An analytical solution for load transfer mechanism of Plum blossom pile foundations

Abstract

Plum blossom pile, a recently developed non-cylindrical pile, exhibits a load transfer mechanism distinct from conventional circular piles due to its unique cross-section. This study presents an analytical approach to characterize its load transfer behavior based on cross-sectional geometry and vertical shearing mechanisms. Based on the equilibrium analytical method, expressions for axial force/dragload and skin friction are derived. The effects of pile-soil shear action, cross-sectional shape, and effective vertical stress were investigated. The results indicate that the convex zone of the pile increases the soil squeezing stress at the pile-soil interface, and the sharp corner zone creates a semi-soil-plug space made up of circular segments, which increases the vertical effective stress of surrounding soil and then increases the skin friction, which means the cross-section of plum blossom pile has a significant enhancement effect on the shaft resistance. The proposed analytical method provides an effective approach for describing the bearing behavior of the plum blossom pile.

Introduction

The pile foundation has the advantages of large stiffness, small deformation, flexible reinforcement depth and strong adaptability to geological conditions. It is favored in soft foundation reinforcement projects. With the large-scale construction of high-speed railways and expressways, the requirements for foundation settlement are also getting higher and higher, which poses new challenges to the design, construction and maintenance of traditional pile foundations and must be dealt with new concepts. In recent years, special-shaped piles have become one of the research hotspots of scholars at home and abroad. For most special-shaped piles, determining the load transfer mechanism of special-shaped pile foundation is always the main issue in the field of foundation reinforcement engineering^1,2,3. Plum blossom piles are designed to enhance the bearing capacity by enlarging the pile perimeter and altering the load transfer mechanism. As a special cross-sectional pile, the analysis of the load transfer mechanism of plum blossom pile foundation needs to be formulated as a pile-soil interaction problem related to the pile section. However, classical methods that are suitable for circular piles are difficult to consider the special interface between the plum blossom pile and the soil. Finding out the analytical solutions of load transfer mechanisms of special-shaped piles by considering the cross-section shape has attracted much more attention of many researchers. In 2014, Lv et al.⁴ developed a 3D FE comparative analysis model to simulate the stress transfer behaviour of X-section cast-in-place pile (XCC pile). The results indicate that the skin friction of XCC pile is superior to circular pile with the same cross section area. Furthermore, the findings of Lv’s study suggest that larger pile lateral surface area will result in greater soil arching effects and nonuniform effective normal stress along the pile depth⁵. Lv’s research was extended by Zhang et al.⁶ as an important contribution to taking into account the influence of the geometrical effect, a very common feature in special-shaped piles is that it significantly influences the load transfer mechanism. Subsequently, case studies and three-dimensional numerical analyses of helical piles subject to impact loading were conducted by Alwalan and El Naggar⁷. The results indicate that the spacing ratio and the selected failure criteria play a role in determining the load transfer mechanism of piles. Li et al.⁸ explored the load transfer mechanism and influencing factors of the bearing capacity of the PHC nodular pile were investigated based on a group of field tests and numerical simulations, and revealed that the ultimate capacity of PHC nodular piles was about 1.23–1.38 times that of the PHC pipe pile after being cured for 40 days. Deng et al.⁹ conducted model tests on the screw groove pile-geogrid composite foundation and discovered that this type of foundation can effectively mitigate uneven settlement in the transverse direction. The screw groove structure on the pile side, combined with the stress diffusion effect of the reinforced cushion, enables the soil surrounding the pile to share the load together. Liu et al.¹⁰ investigated the distribution characteristics of precast pipe pile foundation additional stress as well as the primary sources of settlement deformation induced by new embankment filling. The findings indicate that precast pipe piles effectively transmit embankment loads to deeper bearing layers, thereby mitigating additional stress within shallow soft soil layers of the foundation. Similarly, many studies have been carried out on the load transfer mechanism of other special shaped piles such as barrette piles^11,12, belled piles^13,14,15, pipe piles^16,17,18, tapered piles^18,19,20, tubular piles²¹ and H-piles^22,23.

The studies presented thus far provide evidence that the improvement of pile bearing capacity benefits from the increase of pile skin area and pile end area. Therefore, it is a good way to increase the bearing capacity of a pile by increasing the skin area, end area or both. On the basis of this, in 2023, a new type of cross-sectional shaped pile, plum blossom pile, was designed to enhance the bearing capacity by enlarging the pile perimeter in China. As special cross-sectional piles, the geometric properties of concrete plum blossom piles have been studied and reported by Deng et al.²⁴ which shows that the perimeter of plum blossom pile is 15% to 16% larger than that of the circular pile with the same pile length and the same amount of concrete. Subsequently, a series of model tests of plum blossom piles penetrating transparent soil and numerical simulations were carried out by Li and Deng et al.^2,25. The results show that the compaction transition zones and reaming zones around plum blossom piles occur during pile sinking, among which the soil compaction transition zone of a plum blossom pile is about 1.5 times that of a circular pile with the same cross-sectional area, and the penetration effects of a plum blossom pile are similar to those of a traditional circular pile with the same cross-sectional area in the cavity expansion area. In addition, the pile driving tip resistance, pile shaft and total resistance of plum blossom pile are 1.0 times, 1.38 times and 1.12 times those of circular pile with the same cross-sectional area, respectively.

