Research on optimized design of section parameters of special-shaped pile based on square section

Research on optimized design of section parameters of special-shaped pile based on square section | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Research on optimized design of section parameters of special-shaped pile based on square section De Jiang, Zhenlin Chen, Zhiguang Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5535232/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Pile foundations are widely utilized in the construction of high-rise building foundations; however, traditional square piles may not be sufficiently economical due to their requirement to resist substantial vertical loads with large dimensions and dense arrangements. This study presents a novel special-shaped pile designed for high-rise buildings, which aims to enhance the ultimate bearing capacity by expanding the contact area between the pile and the soil. Based on pile bearing capacity theory, this study analyzes the effects of the number of concave sides and the angle of concavity on the cross-section properties of the special-shaped pile. Additionally, a finite element model built and validated over ABAQUS is employed to simulate the influence of varying numbers of concave edges and different concave angles on vertical load bearing characteristics. The results indicate that the cross-sectional perimeter of the special-shaped pile increases with the number of concave edges and the angle of concavity, measuring 1.01 to 1.78 times that of a square pile with the same cross-sectional area, with the angle of concavity having the most significant effect. The maximum moment of inertia of the cross-section of the special-shaped pile initially increases and then decreases with the number of concave sides, while it consistently increases with the angle of concavity. Notably, for a given angle of concavity, the maximum moment of inertia is greatest for the special-shaped pile with two concave sides. Increasing the number of concave sides and the angle of concavity effectively enhances the side resistance and ultimate bearing capacity of the pile, with the ultimate bearing capacity being 1.05 to 1.92 times that of a square pile with the same cross-sectional area. Physical sciences/Engineering Physical sciences/Engineering/Civil engineering Pile foundation square section section geometric properties vertical bearing capacity special-shaped pile Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1. Introduction Pile foundation is a force-transmitting member that transfers the load of a building to the foundation soil. With the rapid development of the economy, the demand for buildings with higher bearing capacity is increasing, and the traditional pile foundation is difficult to satisfy the safety and economy requirements of buildings at the same time 1 .In response to this problem, a new type of pile foundation, the special-shaped pile, has been developed. By changing the cross-section, it can improve the pile-soil contact area without increasing the amount of concrete, thereby effectively enhancing side resistance and the bearing capacity of the pile foundation. This type of pile foundation is widely used in high-rise buildings, highway construction, soft soil foundation treatment, and other practical projects 2 – 4 . However, the design of special-shaped piles is still immature, and their vertical load bearing characteristics require further study 5 , 6 . In recent years, a series of efforts have been made to study the design and vertical load bearing characteristics of special-shaped piles. Wang et al. 7 examined the design and bearing characteristics of Y-shaped piles, as shown in Fig. 1a. These piles are formed by concaving three circular arcs inward based on a circular cross-section. They proposed a simplified calculation method for bearing capacity. Chen et al. 8 investigated the effects of different template radii and angles on the geometric properties of Y-shaped piles. Their experimental study revealed that the perimeter of the cross-section increased by 1.46 times. Moreover, the vertical ultimate load-carrying capacity improved by approximately 1.5 times compared to circular piles with the same cross-sectional area. Liu et al. 9 – 15 designed XCC piles based on a circular cross-section with four circular arcs concave inwards, as shown in Fig. 1b. They examined the effects of different formwork curvatures and open-arc spacing on the geometric properties of XCC piles. Through model tests, they compared the load-bearing characteristics of XCC piles with that of circular piles. Their findings showed that the vertical load-bearing capacity of XCC piles increased by 1.32 times. Lv et al. 16 – 18 developed a series of numerical solutions to assess the vertical bearing capacity of XCC piles. They also captured the stress transfer mechanism of XCC pile foundations under vertical load. The results indicate that the pile with an XCC cross-section provides greater vertical bearing capacity than a conventional circular pile with the same cross-sectional area. However, the three-dimensional effect is not considered in the analytical model. To address this limitation, Zhou et al. 19 , 20 developed a simplified analytical model based on the CEL approach. This model investigates the effect of the XCC cross-section on the penetration mechanism through finite element limit analysis. Ren et al. 21 designed a five-star pile based on a circular section concave inwards with five circular arcs, as shown in Fig. 1c. They examined the influence of different template arcs and angles on the geometric properties of the five-star pile. Through laboratory tests, they explored its vertical bearing characteristics, finding that the vertical ultimate bearing capacity increased by 2.4 times compared to a circular pile with the same cross-sectional area. Deng et al. 22 designed the plum blossom pile based on a circular cross-section convex with five circular arcs, as shown in Fig. 1d. They conducted preliminary studies through laboratory tests and numerical simulations. These studies examined the effects of the radius of the external tangent circle and the degree of the open arc on the geometric properties of the plum blossom pile cross-section. The existing research mainly focuses on the design of special-shaped piles with circular cross-section, while relatively little attention has been paid to the design of special-shaped piles based on square cross-section. Based on the research results of the existing special-shaped piles, it can be observed that the number of internal concave arcs and the degree of internal concave arcs are crucial parameters affecting the vertical load bearing characteristics of special-shaped piles. Nevertheless, the comparative analyzes of the effects of different numbers of internal concave sides and internal concave angle on the vertical load bearing characteristics of special-shaped piles in the current study remain insufficient. Consequently, this paper integrates two critical cross-section parameters—the number of concave edges and the angle of concavity—that affect the vertical load bearing characteristics of existing special-shaped piles and incorporates the characteristics of square cross-sections to design a novel special-shaped pile. Through theoretical analysis and numerical simulation methods, the study investigates how variations in these parameters influence the geometric properties of the pile cross-section and the vertical load bearing characteristics of the special-shaped pile under constant cross-sectional area conditions. 2. Design of special-shaped pile sections To avoid the influence of anisotropy caused by irregular cross-section, and to incorporate the characteristics of multiple existing special-shaped pile cross-sections, the special-shaped pile cross-section is simplified into an axisymmetric shape. Based on the square cross-section and following the principle of maintaining the same cross-sectional area, n isosceles triangles with an angle 𝜃 are excavated to form the special-shaped pile cross-section proposed in this paper. The cross-section is primarily controlled by three variables: the edge length of the outer square cross-section ( Ln ), the number of concave edges ( n ), and the angle of concavity (𝜃). The schematic diagram of the special-shaped cross-section is shown in Fig. 2. To study the influence of the number of concave sides ( n ) and the concave angle (𝜃) on the cross-sectional properties, the cross-sectional area of the special-shaped pile needs to be determined based on the structural dimensions of piles with square cross-section. By analyzing the dimensions of commonly used square piles in actual projects and referencing relevant specifications 23 , the cross-sectional area of the special-shaped pile is set at 0.16 m 2 . According to the characteristics of square cross-section, the number of concave sides ( n ), satisfies n ≤ 4. when n = 0 or 𝜃 = 0°, the cross section is square. When the concave angle 𝜃 30°, the corners of the cross-section become too sharp, making the design impractical. Therefore, this study selects a range for the concave angle 𝜃 between 5° and 30°. Under the condition of maintaining a constant cross-sectional area, a total of 24 groups of model piles were designed to analyze the geometric properties of the cross-sections and their vertical bearing capacities. The parameters of the model piles are shown in Table 1 . Table 1 Model number and special parameters. Model number 𝜃 n L/mm SSP- 5 − 1 5° 1 404 SSP- 5 − 2 5° 2 409 SSP- 5 − 3 5° 3 414 SSP- 5 − 4 5° 4 419 SSP-10-1 10° 1 409 SSP-10-2 10° 2 419 SSP-10-3 10° 3 429 SSP-10-4 10° 4 441 SSP-15-1 15° 1 414 SSP-15-2 15° 2 430 SSP-15-3 15° 3 447 SSP-15-4 15° 4 468 SSP-20-1 20° 1 420 SSP-20-2 20° 2 442 SSP-20-3 20° 3 469 SSP-20-4 20° 4 502 SSP-25-1 25° 1 426 SSP-25-2 25° 2 457 SSP-25-3 25° 3 496 SSP-25-4 25° 4 548 SSP-30-1 30° 1 432 SSP-30-2 30° 2 474 SSP-30-3 30° 3 531 SSP-30-4 30° 4 615 3. Analysis of geometric properties of special-shaped pile sections When different values of the number of concave edges n and the concave angle 𝜃 of the cross-section are selected, the geometric properties of the special-shaped pile cross-section change, which in turn affects the mechanical properties of the pile. To investigate this, the control variable method is employed to analyze the effects of varying the number of concave edges n and the concave angle 𝜃 on the geometric properties of the special-shaped pile cross-section. 3.1. Section circumference For the special-shaped pile cross-section, the area S is defined as the difference between the area of the outer square cross-section and the total area of n isosceles triangles with specified base angles. The relationship between the side length of the outer square cross-section Ln and the area S of the special-shaped pile, given various cross-sectional parameters, is expressed as follows: $$S=L_{n}^{2} - 0.25nL_{n}^{2}\tan \theta$$ 1 When the cross-sectional area S of the special-shaped pile remains constant, the cross-sectional circumference C is defined as the sum of the waist lengths of n isosceles triangles and the side lengths of the (4-n) outer square segments. The relationship for the cross-sectional circumference C of the special-shaped pile, as a function of the number of concave sides n and the angle of concavity 𝜃, is expressed as follows: $$C\left( {n,\theta } \right)=\left( {4 - n+\frac{n}{{\cos \theta }}} \right)\sqrt {\frac{{4S}}{{4 - n\tan \theta }}}$$ 2 Taking the cross-sectional area S of the special-shaped pile as 0.16 m 2 , the relationship between the cross-sectional perimeter of the special-shaped pile and the cross-sectional parameters, namely the concave angle 𝜃 and the number of concave edges n , can be obtained using Eq. 2 , as shown in Fig. 3 . The perimeter of the cross-section increases with an increase in both the number of concave edges ( n ) and the concave angle (𝜃). When the concave angle (𝜃) is small, the cross-sectional perimeter shows minimal variation with an increasing number of concave edges n . However, as the concave angle gradually increases, the cross-sectional perimeter grows significantly with the increase in the number of concave edges. For example, when the number of concave edges n = 1, the cross-sectional perimeter increases from 1.618 m to 1.795 m as the concave angle 𝜃 increases from 5° to 30°. Similarly, when the concave angle 𝜃 = 5°, the cross-sectional perimeter increases from 1.618 m to 1.682 m as the number of concave edges ( n ) increases from 1 to 4. These results indicate that the concave angle 𝜃 has the most significant effect on the cross-sectional perimeter. Specifically, the perimeter increases by a factor of 1.58 when the number of concave edges is n = 4 and the concave angle 𝜃 changes from 5° to 30°. 