Front. Built Environ. Frontiers in Built Environment Front. Built Environ. 2297-3362 Frontiers Media S.A. 1756475 10.3389/fbuil.2026.1756475 Original Research Undrained capacity of offshore axially loaded piles with the CSUH model incorporating cylindrical cavity theory Jia et al. 10.3389/fbuil.2026.1756475 Jia Xianlin 1 Conceptualization Methodology Validation Writing - original draft Fu Sai 1 Writing – review and editing Pan Kun 2 Data curation Visualization Writing – review and editing Pang Li 3 * Resources Software Supervision Writing – review and editing PowerChina Huadong Engineering Corporation Limited, Hangzhou, China Department of Civil Engineering, Zhejiang University of Technology, Hangzhou, China Department Civil Engineering, Zhejiang University, Hangzhou, China Correspondence: Li Pang, pangli@zju.edu.cn 02 03 2026 2026 12 1756475 28 11 2025 30 01 2026 03 02 2026 Copyright © 2026 Jia, Fu, Pan and Pang. 2026 Jia, Fu, Pan and Pang https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Axially loaded piles are widely employed in deep-water offshore platforms, offshore wind turbine foundations, electrical-transmission towers, and bridge piers, and other structures to support axial loads. This paper presents a semi-analytical solution for predicting the undrained response of axial load piles using the unified hardening model for clays and sands (CSUH), incorporated within the framework of the cylindrical cavity theory. The soil around the pile is idealized as a cylindrical cavity. By combing the stress equilibrium equation, compatibility equation, boundary conditions, and the constitutive model, the soil state around the pile during loading is described as a system of first-order ordinary differential equations (ODEs) with unknown variables, which are solved as an initial value problem. The derived load-displacement (t-z) curve is then incorporated into the compression differential equation to determine the undrained capacity of the axially loaded piles. The proposed approach is validated with finite-element numerical simulation results. Additionally, results from a centrifuge test and a field test are compared with those predicted by the proposed approach. The comparisons demonstrate that the present approach can accurately predict the undrained load-displacement behavior of axially loaded piles, and capture key phenomena observed in pile tests.

 axially loaded piles CSUH model cylindrical cavity theory offshore undrained capacity The author(s) declared that financial support was received for this work and/or its publication. The authors declare that this study received funding from Youth Fund of the National Natural Science Foundation of China Grant No. 52508417, and No. 52508422. The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article, or the decision to submit it for publication. section-at-acceptance Geotechnical Engineering Introduction

Large structures, including deep-water offshore platforms, offshore wind turbine foundations, electrical-transmission towers, and bridge piers, often rely on pile foundations to support axial loads. In recent decades, significant experimental and theoretical progress has been in understanding the axial load capacity of piles (O’Neill, 2001; Randolph, 2003). Nevertheless, current design practices still depend heavily on empirical and semi-empirical correlations, which frequently lead to predictions with considerable scatter (Fleming et al., 2009). In response, researchers have developed various methods to estimate the bearing capacity of axially loaded piles, such as ultimate bearing capacity analyses (Mascarucii et al., 2016; Zhang et al., 2018), load-transfer method (Aguado et al., 2021; Chen et al., 2022; Jiang et al., 2023b; Pang et al., 2024), and numerical simulations (Mascarucii et al., 2014; Gowthaman and Nasvi, 2017; Oustasse et al., 2019). The ultimate bearing capacity approach involves assumptions about interactions among idealized physical models (e.g., wedges, cylinders, or sectors) to simulate pile-soil behavior (Salgado et al., 2011), yet it generally fails to account for the fundamental elastoplastic properties of the soil. The load-transfer method, on the other hand, requires t-z curves (unit shaft resistance versus displacement relationships), which are typically derived from empirical formulas (Randolph and Wroth, 1978; Kraft et al., 1981; Guo and Randolph, 1997). Although numerical simulation offers detailed insights, they tend to be computationally intensive, time-consuming, and require substantial expertise. Consequently, there is a clear need for a relatively accurate semi-analytical approach that can effectively incorporate the elastoplastic characteristics of the soil.

The soil surrounding a pile is commonly idealized as a cylindrical cavity that expands under increasing axial load. As a result, pile-soil interaction is frequently analyzed as a cavity expansion problem (Mascarucii et al., 2016; Zhang et al., 2019; Tan et al., 2021). However, cavity expansion theory has traditionally been applied to penetration process, such as cone penetration tests (CPT) (Zhou et al., 2021; Russell et al., 2023; Mo et al., 2025), jacked pile installation (Wang et al., 2011; Li J. L. et al., 2020), and projectile impact (Deng et al., 2021; Xu et al., 2021). For bored piles, the axial bearing capacity arises from combined contribution of tip resistance and shaft resistance. In long flexible piles, shaft resistance constitutes the primary component of the total capacity, which challenges the direct applicability of conventional cavity expansion theory. In response, Chen et al. (2022) developed a rigorous elastoplastic load-transfer model for axially loaded pile in saturated modified Cam-clay soils. Their approach incorporates concepts from cylindrical cavity expansion theory while accounting for the pile-soil friction, establishing a general framework for predicting the undrained axial capacity of piles. Building on this work, Pang et al. (2024) proposed a semi-analytical solution for predicting the pile-soil response in sands. However, this method does not capture the softening behavior of shaft resistance. To address this limitation, Pang et al. (2023) introduced a two-surface plasticity model, resulting in a more accurate semi-analytical solution for axially loaded piles in sands. Nevertheless, the use of different constitutive models for simulating pile-soil interface behavior and soil response introduces complexity and reduces practical applicability. In offshore engineering, piles are often installed in saturated sands or clays, underscoring the need for a relatively accurate, comprehensive, and practical computational method for predicting pile capacity under undrained conditions.

A unified hardening model for clays and sands (CSUH) was proposed by Yao et al. (2019). The CSUH model is capable predicting the stress-strain behavior of both clays and sands under a wide range of overconsolidation ratios, densities, and stress states. It can also accurately describe triaxial compression, triaxial extension, and three-dimensional responses. The CSUH model and its variants have been employed by several researchers to analytically address cavity expansion problems (Li et al., 2017; Wu and Xu, 2020; Wu and Xu, 2021; Li et al., 2022; Li et al., 2025), and have also been implemented in finite element analyses for predicting embankment behavior (Sun et al., 2004), consolidation settlement (Yao et al., 2010; Feng et al., 2012), and tunnel excavation (Wu et al., 2025), among other applications. However, the application of CSUH model to predict the bearing capacity of pile foundations remains very limited. Only a few studies have utilized its simplified form (the UH model) to numerically simulate pile-soil interaction (Chen et al., 2024; Ma et al., 2024). The present study aims to further extend the application scope of the CSUH model by providing a semi-analytical solution for pile bearing capacity.