Currently, existing experimental studies have predominantly focused on bearing capacity and the geometric properties of plum blossom piles, and the variation in shear action of pile-soil along the depth was not considered, which is inconsistent with the truth. To advance this technique, primarily for the design of the plum blossom piles, it is necessary to describe the load transfer mechanism of the plum blossom pile foundations. This work presents an analytical solution using equilibrium analysis to calculate the load transfer on the plum blossom pile by taking the cross-sectional geometry into account. Additionally, the effect coefficient of shear action of pile-soil on vertical effective stress of soil around pile and the axial force/dragload of the plum blossom pile under concentrated load and uniform surcharge were investigated. Moreover, the load transfer mechanism and influencing factors of the bearing capacity of the plum blossom pile were investigated based on numerical simulations. In other words, the load transfer mechanism between the plum blossom piles and the soil is proposed.

Load transfer mechanism for plum blossom pile foundations

Cross-sectional configuration of plum blossom piles

As shown in Fig. 1, a plum blossom pile is formed by five unique arcs (with a radius of $r_{i}$ and a center of $\theta_{i}$, i = 1, 2, 3), and is governed by two cross-sectional parameters a and θ. Where a and θ are the outsourcing radius and the open arc angle of the cross-section of the plum blossom pile, respectively. The $o_{1}$, $o_{2}$, $o_{3}$, $r_{1}$, $r_{2}$ and $r_{3}$ are the center and radius of the corresponding circle when the arc opening radian θ = 72°, 72° < θ < 180° and θ = 180°, respectively. $s_{1}$ and $s_{2}$ represent the sharp corners of the cross section of plum blossom pile, and z represents the length of plum blossom pile. When θ = 72°, the cross section is circular, when 72° < θ < 180°, the circle center corresponding to the arc opening section is located on the connecting line between the geometric center o₁ of the plum blossom pile section and the center of the circle $o_{3}$ when θ = 180°. When θ = 180°, the section is like a plum blossom. In other words, the five unique arcs of plum blossom pile are semicircular.

Fig. 1

Cross-sectional configuration of plum blossom piles.

When the cross-sectional area of a pile is identical, the intersection points of the circular pile and the plum blossom pile section are defined as dividing points $s_{1}$ and $s_{2}$ (Fig. 1b). These dividing points shift as the outsourcing radius a and the open arc angle θ increase. To establish the geometric relationship between a, θ and the dividing points, and to facilitate subsequent calculation shear effect calculations, the relative positions of the plum blossom pile and the circular pile were analyzed for open arc angle θ = 72°, 108°, 144° and 180° as shown in Fig. 2.

Fig. 2

Geometric position of plum blossom pile and circular pile under different arc opening radian.

Figure 2 indicates that the section size of plum blossom pile increases as the increase of open arc angle. To simplify calculations, only 1/5 of the plum blossom pile cross-section was analyzed due to symmetry. The vertices of the sharp corner zone of the plum blossom pile section are defined as $s_{1}$ and $s_{2}$. The distance from the vertex of the arc $\overset{\lower0.5em\hbox{$\smash{\frown}$}}{AB}$ to the tangent point of the circumscribed circle of the plum blossom pile is denoted as l_i, which values can be determined by using the mathematical software mathematics and are provided in Table 1.

Table 1 Values of λ₁.

When $s_{1} - s_{2}$ = 10 cm, a novel index λ₁ (i.e. l_i/a) based on the outsourcing radius a and the open arc angle θ was proposed, and the specific curve was depicted in Fig. 3, where λ₁ increases as the open arc angle of plum blossom pile θ increases, with an overall upward spiral trend. The correlation of λ₁, a and θ can be fitted using the following equation:

$$\left\{ {\begin{array}{*{20}l} {\theta = 31.86 - 195.21a} \hfill \\ {\lambda_{1} = - 0.47 - 0.47\theta + 0.27\theta^{2} } \hfill \\ \end{array} } \right..$$

(1)

Fig. 3

Curve of λ₁ with the outsourcing radius a of plum blossom pile and the open arc angle θ of plum blossom pile.