3.2. Ratio of same cross section circumference Under vertical loading conditions, when the cross-sectional area of the pile end is constant, the side resistance 𝑄 can be calculated as 24 : $$Q={\tau _f} \cdot Ch$$ 3 Where \({\tau _f}\) is the average side resistance (kPa), C is the circumference of the pile section (m), and h is the pile length (m). From the formula of pile side resistance, it is evident that the side resistance of the special-shaped pile is proportional to the perimeter of its cross-section. Based on the perimeter of the cross-section obtained in Fig. 3 , the ratio curves of the circumference of the special-shaped pile with varying cross-sectional parameters relative to that of the square pile with the same cross-sectional area are plotted. As illustrated in Fig. 4 , the circumference of the special-shaped pile is between 1.01 and 1.78 times that of the square pile with the same cross-sectional area. This indicates that the lateral surface area of the special-shaped pile is 1.01 to 1.78 times greater than that of the square pile with the same cross-sectional area and pile length. Thus, under equivalent conditions, the special-shaped pile exhibits enhanced side resistance and superior vertical bearing capacity compared to the square pile with the same cross-sectional area. 3.3. Analysis of cross-sectional moment of inertia In the design of special-shaped piles, changes in the cross-sectional parameters of the pile also result in variations in the cross-sectional moment of inertia. The cross-sectional moment of inertia is a critical parameter for assessing the bending performance of the pile foundation. When the materials used in the pile are consistent, the bending stiffness of the special-shaped pile primarily depends on the maximum moment of inertia of the cross-section 25 . The calculation results for the maximum moment of inertia of the special-shaped pile section are presented in Fig. 5 . It is evident that the maximum moment of inertia increases with the internal concave angle (𝜃). Due to the varying concave angles, the maximum moment of inertia of the special-shaped pile section exhibits distinct trends as the number of concave sides ( n ) increases. Specifically, when the concave angle 𝜃 < 25°, the maximum moment of inertia shows an initial increase followed by a decrease with the increasing number of concave sides ( n ), with two concave sides yielding the largest moment of inertia, indicating optimal bending resistance. When the concave angle 𝜃 = 30°, the maximum moment of inertia first increases, then decreases, and subsequently increases again, with the moment of inertia of the cross-section of the special-shaped pile being largest for two concave sides. The maximum moment of inertia for the special-shaped pile section is at its minimum when the number of concave sides n = 1 and the concave angle 𝜃 = 5°, with a value of 0.0022 m 4 , which is 1.07 times that of a square pile with the same cross-sectional area. Conversely, the maximum moment of inertia occurs when the number of concave sides n = 2 and the angle of concavity 𝜃 = 30°, yielding a value of 0.00348 m 4 , which is 1.63 times that of a square pile with the same cross-sectional area. Thus, the special-shaped pile possesses a larger moment of inertia than the square pile with the same cross-sectional area, indicating superior bending resistance. 4. Numerical calculation The special-shaped piles theoretically have superior vertical bearing capacity compared to square piles with the same cross-sectional area. To further study the influence of the number of concave sides ( n ) and the angle of concavity (𝜃) on the vertical load bearing characteristics of the special-shaped piles, a numerical calculation model is established using the finite element software ABAQUS. 4.1. Finite element modeling The pile is C30 concrete, which is simulated using a linear elastic model, while the soil is modeled with a Mohr–Coulomb model. The numerical models of both the piles and soil bodies are meshed using eight-node linear hexahedral cells (C3D8R), and the relevant material calculation parameters are shown in Table 2 . The pile-soil interaction is achieved by establishing master-slave contact pairs, with the normal contact surface modeled as hard contact and the tangential direction as Coulomb friction. The friction coefficient remains constant throughout the loading process and is taken as the tangent of the internal friction angle (tan(φ) = 0.45). The geometrical boundary dimensions of the soil are consistent with the literature model, measuring 15 m in length, 15 m in width, and 28 m in depth. To avoid boundary effects, the pile length is set to 14 m. The soil is modeled as free and unconstrained at the top, completely fixed at the bottom, and with no horizontal displacement at the sides. The soil mesh adopts a single-precision radial meshing method to ensure that the mesh is dense near the pile and gradually becomes sparse away from the pile. After conducting a grid sensitivity analysis, it is confirmed that the above meshing method ensures both the calculation accuracy and the convergence of the model. The details of the 3D model and the mesh of the special-shaped pile section are shown in Fig. 6 . Table 2 Material parameters of pile and clay. Material Clay Pile Constitutive model Mohr–Coulomb Linear elasticity Elastic modulus [E/(GPa)] 0.016 20 Density [ρ/(kg·m -3 )] 1800 2300 Poisson’s ratio [v] 0.3 0.2 Internal friction angle [φ/(◦)] 23 - Cohesion [c/(kPa)] 10.9 - Dilatancy angle[ψ/(◦)] 15 - 4.2. Geostress balancing and loading steps In engineering practice, the soil has already experienced consolidation and settlement due to gravity before pile casting. Therefore, in numerical simulation, the initial ground stress should be applied to the soil first to simulate the actual situation. Considering the high accuracy and simplicity of the automatic ground stress balance method, this paper chooses this method to simulate the ground stress state of the soil. The loading steps in the numerical simulation process are as follows: First geostress balance: the soil and pile models are constructed. During the initial geostress equilibrium, the pile units and pile-soil contact pairs are disabled using the birth-death unit method. The horizontal displacement of the soil is restricted, the bottom of the soil is fixed, and gravity is applied to achieve the initial equilibrium of the geostress. Second geostress balance: After the initial geostress balance is completed, the soil units occupied by the piles are removed using the live-dead unit method. The previously disabled pile units and pile-soil contact pairs are then reactivated, and the soil is rebalanced for ground stresses. Applying pile top load: after the second geostress balance is completed, vertical loads are applied to the pile top in a graded manner. 4.3. Validation of the model To ensure that the numerical model developed in this paper can effectively analyze the vertical bearing characteristics of single pile foundations, two cases from published literature are selected for verification. (1) Case 1 The field load test conducted by Chen Zheng et al. 26 was selected to verify the accuracy of the numerical simulation. The schematic diagram of the field load test is shown in Fig. 7 . The parameter settings of the numerical model for the vertical static load test on a single pile, as performed by Chen Zheng et al. 26 , are as follows: the pile is a circular pile, the length of the pile is 18 m, the diameter of the pile is 0.3 m, the modulus of elasticity of the pile is 20 GPa, the Poisson's ratio is 0.2, the density of the soil around the pile is 1800 kg/m 3 , the angle of internal friction is 23°, the angle of dilatancy is 15°, and the cohesive force is 10.9 kPa. The results of the field static load test of the single pile and the numerical simulation are shown in Fig. 8 . When the pile top load is small, the settlement values of the numerical simulation curve and the literature test curve are close to each other; when the load is large, the displacement values of the numerical simulation are slightly larger than the test values. However, the overall trend is in good agreement, which indicates that the numerical model constructed in this paper has high accuracy in assessing the vertical load bearing characteristics of the pile and can realistically simulate the response of the pile in actual projects. (2) Case 2 Since the XCC-shaped pile shares similar cross-sectional geometric characteristics with the special-shaped pile ( n = 4) studied in this paper, the numerical model of the XCC pile established by Lv et al. 27 was used as a validation to assess the reliability of the numerical modeling method for special-shaped piles. Table 3 provides a detailed list of the relevant parameters for the pile and soil used in the validation model The numerical simulation results are shown in Fig. 9. The load-displacement curve obtained from the finite element model developed in this study closely matches the data reported in the literature. Furthermore, the axial force of the XCC pile was extracted using Abaqus software and compared with the data from the literature, showing a similarly close match. An analysis of the error sources suggests that the discrepancies primarily arise from differences in the mesh size used in the two models, which lead to computational errors. Therefore, it can be concluded that the numerical modeling method for special-shaped piles and the axial force extraction approach adopted in this study are both reliable and feasible. Table 3 Material parameters of pile and soil. Material Soil Pile Constitutive model Mohr–Coulomb Linear elasticity Elastic modulus [E/(GPa)] 0.08 45 Density [ρ/(kg·m -3 )] 1490 2700 Poisson’s ratio [v] 0.3 0.35 Internal friction angle [φ/(◦)] 43 - Dilatancy angle[ψ/(◦)] 0 - 5. Analysis of calculation results In order to investigate the effects of the number of concave sides n and the angle of concavity 𝜃 on the vertical load bearing characteristics of piles, we carried out three main research analyzes: (1) load-settlement curves; (2) load sharing of the side and end resistances; (3) distributions of side resistance. 