Introduction

Landslides are widely distributed severe geological disasters that frequently result in casualties, economic losses, and potentially catastrophic outcomes, thereby posing a significant threat to the safety of life and property1,2. In recent years, there has been a growing emphasis on landslide disasters, among which, soil nails are considered an effective measure for slope management and have been widely applied in slope management projects in fields such as railways, highways, hydropower, construction and underground openings3,4,5. The acid rain disaster has been deteriorating for a long time6,7. Under the corrosion and thrust load of acid rain, the cantilever anti-slip pile structure inevitably experiences damage accumulation, resistance attenuation, and performance degradation of the pile body. Cantilever anti-slip piles with degraded service performance have become a major risk source for retaining engineering.

Research methods concerning the bearing characteristics of soil nails include field monitoring of prototype piles and physical tests or numerical simulations of indoor model piles. Guo8 derived calculation formulas for the internal forces and deformations of variable cross-section cantilever soil nails, considering the impact of displacement, based on current standards, and verified them with engineering examples. Yang9 investigated the effects of the embedded length, pile spacing, flexural rigidity of the pile body, and soil properties on the safety factor and load-bearing performance of anti-slip piles using a three-dimensional elastoplastic shear strength reduction method. Matsui10 studied the variation of earth pressure on anti-slip piles with soil displacement, taking into account different pile spacings and pile diameters. Ren11 conducted physical model tests to study the mechanical behavior and moment and deformation characteristics of anti-slip piles with different lengths. Li T12 employed field loading tests and numerical analysis to study the bearing characteristics of CEP piles and straight piles. J.-S. C13 developed a theoretical model to investigate the impact of foundation exposure due to scour on the displacement ductility and bearing capacity of fixed piles. Qu Y14 performed ultimate load tests and numerical simulations on PHC pipe piles to analyze their ultimate bearing capacity characteristics and influencing factors. Peng W15 used the finite element limit analysis (FELA) method to study the influence of slopes on the lateral bearing capacity of large-diameter single piles in cohesive soil. Dekoster M16 simulated the load-displacement curves of reinforced concrete beams with rusted reinforcement in the tension zone using finite element methods and achieved good agreement with experimental results. Feng Z17 conducted numerical simulations to study the vertical and horizontal bearing characteristics of large-diameter hollow composite piles.In finite element numerical simulations, the selection of constitutive models is crucial for accurately representing cracks18. Cohesive zone models (CZMs) can not only predict the response of cracked structures but also adequately simulate the behavior of uncracked structures19, effectively addressing the nonlinear issues at crack tips20, and providing good results for the analysis of structural bearing capacity and damage levels21. Miao Su22 and He P23 employed a fine-scale macroscopic finite element model with an exponential cohesive zone model to simulate the debonding process of materials and interfaces. Luo Y24 analyzed the distribution and failure mode of cracks by globally embedding zero thickness cohesive elements.Mirko M below25 used a dual-cohesive element to describe the dual-scale discontinuities influenced by fibers and fracture interference that affect the overall structural behavior. Ji K26 integrated a concrete damage plasticity model with a cohesive zone model for numerical modeling and calibrated and validated the numerical model based on experimental results. The cohesive zone model has been recognized as a powerful analytical method for nonlinear fracture processes27, playing a vital role in analyzing localization and fracture failure in engineering materials28,29, and can be combined with commercial finite element programs to develop more advanced numerical models30,31,32,33. Currently, scholars mainly focus on the mechanical properties of anti-slip pile structures under landslide thrust.Few studies have involved the the mechanical properties and stability effects of combining cantilever anti-slip pile structures with cohesive force models in acid rain corrosion environments.Therefore, the introduction of the viscous band model is of great significance for in-depth research on the internal forces, deformation, and bearing capacity degradation of soil nails in erosive environments.

This study designed and conducted a cantilever anti-slip pile bearing model test, and established a numerical model of cantilever anti-slip pile based on bilinear viscous element considering the bond slip effect between reinforced concrete. Based on this model, the load displacement behavior, bending moment distribution pattern, and crack distribution pattern of cantilever anti-slip piles were analyzed. Summarized the factors affecting the durability of anti-slip piles in acid rain corrosion environments, and analyzed the bearing characteristics of cantilever anti-slip piles under different corrosion conditions. A formula for calculating the stiffness and bearing capacity of corrosion-resistant steel anti-slip piles considering the degradation effect of bonding performance was proposed, and verified using existing experimental data.

Physical model test of bearing characteristics of cantilever anti-slip pile

Determination of model pile similarity constant and size

Determination of similarity constants

According to the similarity principles34, model tests need to satisfy conditions of geometric similarity, physical similarity, stress similarity, and load similarity. This structural model is designed strictly according to the similarity theory. Considering the experimental site conditions and loading conditions, the geometric similarity constant Cl is selected as 20, and the elastic modulus similarity constant CE is chosen as1.Based on the second law of similarity, other similarity ratios can be calculated as shown in Table 1 below.