When 72° < θ < 180°, connect the points $s_{1}$, $s_{2}$, $o_{i}$, A and B, the angle and labelled $\angle Ao_{i} B$ is $\theta_{i - i}$, $\angle s_{1} o_{i} s_{2}$ is $\theta_{i}$, $\theta_{i - i} /\theta_{i}$ is λ₂, where i = 1, 2, 3. Table 2 shows the corresponding l₂ values. Figures 4 and 5 demonstrate the effect of a and $\theta_{i - i} /\theta_{i}$ on λ₂ when $s_{1} - s_{2}$ = 20 cm. When θ = 180°, the ratio λ₃ = $\overset{\lower0.5em\hbox{$\smash{\frown}$}}{ s_{1} s_{2} } /{\text{s}}_{{1}} {\text{s}}_{2} =\uppi /2$ is a constant.

Table 2 Values of λ₂.

Fig. 4

Effect of a and $\theta_{i - i}$ on λ₂.

Fig. 5

Effect of a and $\theta_{i}$ on λ₂.

In homogeneous soil, the effective vertical stress ($\sigma_{{\text{v}}}^{\prime }$) at depth z is $\sigma_{{\text{v}}}^{\prime } = \gamma^{\prime } z$, where $\gamma^{\prime }$ is the buoyant unit weight. However, the presence of piles implies that the effective vertical stress of the soil is not geostatic ($\sigma_{v}^{\prime } \ne \gamma^{\prime } z$) within a certain range around the piles. In 2002, White analyzed the internal shear stress of tubular piles and found that accurately calculating the external skin friction of tubular piles is challenging due to $\sigma_{v}^{\prime } \ne \gamma^{\prime } z$²⁶. Moreover, Lam²⁷ conducted an equilibrium analysis on the trapped soil element to examine the vertical shearing mechanism of H-piles subject to the surcharge.

Load transfer on plum blossom pile

The load transfer behaviour of the pile foundation can be assumed as an interaction between pile and surrounding soil. It is worth mentioning that there are three main basic assumptions that the theoretical analysis can truly reflect the load transfer phenomenon⁵:

• (1) Pile-soil interaction is limited to a specific range near the surface of the pile.

• (2) Shear effects induced by pile-soil interaction attenuate radially with distance from the plum blossom pile.

• (3) Pile-soil interaction varies at different zones of the plum blossom pile’s cross-section.

Within this framework, the cross-sectional area of plum blossom pile is divided into convex zone and sharp corner zone, as illustrated in Fig. 6a. Furthermore, the soil element free bodies adjacent to convex zone and sharp corner zone of plum blossom pile are established to investigate the load transfer mechanisms specific to each zones. Figure 6b shows simplified vertical equilibrium model of convex zone of plum blossom pile. Here, α represents the size adjustment coefficient of soil unit surrounding the pile.

Fig. 6

Cross section of plum blossom pile.

Consider a simplified vertical equilibrium state of a soil element free body adjacent to plum blossom pile, for which the final vertical stress depends on the interaction of $\sigma_{{\text{v}}}^{\prime }$, τ_s and λτ_s. Based on the equilibrium condition, the equilibrium relationships between $\sigma_{{\text{v}}}^{\prime }$, τ_s and λτ_s are expressed as⁵

$$\frac{{{\text{d}}\sigma_{{\text{v}}}^{{^{\prime } }} }}{{{\text{d}}z}} = \chi \tau_{{\text{s}}} + \gamma^{\prime }$$

(2)

where τ_s represents the skin friction on the pile surface, λ represents the ration of the shear stress, χ represents the effect of shear action of pile-soil on vertical effective stress of soil around pile, and $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ are named in terms of convex zone and sharp corner zone of plum blossom pile.

In Fig. 7, the top and the bottom area of simplified vertical equilibrium model can be expressed as

$$A_{{\text{top-PB}}} = A_{{{\text{bottom}}-{\text{PB}}}} = \frac{{{\uppi }r_{2}^{2} }}{2}\sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)\cos \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right) + \left[ {\left( {\alpha - 1} \right)a - r_{2} } \right]r_{2} \sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right).$$

(3)

Fig. 7

Simplified vertical equilibrium model of sharp corner zone of plum blossom pile.

