based on the measured static load-test Q-s response. Therefore, the empirical methods are retained only as design-stage engineering references, providing a practical background for comparison with commonly used preliminary estimation approaches.

The application of the JGJ94-2008, Meyerhof, Decourt and Schmertmann methods to the test pile is shown in Figure 5 [8,9,10,32].

This study employs four empirical formula methods—JGJ94-2008, Meyerhof, Decourt, and Schmertmann—to calculate the UBC of individual piles [8,9,10,32]. The comparison results between the calculated values obtained using different empirical formula methods for each test pile and the Q_m are shown in Figure 6. The ratio values generally fall within the range of 0.698 to 1.546, indicating that the applicability of empirical formula methods varies across different test piles. From an overall perspective, the JGJ94-2008 method exhibited a pronounced tendency to overestimate in this study, with the calculated values for all nine test piles being no less than the Q_m. The results obtained from the Meyerhof, Decourt and Schmertmann methods exhibit a degree of dispersion across different test piles, with instances of both overestimation and underestimation occurring. Test Piles 3^# to 5^# exhibited varying degrees of overestimation across all four methods (ratios ranging from approximately 1.195 to 1.546), falling within a common overestimation range across all methodologies. For Test Piles 8^# and 9^#, the JGJ94-2008 method yielded values close to the measured results (ratios of approximately 1.028 to 1.039), whilst the other three methods exhibited relatively conservative prediction characteristics (ratios of approximately 0.696 to 0.891).

Calculation results obtained using the JGJ94-2008 method indicate that this approach fails to adequately account for the weakening effect of soft soil layers (specifically silty clay in this study) on the lateral resistance of pile foundations. Consequently, the calculated parameters are excessively high, leading to results that are generally higher than the Q_m from SLT [34,35,36]. Calculations using the Meyerhof method indicate that the fixed empirical parameters employed to determine pile-side resistance and pile-end resistance struggle to reflect the influence of soil layer conditions on bearing capacity contribution as depth varies [37]. Calculations using the Decourt method indicate that the predictive accuracy of this formula diminishes with increasing pile length and diameter [13,37]; the simplified linear empirical model parameters employed by this method are not adjusted for depth or soil layer conditions, making it difficult to adequately reflect variations in the contribution of lateral resistance from the deep pile section [38,39]. Calculations using the Schmertmann method indicate that as pile diameter increases, the degree of soil disturbance changes. Consequently, when employing original empirical coefficients to calculate lateral resistance for large-diameter piles, bearing capacity is overestimated. When pile length exceeds 25 m and penetrates the bearing layer (medium-dense fine sand layer), this method begins to yield markedly conservative results [40].

Although empirical formulas can simplify calculation procedures during the initial stages of engineering projects, their predictive accuracy is significantly influenced by pile length, pile diameter, and soil displacement effects [41,42]. In practical engineering applications, relying on such static estimates as the basis for design would make it difficult to strike a balance between engineering safety and economic efficiency [43]. This study retains these empirical methods as a reference for the contextual design phase, which is widely used in practice, rather than as an equivalent methodological benchmark for the proposed SLT-based interpretation of IDEA.

3.3. Results of Q-S Curve Interpretation Methods (Primary Methodological Comparison)

The application of the Chin, Davisson, Corps of Engineers and Hansen methods to the test pile is shown in Figure 7 [17,18,19,20,21].

This study applied four established Q-s interpretation methods to the nine pile load–settlement curves: two mathematical-model-based approaches, namely the Chin and Davisson methods, and two graphical approaches, namely the Corps of Engineers and Hansen methods. Figure 8 shows the comparison results between the values obtained from different mathematical modeling methods and graphical methods and the Q_m of each test pile. The ratio values generally fall within the range of 0.894 to 1.331, indicating that different curve-fitting criteria yield inconsistent determinations of the limit point. From an overall perspective, the Davisson method demonstrated the highest correlation with Q_m and the lowest dispersion, with a ratio ranging from approximately 0.934 to 1.054. The majority of test piles yielded values close to 1. The prediction results of the Corps of Engineers method exhibit a degree of conservatism, with ratios ranging from 0.894 to 1.033. By contrast, the Chin method exhibited a more pronounced overestimation in this study, with ratios ranging from approximately 1.115 to 1.331. The Hansen method also showed an overall tendency towards larger values, though to a lesser extent than the Chin method, with ratios ranging from approximately 1.121 to 1.200.