Table 1 Similarity ratio of model pile test.
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Determination of model pile dimensions

Based on the geometric similarity ratio Cl of 1:20, the geometric dimensions of the model pile designed for this experiment are b×h = 100 mm×150 mm. Two Φ10 HRB335 grade steel bars are used as longitudinal tensile reinforcement. The compression zone reinforcement and stirrups are made of Φ6 HRB335 grade steel bars, with stirrup dimensions of 60 mm×110 mm, spaced at 106 mm, and a concrete cover thickness of 20 mm. After calculation, the reinforcement ratio of this section meets the code requirements and will not result in over-reinforcement or under-reinforcement failure.

Experimental layout and experimental plan

Experimental raw materials

The model pile is constructed with C30 concrete. In accordance with the “Specification for Mix Proportion Design of Ordinary Concrete“35 and the “Standard for Test Methods of Concrete Physical and mechanical properties“36, the mix ratio for the C30 concrete used in this experiment is water: cement: sand: gravel = 0.49:1:1.37:2.486 (by mass), As shown in Fig. 1. The measured compressive strength of the cube (fcu, k) is 31.3 MPa.

Fig. 1
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Materials used in the experiment.

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Testing equipment

In this experiment, a DH3818Y static strain measuring instrument is used, with the strain gauge for steel bars being model BX120-3AA. The specific parameters are: resistance 120.0 ± 1Ω, sensitivity 2.0 mV/V, grid size 3 mm×2.3 mm, and base size 6.9 mm×3.9 mm. Three vernier calipers are evenly distributed on the compression side with a spacing of 140 mm, with a range of 0 mm to 30 mm and an accuracy of 0.01 mm.The testing equipment used is shown in Fig. 2.

Fig. 2
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Testing Instruments Used in the Experiment.

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Strain gauge arrangement for reinforcement

Strain gauges are arranged on the tensile reinforcement to ensure the accuracy of the experimental data and to prevent damage to the strain gauges on one side. Strain gauges are symmetrically placed at the same positions on both sides of the reinforcement. The arrangement of the strain gauges for the reinforcement is designed based on the bending moment distribution of the cantilever soil nail and is shown in Fig. 3.

Fig. 3
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Layout position of steel bar strain gauges (mm).

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Preparation and maintenance of test piles

The preparation of anti-slip piles for testing is carried out according to the steps of making steel cages→attaching steel strain gauges→making molds→ pouring concrete→ removing molds→ and curing. Water is sprayed and cured under standard conditions for 28 days.

Experimental setup system

Based on the aforementioned experimental plan and instruments, the cantilever soil nail load-bearing characteristic test system is divided into two parts: ①Fixing and loading device, composed of a spiral jack, a thrust distribution frame, an anchorage section steel support, and a displacement meter frame; ②Data acquisition device, consisting of a laptop computer equipped with the DHDAS dynamic model collection and analysis system connected to the DH3818Y static strain measuring instrument, with pressure sensors and steel bar strain gauges connected to the measuring instrument via wires. The cantilever soil nail load-bearing characteristic test setup is shown in Fig. 4;

Fig. 4
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Test System for Bearing Characteristics of Cantilever Anti Sliding Pile.

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Experimental procedure

Conduct the cantilever soil nail load-bearing characteristic test. After system testing, it was confirmed that the loading device and data collection device were functioning properly and preloaded to eliminate the impact of errors.According to the specifications37, apply a formal load of 2 kN per level, When the load value of the pressure sensor is stable, collect the strain data and record the displacement readings on the vernier caliper. When approaching the calculated yield load of the soil nail, slow down the loading speed to obtain an accurate yield load of the soil nail. And after each level of load reaches the design load, maintain the load, observe and map the cracks, and determine the crack dimensions.

Finite element simulation of bearing characteristics of cantilever anti-slip pile

Modeling

Parameters in the Concrete Damaged Plasticity (CDP) model are determined through the constitutive relationships outlined in the “Code for Design of Concrete Structures“37, with the eccentricity ε set to 0.1, and the dilatancy angle taken between 30° and 35° as appropriate. The ratio \(\:{f}_{b0}\)/\(\:{f}_{c0}\), which is the biaxial compressive strength to uniaxial compressive strength of concrete, is taken as the code value of 1.16. K, the parameter affecting the yield surface of concrete, is taken as 2/3. µ, the viscous parameter defined in the CDP model, typically taken as 0.0005 to meet both accuracy and convergence requirements38. The reinforcement is modeled using a bilinear elastoplastic model, representing the stress-strain relationship of the reinforcement with two linear segments. When\(\:\:{\sigma\:<\sigma\:}_{y}\), it is the linear elastic segment with a modulus of elasticity \(\:{E}_{s}\); when \(\:{\epsilon\:>\epsilon\:}_{y}\), it is the linear hardening segment with a modulus of elasticity\(\:\:{E}_{s}^{{\prime\:}}\), which is generally taken as 0.01\(\:{E}_{s}\)39.