5.1. Load-settlement curve The load-settlement curves for both special-shaped piles and square piles under vertical load are illustrated in Fig. 10. These curves exhibit notable similarities, displaying a slow-change behavior. When the load is less than 800 kN, the curves increase approximately linearly. However, when the load exceeds 800 kN, the load-settlement curves of each pile gradually steepen, entering a non-linear stage. This behavior is attributed to the high modulus of elasticity of the pile during loading, resulting in relatively small settlements at the pile top. Simultaneously, the axial force compresses the pile, leading to even smaller settlements at the pile end. Consequently, the pile top load is primarily supported by the pile side resistance, and the pile and soil remain in the elastic deformation stage. Thus, during the initial loading period, the load-settlement curve exhibits linear growth. As the load increases, the settlement of the pile top rises, and the settlement of the pile end also increases. This leads to a gradual increase in end resistance, while the soil at the pile's end begins to undergo plastic deformation. With continued load application, the settlement of the pile top and end increases further, causing the end resistance to rise and the load-settlement curve to start bending. According to current specifications and related calculation methods 28 , when the pile top settles to 40 mm, this load value is considered the ultimate bearing capacity of the special-shaped pile, as depicted in Fig. 11 . From Fig. 11 , it can be observed that the ultimate bearing capacity of the special-shaped pile increases with the number of internal concave sides ( n ) and the internal concave angle (𝜃). When the number of concave sides is n = 1, the ultimate bearing capacity increases from 1120 kN to 1355 kN as the concave angle (𝜃) increases from 5 to 30°, representing a 20.98% increase in ultimate bearing capacity. Similarly, when the concave angle is 𝜃 = 5° and the number of concave sides ( n ) increases from 1 to 4, the ultimate bearing capacity rises from 1120 kN to 1173 kN, showing a 4.7% improvement. These results indicate that changing the concave angle (𝜃) of the cross-section has a more significant effect on the vertical capacity of the special-shaped pile compared to altering the number of concave sides ( n ). When the number of concave sides n = 1 and the concave angle 𝜃 = 5°, the ultimate load capacity of the special-shaped pile is the smallest, measured at 1120 kN. In contrast, when the number of concave sides n = 4 and the concave angle 𝜃 = 30°, the ultimate load capacity of the special-shaped pile reaches its maximum, measured at 2053 kN, which is 1.05 ~ 1.92 times greater than that of a square pile with the same cross-sectional area. The trend of the ultimate bearing capacity of the special-shaped pile is consistent with the trend of its perimeter outlined in Section 3.2 . The perimeter of the special-shaped pile increases by 1.01 ~ 1.78 times compared to a square pile (𝜃 = 0°) with the same cross-sectional area. Furthermore, the increase in ultimate bearing capacity is 1.04 ~ 1.08 times the increase in perimeter, highlighting the prominent effect of expanding the cross-section of the special-shaped pile. 5.2. Load sharing of the side and end resistances According to the analyzes in section 2.2, changes in total side resistance significantly influence the bearing capacity of the special-shaped pile. The pile end axial force is extracted through ABAQUS and the difference between the pile top load and the pile end axial force is the total side resistance. The respective sharing ratios of the total side resistance and end resistance as the pile top load increases are obtained through numerical simulation, as shown in Fig. 12. When the pile top load is less than 600 kN, the deformation of the pile is minimal because the elastic modulus of the pile is substantially larger than that of the soil. This results in a small relative displacement between the pile and the soil, preventing the full utilization of side resistance. As the pile top load increases to 800 kN, the deformation of the pile becomes more pronounced. This leads to an increase in the relative displacement between the pile and the soil, which further activates the side resistance. Simultaneously, the soil at the pile end becomes compressed and densified. At this point, the variation in the number of concave sides ( n ) and the angle of concavity (𝜃) have less effect on the load-sharing ratio of the special-shaped pile. With further increases in the pile top load, the relative displacement between the pile and the soil continues to increase, causing the shear stress at the pile-soil contact surface to gradually reach the soil's shear strength. This leads to the full activation of side resistance, while the additional load on the pile is primarily supported by the soil at the pile's end, resulting in a decrease in the proportion borne by side resistance. This decreasing proportion diminishes at a slower rate as the number of concave edges ( n ) and the angle of concavity (𝜃) increase, eventually leveling off with further increases in the pile top load. This trend occurs because an increase in the number of concave sides ( n ) and the concave angle enhances the contact area between the pile and the soil along the pile's side. Consequently, the ratio of the pile side area to the pile end area increases, improving the proportion of load carried by pile-side friction. Moreover, it is evident that under the same load, the proportion of end resistance in the special-shaped pile is consistently smaller than that of side resistance. This indicates that the top load of the pile is predominantly supported by side resistance, reflecting the bearing characteristics of an end-bearing friction pile. 5.3. Distributions of side resistance The distribution of side resistance in special-shaped piles, relative to the top settlement of the pile under various top loads, is illustrated in Fig. 13. Under vertical loading, the side resistance of special-shaped piles exhibits different responses to settlement. During the initial loading phase, the side resistance of special-shaped piles with varying cross-sectional parameters remains largely consistent at the same settlement and is characterized by slow settlement rates. When the side resistance reaches 670 kN, relative displacement between the pile and the soil begins to occur, leading to an acceleration of settlement. This trend of accelerated subsidence decreases with an increase in the number of concave sides ( n ) and the angle of concavity (𝜃). Additionally, the increase in pile top settlement, accompanied by rising side resistance, demonstrates a tendency to increase proportionally. When the special-shaped pile reached its ultimate bearing capacity (corresponding to a pile top settlement of 40 mm), the side resistance increased from 741 kN to 782 kN as the number of concave sides (n) increased from 1 to 4 at a fixed concave angle ( θ = 5°), representing a 5.5% enhancement in ultimate side resistance. In contrast, when the number of concave sides was held constant ( n = 1), increasing the concave angle ( θ ) from 5° to 30° increased the side resistance from 741 kN to 967 kN, yielding a 30.5% improvement in ultimate side resistance. These findings demonstrate that cross-sectional optimization, achieved by increasing either the number of concave sides ( n ) or the concave angle ( θ ), enhances the side resistance of special-shaped piles, with the concave angle ( θ ) exhibiting a significantly greater influence. For practical applications, maximizing the concave angle ( θ ) is recommended, provided construction feasibility is ensured. 6. Conclusion In this paper, a novel special-shaped pile is designed based on the principle of expanding the perimeter while maintaining the same cross-sectional area. This design combines the characteristics of existing special-shaped pile cross-sections with those of square cross-sections. Theoretical analysis and numerical simulation methods are employed to study the effects of varying the number of concave sides ( n ) from 1 to 4 and adjusting the concave angle (𝜃) from 5° to 30° on the geometric properties and vertical load bearing characteristics of the special-shaped pile. These results are compared and analyzed against those of a square pile with the same cross-sectional area. The following conclusions are drawn: (1) When the concave angle is certain, with the increase of the number of concave edges, the perimeter of the cross-section of the special-shaped pile, the ultimate bearing capacity and side resistance increase, and the maximum moment of inertia of the cross-section increases and then decreases, and the maximum moment of inertia of the cross-section of the special-shaped pile is the largest when the number of concave edges is two. (2) When the number of concave sides is certain, the perimeter, ultimate bearing capacity, and maximum moment of inertia of the cross-section of the special-shaped pile increase with the increase of the concave angle. The increase of section circumference, ultimate bearing capacity and side resistance is more significant, while the increase of maximum moment of inertia of section is relatively insignificant. (3) The special-shaped pile mainly carries the load by lateral friction force, and when the number of inner concave sides is more than 2 and the angle of inner concave is more than 20°, the proportion can reach more than 80%, which shows the bearing characteristics of end-bearing friction pile. (4) Because of the larger pile side area, the special-shaped pile has larger pile side resistance, so the vertical bearing capacity and bending resistance are better than the same cross-section area square piles, the ultimate bearing capacity is 1.05 ~ 1.92 times of the same cross-section area square piles, and the cross-section circumference is 1.01 ~ 1.78 times of the same cross-section area square piles, and the effect of cross-sectional anisotropy is obviously enlarged. Declarations Data A vailability S tatement The datasets during the current study available from the corresponding author on reasonable request. Author Contributions Conceptualization, Z.C.; methodology, Z.C.; software, D.J.; validation, D.J.; formal analysis, D.J.; investigation, D.J.; resources, D.J.; data curation, D.J.; writing—original draft preparation, D.J.; writing—review and editing, D.J.; visualization, Z.Z.; supervision, Z.C.; funding acquisition, Z.C. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by Project of the Natural Science Foundation of Sichuan Province, grant number No. 2023NSFSC0046 and the State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, grant number No. SKLGP2023Z011. Competing interests The authors declare no competing interests. Additional information Correspondence and requests for materials should be addressed to Z.C. Reprints and permissions information is available at www.nature.com/reprints. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References Zhou P., Liu H., Zhou H, et al. Experimental study on the development of surrounding soil stress during XCC pile installation in sand. Acta Geotech . 19, 4017–4035(2024). Khezri, A.; Park, H.; Lee, D. Numerical Study on the Lateral Load Response of Offshore Monopile Foundations in Clay: Effect of Slenderness Ratio. Appl. Sci. 14, 8366(2024). Lv YR; Liu HL; Ding XM; Kong GQ. Field Tests on Bearing Characteristics of X-Section Pile Composite Foundation. Journal of Performance of Constructed Facilities . 26(2), 180–189(2012). Gong WM, Wang ZZ, Dai GL, Liu XG, Zhu MX, Zhao XX, Guo H. Foundations of Yangtze River mainstream bridges in China. Proc Inst Civ Eng-Fo . 173(1),13–24(2020). Wang, XQ, Zhang SM, Huang Y. The Development and Research of Special-Shape Pile. Applied Mechanics and Materials . 578-579, 210 - 213(2014). Yuan BX, Li ZH, Zhao ZQ, et al. Experimental study of displacement field of layered soils surrounding laterally loaded pile based on Transparent Soil. J Soil Sediment . 21,3072-3083(2021). Wang, XQ., Chen, YH., Lin F . Evaluation of ultimate bearing capacity of Y-shaped vibro-pile. J. Cent. South Univ. Technol. 15 (Suppl 2), 186–194(2008). Chen YH, Wang XQ and Liu HL. Research on abnormity characteristic of Y-shaped tube-sinking cast-in-situ pile in highway soft ground. China Journal of Highway and Transport . 