This paper presents a semi-analytical approach for predicting the undrained capacity of axially loaded piles using the unified hardening model for clays and sands (CSUH) in conjunction with cylindrical cavity theory. Leveraging axisymmetric condition, the equilibrium equation, boundary conditions, and the stress-strain relationship prescribed by the CSUH model, the axially loaded pile-soil interaction problem is formulated as a system of first-order ordinary differential equations (ODEs) with respect to unknown variables. This system is solved numerically as an initial value problem using the Runge-Kutta (RK) method to derive t-z curves. The proposed approach is validated against finite-element (FE) simulation results, with comparisons of both t-z curves and load-settlement curves showing good agreement. Furthermore, the validity and predictive capacity of the method are assessed through a centrifuge test and a field test, demonstrating favorable consistency with experimental data. The proposed approach offers provide a generalized framework for analyzing the undrained pile-soil interaction within a largely analytical context. Moreover, a further development of the present approach is expected to predict the drained bearing capacity of the axially pile in sands.

 Description of the problem and basic assumption

This study focuses on the load-settlement behavior of axially loaded piles under undrained conditions, with the core objective being to needs to determine the stress-strain response of the soil surrounding the pile. As illustrated in Figure 1, a bored pile of radius r a is embedded in a semi-infinite soil mass. The marine environment ensures undrained conditions throughout the loading process. An axial load ( Q 0 ) is applied at the pile head, resulting in a settlement ( s 0 ). The soil is modeled as a K 0 -consolidated orthotropic continuous medium with an initial void ratio ( e 0 ), where K 0 = σ h 0 ′ / σ v 0 ′ . Here, σ h 0 ′ and σ v 0 ′ denote the initial effective horizontal and vertical in-situ stresses, respectively, with the vertical effective stress given by σ v 0 ′ = γ s ′ z (where γ s ′ is the effective soil weight of the soil and z is the depth). The infinite soil domain is conceptualized as a stack of thin horizontal discs. Under pile loading, the edge of each disc adjacent to the pile displaces downward due to unit shaft resistance ( τ r z , a ). Since the static load is relatively small at the beginning of loading, the soil around the pile initially deforms elastically. With the increase in the axial load, the soil at pile-soil interface will yield first and then an elastoplastic zone with an outer radius ( r p ) will form in the vicinity of the pile. The surrounding soil experiences deformations under effective radial ( σ r ′ ), hoop ( σ θ ′ ), vertical ( σ z ′ ), and shear ( τ r z ) stresses, corresponding to radial ( ε r ), hoop ( ε θ ), vertical ( ε z ), and shear ( γ r z ) strains, respectively. Due to the shear deformation induced by the pile shaft, excess pore water pressure ( Δ u ) develops under undrained conditions, where Δ u = u − u 0 , and u and u 0 represent the current and initial pore water pressures, respectively.

The schematic diagram for an axially loaded pile and horizontal discs of the soil.

Three-dimensional illustration of a pile foundation embedded in a marine environment with a labeled mud surface, soil thin disc, and indicators for load Q zero, elastoplastic and elastic zones, soil stresses, and coordinate axes x, y, z, theta, and r.

The following assumptions are adopted in this study.

The soil particles are considered uncrushable and exhibit relatively high stiffness. Under undrained condition, the analysis focuses on effective stresses and strains.

The effect of pile installation is not considered in this study, as it falls beyond the scope of the present work. The pile-soil interface is assumed to be perfectly rough, implying that soil displacement at the interface equals that of the pile, as indicated by Chen et al. (2022). It is noted that the perfectly rough assumption may result in overestimation of the load bearing capacity and underestimation of the displacement of the pile. Considering that the majority of the settlement of the pile is occupied by the deformation of the soil in the vicinity of the pile and the compression of the pile, it is also reasonable to regard the pile-soil interface to be perfectly rough in undrained case.

Under increasing axial load, the soil initially deforms elastically. At this state, small-strain conditions are assumed, and the stress-strain relationship follows Hooke’s law. As loading continues and large deformations develop, the soil be-havior is described using the unified hardening model for clays and sands (CSUH) (Yao et al., 2019).

Owing to the axisymmetric nature of the axially loaded pile-soil system, the hoop stress remains constant in the circumferential direction. Furthermore, under the thin disc assumption (Chen et al., 2022), the variations of effective vertical and shear stresses with depth ( d σ z ′ / d z and d τ r z / d z ) can be neglected. The quasi-static stress of an arbitrary soil element around the pile can therefore be described by the following equilibrium in the radial direction: d σ r ′ d r + d u d r + σ r ′ − σ θ ′ r = 0 d τ r z d r + τ r z r = 0 where r is the radial location of an arbitrary soil element; d is the spatial differential of · for a given time (i.e., Eulerian description). Equation 1b can be integrated from the location of the soil element ( r ) to the pile shaft radius ( r a ), and expressed as: τ r z = r a r τ r z , a

Small-strain assumption is adopted for soil at elastic stage, and the geometry can be expressed as (Chen et al., 2022): ε r = − d u r d r ε θ = − u r r γ r z = tan − 1 d u z d r where u z is the shear displacement along the axial direction of the soil element.

To accommodate the large strain for soil at the elastoplastic stage, the natural strain or called logarithmic strain recommended by Collins and Yu (1996) is adopted as follows: ε r = − ln d r d r 0 ε θ = − ln r r 0 γ r z = tan − 1 d u z d r where r 0 is the initial radial location of the soil element.

The strain components confirm the relationship as follows: d ε v = d ε r + d ε θ = 0

Since the thin disc assumption is considered, the increment of the vertical strain ( d ε z ) can be ignored. Moreover, the increment of the volumetric strain ( d ε v ) vanishes and equals to zero under undrained condition in Equation 5.

 Solution to the pile-soil interaction Solution to the elastic stage

The stress-strain relationship of the soil at the elastic stage obeys the Hooke’s law as follows: D ε r e D ε θ e D ε z e D γ r z e = 1 2 G 1 + μ 1 − μ − μ 0 − μ 1 − μ 0 − μ − μ 1 0 0 0 0 2 1 + μ D σ r ′ D σ θ ′ D σ z ′ D τ r z where G is the shear modulus of the soil, which can also be expressed as G = E / 2 1 + μ ; the subscript “e” represents the symbolic variable belongs to the elastic stage, and E is the elastic modulus, given as E = 3 1 − 2 μ 1 + e 0 / κ p ′ + p s , where the meaning of κ and the calculation method for p s are provided in Supplementary Appendix A; D · is the time differential of ( · ) for a given soil particle (i.e., Lagrangian description); μ is the Poisson’s ratio.