Compared to empirical formulas, mathematical models and graphical methods can more intuitively represent the actual stress response within the pile–soil system [44]. Among these models, the Davisson method yielded results most closely aligned with the Q_m, owing to the stability of typical high-strength concrete material parameters in PHC pipe piles [45]. The Chin and Hansen methods exhibit a pronounced tendency to overestimate, meaning they cannot provide effective engineering warnings [44,46,47]. The Corps of Engineers method tends to yield somewhat conservative predictions under conditions where the initial portion of the curve exhibits greater stiffness and settlement progresses at a slower rate, resulting in ultimate bearing capacities slightly lower than Q_m [44,48]. However, these mathematical modeling approaches and graphical methods are fundamentally based on a posteriori analysis of the geometric form of Q-s curves, with their accuracy being highly dependent on the mathematical integrity of the loading process and the degree of idealization of the curve morphology. These limitations motivate the exploration of additional interpretation criteria that may complement existing geometry-based approaches [49].

3.4. Results of IDEA Method and Statistical Comparison of Q-s Curve Interpretation Methods on the Present Datasets

It should be noted that the objective segmented-regression procedure was applied uniformly to all piles without manual adjustment of the breakpoint location. Figure 9 demonstrates the application of the proposed IDEA procedure for identifying failure-related transition loads in the test piles.

As summarized in

Table 5. Summary of segmented-regression-based IDEA interpretation results for the 9 test piles.

Test Pile NO.	Qm (kN)	${\mathit{Q}}_{\mathit{u}\mathit{k}}^{\mathit{I}\mathit{D}\mathit{E}\mathit{A}}$ (kN)	95% CI (kN)	${\mathit{Q}}_{\mathit{u}\mathit{k}}^{\mathit{I}\mathit{D}\mathit{E}\mathit{A}}-{\mathit{Q}}_{\mathit{m}}$ (kN)	Relative Error (%)	k1	k2	∆BIC	Relative CI Width (%)
1#	1332	1324.72	1322.99–1326.28	−7.28	0.55	0.0105	1.998	71.2146	0.25
2#	1332	1329.21	1328.06–1330.48	−2.79	0.21	0.0143	1.9357	76.1245	0.18
3#	1464	1454.53	1450.02–1459.1	−9.47	0.65	0.0115	0.7963	42.4725	0.62
4#	1708	1685.05	1680.16–1690.49	−22.95	1.34	0.0132	0.9397	48.022	0.61
5#	1708	1699.63	1698.72–1700.53	−8.37	0.49	0.005	1.3138	71.1631	0.11
6#	2300	2449.96	2446.99–2452.14	149.96	6.52	0.01	1.1483	93.3877	0.21
7#	2240	2443.65	2441.86–2445.52	203.65	9.09	0.0113	1.3619	84.6018	0.15
8#	2684	2656.92	2655.45–2657.99	−27.08	1.01	0.0086	1.9947	103.5901	0.10
9#	3080	3077.83	3076.67–3078.79	−2.17	0.07	0.0073	2.1206	107.1138	0.07

and illustrated in Figure 10a, the IDEA-estimated capacities were generally close to the reference values for most piles, whereas larger positive deviations were observed for Piles 6^# and 7^#. Figure 10b shows that most piles cluster close to the 1:1 line, while Piles 6^# and 7^# lie above it, indicating overestimation relative to the reference capacities.

As shown in Figure 11a, all ∆BIC values were substantially larger than 10, indicating strong support for the segmented model. Meanwhile, Figure 11b shows that the relative widths of the 95% confidence intervals were all below 0.62%, confirming the numerical stability of the estimated breakpoints. The marked increase from k₁ to k₂ across all piles confirms that the estimated breakpoint corresponds to a substantial change in the incremental work per unit load response.