The non elastic strain under compression and the tensile strain under tension can be calculated according to the following formula38:

$$\:\left\{\begin{array}{c}{\stackrel{\sim}{\epsilon\:}}^{in}={\epsilon\:}_{c}-{\epsilon\:}_{0c}^{el}\\\:{\stackrel{\sim}{\epsilon\:}}_{0c}^{el}=\frac{{\sigma\:}_{c}}{{E}_{0}}\\\:{\stackrel{\sim}{\epsilon\:}}_{c}^{pl}={\stackrel{\sim}{\epsilon\:}}_{c}^{in}-\frac{{d}_{c}}{\left(1-{d}_{c}\right)}\frac{{\sigma\:}_{c}}{{E}_{0}}\end{array}\right.$$
(1)

In the equation: \(\:{\stackrel{\sim}{\epsilon\:}}_{c}^{in}\) represents the non-elastic compressive strain, \(\:{\sigma\:}_{c}\) and \(\:{\epsilon\:}_{c}\) are the compressive stress and compressive strain according to the code, \(\:{\epsilon\:}_{0c}^{el}\) is the elastic compressive strain corresponding to the initial modulus of elasticity, and \(\:{\stackrel{\sim}{\epsilon\:}}_{c}^{pl}\) is the plastic strain under compression.

The tensile strain can be calculated according to the following formula38:

$$\:\left\{\begin{array}{c}{\stackrel{\sim}{\epsilon\:}}_{t}^{ck}={\epsilon\:}_{t}-{\epsilon\:}_{0c}^{el}\\\:{\epsilon\:}_{0t}^{el}=\frac{{\sigma\:}_{t}}{{E}_{0}}\\\:{\stackrel{\sim}{\epsilon\:}}_{t}^{pl}={\stackrel{\sim}{\epsilon\:}}_{t}^{ck}-\frac{{d}_{t}}{\left(1-{d}_{t}\right)}\frac{{\sigma\:}_{t}}{{E}_{0}}\end{array}\right.$$
(2)

In the equation: \(\:{\stackrel{\sim}{\epsilon\:}}_{t}^{ck}\)represents the tensile cracking strain, \(\:{\sigma\:}_{t}\) and \(\:{\epsilon\:}_{t}\) are the tensile stress and tensile strain according to the code, \(\:{\epsilon\:}_{0t}^{el}\) is the elastic tensile strain corresponding to the initial modulus of elasticity, and \(\:{\stackrel{\sim}{\epsilon\:}}_{t}^{pl}\) is the plastic strain under tension.

A discrete model is used for the soil nail modeling, where concrete and longitudinal reinforcement are established as solid elements, and cohesive force elements are generated between the reinforcement and concrete to simulate their bond slip40. Since stirrups are not considered for bond slip, truss elements are used for modeling and then embedded within the concrete elements. The finite element model dimensions are the same as those of the physical model test in Sect. 1, with the model pile cross-section being 100 mm×150 mm, and a height of 1100 mm(As shown in Fig. 5a). The reinforcement is consistent with the model pile, and a comparison model is built using the embedded model with the same material dimensions.By using the finite element software ABAQUS to model the cohesive force model and the embedded model separately, the schematic diagrams of the two models are shown in Fig. 5b and c.

Fig. 5
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Schematic diagram of two different modeling methods and structural reinforcement.

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(created using ABAQUS (version 2021), https://www.3ds.com/products/simulia/abaqus)

The main computational parameters used in the model are shown in Table 2.

Table 2 Main material parameters.
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Zero thickness cohesive elements are inserted through the grid division module of ABAQUS, and the bilinear tension displacement criterion three-dimensional cohesive element (COH3D8) is selected as the cohesive element. Figure 6 is a schematic diagram of the mesh division with inserted cohesive elements in the ABAQUS model.

Fig. 6
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Schematic diagram of cohesive force unit.

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(created using ABAQUS (version 2021), https://www.3ds.com/products/simulia/abaqus)

This paper uses the initial damage criterion of Maximum Nominal Stress (Max Damage) to determine the tensile peak strength of the cohesive elements and adopts a failure criterion based on the failure displacement. Both the maximum stress and failure displacement are derived from the stress-slip data obtained from experiments. Studies have shown that the stiffness of cohesive elements has little effect on the bond strength42, so this paper uniformly adopts 21 MPa/mm. Based on the experiment, the peak bond stress for reinforced concrete is taken as 21.27 MPa, the failure displacement is 8.3 mm, and the density value is 3000 kg/m³.