As the axial load at the pile top increases, the unit shaft resistance ( τ r z ) rises, leading to an increase in shear strain increment ( D γ r z e ). During the elastic stage, no shear dilatancy occurs; the soil element undergoes shear deformation without volume change, a condition referred to as the pure shear state (Chen et al., 2022; Pang et al., 2024). Consequently, the increments of radial, hoop, and vertical stresses and strains are negligible in this stage. No excess pore water pressure develops, and the radial, hoop, and vertical stresses, as well as the pore water pressure, remain at their initial values, expressed as follows: σ r ′ = σ θ ′ = σ h 0 ′ σ z ′ = σ v 0 ′ u = u 0

From Equations 2 and 6, the elastic shear strain ( γ r z e ) of the soil element around the pile can be given as: γ r z e = r a r τ r z , a G

Combined with Equations 3b and 8, the shear displacement ( u z e ) along the axial direction of the soil element at the elastic stage can be obtained as: u z e = ∫ ∞ r ⁡ tan ⁡ γ r z e d r = ∫ ∞ r ⁡ tan ⁡ r a r τ r z , a G d r

The unit shaft displacement ( u r , a ) at elastic stage can be determined by integrating tan ⁡ γ r z e in Equation 9 from far field to the pile shaft, given as: u z , a = ∫ ∞ r a ⁡ tan ⁡ γ r z e d r = ∫ ∞ r a ⁡ tan ⁡ r a r τ r z , a G d r

Equation 10 gives the unit shaft resistance-displacement curve (i.e., t-z curve) of the axially loaded pile at the elastic stage.

 Solution to the elastoplastic stage

The soil around the pile will yield as the unit shaft resistance increases, and CSUH model is adopted in this study to determine the stress-strain response of the soil. A reference yield surface ( f ∼ ) is defined in CSUH model, together with the yield function ( f ), which is given as (Yao et al., 2019): f ∼ = ln 1 + 1 + χ q ∼ 2 M 2 p ∼ ′ 2 − χ q ∼ 2 p ∼ ′ + p s − ln p ∼ x 0 ′ + p s − ε v p c p = 0 f = ln 1 + 1 + χ q 2 M 2 p ′ 2 − χ q 2 p ′ + p s − ln p x 0 ′ + p s − ε v p c p = 0 where p x 0 ′ is the intersection of yield surface with p ′ -axis in initial conditions; ε v p is the volumetric strain of the soil; the calculation method for H , c p , and p s , together with the meaning of model constants ( χ and M ), are provided in Supplementary Appendix A; ∼ represents the symbolic variable belongs to the reference yield surface; where p ′ and q are effective mean stress and deviatoric stress, respectively, defined as: p ′ = 1 3 tr σ ′ q = 3 2 σ i j ′ − p ′ δ i j σ i j ′ − p ′ δ i j where σ ′ is the effective stress tensor; tr means the trace of matrix.

Since normal stresses and excess pore water pressure remain constant at the elastic stage from Equations 7a-c, the stress state when the soil yields can be given as: σ r p ′ = σ θ p ′ = σ h 0 ′ σ z p ′ = σ v 0 ′ u p = u 0 and shear stress at this state can be determined from Equations 11b, 12b, 13a, 13b, as follows: τ r z , p = p 0 ′ 1 3 M 2 ρ − 1 ρ χ + 1 − η 0 2 where p 0 ′ is the initial effective mean stress; η 0 = q 0 / p 0 ′ is the initial stress ratio, and q 0 is the initial deviatoric stress; ρ = p x 0 ′ + p s exp H / c p − p s / p 0 ′ .

Then the radius ( r p ) of the elastoplastic boundary, the shear strain ( γ r z , p ), and shear displacement ( u z , p ) when soil yields can be derived by introducing Equation 13 into Equations 2, 8 and 9, as: r p = τ r z , a τ r z , p r a γ r z , p = τ r z , p G u z , p = ∫ ∞ r p ⁡ tan ⁡ r a r τ r z G d r

The strain increment ( D ε ) of the soil at the elastoplastic stage can be decomposed into the elastic component ( D ε e ) and plastic component ( D ε p ), where D ε e is determined in Equation 6b, and D ε p should be given from the non-associated flow rule provided by the CSUH model (Yao et al., 2019), as: D ε p = 1 K p ∂ f ∂ σ ′ : D σ ′ ∂ g ∂ σ ′ where the colon means the trace of the product of adjacent tensors. The function of the plastic potential surface ( g ) can refer to Supplementary Appendix A. The plastic modulus ( K p ), together with the gradient to the yield surface ( ∂ f / ∂ σ ′ ) and plastic potential surface ( ∂ g / ∂ σ ′ ), are given in the Supplementary Appendix B.

Equation 15 can also be expressed in the matrix form, as: D ε r p D ε θ p D ε z p D γ r z p = 1 K p A r r B r r A r r B θ θ A r r B z A r r B r z A θ θ B r r A θ θ B θ θ A θ θ B z A θ θ B r z A z z B r r A z z B θ θ A z z B z z A z z B r z A r z B r r A r z B θ θ A r z B z z A r z B r z D σ r ′ D σ θ ′ D σ z ′ D τ r z where the matrix elements are given as: A i j = ∂ g ∂ σ i j ′ B i j = ∂ f ∂ σ i j ′

By combining the stress-strain relations of elasticity in Equation 6 and elastoplasticity in Equation 16, the stress increments can be inversely expressed in terms of total strain increments as follows: D σ r ′ D σ θ ′ D σ z ′ D τ r z = 1 T T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 D ε r D ε θ D ε z D γ r z where the detailed expressions of matrix elements are given in Supplementary Appendix C.

Since the loading condition is undrained, Equation 18 can be simplified as (Chen et al., 2022): D σ r ′ D σ θ ′ D σ z ′ D τ r z = 1 T T 11 T 12 T 13 T 14 T 21 T 22 T 23 T 24 T 31 T 32 T 33 T 34 T 41 T 42 T 43 T 44 0 0 0 D γ r z

Equation 19 formulates the governing equations in terms of Lagrangian description for the stress-strain relations of an arbitrary soil element around the pile at the elastoplastic stage. Since the soil element at the same thin disc experiences the same stress and strain paths during the undrained loading process, the problem can be regarded as a typical initial value problem, with the initial values of variables given by Equations 1a-d.