To ensure methodological comparability, the primary statistical comparison was restricted to methods that infer ultimate capacity from the same Q-s curves, namely the Chin, Davisson, Corps of Engineers, Hansen, and IDEA methods. The empirical formula based on SPT is retained solely as a reference during the conceptual design phase; consequently, it has not been included in the main COV/MAPE/RMSE comparisons, as its input data and methodological objectives differ fundamentally from those of the Q-s curve interpretation method.

As shown in

Table 6. Descriptive statistical comparison of Q-s interpretation methods against the JGJ 106-2014 reference loads.

Method	Mean Ratio	COV	MAPE (%)	RMSE (kN)	Mean Bias (kN)
Chin	1.1961	0.0625	20.4	345.18	385.68
Davisson	1.0081	0.0312	2.45	46.82	22.36
Corps of Engineers	0.9687	0.0414	3.87	101.65	−67.13
Hansen	1.1539	0.0258	15.39	314.92	301.58
IDEA	1.0126	0.0374	2.21	85.28	30.39

, among the five Q-s curve interpretation methods, the IDEA and Davisson methods showed the closest overall agreement with the reference capacities which were interpreted from the static load test according to JGJ106-2014. The MAPE obtained using the IDEA method was the lowest (2.21%), whilst the coefficient of variation and RMSE values obtained using the Davisson method were slightly lower. In contrast, the Chin and Hansen methods exhibited a marked tendency to systematically overestimate, whereas the US Army Corps of Engineers method was generally more conservative. It should be noted that Table 6 provides descriptive error statistics rather than a formal significance test among the interpretation methods. The 95% confidence intervals reported for IDEA quantify the numerical uncertainty of the segmented-regression breakpoint, whereas the Chin, Davisson, Corps of Engineers, and Hansen methods are deterministic point-estimate procedures under a fixed Q-s record. Therefore, the small difference between the MAPE of IDEA (2.21%) and that of Davisson (2.45%) should not be interpreted as statistically significant evidence of superiority. The main conclusion is limited to stating that IDEA and Davisson provided comparably close agreement with the JGJ 106-2014 reference interpretation on the present dataset, with IDEA showing a slightly lower MAPE and Davisson showing lower COV and RMSE.

Although the IDEA method produced the lowest MAPE among the five Q-s interpretation methods considered, it should not be regarded as uniformly unbiased for all individual records. In particular, Piles 6^# and 7^# showed larger positive deviations from the JGJ 106-2014 reference capacities. This indicates that, even when the segmented model is statistically well supported, the IDEA breakpoint may shift upward if the transition region is represented by relatively few loading stages or if the abrupt deformation response develops very close to the final loading levels. Therefore, IDEA is better characterized as a competitive and promising interpretation tool within the examined dataset, rather than as a universally superior capacity interpretation method.

3.5. Sensitivity of the IDEA Breakpoint to Measurement Noise and Data Density

To directly address the concern that the identified IDEA breakpoint may be affected by settlement measurement errors and by the density of loading points, an additional sensitivity analysis was carried out for all nine pile tests. It is worth noting that all perturbed or reduced datasets were processed using the exact same formal breakpoint identification procedure as that employed in the main IDEA analysis. Specifically, this involved reconstructing cumulative work from the original Q-s data, deriving the IDER-Q curve, and identifying breakpoints using a single-breakpoint continuous piecewise regression model.

Because the in-house static load tests followed a fixed loading scheme, with the load increment specified according to the testing procedure, the independent influence of different prescribed load-increment sizes cannot be isolated from the available experimental records. To avoid introducing unsupported assumptions, the present study evaluates this issue indirectly through data-density reduction. Retaining fewer loading points can be regarded as an approximate representation of a coarser loading record, although it is not identical to performing a new static load test with different load increments.