Analysis of experimental and simulation results

Load pile top displacement curve and bending moment analysis

Load-displacement curve analysis

Figure 7 compares the load-settlement curves of the soil nail test results with those of the fully bonded embedded model and the bond-slip model based on the cohesive zone model. The shapes of the three curves are essentially consistent, with both numerical models having similar bending stiffness, but both are larger compared to the experimental values. The entire process of numerical simulation of loading is consistent with the form of the experimental study mentioned earlier, divided into three stages:①Pre-cracking stage, in this stage, the displacement increases proportionally with the increase in load, which is the elastic stage.②Crack propagation stage, when the crack extends above the tensile reinforcement, the tensile force originally borne by the concrete is shared with the tensile reinforcement, causing the overall displacement of the soil nail to continue to increase and not remain stable.③Failure stage, at this point, the stiffness of the soil nail decreases significantly, and the load-bearing capacity of the soil nail basically reaches its limit. It can be observed that the settlement at the top of the pile increases significantly under almost constant load, indicating that the cantilever soil nail is approaching failure. The yield load of the fully bonded embedded model is 21.2 kN, the yield load of the cohesive force bond-slip model is 19.1 kN, and the experimental yield load is 18.5 kN.

Fig. 7
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Load displacement curve of cantilever Anti-slip pile at pile top.

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In comparing the fully bonded embedded model and the bond-slip model based on the cohesive zone model, the latter’s yield load is closer to the experimental curve, the embedded model and the cohesive zone model completely coincide in the pre-cracking stage, after the load is applied to 12 kN, the stiffness of the cohesive zone model begins to decrease and approaches the experimental curve, after the tensile reinforcement yields, the stiffness of the soil nail considering bond slip is smaller compared to the experimental values, but the stiffness of the fully bonded embedded model is too high. This is because, in the actual loading process of the soil nail, when the load level is low, there is basically no slippage between the reinforcement and concrete inside the soil nail, so the calculated values of the two models are close, when the load level is high, slippage will occur between the reinforcement and concrete, further reducing the load-bearing capacity of the soil nail, and the fully bonded embedded model ignores this process, leading to its load-bearing capacity being greater than the actual experiment. The bond-slip model based on the cohesive zone model addresses this issue, so it is closer to the experimental curve.The RMSE of the two models and experimental values are: RMSEem=2.7; RMSEcm=0.6. It also indicates that the results of the cohesive force model fit better with the experimental results.

Bending moment analysis

Fig. 8
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Variation of bending moment of Anti-slip pile body with load.

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As shown in Fig. 8, reveals that the bending moments are concentrated in the range from 200 mm below to 400 mm above the slip surface, with the maximum bending moment at the slip surface reaching 5.22 × 103 N·m, which is close to the ultimate bending moment value of 5.17 × 103 N·m calculated according to the specification《GB50010-2010》. During the loading process, as the load increases, the increments of bending moments at the top and bottom anchorage sections of the anti-sliding pile is not significant, but the increase at the slip surface is pronounced, with the bending moment curve prominently convex outward, exhibiting a distinct “single peak” phenomenon.This is because for the cantilever anti-slip piles, the load of the cantilever section will be transmitted to the sliding surface through the pile body, and the force of the anchorage section will also concentrate on the sliding surface, causing stress to accumulate at the sliding surface and resulting in the peak of bending moment.Consistent with the bending moment distribution calculated from the experiment. The embedded model’s bending moment data deviates significantly from the experimental data, while the cohesion model’s bending moments match the experiment well. Although both models’ bending moment distributions are similar to the experiment, with bending moments increasing with load and peaking at the slip surface, the embedded model’s peak bending moment has a larger error compared to the experimental value. Therefore, the cohesion model using solid elements can accurately reflect the actual bending moment distribution of the anti-sliding pile and is more suitable for internal force analysis.

Crack development process

As depicted in the comparison between the experiment and the model (Fig. 9), the cracks on the anti-sliding pile body are relatively evenly distributed in the tensile zone, with a total of five main cracks appearing horizontally between 110 mm above and 150 mm below the slip surface, with the largest crack located between 60 mm and 80 mm above the pile body, reaching a maximum width of 2.1 mm.The lengths from top to bottom are 9 mm, 19.41 mm, 15.53 mm, 16.82 mm, and 25.88 mm, respectively.The overall number of cracks in the experimental pile is fewer compared to the damage zones of the two finite element models. The damage zones of the cohesion model are distributed from 150 mm above to 140 mm below the slip surface (Fig. 9a), while the embedded model’s crack distribution is similar to that of the cohesion model but with fewer cracks, distributed from 140 mm above to 110 mm below the slip surface (Fig. 9b). The cohesion model calculates a greater number of damage zones, all of which include the development path of the experimental pile’s cracks, covering a broader range of areas prone to cracking, thus making the cohesion model more representative of the actual conditions.

As the loading progresses, there are differences in the stress-strain distribution at different positions of the pile body. Due to the combined action of thrust and bending moment, the stress concentration in the middle area of the pile body is more obvious, and microcracks inside the concrete are more likely to initiate and propagate here. The crack extends from the tensile zone to the compressive zone of the anti-slip pile, and the direction of crack extension gradually approaches the sliding surface from top to bottom. Finally, the crack morphology is a “radial” crack from the compressive zone to the tensile zone of the anti-slip pile. Unfold the cracks as a whole to present an olive shaped distribution with a wide center and narrow top and bottom. The crack widths from top to bottom are 0.4 mm, 2.1 mm, 1.7 mm, 1.1 mm, and 0.8 mm respectively.