The inner and outer surface area of simplified vertical equilibrium model of convex zone can be expressed as Eqs. (4) and (5):

$$A_{{{\text{inner}}}} = \theta_{i} \lambda_{2} r_{2} {\text{d}}z$$

(4)

$$A_{{{\text{outer}}}} = \alpha \theta_{i} \lambda_{2} r_{2} {\text{d}}z.$$

(5)

In convex zone of plum blossom pile, the coefficient $\chi_{{{\text{PB}}}}$ can be given by Eq. (6):

$$\chi_{{{\text{PB}}}} = \frac{{\left( {\alpha - 1} \right)\lambda \theta_{i} \lambda_{2} r_{2} {\text{d}}z}}{{\frac{{{\uppi }r_{2}^{2} }}{2}\sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)\cos \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right) + \left[ {\left( {\alpha - 1} \right)a - r_{2} } \right]r_{2} \sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)}}.$$

(6)

Similarly, for the sharp corner zone of the plum blossom pile, $A_{{{\text{top}}-{\text{PS}}}}$ and $A_{{{\text{bottom}}-{\text{PS}}}}$ as follows:

$$A_{{{\text{top}}-{\text{PS}}}} = A_{{{\text{bottom}}-{\text{PS}}}} = \theta_{i} r_{2} + \left( {a - r_{2} } \right)r_{2} \sin \left( {\frac{{\theta_{i} }}{2}} \right) - \frac{{{\uppi }r_{2}^{2} }}{2}\sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)\cos \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right) + \left[ {r_{2} - \left( {\alpha - 1} \right)a} \right]r_{2} \sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right).$$

(7)

The inner and outer surface area of simplified vertical equilibrium model of sharp corner zone can be expressed as Eqs. (8) and (9):

$$A_{{{\text{inner}}}} = \frac{{a - l_{i} }}{{\lambda_{2} }}{\text{d}}\left( {\theta_{i} - \theta_{i - i} } \right){\text{d}}z$$

(8)

$$A_{{{\text{outer}}}} = \frac{{\alpha \left( {a - l_{i} } \right)}}{{\lambda_{2} }}{\text{d}}\left( {\theta_{i} - \theta_{i - i} } \right){\text{d}}z.$$

(9)

Furthermore, the coefficient $\chi_{{{\text{PS}}}}$ can find as

$$\chi_{{{\text{PS}}}} = \frac{{\frac{{\left( {\alpha - 1} \right)\left( {a - l_{i} } \right)}}{{\lambda_{2} }}{\text{d}}\left( {\theta_{i} - \theta_{i - i} } \right){\text{d}}z\lambda }}{{\theta_{i} r_{2} + \left( {a - r_{2} } \right)r_{2} \sin \left( {\frac{{\theta_{i} }}{2}} \right) - r_{2}^{2} \lambda_{3} \sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)\cos \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right) + \left[ {r_{2} - \left( {\alpha - 1} \right)a} \right]r_{2} \sin \left( {\frac{{\theta_{i} \lambda_{2} }}{2}} \right)}}.$$

(10)

Integrating Eq. (2) at z = 0, the vertical effective stress ($\sigma^{\prime}_{{{\text{v}}-{\text{PB}}}}$) of soil around the plum blossom pile in convex zone can be written as

$$\sigma_{{{\text{v}}-{\text{PB}}}}^{\prime } = \left( {\sigma_{{{\text{v}}0}} + \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} K\tan \varphi }}} \right)e^{{\chi_{{{\text{PB}}}} K\tan \varphi z}} - \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} K\tan \varphi }}$$

(11)

Substituting the Eq. (11) into the Coulomb equation $\tau_{{\text{s}}} = c^{\prime} + K\sigma^{\prime}_{{\text{v}}} \tan \delta$⁵, the ultimate skin friction of plum blossom pile on the convex zone is given by:

$$\tau_{{{\text{s}}-{\text{PB}}}} = K\sigma_{{{\text{v}}-{\text{PB}}}}^{\prime } \tan \varphi = \left( {\sigma_{{{\text{v}}0}} K\tan \varphi + \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} }}} \right)e^{{\chi_{{{\text{PB}}}} K\tan \varphi z}} - \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} }}.$$

(12)

By using the same procedure, the ultimate skin friction of plum blossom pile on the sharp corner zone can be expressed as follows

$$\tau_{{{\text{s}}-{\text{PS}}}} = K\sigma_{{{\text{v}}-{\text{PS}}}}^{\prime } \tan \varphi = \left( {\sigma_{{{\text{v}}0}} K\tan \varphi + \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PS}}}} }}} \right)e^{{\chi_{{{\text{PS}}}} K\tan \varphi z}} - \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PS}}}} }}$$

(13)

where $c^{\prime}$, K and φ are the effective cohesion, the coefficient of the earth pressure at rest and it’s friction angle, respectively. Integrating over the inner surface of simplified vertical equilibrium model, we have

$$Q_{{{\text{s}}-{\text{PB}}}} = \int\limits_{0}^{{A_{{{\text{inner}}}} }} {\left[ {\left( {\sigma_{{{\text{v}}0}} K\tan \varphi + \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} }}} \right)e^{{\chi_{{{\text{PB}}}} K\tan \varphi z}} - \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PB}}}} }}} \right]} .$$

(14)

Similarly, one can find $Q_{{{\text{s}}-{\text{PS}}}}$ as:

$$Q_{{{\text{s}}-{\text{PS}}}} = \int\limits_{0}^{{A_{{{\text{inner}}}} }} {\left[ {\left( {\sigma_{{{\text{v}}0}} K\tan \varphi + \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PS}}}} }}} \right)e^{{\chi_{{{\text{PS}}}} K\tan \varphi z}} - \frac{{\gamma^{\prime } }}{{\chi_{{{\text{PS}}}} }}} \right]}$$

(15)

where $Q_{{{\text{s}}-{\text{PB}}}}$ and $Q_{{{\text{s}}-{\text{PS}}}}$ represents, respectively, the total force generated by skin friction on the convex zone and sharp corner zone.