In the noise analysis, the incremental settlement between adjacent loading stages was perturbed by independent random errors of ±1%, ±2% and ±5%, and 1000 Monte Carlo simulations were generated for each pile and each perturbation level. The resulting breakpoint shifts were consistently very small. As shown in Figure 12, the median relative shift of the IDEA breakpoint remained close to zero at all noise levels, with the overall median values increasing only slightly from approximately 0.01% under the ±1% perturbation to about 0.02% under ±2% and about 0.04% under ±5%. Even the most unfavorable simulated cases remained below about 0.15%. These results indicate that the IDEA breakpoint is practically insensitive to moderate settlement measurement noise.

A different trend was observed for the data-density analysis. When every second loading point was retained, the successfully identified breakpoints generally remained close to the full-data estimates, and most relative deviations were below about 0.5%, although several piles showed somewhat larger deviations approaching 2%. However, under the more severe reduction in which only every third loading point was retained, breakpoint identification became unstable and failed for all piles. By contrast, removing one early-stage point had a negligible influence on the identified breakpoint, with relative deviations typically below 0.2%. The strongest effect was observed when one point near the pre-failure stage was removed: in that case, all piles still yielded identifiable breakpoints, but the relative deviations increased markedly, ranging from about 6.34% to 17.94%, as shown in Figure 13.

In the present dataset, the median breakpoint shift under ±1%, ±2%, and ±5% settlement perturbations remained very small, whereas a severe reduction in loading points caused unstable breakpoint identification or substantial deviations when pre-failure information was removed. In particular, removing an early-stage point had a negligible influence, whereas removing a point close to the pre-failure stage produced much larger breakpoint deviations. This confirms that reliable IDEA interpretation requires sufficiently dense loading stages around the transition region and adequate coverage of the pre-failure response branch.

From an engineering perspective, the sensitivity results indicate that IDEA should not be applied to a Q-s record that contains only the pre-transition branch and the first post-transition point. For practical static load testing, at least two to three recorded loading stages should preferably be available within or immediately around the pre-failure transition zone. If a test is terminated immediately after the first code-specified settlement or failure-related criterion is reached, the critical post-transition points required for stable segmented-regression identification may be removed. Therefore, when IDEA is intended for post-test interpretation, the loading record should provide sufficient resolution around the last stable loading stage, the transition stage, and the initial post-transition response, while still satisfying safety and code requirements.

3.6. External Validation and Comparative Evaluation on Independent Published Q-S Datasets

To provide a preliminary external check on transferability beyond the in-house dataset, published PHC pile load–settlement curves from two sources (Wei et al. [11]; Li and Li [50]) were reanalyzed in the present study. For direct cross-method comparability, a common 10-case subset was retained for the external assessment reported herein. For consistency with the in-house interpretation basis, the external reference capacities Qref were not simply taken as the terminal applied loads or maximum loads reported in the source studies. Instead, the published external Q-s curves were digitized and reinterpreted in the present study using the same abrupt-settlement criterion consistent with JGJ 106-2014 [31]. Specifically, when the settlement response at a loading stage showed an abrupt increase relative to the previous stage, the preceding stable load level was adopted as Qref. This criterion is consistent with the code-based rule that when the pile-head settlement under a given load is greater than twice that under the previous load and does not reach the relative stability requirement within 24 h, the previous load should be adopted as the vertical compressive bearing capacity. In addition, four established Q-s interpretation methods, namely the Chin, Davisson, Corps of Engineers, and Hansen methods, were also applied to the same external cases according to the procedures defined in Section 2.2.2, so that the external performance of IDEA could be evaluated against widely used existing methods on an identical dataset.

It should be noted that the original settlement–time records of the external datasets were not available in the published papers. Therefore, the 24 h relative-stability condition specified in JGJ 106-2014 could not be independently verified for all external cases. The external Qref values used here should therefore be understood as curve-based, code-consistent reinterpretations from digitized Q-s curves, rather than strictly code-certified capacities. Accordingly, the external comparison is intended as a preliminary transferability check under a unified interpretation basis.

Table 7. Basic information and reference-capacity determination basis of the external PHC pile load–settlement datasets [11,50].