Fig. 9
figure 9

Comparison of experimental cracks, cohesive forces, and embedded model damage zones.

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Stress and strain distribution of concrete and reinforcement

Fig. 10
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Concrete stress and steel stress pile top displacement curves in the tensile and compressive zones at the sliding surface.

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Figures 10(a)-(d) present the stress curves of concrete in the tension and compression zones at the slip surface, as well as the stress curves of reinforcement in both tension and compression zones within the finite element model of the cantilever anti-sliding pile, as the displacement at the pile top changes. Simulation data indicates that when the displacement at the top of the anti-sliding pile reaches 0.23 mm, the maximum stress in the concrete of the tension zone at the slip surface reaches 1.21 MPa (Fig. 10a), with the cracking load being 4.3kN. When the displacement at the pile top reaches approximately 3.6 mm, the maximum stress in the concrete of the compression zone at the slip surface reaches 42.3 MPa (Fig. 10b), after which the concrete is crushed. The tensile longitudinal reinforcement inside the anti-sliding pile yields when the displacement at the pile top reaches 3.6 mm (Fig. 10c). As the load continues to increase and the displacement at the pile top reaches 9 mm, the reinforcement in the compression zone also reaches the yield stress (Fig. 10 d), at which point the load-bearing capacity of the anti-sliding pile begins to drop sharply.

Analysis of bond stress and displacement

Figure 11 shows the stress and axial displacement distribution of the reinforced concrete bonding layer in the tensile zone under different load levels. It can be seen from Fig. 11a that with the increase of load, the stress of the bonding layer increases nonlinearly, and there is a peak stress point in both cases. The position of the peak stress point gradually moves from the sliding surface to the pile bottom with the increase of load. The uneven stress phenomenon of the bonding layer in the anchorage section will increase with the increase of load. From Fig. 11b, it can be seen that when the ant-slip pile structure is under stress, the slip amount of the internal tensile steel bars is much greater than that of the compressive steel bars, and there is no slip phenomenon in the anchorage section. The slip of the bonding layer in the loaded section will increase with the increase of load, while the slip amount in the bonding layer of the anchorage section will also increase with the increase of load.

Fig. 11
figure 11

Stress and axial displacement distribution of reinforced concrete bonding layer in tension zone under different load levels.

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Analysis of the load-carrying performance of cantilever anti-sliding piles in acid rain corrosion environment

The impact of concrete strength and reinforcement corrosion on the load-carrying performance of anti-sliding piles in acid rain corrosion environment

The main factors affecting the durability of anti-sliding piles in an acid rain corrosion environment are threefold: concrete strength, mechanical properties of reinforcement, and bond strength between reinforcement and concrete. In the early stages of acid rain corrosion, the strength of concrete will change, but the internal reinforcement is protected by the concrete layer and has not yet corroded. In this case, only the change in pile body concrete strength affecting the load-carrying capacity of the anti-sliding pile needs to be considered. At present, scholars have conducted simulated acid rain corrosion tests on anti-slip pile concrete by configuring simulated acid rain solutions43. Dry wet cycles are used to simulate the corrosion environment of anti-slip piles in practical engineering, and the deterioration of concrete compressive strength is tested by indoor soaking and drying alternating method with a cycle of two days. This paper calculates the CDP model parameters based on experimental data (Table 3) and establishes a numerical model of anti-sliding piles in an acid rain corrosion environment.

Table 3 Compressive strength values of concrete at different corrosion ages.
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Based on the experimental study of the bond performance between steel bars and concrete in an acid rain corrosion environment, numerical simulation studies were conducted on cantilever anti-sliding piles with corrosion rates of 0.32%, 0.86%, 2.68%, 5.87%, and 6.79%.The key to establishing a model of anti-sliding piles with corroded reinforcement in an acid rain corrosion environment lies in the construction of the post-corroded steel bar constitutive model and the selection of bonding parameters for the interface layer between post-corroded steel bars and concrete. Based on the research results of accelerated corrosion tests of reinforcement in an acid rain environment44, a bilinear stress-strain relationship with corrosion rate for numerical simulation was established. It is also assumed that the reinforcement corrodes uniformly in the acid rain corrosion environment, with the mass corrosion rate being the same as the cross-sectional corrosion rate. The stress-strain curve expression for corroded reinforcement is as follows:

$$\:\left\{\begin{array}{c}{f}_{y\eta\:}=\left(1-1.007{\eta\:}_{s}\right){f}_{y0}\\\:{E}_{s\eta\:}=\left(1-0.955{\eta\:}_{s}\right){E}_{s0}\end{array}\right.$$
(3)

In the equation: \(\:{f}_{y\eta\:}\)is yield strength of corroded reinforcement. \(\:{E}_{s\eta\:}\) is elastic modulus of corroded reinforcement.

According to the stress-strain curve expression of corroded steel bars, the material parameters of the steel bars in the calculation model are shown in the following Tables 4 and 5.