21(5), 9-25(2008). Liu HL, ZHOU H, KONG GQ. XCC pile installation effect in soft soil ground: A simplified analytical model. Comput . Geotech . ,62 , 268－282(2014). Zhou H, Liu HL, Ding XM, Kong GQ. A p-y curve model for laterally loaded XCC pile in soft clay. Acta Geotech . 15,3229–3242(2020). Zhou P, Liu HL, Zhou H, Cao GW, Ding XM. A lateral soil resistance model for XCC pile in soft clay considering the effect of the geometry of cross section. Acta Geotech . 17(10) ,4681–4697(2022). Zhou P, Liu HL, Zhou H, Cao GW, Ding XM. A simplified analysis approach for the effect of the installation of adjacent XCC pile on the existing single XCC pile in undrained clay. Acta Geotech . 17(12) ,5799–5519(2022). Zhou H, Liu HL, Li YZ, Ding XM. Limit lateral resistance of XCC pile group in undrained soil. Acta Geotech . 15(6),1673–1683(2020). Zhou H, Yuan JR, Liu HL, Kong GQ. Analytical model for evaluating XCC pile shaft capacity in soft soil by incorporating penetration effects. Soils Found . 58(5),1093–1112(2018). Li XC, Zhou H, Liu HL, Chen ZX. Three-dimensional analytical continuum model for axially loaded noncircular piles in multilayered elastic soil. Int J Numer Anal Met . 45,2654–2681(2021). Lv YR, Zhang DD. Geometrical effects on the load transfer mechanism of pile groups: Three-dimensional numerical analysis. Can Geotech J . 55,749–757(2018). Lv YR, Liu HL, Ng CWW, Ding XM, Gunawan A. Three-dimensional numerical analysis of the stress transfer mechanism of XCC piled raft foundation. Comput . Geotech . 55,365–377(2014). Lv YR, Liu HL, Ng CWW, Gunawan A, Ding XM. A modified analytical solution of soil stress distribution for XCC pile foundations. Acta Geotech. 9(3),529–546(2014). Zhou H, Liu HL, Yuan JR, Chu J. Numerical simulation of XCC pile penetration in undrained clay. Comput . Geotech . 106,18–41(2019). Zhou H, Liu HL, Wang LH, Kong GQ. Finite element limit analysis of ultimate lateral pressure of XCC pile in undrained clay. Comput . Geotech . 95,240–246(2018). Ren LW, Zhan JF, Yang QW. Section optimization and model test study on bearing mechanisms of five-star-shaped pile. Rock Soil Mech . 38 (10), 2855-2864(2017). Li L, Deng, YS. Strengthening mechanism of plum blossom pile composite foundation. Acta Geotech . 19, 4791–4808(2024). JGJ/T 405-2017; Technical Specification for Prestressed Concrete Shaped Precast Piles. China Architecture & Building Press: Beijing, China (2017). Mohamed Ashour, M. ASCE, Amr Helal. Contribution of Vertical Skin Friction to the Lateral Resistance of Large-Diameter Shafts. Journal of Bridge Engineering . 19(2),289-302(2013). Gil-Martín, L.M., Fernández-Ruiz, M.A., & Hernández-Montes E. Effective Moment of Inertia of Reinforced Concrete Piles. ACI Structural Journal . ,119(5),167-178(2022). Chen Z, Su RJ. The Influence of Different Loading Sequence on Bearing Behavior of Micropile. Archit. Sci. , 27(03),45-48+54(2011). Lv YR, Zhang DD. Geometrical effects on the load transfer mechanism of pile groups: three-dimensional numerical analysis. Can. Geotech. J . 55(5): 749-757(2018). JGJ94-2008; Technical specification for building pile foundations. China Architecture & Building Press: Beijing, China(2008). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 28 Apr, 2025 Reviewers agreed at journal 23 Apr, 2025 Reviews received at journal 23 Apr, 2025 Reviewers agreed at journal 23 Apr, 2025 Reviewers invited by journal 23 Apr, 2025 Submission checks completed at journal 18 Apr, 2025 First submitted to journal 05 Apr, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5535232","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":447166222,"identity":"a6489300-b2d1-4d34-b079-e61dde668e50","order_by":0,"name":"De Jiang","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"De","middleName":"","lastName":"Jiang","suffix":""},{"id":447166223,"identity":"3f8f6ffe-fdd4-4434-8a40-12a5cfee5056","order_by":1,"name":"Zhenlin Chen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVRIiWNgGAWjYBACfoaDjQ8+VNjIEa9FsvFws+GMM2nGxGsxaD7eJs3bdjixgXgtbAebDXjOMKf3HU9g/PAxhwgt5jxAv0hUsOXOPPOAWXLmNiK0WM4A2mJwhid3w40ENmZeYrQY3H/YJpHYJpFuQLyWAwfbJA62GSQQr0Wy4WCzYcOZBMOZZx42E+cXfobjDx//qfgvz3c8+eCHj8RoQYADJEQNTEsCqTpGwSgYBaNgpAAAZcZB5M3s330AAAAASUVORK5CYII=","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Zhenlin","middleName":"","lastName":"Chen","suffix":""},{"id":447166225,"identity":"194b1d21-a167-41ba-819b-7c808ec2511e","order_by":2,"name":"Zhiguang Zhang","email":"","orcid":"","institution":"Chengdu University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Zhiguang","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2024-11-27 12:08:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5535232/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5535232/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":81261206,"identity":"5d542376-c8fb-46ce-9588-9d7aa13e6f1e","added_by":"auto","created_at":"2025-04-24 06:20:33","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":44982,"visible":true,"origin":"","legend":"\u003cp\u003eSome typical cross sections of special-shaped pile:(a) Y-shaped pile\u003csup\u003e6\u003c/sup\u003e, (b) X-shaped pile\u003csup\u003e8\u003c/sup\u003e, (c) five-star shaped pile\u003csup\u003e20\u003c/sup\u003e, (d) plum blossom pile\u003csup\u003e21\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/9600f0a9c37fb5fc20486025.png"},{"id":81260058,"identity":"dad3b4bf-a16f-43f9-8b8e-cf938de87517","added_by":"auto","created_at":"2025-04-24 06:04:33","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":44979,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic cross-section of special-shaped piles with different cross-section parameters:(a) the number of concave edges \u003cem\u003en\u003c/em\u003e=1, (b) the number of concave edges \u003cem\u003en\u003c/em\u003e=2, (c) the number of concave edges \u003cem\u003en\u003c/em\u003e=3, (d) the number of concave edges \u003cem\u003en\u003c/em\u003e=4.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/9a1d32eb5eb93cfb7e077933.png"},{"id":81260056,"identity":"e09e9070-42f8-4609-ab58-40768ef36047","added_by":"auto","created_at":"2025-04-24 06:04:33","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":181747,"visible":true,"origin":"","legend":"\u003cp\u003eInfluence of \u003cem\u003en\u003c/em\u003e, 𝜃 on section circumference.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/c7dc7ca532f6caf15ca71a22.png"},{"id":81260083,"identity":"5f4011c8-6cac-4fe5-bee4-51739a1391fd","added_by":"auto","created_at":"2025-04-24 06:04:35","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":90049,"visible":true,"origin":"","legend":"\u003cp\u003eInfluence of \u003cem\u003en\u003c/em\u003e, 𝜃 on the perimeter ratio of same cross section circumference.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/79694658a808ff2bf41399b8.png"},{"id":81260874,"identity":"bce60d92-d4fb-45e9-950d-c2cbbcd7612c","added_by":"auto","created_at":"2025-04-24 06:12:33","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":102960,"visible":true,"origin":"","legend":"\u003cp\u003eChanges of maximum moment of inertia of pile section affected by \u003cem\u003en\u003c/em\u003e, 𝜃 of special-shaped pile.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/66bfac32b81db4c571d72a13.png"},{"id":81260060,"identity":"27aace46-5e41-45bd-9546-e68060540710","added_by":"auto","created_at":"2025-04-24 06:04:33","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":307828,"visible":true,"origin":"","legend":"\u003cp\u003eView of numerical model.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/4e5976166dd14243858d86b1.png"},{"id":81261207,"identity":"1ef75ed0-fcb3-4313-a238-1d92f3a7ba93","added_by":"auto","created_at":"2025-04-24 06:20:33","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":148359,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of field static load test loading.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/6f88fbcdd78503f343040c79.png"},{"id":81260055,"identity":"f305ef9d-43d2-4f66-816b-524d4bd2faea","added_by":"auto","created_at":"2025-04-24 06:04:33","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":39031,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of measured and simulation load–displacement curves.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/87d0aecffeba9b9168a2518e.png"},{"id":81260062,"identity":"51053c50-ba3b-49e3-ae6c-366920fb8c58","added_by":"auto","created_at":"2025-04-24 06:04:33","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":95587,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of measured and simulation: (a)load–settlement curves; (b) axial force.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/b99393ac401a12488a7a77bd.png"},{"id":81260069,"identity":"5984e99f-a1c1-465d-9dfa-1e7170f6129f","added_by":"auto","created_at":"2025-04-24 06:04:34","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":229267,"visible":true,"origin":"","legend":"\u003cp\u003eLoad–settlement curves of piles: (a) the number of concave edges \u003cem\u003en\u003c/em\u003e= 1, (b) the number of concave edges \u003cem\u003en \u003c/em\u003e= 2, (c) the number of concave edges \u003cem\u003en \u003c/em\u003e= 3, (d) the number of concave edges \u003cem\u003en \u003c/em\u003e= 4.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/2a23f2ecc95d76c45f9e6e34.png"},{"id":81260893,"identity":"b8c29d4b-7190-434a-84af-334d7d909685","added_by":"auto","created_at":"2025-04-24 06:12:36","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":70767,"visible":true,"origin":"","legend":"\u003cp\u003eUltimate bearing capacity of special-shaped piles with different cross-sectional parameters.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/a7e55561cb3aae4c7cfa82f9.png"},{"id":81260876,"identity":"9c1f83f5-34b0-47f5-9480-4065f62e195d","added_by":"auto","created_at":"2025-04-24 06:12:33","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":360546,"visible":true,"origin":"","legend":"\u003cp\u003eLoad-sharing of side resistance and end resistance: (a) the number of concave edges \u003cem\u003en \u003c/em\u003e= 1, (b) the number of concave edges \u003cem\u003en\u003c/em\u003e = 2, (c) the number of concave edges \u003cem\u003en\u003c/em\u003e= 3, (d) the number of concave edges \u003cem\u003en\u003c/em\u003e = 4.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/38afc9a8632d6af1e38bb2fa.png"},{"id":81260074,"identity":"7d60f8c4-f0a4-4dd7-a963-f11e9f8b32df","added_by":"auto","created_at":"2025-04-24 06:04:34","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":218076,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship curves between total side resistance and top settlement of pile: (a) the number of concave edges \u003cem\u003en \u003c/em\u003e= 1, (b) the number of concave edges \u003cem\u003en \u003c/em\u003e= 2, (c) the number of concave edges \u003cem\u003en \u003c/em\u003e= 3, (d) the number of concave edges \u003cem\u003en \u003c/em\u003e= 4.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/f75cb070e7ec435a9b02f970.png"},{"id":81261698,"identity":"1f75c41e-7eb2-46b1-9521-dba8e8d43a61","added_by":"auto","created_at":"2025-04-24 06:28:42","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2638222,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5535232/v1/5e6d8647-67bb-4008-b54a-a69ce7bf5723.