Test Pile No.	Source	Length (m)	Diameter (mm)	Thickness (mm)	Bearing Soil Layer	Source-Reported SLT Value (kN)	Adopted Qref in This Study (kN)	Qref Determination Basis
C-1	Wei et al. [11]	21	500	125	Medium sand	5600	5600	Reinterpreted from digitized Q-s curve; same as source-reported value
C-2	Wei et al. [11]	21	500	125	Medium sand	4900	4900
D-1	Wei et al. [11]	12.5	500	125	Coarse sand	3500	3500
D-3	Wei et al. [11]	12.5	500	125	Coarse sand	4000	4000
D-2	Wei et al. [11]	12.5	500	125	Coarse sand	3500	3000	Preceding load before abrupt settlement response identified from digitized Q-s curve
K-1	Wei et al. [11]	21	500	125	Medium sand	4500	4000
3#	Li and Li [50]	24	500	125	Strongly weathered mudstone	4800	4800	Source Q-s interpretation; consistent with abrupt-settlement criterion
5#	Li and Li [50]	25	500	125	Strongly weathered mudstone	5200	5200
6#	Li and Li [50]	25	500	125	Strongly weathered mudstone	5600	5600
7#	Li and Li [50]	25	400	95	Strongly weathered mudstone	2400	2400

Note: Q_ref denotes the reference capacity adopted in the present external comparison. For consistency with the in-house Q_m interpretation, the external Q_ref values were reinterpreted from the digitized Q-s curves using an abrupt-settlement criterion consistent with JGJ 106-2014. When an abrupt settlement increase was identified at a given loading stage, the preceding load level was adopted as Q_ref. Because the original settlement–time records were not available for all external cases, the 24 h relative-stability condition could not be independently verified; therefore, these Qref values are treated as curve-based code-consistent reinterpretations rather than strictly code-certified capacities.

summarizes the basic information of the external PHC pile cases used in this comparative assessment. These external cases cover pipe piles reported under different bearing-layer conditions, including medium sand, coarse sand, and strongly weathered mudstone, and therefore provide a broader validation context than the present in-house dataset alone. The published Q-s curves were digitized from the original figures using a standard curve-digitization procedure, and the extracted coordinates were checked manually before reanalysis.

The IDEA-based external interpretation results are listed in

Table 8. Comparison between reinterpreted external reference capacities and IDEA-derived transition loads.

Test Pile NO.	Qref (kN)	${\mathit{Q}}_{\mathit{u}\mathit{k}}^{\mathit{I}\mathit{D}\mathit{E}\mathit{A}}$(kN)	$\mathbf{Ratio} {\mathit{Q}}_{\mathit{u}\mathit{k}}^{\mathit{I}\mathit{D}\mathit{E}\mathit{A}}$/Qref	Error (kN)	Absolute Percentage Error (%)
C-1	5600	5592.65	0.9987	−7.35	0.13%
C-2	4900	4858.43	0.9915	−41.57	0.85%
D-1	3500	3459.73	0.9885	−40.27	1.15%
D-2	3000	2963.83	0.9879	−36.17	1.21%
D-3	4000	3885.22	0.9713	−114.78	2.87%
K-1	4000	3951.09	0.9878	−48.91	1.22%
3#	4800	4788.96	0.9977	−11.04	0.23%
5#	5200	5195.38	0.9991	−4.62	0.09%
6#	5600	5569.04	0.9945	−30.96	0.55%
7#	2400	2383.51	0.9931	−16.49	0.69%

Note: Q_ref denotes the reference capacity reinterpreted in the present study from the digitized external Q-s curves using a uniform abrupt-settlement criterion consistent with JGJ 106-2014. ${\mathrm{Q}}_{\mathrm{u}\mathrm{k}}^{\mathrm{I}\mathrm{D}\mathrm{E}\mathrm{A}}$ denotes the transition load interpreted using the proposed IDEA method. For the 10 external cases, MR = 0.9910, COV = 0.8243%, MAPE = 0.8987%, RMSE = 46.53 kN, and MB = −35.22 kN.