From Equation 19, the shear strain increment can be given as: D γ r z = T T 44 D τ r z

For an arbitrary soil element at the elastoplastic stage, the shear strain can be determined by adding up the shear strain ( γ r z , p ) from Equation 14b when the soil yields and the integration of Equation 20 from τ r z , p in Equation 13b to the current shear stress ( τ r z ) in Equation 2, as: γ r z = γ r z , p + ∫ τ r z , p τ r z T T 44 D τ r z

Combined with Equation 3b and 21, the shear displacement ( u z p ) along the axial direction of the soil element at the elastoplastic stage can be obtained as: u z p = ∫ r p r ⁡ tan ⁡ γ r z d r = ∫ r p r ⁡ tan ⁡ γ r z , p + ∫ τ r z , p τ r z T T 44 D τ r z d r

Combined with Equation 10, the unit shaft displacement ( u z , a ) can be determined by integrating tan ⁡ γ r z in Equation 22 from r p to the pile shaft, given as: u z , a = ∫ ∞ r p ⁡ tan ⁡ r a r τ r z , a G d r + ∫ r p r a ⁡ tan ⁡ γ r z , p + ∫ τ r z , p τ r z , a T T 44 D τ r z d r

Substituting Equation 20 to Equation 19, normal stress increments are given as: D σ r ′ = T 14 T 44 D τ r z D σ θ ′ = T 24 T 44 D τ r z D σ z ′ = T 34 T 44 D τ r z

Then, integrating Equations 24a-c from τ r z , p expressed by Equation 13d and current shear stress given by Equation 2, the normal stresses are determined as: σ r ′ = σ h 0 ′ + ∫ τ r z , p τ r z T 14 T 44 D τ r z σ θ ′ = σ h 0 ′ + ∫ τ r z , p τ r z T 24 T 44 D τ r z σ z ′ = σ v 0 ′ + ∫ τ r z , p τ r z T 34 T 44 D τ r z

Since the initial values of σ r ′ and σ θ ′ at the elastoplastic stage are determined by Equation 13a as σ r p ′ = σ θ p ′ = σ h 0 ′ , matrix elements A r r and A θ θ in Equation 16 determined by Equation B.2 can be deduced to be equal. Thus, matrix elements T 14 and T 24 in Equation 19 are also equal from Equations C.10 and (C.14) as T 14 = T 24 . Hence, from Equations 25a and 25b, it can be known that the effective radial stress ( σ r ′ ) is always equal to the effective hoop stress ( σ θ ′ ) at the elastoplastic stage, which gives: σ r ′ = σ θ ′

Substituting Equation 26 into Equation 1a and considering the self-similar behavior, it can be derived that the total radial stress ( σ r ) keeps unchanged in the same horizontal disc ( d σ r / d r = d σ r ′ / d r + d u / d r = 0 ). Hence, combined with Equations 13a and 13c, the excess pore water pressure ( Δ u ) of the soil element at the elastoplastic stage can be given as: Δ u = u − u 0 = σ r − σ r ′ − u 0 = σ h 0 ′ − σ r ′ where σ r p is the total radial stress when the soil element yields, and σ r = σ r p for the soil element at the elastoplastic stage.

Introducing Equation 25a into Equation 27, the excess pore water pressure ( Δ u ) at the elastoplastic stage can be obtained as: Δ u = − ∫ τ r z , p τ r z T 14 T 44 D τ r z

Finally, normal stresses ( σ r ′ , σ θ ′ , and σ z ′ ), shear stress ( τ r z ), and excess pore water pressure ( Δ u ) at the elastoplastic stage are given by Equations 25a-c, 2, and 28, respectively. Equation 23 gives the unit shaft resistance-displacement curve (i.e., t-z curve) of the axially loaded pile at the elastoplastic stage.

 Monotonic t-z curve

Equations 10 and 23 provide shaft resistance-displacement curve (t-z) curves of the axially loaded pile, which can be summarized as: u z , a = ∫ ∞ r a ⁡ tan ⁡ r a r τ r z , a G d r , τ r z , a ≤ τ r z , a p u z , a = ∫ ∞ r p ⁡ tan r a r τ r z , a G d r + ∫ r p r a ⁡ tan ⁡ γ r z , p + ∫ τ r z , p τ r z , a T T 44 D τ r z d r , τ r z , a > τ r z , a p where τ r z , a p is the shear stress at the pile-soil interface when the soil element adjacent to the pile surface yields.

Figure 2 shows that under initial loading, the shaft resistance ( τ r z , a ) induces a linear segment in the t-z curve, indicating elastic behavior. As τ r z , a increases beyond the elastoplastic threshold ( τ r z , a p ), the unit shear displacement ( u z , a ) evolves nonlinearly. Ultimately, the t-z curve reaches a critical state where u z , a continues to increase while τ r z , a remains constant at its ultimate value ( τ r z , a u ). It is important to note that, unlike traditional t-z curves where τ r z , a is expressed as a function of u z , a , the curve derived from Equation 29 and shown in Figure 2 treats τ r z , a as the independent variable and u z , a as the dependent variable.

Schematic diagram of the present t-z curve of the axially loaded pile at different stages.

Line graph showing uz,a on the vertical axis and τrz,a on the horizontal axis, divided into three regions: Elastic, Elastoplastic, and Critical. τrz,ap and τrz,au are marked; the curve increases linearly in the Elastic region, then curves upward in the Elastoplastic region, and rises steeply in the Critical region. Dotted lines delineate each region. Solving procedure

The pile-soil interaction for a segment of the investigated pile can be characterized by the proposed t-z curve given in Equation 29. This t-z curve is incorporated into the axial force equilibrium equation of the pile to predict the load-settlement ( Q - s ) behavior of a rough-surfaced pile embedded in undrained soil. In this context, the settlement ( s ) of each pile segment corresponds to the shear displacement ( u z , a ) at the pile-soil interface. For clarity, all subsequent references to settlements in this paper are denoted by ‘ s ’. The equilibrium of axial forces acting on an analytical element of the pile at depth ( z ) can be expressed as follows: d d z E p A p U p d s d z − τ r z , a = 0 where E p is the elastic modulus of the pile; A p = π r a 2 is the area of the pile section; U p = 2 π r a is the perimeter of the pile section; s is the settlement of the pile segment. The widely used finite-difference method (FDM) is used in this study to solve the nonhomogeneous nonlinear differential equation of Equation 30 (Reese et al., 2014; Jiang et al., 2023a; Pang et al., 2023). With the FDM, the investigated pile is divided into n segments with equal length of h = L / n , and Equation 30 can be expressed as: s i − 1 − 2 s i + s i + 1 h 2 − U p E p A p τ r z , a = 0 where the subscript ‘ i ’ denotes the i th node along the pile. Substituting values of i from 0 to n into Equation 31 yields a system of ( n + 1 ) equations corresponding to the discrete nodes. Note that two virtual nodes are introduced–one above the pile head and one below the pile tip - as illustrated in Figure 3.