When the concentrated load ($Q_{0}$) works, at the depth z = 0, the boundary conditions yield $Q = Q_{0}$ and $\sigma_{{{\text{v0}}}} = 0$. The axial force of the plum blossom pile can be written as

$$Q_{1} = Q_{0} - 5\left( {Q_{{\text{s-PB}}} + Q_{{\text{s-PS}}} } \right).$$

(16)

When the uniform surcharge ($\sigma_{{{\text{v0}}}}$) works, at the depth z = 0, the boundary conditions yield $Q_{0} = 0$ and $\sigma_{{\text{v}}}^{\prime } = \sigma_{{{\text{v}}0}}$. The dragload can be expressed as

$$Q_{2} = 5\left( {Q_{{\text{s-PB}}} + Q_{{\text{s-PS}}} } \right).$$

(17)

Verification and discussion

Verification by a small-scale model test in homogeneous soil

To verify the accuracy of the present solution, a small-scale model test in homogeneous soil was carried out in the laboratory of the School of Architectural and Civil Engineering, Xi’an University of Science and Technology. A testing system of pile-soil model experiment was developed, which consists of five parts: model box, loading system, loess sample, strain acquisition system, and a computer (equipped with the software for data analysis), as shown in Fig. 8.

Fig. 8

Testing system of pile-soil model experiment.

This laboratory model test is based on a foundation project of the Xi’an-Ankang high-speed railway. The loess samples, collected from the project site, were subsequently sieved to a 2 mm particle size to produce a fine-grained soil for testing. The basic physical properties of loess samples are shown in Table 3. The pile length is 8 m, the pile diameter is 0.5 m. In light of test device and site condition, with reference to the research by Zhu et al.^28,29, the geometric length, elasticity modulus, and density were taken as basic physical parameters, and the geometric similarity ratio was finally determined to be 10, that is, the outsourcing radius of plum blossom pile a = 25 mm, the open arc angle θ = 72° mm and pile length L = 800 mm, respectively. The structure of the model box was composed of profile steel and tempered glass and its two sides were made transparent to facilitate the observation of the test process, the bottom side was composed of a ribbed steel plate, and the dimension of the model box was length × width × height = 4000 mm × 2000 mm × 1700 mm (length × width × height).

Table 3 Physical parameters of soils.

To eliminate the influence of the compactness of soil on the bearing performance of piles, the foundation soil was filled in layers, i.e. mark the scale in the vertical direction around the model box, transport the model soil in plastic buckets, pack 300 mm high model soil each time, compact to 250 mm, and repeat this procedure many times, to ensure that the density of soil is basically uniform (the compaction coefficient > 0.9). The maximum vertical load of $Q_{0}$ = 16.0kN was gradually applied to the plum blossom pile head through eight loading steps. The developed axial force was continuously monitored using strain gauges, which were used to calibrate the calculated axial force. In this work, a load of $Q_{0}$ = 16.0kN is selected for analysis. More details about laboratory model tests are refer to previous relevant work^29,30.

Figure 9 demonstrates the measured and calculated axial force of the test plum blossom pile. From Eqs. (14) and (15), the axial force was found to scale with coefficient χ, which is related to the pile performance, the soil properties and the pile-soil interface (Eqs. 6 and 10). Therefore, χ should be determined through a series of tests and numerical simulations, which are not included in this work. In this section, ten χ values were considered for analysis. They were 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. As a reference, the estimated axial force for the geostatic conditions $Q = A_{{\text{s}}} \gamma^{\prime } hK\tan \delta$ is also plotted where A_s is the pile surface area above depth h. As shown in Fig. 9, with the increase of χ, the calculated axial force decreases, which indicates the vertical shearing effect is greater, the effective vertical stress of the surrounding soil increases, which results in a greater axial force in the pile shaft. It can be observed that the calculated axial force is consistent with the measured value of laboratory model test.

Fig. 9

Comparison between the measured and the calculated axial force.