Table 4 Modeling parameters of steel bars under different corrosion rates.
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The maximum nominal stress criterion, Maxs Damage, is used to determine the tensile peak strength of the cohesive elements, and a damage criterion based on the failure displacement is employed. Both the maximum stress and failure displacement are obtained from the stress-slip curves derived from experiments45.

Table 5 Basic parameters of reinforced concrete bonding layer under different corrosion rates.
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Finite element results and analysis

load-displacement curves and yield load variations of eroded concrete anti-sliding piles

As shown in Fig. 12 a, under the action of acid rain corrosion, considering only the decrease in concrete strength, in the initial stage of loading, the load-displacement curves of anti-sliding piles with different acid rain corrosion ages basically coincide, and their cracking loads are essentially the same. As the load increases and the tensile zone concrete cracks, anti-sliding piles with longer corrosion ages exhibit greater lateral displacement, indicating a decrease in the bending stiffness of the anti-sliding piles. When the reinforcement in the tensile zone reaches the yield state, the yield load of anti-sliding piles with lower concrete strength also decreases. When the acid rain corrosion cycle reaches 30 days, the concrete strength increases by 11.14%, and the yield load of the anti-slip pile is 19.3kN, which is an increase of 0.85% compared to the uneroded anti-sliding pile. As the corrosion time continues to increase, the concrete strength begins to decline. When the corrosion cycle reaches 120 days, the concrete strength decreases by 23.51%, and its ultimate load capacity drops from 19.1kN to 18.9kN, a decrease of 1.53%. Based on the simulation results, the variation rate of the yield load of anti-sliding piles under different corrosion times is plotted in Fig. 12b. It can be seen that in the early stages of corrosion, due to the increase in concrete strength, the load-bearing capacity of the anti-sliding piles also increases to some extent. As the concrete strength decreases, their load-bearing capacity also declines, but the change in load-bearing capacity is small. The study indicates that the impact of concrete strength on the load-bearing capacity of the anti-sliding piles is relatively low.

Fig. 12
figure 12

Load displacement curve and yield load variation of eroded concrete Anti-slip pile.

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Load-displacement curves and yield load variations of corroded reinforcement concrete anti-sliding piles

Figure 13a presents the load-displacement curves of anti-sliding piles at various corrosion rates. In the initial stage of loading, the load-displacement curves of anti-sliding piles with different corrosion rates are essentially linear, with their cracking loads being close. As the load continues to increase and the tensile zone concrete of the anti-sliding pile cracks, the bending stiffness of the anti-sliding pile decreases with the increase in the corrosion rate. Continuing to load, as the reinforcement in the tensile zone of the anti-sliding pile reaches the yield state, the yield load of the anti-sliding pile decreases with the increase in the corrosion rate. When the corrosion rate of the reinforcement reaches 0.32%, the yield load of the anti-sliding pile is 19.0kN, a decrease of 0.52% compared to the non-corroded anti-sliding pile. When the corrosion rate of the reinforcement reaches 2.68%, the yield load of the anti-sliding pile decreases by 3.98%. When the corrosion rate of the reinforcement reaches 6.79%, the yield load of the anti-sliding pile decreases by 8.01%.The relationship between the yield load variation rate of the anti-sliding pile and the corrosion rate is shown in Fig. 13b. Due to the strong yield deformation ability of steel bars as the main load-bearing component, the failure of anti-slip piles still exhibits obvious ductile failure characteristics under different corrosion rates.

Fig. 13
figure 13

Load displacement curve and yield load variation of corroded reinforced concrete Anti-slip pile.

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Analysis of stress and axial displacement in the bond layer of corroded reinforcement concrete anti-sliding piles

Figure 14 illustrates the stress distribution in the bond layer between corroded reinforcement and concrete in the compression and tension zones, For the bond layer in the compression zone (Fig. 14a), the stress in the bond layer gradually increases from the bottom of the pile to the slip surface of the pile body, which is consistent with practical observations, from the slip surface to the top of the pile, the axial stress in the bond layer is unevenly distributed, but generally shows a decreasing trend, when the reinforcement corrodes, corrosion will occur unevenly on every part of the surface of the steel reinforcement, causing a serious decrease in the bonding performance of the local area of the steel reinforcement.The uneven distribution of axial stress in the bond layer becomes more pronounced, the peak bond stress without corrosion is 13.9 MPa, and when the corrosion rate reaches 6.79%, the peak stress decreases to 6.2 MPa; For the bond layer in the tension zone (Fig. 14b), the stress in the bond layer gradually increases from the bottom of the pile to the slip surface of the pile body, reaches a peak near the slip surface, and then drops sharply. The stress distribution trend of the bond layer at different corrosion rates is essentially the same; the peak stress without corrosion is 21.8 MPa, and when the corrosion rate reaches 6.79%, the stress decreases to 8.3 MPa. The research results indicate that the bond layer stress between the concrete and reinforcement of the anti-sliding pile decreases with an increase in the corrosion rate.

Fig. 14
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Stress distribution of reinforced concrete bonding layer in compression and tension zones with different corrosion rates.