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Research on optimized design of section parameters of special-shaped pile based on square section","fulltext":[{"header":"1. Introduction","content":"\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003ePile foundation is a force-transmitting member that transfers the load of a building to the foundation soil. With the rapid development of the economy, the demand for buildings with higher bearing capacity is increasing, and the traditional pile foundation is difficult to satisfy the safety and economy requirements of buildings at the same time\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e.In response to this problem, a new type of pile foundation, the special-shaped pile, has been developed. By changing the cross-section, it can improve the pile-soil contact area without increasing the amount of concrete, thereby effectively enhancing side resistance and the bearing capacity of the pile foundation. This type of pile foundation is widely used in high-rise buildings, highway construction, soft soil foundation treatment, and other practical projects\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. However, the design of special-shaped piles is still immature, and their vertical load bearing characteristics require further study \u003csup\u003e\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\n \u003cp\u003eIn recent years, a series of efforts have been made to study the design and vertical load bearing characteristics of special-shaped piles. Wang et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e examined the design and bearing characteristics of Y-shaped piles, as shown in Fig. 1a. These piles are formed by concaving three circular arcs inward based on a circular cross-section. They proposed a simplified calculation method for bearing capacity. Chen et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e investigated the effects of different template radii and angles on the geometric properties of Y-shaped piles. Their experimental study revealed that the perimeter of the cross-section increased by 1.46 times. Moreover, the vertical ultimate load-carrying capacity improved by approximately 1.5 times compared to circular piles with the same cross-sectional area. Liu et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e designed XCC piles based on a circular cross-section with four circular arcs concave inwards, as shown in Fig. 1b. They examined the effects of different formwork curvatures and open-arc spacing on the geometric properties of XCC piles. Through model tests, they compared the load-bearing characteristics of XCC piles with that of circular piles. Their findings showed that the vertical load-bearing capacity of XCC piles increased by 1.32 times. Lv et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e developed a series of numerical solutions to assess the vertical bearing capacity of XCC piles. They also captured the stress transfer mechanism of XCC pile foundations under vertical load. The results indicate that the pile with an XCC cross-section provides greater vertical bearing capacity than a conventional circular pile with the same cross-sectional area. However, the three-dimensional effect is not considered in the analytical model. To address this limitation, Zhou et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e developed a simplified analytical model based on the CEL approach. This model investigates the effect of the XCC cross-section on the penetration mechanism through finite element limit analysis. Ren et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e designed a five-star pile based on a circular section concave inwards with five circular arcs, as shown in Fig. 1c. They examined the influence of different template arcs and angles on the geometric properties of the five-star pile. Through laboratory tests, they explored its vertical bearing characteristics, finding that the vertical ultimate bearing capacity increased by 2.4 times compared to a circular pile with the same cross-sectional area. Deng et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e designed the plum blossom pile based on a circular cross-section convex with five circular arcs, as shown in Fig. 1d. They conducted preliminary studies through laboratory tests and numerical simulations. These studies examined the effects of the radius of the external tangent circle and the degree of the open arc on the geometric properties of the plum blossom pile cross-section.\u003c/p\u003e\n \u003cp\u003eThe existing research mainly focuses on the design of special-shaped piles with circular cross-section, while relatively little attention has been paid to the design of special-shaped piles based on square cross-section. Based on the research results of the existing special-shaped piles, it can be observed that the number of internal concave arcs and the degree of internal concave arcs are crucial parameters affecting the vertical load bearing characteristics of special-shaped piles. Nevertheless, the comparative analyzes of the effects of different numbers of internal concave sides and internal concave angle on the vertical load bearing characteristics of special-shaped piles in the current study remain insufficient.\u003c/p\u003e\n \u003cp\u003eConsequently, this paper integrates two critical cross-section parameters\u0026mdash;the number of concave edges and the angle of concavity\u0026mdash;that affect the vertical load bearing characteristics of existing special-shaped piles and incorporates the characteristics of square cross-sections to design a novel special-shaped pile. Through theoretical analysis and numerical simulation methods, the study investigates how variations in these parameters influence the geometric properties of the pile cross-section and the vertical load bearing characteristics of the special-shaped pile under constant cross-sectional area conditions.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"2. Design of special-shaped pile sections","content":"\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eTo avoid the influence of anisotropy caused by irregular cross-section, and to incorporate the characteristics of multiple existing special-shaped pile cross-sections, the special-shaped pile cross-section is simplified into an axisymmetric shape. Based on the square cross-section and following the principle of maintaining the same cross-sectional area, \u003cem\u003en\u003c/em\u003e isosceles triangles with an angle 𝜃 are excavated to form the special-shaped pile cross-section proposed in this paper. The cross-section is primarily controlled by three variables: the edge length of the outer square cross-section (\u003cem\u003eLn\u003c/em\u003e), the number of concave edges (\u003cem\u003en\u003c/em\u003e), and the angle of concavity (𝜃). The schematic diagram of the special-shaped cross-section is shown in Fig. 2.\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eTo study the influence of the number of concave sides (\u003cem\u003en\u003c/em\u003e) and the concave angle (𝜃) on the cross-sectional properties, the cross-sectional area of the special-shaped pile needs to be determined based on the structural dimensions of piles with square cross-section. By analyzing the dimensions of commonly used square piles in actual projects and referencing relevant specifications\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e, the cross-sectional area of the special-shaped pile is set at 0.16 m\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e. According to the characteristics of square cross-section, the number of concave sides (\u003cem\u003en\u003c/em\u003e), satisfies \u003cem\u003en\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;4. when \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 or 𝜃 = 0\u0026deg;, the cross section is square. When the concave angle 𝜃 \u0026lt; 5\u0026deg;, the difference in the side length of the outer square cross-section of the special-shaped cross-section compared to the square cross-section is negligible. Conversely, when 𝜃 \u0026gt; 30\u0026deg;, the corners of the cross-section become too sharp, making the design impractical. Therefore, this study selects a range for the concave angle 𝜃 between 5\u0026deg; and 30\u0026deg;. Under the condition of maintaining a constant cross-sectional area, a total of 24 groups of model piles were designed to analyze the geometric properties of the cross-sections and their vertical bearing capacities. The parameters of the model piles are shown in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eModel number and special parameters.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eModel number\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e𝜃\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003en\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eL/mm\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP- 5\u0026thinsp;\u0026minus;\u0026thinsp;1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e404\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP- 5\u0026thinsp;\u0026minus;\u0026thinsp;2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e409\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP- 5\u0026thinsp;\u0026minus;\u0026thinsp;3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e414\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP- 5\u0026thinsp;\u0026minus;\u0026thinsp;4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e419\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-10-1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e409\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-10-2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e419\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-10-3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e429\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-10-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e441\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-15-1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e414\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-15-2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e430\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-15-3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e447\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-15-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e468\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-20-1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e420\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-20-2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e442\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-20-3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e469\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-20-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e502\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-25-1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e426\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-25-2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e457\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-25-3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e496\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-25-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e548\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-30-1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e432\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-30-2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e474\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-30-3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e531\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSSP-30-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e615\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"3. Analysis of geometric properties of special-shaped pile sections","content":"\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eWhen different values of the number of concave edges \u003cem\u003en\u003c/em\u003e and the concave angle 𝜃 of the cross-section are selected, the geometric properties of the special-shaped pile cross-section change, which in turn affects the mechanical properties of the pile. To investigate this, the control variable method is employed to analyze the effects of varying the number of concave edges \u003cem\u003en\u003c/em\u003e and the concave angle 𝜃 on the geometric properties of the special-shaped pile cross-section.