Verification by a case study

To further verify the precision of the method presented in this paper and to investigate the vertical shearing effect of the plum blossom pile. The parameters, as shown in Tables 4 and 5, were utilized to calculate the axial force/dragload of a plum blossom pile. The parameter H represents the length of the pile, the radius of the external tangent circle and cross-section outsourcing circle of the cross-section is represented by a/R, the area of the cross-section is represented by A, and $a^{\prime }$ represents the equivalent radius of the equal cross-section circular piles, $\theta_{{\text{p}}}$ represents the open arc angle of the plum blossom pile. As a reference, the measured dragload of a circular pile obtained from a laboratory model test^5,30 and the calculated dragload using the equation $Q = A_{{\text{s}}} \left( {\gamma^{\prime } + \vartriangle p} \right)hK\tan \delta$, where $A_{{\text{s}}}$ is the pile surface area above depth z. In this work, a load of $\sigma_{{{\text{v0}}}}$ = 45 kPa and $Q_{0}$ = 90 kN are selected for analysis.

Table 4 Cross-sectional parameters of piles.

Table 5 Parameters of soil.

Figure 10 demonstrates the calculated and measured dragload of the test plum blossom pile. The dragload is influenced by the coefficient χ, as indicated by Eqs. (14), (15), and (17). To investigate this influence, analyses were performed using eleven χ values: 0 (replaced by 1 × 10⁻⁶ since the denominator $\chi_{{{\text{PB}}}}$, $\chi_{{{\text{PS}}}}$ in Eqs. (14) and (15) cannot be zero), 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0. The calculated dragload decreases significantly as |χ| increases. This occurs because a higher |χ| value indicates greater shear action exerted by the pile on the surrounding soil. This intensified shear action leads to a larger reduction in the vertical effective stress of the soil around the pile. Consequently, the smaller vertical effective stress results in the development of less negative skin friction, thereby reducing the dragload. Comparison of the calculated results with the theoretical dragload formula shows that the results of this study closely match the results obtained from equation $Q = A_{{\text{s}}} \left( {\gamma^{\prime } + \vartriangle p} \right)hK\tan \delta$.

Fig. 10

Comparison between the measured and the calculated dragload, $\sigma_{{{\text{v0}}}}$ = 45 kPa.

As shown in Fig. 10, the change rule of dragload of plum blossom pile was obviously different from dragload of circular pile when |χ| is within the range of 0 ~ 1.0. With the increase of |χ|, the influence of the pile-soil shear action on dragload has decreased, and the influence is smaller and smaller as |χ| increases. This is because the interaction of pile-soil decreases continuously as |χ| increases.With the increase of |χ|, the vertical effective stress of the soil around the plum blossom pile has decreased continuously and the degree of weakening of dragload decreases as |χ| increases. This phenomenon is likely due to the unique surface features of the plum blossom pile, which include convex and sharp corner zones. The convex zone increases the soil squeezing stress at the pile-soil interface, while the sharp corner zone creates a semi-soil-plug space made up of circular segments. This space often experiences a soil plug effect, leading to a reduction in the effective stress of some soil around the pile in the sharp corner zone. Due to the soil squeezing stress in the convex zone and the soil plug effect on the sharp corner zone surface, the dragload under the plum blossom pile is reduced. However, this reduction is less significant than that of the circular pile, which may be due to the fact that the cross-sectional perimeter of plum blossom pile is 1.15–1.16 times that of circular piles under the same cross-sectional area and pile length.

Figure 11 shows the axial force distribution of plum blossom pile and circular pile with same cross section area under the concentrated load at the top of the pile. The black solid dots in the Fig. 11a are the results of the circular pile model tests carried out by Lv⁵, which are used to verify the reliability of the formulae presented in this paper, and the dotted line represents the calculated results without considering the shear action of pile soil. To investigate the impact of shear action on the bearing characteristics of plum blossom pile, an analysis of pile axial force is conducted using shear action coefficients χ of 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0.

Fig. 11

Axial force $Q_{0}$ = 90 kN.

It can be seen in Fig. 11a that in the direction of the depth of the pile body the axial force of the pile body decreases with the increase in shear action coefficients. This is due to the fact that, at the top of the pile, it is subjected to a vertical concentrated load. The plum blossom pile will cause settlement in the soil around it, which will be less than the settlement of the pile itself. This results in a positive shear action coefficient. With the gradual increase of the shear coefficient, the skin friction resistance of the pile-soil interface gradually increases, and the proportion of the upper load acting on the pile body used to overcome the lateral friction resistance increases, in other words, more upper load is transferred to the pile peripheral soil through the lateral friction resistance of the pile body, and thus the pile body axial force decreases. Comparing the calculation results of pile body axial force without considering the pile-soil shear effect and the test results of the existing literature, it can be seen that the trends of the pile body axial force obtained by the three calculation methods with the buried depth of the plum blossom pile basically coincide with each other, which verifies the reliability of the method in this paper.