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Figure 15 illustrates the distribution of axial displacement along the pile body in the bond layer between concrete and reinforcement in the tension and compression zones of the cantilever anti-sliding pile; For the axial displacement of the bond layer between concrete and reinforcement in the tension zone (Fig. 15a), there is essentially no slip between the reinforcement and concrete below the slip surface, but a rapid increase occurs at the slip surface. The maximum slip of the bond layer between uncorroded reinforcement and concrete in the tension zone is 2.75 mm, and the slip increases with the increase of the corrosion rate. When the corrosion rate reaches 6.79%, the slip of the bond layer in the loaded section reaches 4.63 mm; For the axial displacement of the bond layer between concrete and reinforcement in the compression zone (Fig. 15b), the overall slip pattern is similar to that of the tension zone, but the direction of slip is opposite and the overall slip is much less than that in the tension zone. The maximum slip of the bond layer between uncorroded reinforcement and concrete in the compression zone is 0.34 mm, but the slip gradually decreases with the increase of the corrosion rate. When the corrosion rate reaches 6.79%, the slip is 0.21 mm.The research results indicate that the bond slip phenomenon between the tensile reinforcement and concrete in the anti-sliding pile will increase with the increase of the corrosion rate. The change in slip is not significant at low corrosion rates, which is due to an initial increase in the bond strength between reinforcement and concrete at the beginning of corrosion, hence the slow growth of slip in the bond layer during the early stages of corrosion.

Fig. 15
figure 15

Distribution diagram of axial displacement of reinforced concrete bonding layer in tensile and compressive zones with different corrosion rates.

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Strain analysis of concrete and reinforcement at the slip surface of corroded anti-sliding piles

Fig. 16
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Curve of concrete strain and steel bar strain with load variation under different corrosion rates.

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Figure 16 shows the variation curves of concrete strain and reinforcement strain at the slip surface of anti-sliding piles under different corrosion rates with load. When the pile top displacement reaches 4 mm, the strain of uncorroded reinforcement is 1.61 × 10−3, and the strain of concrete is 3.35 × 10−3, with a ratio of 2.08 between them; When the corrosion rate is 0.32%, the strain of reinforcement is 1.55 × 10−3, and the strain of concrete is 3.27 × 10−3, with a concrete-to-reinforcement strain ratio of 2.11; When the corrosion rate is 0.86%, the strain of reinforcement is 1.54 × 10−3, and the strain of concrete is 3.17 × 10−3, with a concrete-to-reinforcement strain ratio of 2.06; When the corrosion rate is 2.68%, the strain of reinforcement is 1.48 × 10−3, and the strain of concrete is 2.81 × 10−3, with a concrete-to-reinforcement strain ratio of 1.90; When the corrosion rate is 5.87%, the strain of reinforcement is 1.49 × 10−3, and the strain of concrete is 3.46 × 10 − 3, with a concrete-to-reinforcement strain ratio of 2.88; When the corrosion rate is 6.79%, the strain of reinforcement is 1.42 × 10−3, and the strain of concrete is 4.92 × 10−3, with a concrete-to-reinforcement strain ratio of 3.46.

In the early stage of loading before the tensile reinforcement reaches the yield state, the strains of reinforcement and concrete at different corrosion rates are essentially the same. Both reinforcement strain and concrete strain increase with the increase of load, with concrete strain always being greater than reinforcement strain. Moreover, the strain ratio of concrete to reinforcement increases with the increase of the corrosion rate, indicating that the corrosion rate has a significant impact on the strain coordination of reinforced concrete. After the reinforcement in the anti-sliding pile structure corrodes, the strain coordination between the internal reinforcement and concrete deteriorates, further affecting the load-bearing capacity of the anti-sliding pile structure.

Analysis of factors affecting the bearing performance of Anti-Slip piles

Sensitivity analysis is conducted on the three factors affecting the durability of anti-sliding piles in an acid rain corrosion environment, considering the changes in the load-bearing capacity of anti-sliding piles when the compressive strength of concrete, mechanical properties of reinforcement, and bond strength decrease by 5%, 10%, 15%, and 20%, respectively, and simulating based on the Table 6.

Table 6 Material parameters of Anti-Sliding pile model at different strength reduction Rates.
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The numerical simulation data is plotted as shown in Fig. 17.

Fig. 17
figure 17

The degree of influence of different influencing factors on the bearing capacity of anti sliding piles.