Schematic representation of finite-difference method.

Diagram of a vertically embedded pile subdivided into n discrete segments, labeled from zero at the top to n at the bottom, with forces and shear stresses τ_rz,a indicated along its length, and enlarged view of segment i showing applied forces Q_(i-1) and Q_i with segment height h equal to L divided by n.

As a result, the system comprises ( n + 1 ) equations with n + 3 unknown variables. To achieve closure, boundary conditions must be imposed. An axial load ( Q 0 ) is applied at the pile head, resulting in a settlement ( s 0 ), while the pile tip experiences a base resistance ( Q n ) and a settlement ( s n ). The boundary conditions at the top and bottom of the pile can therefore be expressed as follows: E p A p d s d z z = 0 = E p A p s 0 − s 2 2 h = Q 0 E p A p d s d z z = n = E p A p s n − 2 − s n 2 h = Q n

Since the present study primarily focuses on shaft resistance, long flexible piles are of greater relevance. In such piles, the base resistance contributes only marginally to the overall bearing capacity due to the predominant mobilization of skin friction along the shaft (Nanda and Patra, 2014). For simplicity, an elastic base resistance is adopted, which can be estimated as follows (Randolph et al., 2011): Q n = 4 s n r a G b 1 − μ where G b is the shear modulus at the pile base, which can be determined as G = E / 2 1 + μ and E = 3 1 − 2 μ 1 + e 0 / κ p ′ + p s from Equation 6. In this way, the solving procedure for the load-settlement response of axially loaded piles is illustrated in Figure 4.

Schematic representation of the solving procedure for load-settlement response of axially loaded piles.

Flowchart diagram illustrating a computational process for analyzing soil-pile interactions. The left section outlines steps for finite difference method calculations, including input parameters, segment division, stress state initialization, matrix formation, iterative calculations, and convergence checks. The right section details steps for constitutive modeling, including additional parameters, initial state assignments, elastoplastic zone iterations, matrix updates, stress updates, and convergence. Central output includes Q0 and s0 at the pile head. Validation of the approach Solutions against numerical simulation t-z curve

This- subsection validates the t-z curve derived in this study against finite element method (FEM) simulation performed by Chen et al. (2022). A finite element model was developed to simulate the load-displacement behavior of a thin horizontal soil disc located far from the ground surface and the pile tip. To ensure consistency with the simulation, in which the soil was modelled using the modified Cam-clay (MCC) model, the CSUH model employed in this study was reduced to the MCC model by setting p s = 0 , χ = 0 , and m = 0 . The remaining soil parameters and model constants are listed in Table 1. Chen et al. (2022) performed two sets of numerical simulations and four sets of analytical calculations. As shown in Figure 5, the radial and hoop effective stresses remain equal ( σ r ′ = σ θ ′ ) throughout the loading process, in accordance with Equation 26. Figure 6 depicts shear stress-displacement curve with changes of overconsolidation ratio ( R ), which shows good agreements. Furthermore, from Figure 7 indicates that the development of excess pore water pressure ( Δ u ) is exactly opposite to that of effective radial stress ( σ r ′ ), consistent with the relations expressed in Equations 25a and 28. These results confirm that when the CSUH model is degraded to the MCC model, the proposed approach yields outcomes consistent with the numerical simulations.

Soil parameters and model constants (Chen et al., 2022).

Symbol Numerical model Analytical model
I II I II III IV
R (−) 1 3 1.5 2 4 5
σ v 0 ′ (kPa) 160 120 150 140 110 100
u 0 (kPa) 100 100 100 100 100 100
K 0 (−) 0.625 1 0.7 0.786 1.136 1.3
p 0 ′ (kPa) 120 120 120 120 120 120
q 0 (kPa) 60 0 45 30 15 30
e 0 (−) 0.92 0.78 0.87 0.83 0.75 0.72

M = 1.2 , λ = 0.15 , κ = 0.03 , and μ = 0.278 .

Distribution of stress around the pile at the instant the soil reaches critical state.

Line graph comparing normalized stress components and displacement versus normalized radial location, with four curves representing different parameters: σ̄z/p0, σ̄r/p0 and σ̄θ/p0, τ̄rz/p0, and ū/p0. Open circles denote simulation model data, and solid lines indicate the proposed approach. All curves decrease or increase monotonically, and the x-axis uses a logarithmic scale.

t-z curves with different overconsolidation ratios.

Line graph comparing normalized shear stress versus normalized displacement for different R values. Black lines represent the analytical model, red lines represent the proposed approach, and open circles represent numerical model data. Increasing R values yield higher normalized shear stress plateaus, with R equal to one through four shown. The proposed approach closely matches the numerical model and analytical model trends.

Development of excess pore pressure with displacement.

Line graph compares normalized excess pore pressure versus normalized displacement for five R values: R = 1, 1.5, 2, 3, and 4, with results from a numerical model (circles), analytical model (black lines), and proposed approach (red lines). Load-settlement response

This subsection validates the load-settlement response derived in this study with finite element method (FEM) simulation performed by Chen et al. (2024). The analysis considers a concrete solid pile embedded in normally consolidated saturated clay, with the water table level at the ground surface. Relevant pile and soil parameters, along with model constants, are summarized in Table 2. Additional simulation parameters include a Poisson’s ratio of 0.2 for the pile and a pile-soil interface friction coefficient of 0.577. As the soil consists of normally consolidated clay, the CSUH model can be reduced to the modified Cam-clay (MCC) model for this validation.

Pile and soil parameters and model constants.

Materials Parameters Value
Pile
​ L (m) 10.0
​ r a (m) 0.25
​ E p (GPa) 20.0
Soil
​ e 0 (−) 2.0
​ γ s ′ (kN/m3) 8.00
​ μ (−) 0.35
​ λ (−) 0.20
​ κ (−) 0.04
​ M 1.20

As shown in Figure 8, the numerical simulation results indicate a nearly elastic development of the load-settlement curve until the load reaches approximately 180 kN, after which the response becomes nonlinear and ultimately approaches an ultimate capacity of about 230 kN. In contrast, the proposed approach yields a nonlinear development throughout the loading process. Up to a pile-head load of around 150 kN, the settlement predicted by the proposed method is smaller than that from the numerical simulation. Beyond this point, the trend reverses. Ultimately, the proposed approach predicts a higher bearing capacity than the numerical simulation. This discrepancy may be attributed to the relatively low flexibility of the pile, which has a slenderness ratio ( L / D ) of 20 and an elastic modulus of 20 GPa. In this study, the pile-tip load-displacement behavior is modelled using Equation 33, which allows the soil near the pile tip to mobilize greater resistance, thereby increasing the overall estimated bearing capacity. Furthermore, as illustrated in Figure 9, the proposed method provides a satisfactory prediction of the excess pore water pressure distribution.