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e3.1. Section circumference\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eFor the special-shaped pile cross-section, the area \u003cem\u003eS\u003c/em\u003e is defined as the difference between the area of the outer square cross-section and the total area of \u003cem\u003en\u003c/em\u003e isosceles triangles with specified base angles. The relationship between the side length of the outer square cross-section \u003cem\u003eLn\u003c/em\u003e and the area \u003cem\u003eS\u003c/em\u003e of the special-shaped pile, given various cross-sectional parameters, is expressed as follows:\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$S=L_{n}^{2} - 0.25nL_{n}^{2}\\tan \\theta$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eWhen the cross-sectional area S of the special-shaped pile remains constant, the cross-sectional circumference \u003cem\u003eC\u003c/em\u003e is defined as the sum of the waist lengths of \u003cem\u003en\u003c/em\u003e isosceles triangles and the side lengths of the (4-n) outer square segments. The relationship for the cross-sectional circumference \u003cem\u003eC\u003c/em\u003e of the special-shaped pile, as a function of the number of concave sides \u003cem\u003en\u003c/em\u003e and the angle of concavity 𝜃, is expressed as follows:\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$C\\left( {n,\\theta } \\right)=\\left( {4 - n+\\frac{n}{{\\cos \\theta }}} \\right)\\sqrt {\\frac{{4S}}{{4 - n\\tan \\theta }}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n \u003c/div\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eTaking the cross-sectional area S of the special-shaped pile as 0.16 m\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e, the relationship between the cross-sectional perimeter of the special-shaped pile and the cross-sectional parameters, namely the concave angle 𝜃 and the number of concave edges \u003cem\u003en\u003c/em\u003e, can be obtained using Eq. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. The perimeter of the cross-section increases with an increase in both the number of concave edges (\u003cem\u003en\u003c/em\u003e) and the concave angle (𝜃). When the concave angle (𝜃) is small, the cross-sectional perimeter shows minimal variation with an increasing number of concave edges \u003cem\u003en\u003c/em\u003e. However, as the concave angle gradually increases, the cross-sectional perimeter grows significantly with the increase in the number of concave edges. For example, when the number of concave edges \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, the cross-sectional perimeter increases from 1.618 m to 1.795 m as the concave angle 𝜃 increases from 5\u0026deg; to 30\u0026deg;. Similarly, when the concave angle 𝜃 = 5\u0026deg;, the cross-sectional perimeter increases from 1.618 m to 1.682 m as the number of concave edges (\u003cem\u003en\u003c/em\u003e) increases from 1 to 4. These results indicate that the concave angle 𝜃 has the most significant effect on the cross-sectional perimeter. Specifically, the perimeter increases by a factor of 1.58 when the number of concave edges is \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4 and the concave angle 𝜃 changes from 5\u0026deg; to 30\u0026deg;.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e3.2. Ratio of same cross section circumference\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eUnder vertical loading conditions, when the cross-sectional area of the pile end is constant, the side resistance 𝑄 can be calculated as\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e:\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$Q={\\tau _f} \\cdot Ch$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau _f}\\)\u003c/span\u003e\u003c/span\u003e is the average side resistance (kPa), \u003cem\u003eC\u003c/em\u003e is the circumference of the pile section (m), and \u003cem\u003eh\u003c/em\u003e is the pile length (m).\u003c/p\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eFrom the formula of pile side resistance, it is evident that the side resistance of the special-shaped pile is proportional to the perimeter of its cross-section. Based on the perimeter of the cross-section obtained in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, the ratio curves of the circumference of the special-shaped pile with varying cross-sectional parameters relative to that of the square pile with the same cross-sectional area are plotted. As illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e, the circumference of the special-shaped pile is between 1.01 and 1.78 times that of the square pile with the same cross-sectional area. This indicates that the lateral surface area of the special-shaped pile is 1.01 to 1.78 times greater than that of the square pile with the same cross-sectional area and pile length. Thus, under equivalent conditions, the special-shaped pile exhibits enhanced side resistance and superior vertical bearing capacity compared to the square pile with the same cross-sectional area.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3. Analysis of cross-sectional moment of inertia\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eIn the design of special-shaped piles, changes in the cross-sectional parameters of the pile also result in variations in the cross-sectional moment of inertia. The cross-sectional moment of inertia is a critical parameter for assessing the bending performance of the pile foundation. When the materials used in the pile are consistent, the bending stiffness of the special-shaped pile primarily depends on the maximum moment of inertia of the cross-section\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\n \u003cp\u003eThe calculation results for the maximum moment of inertia of the special-shaped pile section are presented in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. It is evident that the maximum moment of inertia increases with the internal concave angle (𝜃). Due to the varying concave angles, the maximum moment of inertia of the special-shaped pile section exhibits distinct trends as the number of concave sides (\u003cem\u003en\u003c/em\u003e) increases. Specifically, when the concave angle 𝜃 \u0026lt; 25\u0026deg;, the maximum moment of inertia shows an initial increase followed by a decrease with the increasing number of concave sides (\u003cem\u003en\u003c/em\u003e), with two concave sides yielding the largest moment of inertia, indicating optimal bending resistance. When the concave angle 𝜃 = 30\u0026deg;, the maximum moment of inertia first increases, then decreases, and subsequently increases again, with the moment of inertia of the cross-section of the special-shaped pile being largest for two concave sides. The maximum moment of inertia for the special-shaped pile section is at its minimum when the number of concave sides \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 and the concave angle 𝜃 = 5\u0026deg;, with a value of 0.0022 m\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, which is 1.07 times that of a square pile with the same cross-sectional area. Conversely, the maximum moment of inertia occurs when the number of concave sides \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2 and the angle of concavity 𝜃 = 30\u0026deg;, yielding a value of 0.00348 m\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, which is 1.63 times that of a square pile with the same cross-sectional area. Thus, the special-shaped pile possesses a larger moment of inertia than the square pile with the same cross-sectional area, indicating superior bending resistance.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e"},{"header":"4. Numerical calculation","content":"\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eThe special-shaped piles theoretically have superior vertical bearing capacity compared to square piles with the same cross-sectional area. To further study the influence of the number of concave sides (\u003cem\u003en\u003c/em\u003e) and the angle of concavity (𝜃) on the vertical load bearing characteristics of the special-shaped piles, a numerical calculation model is established using the finite element software ABAQUS.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1. Finite element modeling\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eThe pile is C30 concrete, which is simulated using a linear elastic model, while the soil is modeled with a Mohr\u0026ndash;Coulomb model. The numerical models of both the piles and soil bodies are meshed using eight-node linear hexahedral cells (C3D8R), and the relevant material calculation parameters are shown in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. The pile-soil interaction is achieved by establishing master-slave contact pairs, with the normal contact surface modeled as hard contact and the tangential direction as Coulomb friction. The friction coefficient remains constant throughout the loading process and is taken as the tangent of the internal friction angle (tan(\u0026phi;)\u0026thinsp;=\u0026thinsp;0.45). The geometrical boundary dimensions of the soil are consistent with the literature model, measuring 15 m in length, 15 m in width, and 28 m in depth. To avoid boundary effects, the pile length is set to 14 m. The soil is modeled as free and unconstrained at the top, completely fixed at the bottom, and with no horizontal displacement at the sides. The soil mesh adopts a single-precision radial meshing method to ensure that the mesh is dense near the pile and gradually becomes sparse away from the pile. After conducting a grid sensitivity analysis, it is confirmed that the above meshing method ensures both the calculation accuracy and the convergence of the model. The details of the 3D model and the mesh of the special-shaped pile section are shown in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eMaterial parameters of pile and clay.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMaterial\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eClay\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePile\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConstitutive model\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMohr\u0026ndash;Coulomb\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLinear elasticity\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eElastic modulus [E/(GPa)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDensity [\u0026rho;/(kg\u0026middot;m\u003csup\u003e-3\u003c/sup\u003e)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1800\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2300\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePoisson\u0026rsquo;s ratio [v]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInternal friction angle [\u0026phi;/(◦)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCohesion [c/(kPa)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDilatancy angle[\u0026psi;/(◦)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2. Geostress balancing and loading steps\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eIn engineering practice, the soil has already experienced consolidation and settlement due to gravity before pile casting. Therefore, in numerical simulation, the initial ground stress should be applied to the soil first to simulate the actual situation. Considering the high accuracy and simplicity of the automatic ground stress balance method, this paper chooses this method to simulate the ground stress state of the soil. The loading steps in the numerical simulation process are as follows:\u003c/p\u003e\n \u003c/div\u003e\n \u003cul\u003e\n \u003cli\u003e\n \u003cp\u003eFirst geostress balance: the soil and pile models are constructed. During the initial geostress equilibrium, the pile units and pile-soil contact pairs are disabled using the birth-death unit method. The horizontal displacement of the soil is restricted, the bottom of the soil is fixed, and gravity is applied to achieve the initial equilibrium of the geostress.