, and the corresponding parity plot is shown in Figure 14. Overall, the IDEA-derived capacities remained very close to the adopted reinterpreted external reference capacities for all ten external cases. The ratio $Q_{uk}^{IDEA}$/Q_ref ranged from 0.9713 to 0.9991, indicating consistently small deviations. The overall statistical indicators were MR = 0.9910, COV = 0.8243%, MAPE = 0.8987%, RMSE = 46.53 kN, and MB = −35.22 kN. Among the ten external cases, nine showed absolute percentage errors no greater than 1.22%, while the largest deviation occurred for case D-3, with an absolute percentage error of 2.87%. These results indicate that the IDEA procedure can be transferred to independently published PHC pile Q-s curves with limited overall deviation on the present external subset.

To place the external IDEA performance in a broader methodological context, the same 10-case subset was further interpreted using the Chin, Davisson, Corps of Engineers, and Hansen methods. The pile-by-pile external reinterpretation results are summarized in

Table 9. External reinterpretation results of established Q-s methods for the common 10-case subset.

Test Pile NO.	Qref (kN)	Chin (kN)	Davisson (kN)	Corps of Engineers (kN)	Hansen (kN)
C-1	5600	6694.56	5716.39	4524.00	6314.98
C-2	4900	6075.69	5028.01	4238.19	5600.24
D-1	3500	3806.22	3477.87	3096.9	4014.33
D-2	3000	2468.96	3067.20	2867.57	3501.70
D-3	4000	4781.96	3672.14	3473.07	4510.66
K-1	4000	5688.19	4204.41	3767.27	4500.21
3#	4800	5503.17	4812.53	3554.99	5184.71
5#	5200	6007.87	5240.67	3710.8	5584.82
6#	5600	6312.90	5629.31	4051.7	5993.60
7#	2400	3324.42	2404.30	4339.76	2670.72

, and the overall statistical comparison is presented in

Table 10. Statistical comparison of IDEA and established Q-s interpretation methods on the common external dataset.

Method	Mean Ratio	COV (%)	MAPE (%)	RMSE (kN)	Mean Bias (kN)
Chin	1.1778	14.02	21.32	944.85	766.39
Davisson	1.0050	3.45	2.26	136.71	25.28
Corps of Engineers	0.9310	34.39	23.07	1097.05	−537.58
Hansen	1.1175	2.95	11.75	505.28	487.60
IDEA	0.991	0.8243	0.8987	46.53	−35.22

. Among the four established methods, the Davisson method showed the closest agreement with the adopted reinterpreted external reference capacities, with MR = 1.0050, COV = 3.45%, MAPE = 2.26%, RMSE = 136.71 kN, and MB = 25.28 kN. By contrast, the Chin method showed a clear tendency to overestimate the reference capacities, with MR = 1.1778 and MAPE = 21.32%. The Hansen method also exhibited an overall positive bias, with MR = 1.1175, MAPE = 11.75%, and MB = 487.60 kN. The Corps of Engineers method yielded an overall negative bias, with MR = 0.9310 and MB = −537.58 kN, and also showed the largest dispersion among the four established methods in the present external subset.

The differences observed among the external interpretation methods are broadly consistent with previous comparative studies. Earlier investigations have shown that different interpretation criteria applied to the same load–settlement record may produce substantially different ultimate capacities, and that the discrepancy may become pronounced when the curve shape or the loading range does not equally satisfy all method-specific assumptions [22,23,49]. In particular, previous comparisons have frequently reported that Davisson tends to provide relatively stable or conservative estimates, whereas Chin-type extrapolation methods may yield larger interpreted capacities [22,23]. The present external reanalysis of published PHC pile cases shows a similar general pattern: Davisson remained the strongest established comparator, Chin and Hansen tended to give larger capacities, and the Corps of Engineers method displayed greater variability in the present external subset.