Pile-head load-settlement curves.

Graph showing pile-head load in kilonewtons on the x-axis and pile-head settlement in meters on the y-axis, with simulation model data as open circles and proposed approach as a solid curve indicating a nonlinear increasing settlement trend.

Distributions of excess pore water pressure.

Line graph comparing excess pore water pressure in kilopascals against normalized radial location. The simulation model is represented by circles and the proposed approach by a solid line. Both data sets show a rapid decrease in excess pore water pressure as normalized radial location increases, eventually approaching zero beyond a value of four. Solutions against centrifuge and field tests Centrifuge test

This subsection validates the pile load-settlement curve derived in this study against centrifuge tests conducted by Li L. et al. (2020). The experimental program included three pile diameters: 0.95 m, 1.27 m, and 1.90 m. All piles had an embedment depth of 4 m and a Young’s modulus of 200 GPa. The soil deposit consisted of fine-grained silica sand prepared in a loose state. A comparison shows that the mechanical properties of this sand are similar to those of Toyoura sand (Verdugo and Ishihara, 1996); therefore, the model constants for loose Toyoura sand within the CSUH framework, as provided by Yao et al. (2019), were adopted in the analysis (see Table 3). The de-air water is instilled from the bottom of the rigid container, and the suction is on for 24 h to reach saturation.

Model constants for Toyoura sand in the CSUH model (Yao et al., 2019).

M λ κ μ N e c 0 χ m
1.25 0.135 0.04 0.3 1.973 0.934 0.4 1.8

Figure 10 presents a comparison between the centrifuge test results and the predictions from the proposed approach. The analytical method captures a wider range of response intervals than the experimental measurements. The proposed approach yields relatively accurate predictions for the large pile-diameter ( 2 r a = 1.90 ), as Equation 33 effectively models the pile-tip response under small base settlement. However, for the smaller pile diameter ( 2 r a = 0.95 ), the proposed method predicts a more rapid development of pile-head settlement with increasing load. This discrepancy arises from the assumption of a perfectly rough pile-soil interface in the proposed model, whereas some relative displacement likely occurred at the interface in the centrifuge test. In the analytical model, soil deformation around the pile is constrained to match the pile shaft displacement, leading to a stiffer load-settlement response. Overall, the results from the proposed approach show good agreements with the centrifuge test data.

Pile-head load-settlement curves of pile models and from proposed approach.

Scatter plot with three solid curves representing proposed approach predictions for different 2ra values (1.90, 1.27, 0.95) and open circles showing centrifuge test data, plotting pile-head load Q0 (kN) versus pile-head settlement s0 (m). Field test

This subsection validates the pile load-settlement curve obtained in this study using field test data reported by Briaud and Tucker (1989). As illustrated in Figure 11, the site stratigraphy consists of sandy gravel with particles up to 10 cm in size within the top 1.37 m, underlain by a hydraulic fill of clean sand extending to a depth of 12.20 m. Below this depth, layers of medium stiff to stiff silty clay are interbedded with sand down to bedrock found around 14.33 m. The water table is located 2.4 m below the ground surface. The pile is a closed-end steel pipe with an outer diameter of 273 mm and a wall thickness of 9.3 mm, filled with concrete having a Young’s modulus of 42.1 GPa. The total pile length is 10.5 m, with an embedded depth 9.15 m. Since the proposed approach is designed to model the response of pile segments away from the head and tip, and the majority of the pile is embedded in saturated sand, the method is generally applicable for this prediction.

Schematic representation of test condition.

Geotechnical cross-sectional diagram showing a vertical shaft penetrating three soil layers: sandy gravel fill to 1.37 meters, clean sand (hydraulic fill) to 12.20 meters, and sand with stiff silty clay from 12.20 to 14.33 meters. A water table is marked at 2.40 meters. Red arrow labeled Q0 indicates load at the top.

The hydraulic fill medium-dense sand has 80% of particles (by weight) smaller than 1 mm, a dry unit weight of 15.7 kN/m3, and a friction angle of 35.4°. A comparison shows that its mechanical properties are similar to those of Tokachi sand (Ling et al., 2010). Therefore, model constants for medium dense Tokachi sand within CSUH model, as provided by Ling and Yang. (2006), are adopted in this analysis (see Table 4).

Model constants for Tokachi sand in the CSUH model (Ling and Yang, 2006).

M λ κ μ N e c 0 χ m
1.60 0.037 0.04 0.17 2.29 1.080 0.4 1.46

Sharo et al. (2022) also predicted the load-settlement behavior for this case using a tri-linear softening model, with a bilinear load-transfer function for the tip resistance. The ultimate tip settlement was assumed to be 1 mm; before and after this value, the initial and secondary stiffness values were taken as 714.869 A p kN and 158.42 A p kN, respectively. This same tip resistance model is adopted herein in place of Equation 10.

As shown in Figure 12, the proposed approach provides a more accurate prediction of the total resistance compared to the method of Sharo et al. (2022). With increasing elastic tip resistance, the slope of the tip resistance-settlement curve becomes exceeds that of the total resistance-settlement curve, leading to a reduction in shaft resistance and exhibiting a softening response.

Total, shaft, and load-settlement curve.

Line graph comparing pile-head load versus pile-head settlement with three lines for total resistance, shaft resistance, and end resistance, showing field test data as circles, previous predictions as a black line, and a proposed approach as a red line. Conclusion

The present study proposes a semi-analytical approach to predict the undrained load-settlement response of offshore axially loaded piles using the unified hardening model for clay and sand (CSUH). The soil around the pile is idealized as a cylindrical cavity. A system of first-order ordinary differential equations (ODEs) with unknown variables is established and solved numerically as an initial value problem. By integrating the derived t-z curve with the differential equation for axial compression, the load-settlement behavior of the pile is obtained using finite difference method. The proposed approach is validated through numerical simulations, centrifuge tests, and field tests, all of which show good agreement with the predicted results. The following conclusions can also be drawn.