\u003c/p\u003e\n \u003c/li\u003e\n \u003cli\u003e\n \u003cp\u003eSecond geostress balance: After the initial geostress balance is completed, the soil units occupied by the piles are removed using the live-dead unit method. The previously disabled pile units and pile-soil contact pairs are then reactivated, and the soil is rebalanced for ground stresses.\u003c/p\u003e\n \u003c/li\u003e\n \u003cli\u003e\n \u003cp\u003eApplying pile top load: after the second geostress balance is completed, vertical loads are applied to the pile top in a graded manner.\u003c/p\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003e4.3. Validation of the model\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eTo ensure that the numerical model developed in this paper can effectively analyze the vertical bearing characteristics of single pile foundations, two cases from published literature are selected for verification.\u003c/p\u003e\n \u003cp\u003e(1) Case 1\u003c/p\u003e\n \u003cp\u003eThe field load test conducted by Chen Zheng et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e was selected to verify the accuracy of the numerical simulation. The schematic diagram of the field load test is shown in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e. The parameter settings of the numerical model for the vertical static load test on a single pile, as performed by Chen Zheng et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e, are as follows: the pile is a circular pile, the length of the pile is 18 m, the diameter of the pile is 0.3 m, the modulus of elasticity of the pile is 20 GPa, the Poisson\u0026apos;s ratio is 0.2, the density of the soil around the pile is 1800 kg/m\u003csup\u003e3\u003c/sup\u003e, the angle of internal friction is 23\u0026deg;, the angle of dilatancy is 15\u0026deg;, and the cohesive force is 10.9 kPa. The results of the field static load test of the single pile and the numerical simulation are shown in Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e. When the pile top load is small, the settlement values of the numerical simulation curve and the literature test curve are close to each other; when the load is large, the displacement values of the numerical simulation are slightly larger than the test values. However, the overall trend is in good agreement, which indicates that the numerical model constructed in this paper has high accuracy in assessing the vertical load bearing characteristics of the pile and can realistically simulate the response of the pile in actual projects.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003e(2) Case 2\u003c/p\u003e\n \u003cp\u003eSince the XCC-shaped pile shares similar cross-sectional geometric characteristics with the special-shaped pile (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4) studied in this paper, the numerical model of the XCC pile established by Lv et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e was used as a validation to assess the reliability of the numerical modeling method for special-shaped piles. Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e provides a detailed list of the relevant parameters for the pile and soil used in the validation model The numerical simulation results are shown in Fig. 9. The load-displacement curve obtained from the finite element model developed in this study closely matches the data reported in the literature. Furthermore, the axial force of the XCC pile was extracted using Abaqus software and compared with the data from the literature, showing a similarly close match. An analysis of the error sources suggests that the discrepancies primarily arise from differences in the mesh size used in the two models, which lead to computational errors. Therefore, it can be concluded that the numerical modeling method for special-shaped piles and the axial force extraction approach adopted in this study are both reliable and feasible.\u003c/p\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eMaterial parameters of pile and soil.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMaterial\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSoil\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePile\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eConstitutive model\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMohr\u0026ndash;Coulomb\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLinear elasticity\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eElastic modulus [E/(GPa)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDensity [\u0026rho;/(kg\u0026middot;m\u003csup\u003e-3\u003c/sup\u003e)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1490\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2700\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePoisson\u0026rsquo;s ratio [v]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInternal friction angle [\u0026phi;/(◦)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDilatancy angle[\u0026psi;/(◦)]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5. Analysis of calculation results","content":"\u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eIn order to investigate the effects of the number of concave sides \u003cem\u003en\u003c/em\u003e and the angle of concavity 𝜃 on the vertical load bearing characteristics of piles, we carried out three main research analyzes: (1) load-settlement curves; (2) load sharing of the side and end resistances; (3) distributions of side resistance.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n \u003ch2\u003e5.1. Load-settlement curve\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eThe load-settlement curves for both special-shaped piles and square piles under vertical load are illustrated in Fig.\u0026nbsp;10. These curves exhibit notable similarities, displaying a slow-change behavior. When the load is less than 800 kN, the curves increase approximately linearly. However, when the load exceeds 800 kN, the load-settlement curves of each pile gradually steepen, entering a non-linear stage. This behavior is attributed to the high modulus of elasticity of the pile during loading, resulting in relatively small settlements at the pile top. Simultaneously, the axial force compresses the pile, leading to even smaller settlements at the pile end. Consequently, the pile top load is primarily supported by the pile side resistance, and the pile and soil remain in the elastic deformation stage. Thus, during the initial loading period, the load-settlement curve exhibits linear growth. As the load increases, the settlement of the pile top rises, and the settlement of the pile end also increases. This leads to a gradual increase in end resistance, while the soil at the pile\u0026apos;s end begins to undergo plastic deformation. With continued load application, the settlement of the pile top and end increases further, causing the end resistance to rise and the load-settlement curve to start bending. According to current specifications and related calculation methods\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e, when the pile top settles to 40 mm, this load value is considered the ultimate bearing capacity of the special-shaped pile, as depicted in Fig. \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eFrom Fig. \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e, it can be observed that the ultimate bearing capacity of the special-shaped pile increases with the number of internal concave sides (\u003cem\u003en\u003c/em\u003e) and the internal concave angle (𝜃). When the number of concave sides is \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, the ultimate bearing capacity increases from 1120 kN to 1355 kN as the concave angle (𝜃) increases from 5 to 30\u0026deg;, representing a 20.98% increase in ultimate bearing capacity. Similarly, when the concave angle is 𝜃 = 5\u0026deg; and the number of concave sides (\u003cem\u003en\u003c/em\u003e) increases from 1 to 4, the ultimate bearing capacity rises from 1120 kN to 1173 kN, showing a 4.7% improvement. These results indicate that changing the concave angle (𝜃) of the cross-section has a more significant effect on the vertical capacity of the special-shaped pile compared to altering the number of concave sides (\u003cem\u003en\u003c/em\u003e).\u003c/p\u003e\n \u003cp\u003eWhen the number of concave sides \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 and the concave angle 𝜃 = 5\u0026deg;, the ultimate load capacity of the special-shaped pile is the smallest, measured at 1120 kN. In contrast, when the number of concave sides \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4 and the concave angle 𝜃 = 30\u0026deg;, the ultimate load capacity of the special-shaped pile reaches its maximum, measured at 2053 kN, which is 1.05\u0026thinsp;~\u0026thinsp;1.92 times greater than that of a square pile with the same cross-sectional area. The trend of the ultimate bearing capacity of the special-shaped pile is consistent with the trend of its perimeter outlined in Section \u003cspan class=\"InternalRef\"\u003e3.2\u003c/span\u003e. The perimeter of the special-shaped pile increases by 1.01\u0026thinsp;~\u0026thinsp;1.78 times compared to a square pile (𝜃 = 0\u0026deg;) with the same cross-sectional area. Furthermore, the increase in ultimate bearing capacity is 1.04\u0026thinsp;~\u0026thinsp;1.08 times the increase in perimeter, highlighting the prominent effect of expanding the cross-section of the special-shaped pile.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n \u003ch2\u003e5.2. Load sharing of the side and end resistances\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eAccording to the analyzes in section 2.2, changes in total side resistance significantly influence the bearing capacity of the special-shaped pile. The pile end axial force is extracted through ABAQUS and the difference between the pile top load and the pile end axial force is the total side resistance. The respective sharing ratios of the total side resistance and end resistance as the pile top load increases are obtained through numerical simulation, as shown in Fig.\u0026nbsp;12.\u003c/p\u003e\n \u003cp\u003eWhen the pile top load is less than 600 kN, the deformation of the pile is minimal because the elastic modulus of the pile is substantially larger than that of the soil. This results in a small relative displacement between the pile and the soil, preventing the full utilization of side resistance. As the pile top load increases to 800 kN, the deformation of the pile becomes more pronounced. This leads to an increase in the relative displacement between the pile and the soil, which further activates the side resistance. Simultaneously, the soil at the pile end becomes compressed and densified. At this point, the variation in the number of concave sides (\u003cem\u003en\u003c/em\u003e) and the angle of concavity (𝜃) have less effect on the load-sharing ratio of the special-shaped pile.