Figure 11b shows that the theoretical axial force of the circular pile body is smaller than the test value when the pile depth is not greater than 4 m. This may be due to the decrease in shear effect of the pile-soil contact surface with increasing pile burying depth. The test piles are short, which could explain the larger test result. From the analysis of Fig. 12a, b, it is evident that the calculation results presented in this paper are in agreement with both the test results and the calculation results that do not consider the shear effect. In addition, it was observed that the reduction rate of the axial force curve of the plum blossom pile is smaller than that of the circular pile with same cross section area. The plum blossom pile exhibits a clear shear effect, which may be attributed to the soil plug effect and the tip angle effect at the pile tip, and these effects increase the positive skin friction on the pile side, which results also prove that the shear action between the plum blossom pile and the soil is more complex than that of the circular pile with the same cross-sectional area.

Fig. 12

Effects of α, λ on $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ under the axial force.

Parametric study and discussion

Analysis of the coefficient $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$

The coefficient of pile-soil shear action χ indicates the difference in shear stresses on the inner and outer faces of the soil around the pile under the action of vertical load at the top of the pile (or the action of vertical homogeneous load in the soil around the pile), and the pile-soil shear action is affected by the characteristic (shape, size, etc.) of the pile-soil contact surface⁶. In this study, the effects of the shear stress coefficient (λ) and the expansion coefficient of the inner and outer surfaces of the soil unit around the pile (α) are investigated principally on the coefficient $\chi_{{{\text{PB}}}}$ in the convex zone and the coefficient $\chi_{{{\text{PS}}}}$ in the sharp corner zone of the plum blossom pile. By considering the pile-soil shear action behavior under the different load to investigate the vertical bearing characteristics of the plum blossom pile.

The coefficients $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ in the convex zone and sharp corner zone of the plum blossom pile, which are related to λ and α, were found using Eqs. (6) and (10). Determining $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ requires a series of tests, which were not included in this work. By substituting the calculation properties shown in Table 6 into Eqs. (6) and (10). The evolution of the shear stress coefficient (λ) for six α values (1, 2, 3, 4, 5, 6) and six λ values (0, 0.2, 0.4, 0.6, 0.8, 1.0) as shown in Figs. 12 and 13.

Table 6 Calculation properties.

Fig. 13

Effects of α, λ on $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ under the uniform surcharge.

The coefficients $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ obviously increases as λ (from 0 to 1) when the pile top is under the axial force. What is interesting in Fig. 13 is that the coefficients $\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$ remain stable with the increase of α when λ is close to 0. This is due to the vertical shear effect is smaller, the expansion coefficient of the inner and outer surfaces of the soil unit around the pile become is limited. This observation can be explained by the geometrical effect of the plum blossom pile. The same variation tendencies have been noticed for the coefficients of pile-soil shear action ($\chi_{{{\text{PB}}}}$ and $\chi_{{{\text{PS}}}}$) under the uniform surcharge (Fig. 13) but, in the opposite direction.

Effects of coefficients χ on vertical effective stresses, negative skin friction and dragload of pile when σ _v0 = 45 kPa

Figure 14 demonstrates the calculated effective vertical stress, negative skin friction and dragload of plum blossom pile. For this analysis, eleven χ values were considered: χ = 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5 and 5.0. It can be seen that the effective vertical stress of the soil elements and the negative skin friction of plum blossom pile at the depth z = 0 are greater than 0, differs from that, the dragload of plum blossom pile is equal to 0. When χ = 0 (i.e. without considering the vertical shearing effect), the vertical effective stress of the soil elements surrounding plum blossom pile $\sigma_{{\text{v}}}^{\prime } = \gamma^{\prime } z$ is an inclined straight line with a slope of $\gamma^{\prime }$, and the vertical effective stress increases linearly with the increase of the normalized depth z/H, when χ ≠ 0 (i.e., considering the vertical shearing effect), the vertical effective stress of the soil elements surrounding plum blossom pile decreases linearly with the increase of the normalized depth z/H, and with the increase of |χ|, the vertical effective stress of the soil elements surrounding pile gradually decreases, indicating that the increase of the shear action coefficient causes the damping of the soil elements surrounding pile. And with the increase of |χ|, the vertical effective stress of the soil elements surrounding plum blossom pile gradually decreases, indicating that the increase of the shear coefficient causes the attenuation of the vertical effective stress.

Fig. 14

Computed distributions of effective vertical stress, negative skin friction and dragload of plum blossom pile when $\sigma_{{{\text{v}}0}} = 45\;{\text{kPa}}$.