Under undrained shear loading, the effective radial stress remains equal to the effective hoop stress throughout the loading process. Furthermore, the development of excess pore water pressure is opposite to that of the effective radial stress.

The proposed approach accurately predicts the undrained load-settlement behavior of axially loaded piles with varying diameters. The analytical method captures a wider range of response intervals in the load–settlement curve compared to experimental measurements.

The assumption of elastic behavior in the pile base response leads to a softening effect in the shaft resistance. However, this may result in less accurate predictions of pile–soil interaction for piles with relatively small diameters.

It is also noted that the present approach is more applicable to the case that the pile is relatively long, which can fully leverage the role of pile shaft resistance in bearing capacity. This means for the piles with large diameters, such as offshore wind turbines (OWTs), which mainly relying on the foundation reaction force of the soil at pile bottom to bear axial loads, the present approach in this paper may not work so well.

 Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

 Author contributions

XJ: Conceptualization, Methodology, Validation, Writing – original draft. SF: Writing – review and editing. KP: Data curation, Visualization, Writing – review and editing. LP: Resources, Software, Supervision, Writing – review and editing.

 Acknowledgements

The authors acknowledge PowerChina Huadong Engineering Corporation Limited, Zhejiang University of Technology, and Zhejiang Univeristy, for awarding a research grant to the team through which this study was carried out.

 Conflict of interest

Authors XJ and SF were employed by PowerChina Huadong Engineering Corporation Limited.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

 Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbuil.2026.1756475/full#supplementary-material