\u003c/p\u003e\n \u003cp\u003eWith further increases in the pile top load, the relative displacement between the pile and the soil continues to increase, causing the shear stress at the pile-soil contact surface to gradually reach the soil\u0026apos;s shear strength. This leads to the full activation of side resistance, while the additional load on the pile is primarily supported by the soil at the pile\u0026apos;s end, resulting in a decrease in the proportion borne by side resistance. This decreasing proportion diminishes at a slower rate as the number of concave edges (\u003cem\u003en\u003c/em\u003e) and the angle of concavity (𝜃) increase, eventually leveling off with further increases in the pile top load. This trend occurs because an increase in the number of concave sides (\u003cem\u003en\u003c/em\u003e) and the concave angle enhances the contact area between the pile and the soil along the pile\u0026apos;s side. Consequently, the ratio of the pile side area to the pile end area increases, improving the proportion of load carried by pile-side friction.\u003c/p\u003e\n \u003cp\u003eMoreover, it is evident that under the same load, the proportion of end resistance in the special-shaped pile is consistently smaller than that of side resistance. This indicates that the top load of the pile is predominantly supported by side resistance, reflecting the bearing characteristics of an end-bearing friction pile.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n \u003ch2\u003e5.3. Distributions of side resistance\u003c/h2\u003e\n \u003cdiv class=\"BlockQuote\"\u003e\n \u003cp\u003eThe distribution of side resistance in special-shaped piles, relative to the top settlement of the pile under various top loads, is illustrated in Fig.\u0026nbsp;13. Under vertical loading, the side resistance of special-shaped piles exhibits different responses to settlement. During the initial loading phase, the side resistance of special-shaped piles with varying cross-sectional parameters remains largely consistent at the same settlement and is characterized by slow settlement rates. When the side resistance reaches 670 kN, relative displacement between the pile and the soil begins to occur, leading to an acceleration of settlement. This trend of accelerated subsidence decreases with an increase in the number of concave sides (\u003cem\u003en\u003c/em\u003e) and the angle of concavity (𝜃). Additionally, the increase in pile top settlement, accompanied by rising side resistance, demonstrates a tendency to increase proportionally.\u003c/p\u003e\n \u003cp\u003eWhen the special-shaped pile reached its ultimate bearing capacity (corresponding to a pile top settlement of 40 mm), the side resistance increased from 741 kN to 782 kN as the number of concave sides (n) increased from 1 to 4 at a fixed concave angle (\u003cem\u003e\u0026theta;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5\u0026deg;), representing a 5.5% enhancement in ultimate side resistance. In contrast, when the number of concave sides was held constant (\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1), increasing the concave angle (\u003cem\u003e\u0026theta;\u003c/em\u003e) from 5\u0026deg; to 30\u0026deg; increased the side resistance from 741 kN to 967 kN, yielding a 30.5% improvement in ultimate side resistance. These findings demonstrate that cross-sectional optimization, achieved by increasing either the number of concave sides (\u003cem\u003en\u003c/em\u003e) or the concave angle (\u003cem\u003e\u0026theta;\u003c/em\u003e), enhances the side resistance of special-shaped piles, with the concave angle (\u003cem\u003e\u0026theta;\u003c/em\u003e) exhibiting a significantly greater influence. For practical applications, maximizing the concave angle (\u003cem\u003e\u0026theta;\u003c/em\u003e) is recommended, provided construction feasibility is ensured.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eIn this paper, a novel special-shaped pile is designed based on the principle of expanding the perimeter while maintaining the same cross-sectional area. This design combines the characteristics of existing special-shaped pile cross-sections with those of square cross-sections. Theoretical analysis and numerical simulation methods are employed to study the effects of varying the number of concave sides (\u003cem\u003en\u003c/em\u003e) from 1 to 4 and adjusting the concave angle (\u0026#120579;) from 5\u0026deg; to 30\u0026deg; on the geometric properties and vertical load bearing characteristics of the special-shaped pile. These results are compared and analyzed against those of a square pile with the same cross-sectional area. The following conclusions are drawn:\u003c/p\u003e \u003cp\u003e(1) When the concave angle is certain, with the increase of the number of concave edges, the perimeter of the cross-section of the special-shaped pile, the ultimate bearing capacity and side resistance increase, and the maximum moment of inertia of the cross-section increases and then decreases, and the maximum moment of inertia of the cross-section of the special-shaped pile is the largest when the number of concave edges is two.\u003c/p\u003e \u003cp\u003e(2) When the number of concave sides is certain, the perimeter, ultimate bearing capacity, and maximum moment of inertia of the cross-section of the special-shaped pile increase with the increase of the concave angle. The increase of section circumference, ultimate bearing capacity and side resistance is more significant, while the increase of maximum moment of inertia of section is relatively insignificant.\u003c/p\u003e \u003cp\u003e(3) The special-shaped pile mainly carries the load by lateral friction force, and when the number of inner concave sides is more than 2 and the angle of inner concave is more than 20\u0026deg;, the proportion can reach more than 80%, which shows the bearing characteristics of end-bearing friction pile.\u003c/p\u003e \u003cp\u003e(4) Because of the larger pile side area, the special-shaped pile has larger pile side resistance, so the vertical bearing capacity and bending resistance are better than the same cross-section area square piles, the ultimate bearing capacity is 1.05\u0026thinsp;~\u0026thinsp;1.92 times of the same cross-section area square piles, and the cross-section circumference is 1.01\u0026thinsp;~\u0026thinsp;1.78 times of the same cross-section area square piles, and the effect of cross-sectional anisotropy is obviously enlarged.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003eA\u003c/strong\u003e\u003cstrong\u003evailability\u003c/strong\u003e\u003cstrong\u003e\u0026nbsp;S\u003c/strong\u003e\u003cstrong\u003etatement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets during the current study available from the corresponding author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConceptualization, Z.C.; methodology, Z.C.; software, D.J.; validation, D.J.; formal analysis, D.J.; investigation, D.J.; resources, D.J.; data curation, D.J.; writing\u0026mdash;original draft preparation, D.J.; writing\u0026mdash;review and editing, D.J.; visualization, Z.Z.; supervision, Z.C.; funding acquisition, Z.C. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research was funded by Project of the Natural Science Foundation of Sichuan Province, grant number No. 2023NSFSC0046 and the State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, grant number No. SKLGP2023Z011.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAdditional information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCorrespondence\u003c/strong\u003e and requests for materials should be addressed to Z.C.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReprints and permissions information\u003c/strong\u003e is available at www.nature.com/reprints.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePublisher\u0026rsquo;s Note\u003c/strong\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003eSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eZhou P., Liu H., Zhou H, et al. 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The Influence of Different Loading Sequence on Bearing Behavior of Micropile.\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003cem\u003eArchit. Sci.\u003c/em\u003e, 27(03),45-48+54(2011).\u003c/li\u003e\n \u003cli\u003eLv YR, Zhang DD. Geometrical effects on the load transfer mechanism of pile groups: three-dimensional numerical analysis.\u003cem\u003e\u0026nbsp;Can. Geotech. J\u003c/em\u003e. 55(5): 749-757(2018).\u003c/li\u003e\n \u003cli\u003eJGJ94-2008; Technical specification for building pile foundations. China Architecture \u0026amp; Building Press: Beijing, China(2008).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Pile foundation, square section, section geometric properties, vertical bearing capacity, special-shaped pile","lastPublishedDoi":"10.21203/rs.3.rs-5535232/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5535232/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePile foundations are widely utilized in the construction of high-rise building foundations; however, traditional square piles may not be sufficiently economical due to their requirement to resist substantial vertical loads with large dimensions and dense arrangements. This study presents a novel special-shaped pile designed for high-rise buildings, which aims to enhance the ultimate bearing capacity by expanding the contact area between the pile and the soil. Based on pile bearing capacity theory, this study analyzes the effects of the number of concave sides and the angle of concavity on the cross-section properties of the special-shaped pile. Additionally, a finite element model built and validated over ABAQUS is employed to simulate the influence of varying numbers of concave edges and different concave angles on vertical load bearing characteristics. The results indicate that the cross-sectional perimeter of the special-shaped pile increases with the number of concave edges and the angle of concavity, measuring 1.01 to 1.78 times that of a square pile with the same cross-sectional area, with the angle of concavity having the most significant effect. The maximum moment of inertia of the cross-section of the special-shaped pile initially increases and then decreases with the number of concave sides, while it consistently increases with the angle of concavity. Notably, for a given angle of concavity, the maximum moment of inertia is greatest for the special-shaped pile with two concave sides. Increasing the number of concave sides and the angle of concavity effectively enhances the side resistance and ultimate bearing capacity of the pile, with the ultimate bearing capacity being 1.05 to 1.92 times that of a square pile with the same cross-sectional area.\u003c/p\u003e","manuscriptTitle":"Research on optimized design of section parameters of special-shaped pile based on square section","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-24 06:04:28","doi":"10.21203/rs.3.rs-5535232/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2025-04-29T01:11:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"13167318857089339859687577631179174936","date":"2025-04-24T01:15:41+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-04-23T14:38:45+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"48180481842110571131428139312766150877","date":"2025-04-23T14:28:03+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-04-23T13:59:18+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-04-18T04:42:54+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-04-05T08:59:07+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"54c40dd2-bb63-4872-b649-1c777e2b10c8","owner":[],"postedDate":"April 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":47597026,"name":"Physical sciences/Engineering"},{"id":47597027,"name":"Physical sciences/Engineering/Civil engineering"}],"tags":[],"updatedAt":"2025-05-07T08:08:45+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-24 06:04:28","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5535232","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5535232","identity":"rs-5535232","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}