Without considering the vertical shearing effect, the negative skin friction of pile $\tau_{{\text{s}}} = \gamma^{\prime } K\tan \delta$ is a decreasing straight line with a slope of $K\tan \delta$ in the normalized depth z/H. As the coefficient |χ| increases, the negative skin friction on the pile gradually decreases. While dragload increases linearly with normalized depth z/H, but decreases gradually as the |χ| increases. This confirms that dragload diminishes as |χ| increases when vertical shearing effects are considered. Compared with the vertical effective stress of soil elements around plum blossom pile and the negative skin friction of plum blossom pile, the coefficient χ has a relatively smaller effect on dragload. A possible explanation is that dragload depends not only on vertical effective stress but also on the distinct cross-sectional geometry of the plum blossom pile.

Under the given applied load $Q_{0}$ = 90 kN, the computed effective vertical stress, positive skin friction and axial force of plum blossom pile are shown in Fig. 15a–c, respectively.

Fig. 15

Computed distributions of effective vertical stress, positive skin friction and axial force of plum blossom pile when $Q_{0}$ = 90 kN.

It can be observed that the effective vertical stress and positive skin friction exhibit the same trends: the two values are zero at the top plane of the pile, then increase as the coefficients χ are increased. The vertical effective stress and positive skin friction increase linearly with the increase of normalized depth z/H, and gradually increase with the increase of coefficient χ, which indicates that the increase of coefficient is helpful to increase the vertical effective stress and then increase the positive skin friction. Without considering the vertical shearing effect, the axial force of plum blossom pile decreases linearly in the normalized depth z/H and peaked in z/H = 0. Considering the vertical shearing effect, with the increase of |χ|, the axial force of plum blossom pile gradually decreases. When χ > 1.0, the axial force of plum blossom pile at the same pile depth is generally less than that without considering the vertical shearing effect.

Effect of geometric characteristics of the cross section of plum blossom pile on dragload and axial force

From the geometric analysis of plum blossom pile in “Load transfer mechanism for plum blossom pile foundations” section, it can be seen that the dragload and axial force of plum blossom pile (or circular pile) are expressed by geometric characteristics (the outsourcing radius a (or circles radius a) and the open arc angle θ). To reveal the influence of geometric characteristics (a and θ) on vertical shear action (the dragload and axial force of plum blossom pile and circular pile), a comparative analysis was conducted between circular piles with the same cross section area and plum blossom piles in this section. Table 7 shows the pile cross-section parameters and calculation parameters of the soil around the pile. Under the load of $\sigma_{{{\text{v0}}}}$ = 45 kPa and $Q_{0}$ = 90 kN, the distributions of computed dragload and axial force are shown in Fig. 16.

Table 7 Calculation properties.

Fig. 16

Effects of pile cross-section parameters a and θ on dragload and axial force of pile.

It can be seen that when the soil around the pile is subjected to vertical uniform load of 45 kPa, the dragload increases with the increase of the outsourcing radius a, and decreases with the increase of the open arc angle θ. When a < 2 m, the dragload remains steady with the increase of the open arc angle, and the dragload of the circular pile is found to be obviously less than the dragload of the plum blossom pile, which is consistent with the conclusion of “Verification by a case study” section. The probable reason for the higher dragload values of plum blossom pile is the greater skin friction caused. When $Q_{0}$ = 90 kN and the open arc angle θ is close to π, the axial force of the pile increases with the increase of the outsourcing radius a. When the outsourcing radius is a fixed, the axial force of the pile decreases with the increase of the open arc angle, and when the outsourcing radius is within a limited range (a < 2 m), the axial force of the pile decreases rapidly with the increase of the outsourcing radius. In addition, the axial force of the plum blossom pile is always greater than that of the circular pile with the same cross-sectional area. This is because plum blossom pile has larger pile side surface area than circular pile under the same pile length and cross section area, that is, it has greater pile skin friction. The present conclusion verifies the theoretical analysis result of plum blossom pile with larger pile side surface area than circular pile with the same cross section area in “Load transfer mechanism for plum blossom pile foundations” section.

Conclusions

The load transfer mechanism for a plum blossom pile foundation, which is a new type of foundation, is formulated algebraically. An analytical solution dependent on the cross-sectional geometry and the vertical shearing mechanism is derived by means of equilibrium analysis to calculate the effective vertical stress of the surrounding soil, the axial force/dragload and skin friction of the plum blossom piles. Additionally, the effect of shear action of pile-soil on vertical effective stress of soil around pile is proposed. The calculated axial force and the dragload due to the skin friction are calibrated using the results of small-scale model tests in homogeneous soil and a case study solution, respectively. Good agreements indicate that the derivation procedures are valid and that valid formulae are obtained. Because this analytical solution dependent on the cross-sectional geometry and the vertical shearing mechanism, this solution is beneficial for the design of the plum blossom piles and provides an efficient tool for most practical situations. For the dynamic load and other loading conditions, its load transfer mechanism may vary. The above-mentioned problems can be further discussed in a follow-up study to improve the theoretical system of the plum blossom pile.