References Aguado M. B. Sánchez F. E. Pérez E. S. (2021). Model uncertainty for settlement prediction on axially loaded piles in hydraulic fill built in marine environment. J. Mar. Sci. Eng. 9 (1), 63. 10.3390/jmse9010063 Briaud J. L. Tucker L. M. (1989). “Axially loaded 5 pile group and single pile in sand,” in 12th international conference on soil mechanics and Foundation Engineering. Chen H. H. Li L. Li J. P. Sun D. A. (2022). A rigorous elastoplastic load-transfer model for axially loaded pile installed in saturated modified Cam-clay soils. Acta Geotech. 17, 635–651. 10.1007/s11440-021-01248-z Chen Q. Zheng H. Tian D. S. (2024). Application of dimension extending technique to unified hardening model. Appl. Sci. 14, 5677. 10.3390/app14135677 Collins I. F. Yu H. S. (1996). Undrained cavity expansion in critical state soils. Int. J. Numer. Anal. Methods Geomech. 20 (4), 489–516. 10.1002/(sici)1096-9853(199607)20:7<489::aid-nag829>3.0.co;2-v Deng Y. J. Chen X. W. Song W. J. (2021). Dynamic cavity-expansion penetration model of elastic-cracked-crushed response for reinforced-concrete targets. Int. J. Impact Eng. 157, 103981. 10.1016/j.ijimpeng.2021.103981 Feng X. Yao Y. P. Li C. L. (2012). Application of UH model to analysis of deformation of double-layer subgrade. Chin. J. Geotech. Eng. 34 (5), 805–812. (in Chinese). Fleming K. Weltman A. Randolph M. Weltan A. J. (2009). Piling engineering. London: Taylor & Francis. Gowthaman S. Nasvi M. C. M. (2017). 2D and 3D numerical simulation of load-settlement behavior of axially loaded pile foundations. Am. J. Civ. Eng. Archit. 5 (5), 187–195. 10.12691/ajcea-5-5-2 Guo W. D. Randolph M. F. (1997). Vertically loaded piles in non-homogeneous media. Int. J. Numer. Methods Geomech. 21 (8), 507–538. 10.1002/(sici)1096-9853(199708)21:8<507::aid-nag888>3.0.co;2-v Jiang C. Deng L. Pang L. (2023a). Load transfer analysis of vertically loaded piles in sea reclamation area considering the effect of the pile-gravel interface roughness. Ocean. Eng. 271, 113742. 10.1016/j.oceaneng.2023.113742 Jiang C. Shi Z. X. Pang L. (2023b). Analytical solution for negative skin friction in offshore wind power pile foundations on artificial islands under influence of soil consolidation. J. Mar. Sci. Eng. 11 (5), 1071. 10.3390/jmse11051071 Kraft L. M. Ray R. P. Kagawa T. (1981). Theoretical t-z curves. J. Geotech. Engrg. Div. 107 (11), 1543–1561. 10.1061/ajgeb6.0001207 Li L. Li J. P. Sun D. A. Gong W. (2017). Unified solution to drained expansion of a spherical cavity in clay and sand. Int. J. Geomech. 17 (8), 04017028. 10.1061/(asce)gm.1943-5622.0000909 Li J. L. Wang X. F. Guo Y. Yu X. B. (2020). The loading behavior of innovative monopile foundations for offshore wind turbine based on centrifuge experiments. Renew. Energy 152, 1109–1120. 10.1016/j.renene.2020.01.112 Li L. Li J. P. Sun D. A. Gong W. (2020). A feasible approach to predicting time-dependent bearing performance of jacked piles from CPTu measurements. Acta Geotech. 15, 1935–1952. 10.1007/s11440-019-00875-x Li J. P. Liu G. Y. Zhou P. (2022). A semi-analytical solution for cavity undrained expansion in over-consolidated soils based on similarity transform theory. Rock Soil Mech. 43 (3), 582–590. (in Chinese). 10.16285/j.rsm.2021.0830 Li J. B. Li M. G. Chen J. J. Xiao X. (2025). Drained cavity expansion analysis with a unified hardening model for clays and sands. J. Eng. Mech. 151 (9), 04025039. 10.1061/jenmdt.emeng-8296 Ling H. I. Yang S. T. (2006). Unified sand model based on the critical state and generalized plasticity. J. Eng. Mech. 132 (12), 1380–1391. 10.1061/(asce)0733-9399(2006)132:12(1380) Ling H. I. Yang S. T. Leshchinsky D. Liu H. Burke C. (2010). Finite-element simulations of full-scale modular-block reinforced soil retaining walls under earthquake loading. J. Eng. Mech. 136 (5), 653–661. 10.1061/(asce)em.1943-7889.0000108 Ma Z. R. Liu Z. T. Wang J. X. (2024). Application of unified hardening model in soil-pile analysis of offshore wind power large-diameter monopiles. Energy Constr. 11 (2), 68–81. (in Chinese). 10.16516/j.ceec.2024.2.07 Mascarucii Y. Miliziano S. Mandolini A. (2014). A numerical approach to estimate shaft friction of bored piles in sands. Acta Geotech. 9 (3), 547–560. 10.1007/s11440-014-0305-4 Mascarucii Y. Miliziano S. Mandolini A. (2016). 3M analytical: evaluation of shaft friction of bored piles in sands. J. Geotech. Geoenviron. Eng. 142 (3), 04015086. Mo P. Q. Cai G. J. Wang K. J. Eslami A. Yu H. S. (2025). Cavity expansion-based interpretation of piezocone penetration test (CPTu) data in clays. Géotechnique 75 (2), 258–277. 10.1680/jgeot.23.00045 Nanda S. Patra N. R. (2014). Theoretical load-transfer curves along piles considering soil nonlinearity. J. Geotech. Geoenviron. Eng. 140 (1), 91–101. 10.1061/(asce)gt.1943-5606.0000997 Oustasse A. S. Déthié S. A. R. R. Makhaly B. A. Atsou E. T. (2019). Numerical modelling of the behavior of a pile foundation under axial load with ansys. Am. J. Mater. Sci. Eng. 7 (1), 12–20. 10.12691/ajmse-7-1-3 O’Neill M. W. (2001). Side resistance in piles and drilled shafts. J. Geotech. Geoenviron. Eng. 127 (1), 3–16. 10.1061/(asce)1090-0241(2001)127:1(3) Pang L. Jiang C. Zhang C. Y. (2023). An analytical method for predicting the soil-structure interaction of axially loaded piles in sands incorporating the two-surface plasticity model. Eng. Strut. 295 (15), 11684. 10.1016/j.engstruct.2023.116841 Pang L. Jiang C. Zhang C. Y. (2024). A rigorous analytical approach to predicting the load-settlement behavior of axially loaded piles embedded in sands incorporating the SANISAND model. Acta Geotech. 19, 2823–2840. 10.1007/s11440-023-02074-1 Randolph M. F. (2003). Science and empiricism in pile foundation design. Géotechnique 53 (10), 847–875. 10.1680/geot.53.10.847.37518 Randolph M. F. Wroth C. P. (1978). Analysis of deformation of vertically loaded piles. J. Geotech. Eng. Div. 104 (12), 1465–1487. 10.1061/ajgeb6.0000729 Randolph M. Gourvenec S. (2011). “Offshore geotechnical engineering. Spon,”. New York. Reese L. C. Wang S. T. Arrellaga J. A. (2014). TZPILE 2014 user’s manual – analysis of load versus settlement for an axially loaded deep foundation. Austin, TX: Ensoft Inc. Russell A. R. Wang Y. Z. Khalili N. (2023). Cavity expansion in partially drained soils and a comparison to CPT results. Comput. Geotech. 158, 105381. 10.1016/j.compgeo.2023.105381 Salgado S. Woo S. I. Kim D. (2011). “Development of load and resistance factor design for ultimate and serviceability limit states of transportation structure foundations,” in Joint transportation research program publication no. FHWA/IN/JTRP-2011/03. West Lafayette: Purdue University Press. Sharo A. Al-Shorman B. Baker M. B. (2022). New approach for predicting the load-displacement curve of axially loaded piles in sand. Case Stud. Constr. Mater. 17, e01674. 10.1016/j.cscm.2022.e01674 Sun D. A. Matsuoka H. Yao Y. P. Ishii H. (2004). An anisotropic hardening elastoplastic model for clays and sands and its application to FE analysis. Comput. Geotech. 31 (1), 37–46. 10.1016/j.compgeo.2003.11.003 Tan X. Feng L. J. Hu Z. B. Zhao M. (2021). Failure modes and ultimate bearing capacity of the isolated stone column in soft soil. Bull. Eng. Geol. Environ. 80, 2629–2642. 10.1007/s10064-020-02066-5 Verdugo R. Ishihara K. (1996). The steady state of sandy soils. Soils Found. 36 (2), 81–91. 10.3208/sandf.36.2_81 Wang J. Hao W. Zhang C. (2011). “Computation of pile-sinking resistance of jacked pile based on cavities expansion theory,” in Advances in computer science, intelligent system and environment. Advances in intelligent and soft computing. Editors Jin D. Lin S. (Berlin, Heidelberg: Springer), 105, 563–568. 10.1007/978-3-642-23756-0_90 Wu X. T. Xu Y. F. (2020). Elasto-plastic solution for drained cavity expansion in over-consolidated soil incorporating three-dimensional unified hardening model. Chin. J. Geotech. Eng. 42 (10), 1903–1913. (in Chinese). 10.11779/CJGE202010016 Wu X. T. Xu Y. F. (2021). Undrained unified solutions to cylindrical cavity expansion in soils and sands based CSUH model. Chin. J. Geotech. Eng. 43 (6), 1019–1028. (in Chinese). 10.11779/CJGE202106005 Wu X. T. Yao Y. P. Wei R. (2025). Numerical simulation of soil deformation induced by tunnel construction with unified hardening model. Rock Soil Mech. 46 (3), 1013–1024. (in Chinese). Xu S. L. Zhou F. Li Q. H. Wu P. (2021). A novel dynamic cavity expansion model to predict the resistance of reactive powder concrete (RPC) against projectile impact. Compos. Part. B-Eng. 223, 109107. 10.1016/j.compositesb.2021.109107 Yao Y. P. Feng X. Huang X. (2010). Application of UH model to finite element analysis. Rock Soil Mech. 31 (1), 237–245. (in Chinese). Yao Y. P. Liu L. Luo T. Tian Y. Zhang J. M. (2019). Unified hardening (UH) model for clays and sands. Comput. Geotech. 110, 326–343. 10.1016/j.compgeo.2019.02.024 Zhang Q. Q. Feng R. F. Liu S. W. Li X. M. (2018). Estimation of uplift capacity of a single pile embedded in sand considering arching effect. Int. J. Geomech. 19 (9), 06018021. 10.1061/(asce)gm.1943-5622.0001240 Zhang Q. Q. Feng R. F. Yu Y. L. Liu S. w. Qian J. g. (2019). Simplified approach for prediction of nonlinear response of bored pile embedded in sand. Soils Found. 59, 1562–1578. 10.1016/j.sandf.2019.07.011 Zhou H. Liu H. L. Wang Z. L. Tong L. (2021). Analytical solution for cavity expansion in rate-dependent and strain-softening clay and its application for CPT tests. J. Eng. Mech. 147 (3), 04021004. 10.1061/(asce)em.1943-7889.0001911

Edited by: Sandip Mondal, Chandigarh University, India

Reviewed by: Barnali Debnath, National Institute of Technology Agartala, India

Haoyong Qian, Tsinghua University, China