When the common external dataset was used for direct comparison, IDEA yielded the smallest overall deviation on the present common external subset. Therefore, the revised external assessment no longer relies solely on an IDEA-only application to published data, but instead demonstrates that the proposed method remains competitive when benchmarked directly against established Q-s interpretation methods on independently published PHC pile records. Nevertheless, the external assessment should still be interpreted as preliminary. The number of available published PHC pile cases is limited, and the external Q-s curves were reconstructed from published figures rather than obtained from original measurement files. Consequently, digitization uncertainty and possible differences in testing procedures may affect the comparison. The present external results therefore support the transferability and competitiveness of IDEA within the examined external subset, but they do not constitute definitive evidence of universal comparative performance.

3.7. Limitations and Recommended Scope of Application

Several limitations should be recognized. First, the in-house dataset includes only nine PHC pipe piles from one project site, and the observed performance may therefore reflect site-specific soil conditions and loading responses. Second, the reference capacities used in the main comparison were interpreted from the same static load tests according to JGJ 106-2014; hence, the comparison evaluates agreement with an accepted interpretation benchmark rather than independent prediction accuracy. Third, the external assessment was limited to a common ten-case subset of published PHC pile records, and some Q-s curves were digitized from figures, introducing unavoidable reconstruction uncertainty. Fourth, the work-based IDEA indicator does not strictly separate elastic and plastic energy components and cannot fully represent path-dependent or cyclic pile–soil interaction. Finally, the sensitivity analysis indicates that the method requires sufficiently dense loading records and adequate coverage of the pre-failure response branch. Broader validation across additional pile types, soil profiles, and loading protocols is still required before a stronger generalization can be made.

4. Conclusions

This study investigated nine full-scale PHC pipe piles at a site in the lower reaches of the Yangtze River, China, and examined a segmented-regression-based IDEA procedure for interpreting failure-related transition loads from static load-test Q-s curves. Within the scope of the present datasets, the following conclusions may be drawn:

• The proposed IDEA procedure provides an objective and reproducible framework for interpreting PHC pile Q-s curves. By reconstructing cumulative work from the measured load–settlement response and fitting the IDER-Q relationship with a one-break continuous segmented-regression model, the transition load can be identified without manual breakpoint selection. Bootstrap confidence intervals and ΔBIC values further provide numerical support for the identified breakpoint.
• For the nine PHC test piles considered in this study, the segmented model was strongly supported and numerically stable. All piles showed a clear preference for the segmented model over a single-line model, and the relative widths of the 95% confidence intervals remained small. The marked increase from the pre-breakpoint slope to the post-breakpoint slope indicates that the identified breakpoint corresponds to a pronounced change in the work-based response of the tested pile system.
• Among the five Q-s curve interpretation methods compared on the present dataset, IDEA and Davisson showed the closest overall agreement with the reference capacities interpreted from the static load tests according to JGJ 106-2014. IDEA yielded the smallest MAPE, whereas Davisson showed slightly lower COV and RMSE. These results indicate that IDEA is a competitive and promising interpretation method within the examined PHC pile cases, but they do not imply universal superiority over existing interpretation criteria.
• The IDEA breakpoint was insensitive to moderate settlement measurement noise but sensitive to data completeness near the failure-related transition stage. The perturbation analysis showed only very small breakpoint shifts under ±1%, ±2%, and ±5% settlement noise. In contrast, reducing the density of loading points or removing pre-failure information could lead to unstable identification or substantial breakpoint deviations. Therefore, reliable application of IDEA requires not only generally dense loading records, but preferably at least two to three recorded loading stages within or immediately around the pre-failure transition zone. The record should include the last stable loading stage, the failure-related transition stage, and the initial post-transition response; otherwise, early termination may lead to unstable or biased breakpoint identification.
• The preliminary external application to a common ten-case subset of published PHC pile datasets provided initial support for the transferability of IDEA. However, because the internal dataset is limited to one field condition and the external cases remain limited and partly reconstructed from published figures, the present evidence should be regarded as preliminary. Further validation using original test records from broader pile types, soil conditions, and loading schemes is necessary before stronger general conclusions can be drawn.