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When the concrete strength decreases by 25%, the load-bearing capacity of the anti-sliding pile remains essentially unchanged, indicating that the concrete strength has a minimal impact on the load-bearing capacity of the anti-sliding pile. When considering only the degradation of bond slip between reinforcement and concrete, the load-bearing capacity of the anti-sliding pile decreases by 10% when the bond strength decreases by 25%. This is because when tensile damage occurs in the tensile zone concrete, the reduction in bond strength leads to rapid development of the damaged area, resulting in a decrease in the load-bearing capacity of the anti-sliding pile. The degradation of bond slip has a greater impact on the load-bearing performance of the anti-sliding pile than the deterioration of concrete strength. When considering only the loss of reinforcement strength, and when the strength decreases to 25%, its load-bearing capacity decreases by 29%. The reinforcement strength has a significant impact on the load-bearing capacity of the anti-sliding pile. In summary, the strength of the reinforcement and the bond performance between reinforcement and concrete are the main factors causing the decrease in the load-bearing capacity of the anti-sliding pile. When reinforcement corrodes, the combination of the degradation of corroded reinforcement and the bond performance of reinforcement and concrete leads to the decrease in the load-bearing capacity of the anti-sliding pile, with the decrease in reinforcement strength being the dominant factor.

Deformation performance analysis of anti-slip pile with corroded reinforcement

Calculation of bending stiffness of anti-slip pile with corroded reinforcement

This paper develops a stiffness calculation formula for corroded anti-slip piles by employing a reduction method for the short-term stiffness of the components.

$$\:{B}_{\eta\:}={k}_{b}{B}_{s}$$
(4)

In the equation: \(\:{B}_{\eta\:}\) is the short-term stiffness of corroded reinforced anti-slip pile; \(\:{k}_{b}\) is the degradation coefficient of bending stiffness; \(\:{B}_{s}\) is the short-term bending stiffness value of anti-slip pile without corrosion.

According to the internal force calculation method and requirements of cantilever pile structure in code for design of anti-slip pile treatment engineering46, the internal force of pile body at and above the sliding surface is calculated as cantilever beam according to the landslide design thrust and resistance in front of the pile. At this time, the pile top displacement of cantilever anti-slip pile can be calculated according to the following formula:

$$\:f=\frac{{ql}^{4}}{{8B}_{s}}$$
(5)

In the equation: \(\:f\) is the top displacement of cantilever anti-slip pile, \(\:l\) is the length of cantilever section of anti-slip pile, \(\:{B}_{s}\) is the short-term stiffness of cantilever anti-slip pile.

The ratio of the displacement at the top of the corroded anti-slip pile to that of the non-corroded anti-slip pile is the stiffness degradation coefficient of the anti-slip pile structure caused by steel corrosion.

$$\:{k}_{b}=\frac{{B}_{\eta\:}}{{B}_{s}}=\left\{\frac{{8f}_{\eta\:}}{{q}_{\eta\:}{l}^{4}}/\frac{{8f}_{s}}{{q}_{s}{l}^{4}}\right\}=\frac{{q}_{s}{f}_{\eta\:}}{{{f}_{s}q}_{\eta\:}}$$
(6)

According to formula (6), the numerical model of anti-slip pile with corroded reinforcement under the corrosion rate of 0–15% is established. The relationship between the degradation coefficient of flexural stiffness of anti-slip pile with corroded reinforcement and the corrosion rate of reinforcement is calculated and verified by combining with the test data47,48,49,50,51. The results are shown in Fig. 18.

Fig. 18
figure 18

The relationship between corrosion rate and degradation coefficient of flexural stiffness and the verification calculation results.

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It can be seen that the degradation coefficient of flexural stiffness of reinforced concrete beams decreases with the increase of corrosion rate. The average ratio of the calculated value of stiffness degradation coefficient to the test value is 0.981, and the standard deviation is 0.087. The calculated value is in good agreement with the test value as a whole. When the corrosion rate is low, the coincidence degree of the calculated results is higher, but for the reinforced concrete members with serious corrosion, the coincidence degree between the calculated results and the experimental values decreases.

Calculation of flexural bearing capacity of anti-slip piles with corroded reinforcement

For the non-corroded reinforced concrete anti-slip pile, when the longitudinal compression reinforcement is included, the bending bearing capacity of its normal section is calculated as follows:

$$\:{M}_{y}={\alpha\:}_{1}{f}_{c}bx\left({h}_{0}-\frac{x}{2}\right){+f}_{y}^{{\prime\:}}{A}_{s}^{{\prime\:}}\left({h}_{0}-{a}_{s}^{{\prime\:}}\right)$$
(7)

Therefore, the flexural bearing capacity of the normal section of the corroded reinforced anti-slip pile is:

$$\:{M}_{y}^{{\prime\:}}={k}_{m}\left({M}_{y}\right)$$
(8)
$$\:x=\frac{{{f}_{y\eta\:}A}_{s\eta\:}-{f}_{y}^{{\prime\:}}{A}_{s}^{{\prime\:}}}{{\alpha\:}_{1}{f}_{c}b}$$
(9)

In the equation:\(\:\:{\alpha\:}_{1}\) is coefficient, taken as 1 when the concrete strength grade does not exceed C50; \(\:{A}_{s}^{{\prime\:}}\) is Sectional area of longitudinal reinforcement in compression zone;\(\:{a}_{s}^{{\prime\:}}\) is the distance from the reasonable point of longitudinal reinforcement in the compression zone to the compression edge of the section; \(\:b\) is width of rectangular section; \(\:{f}_{c}\) is design value of concrete axial compressive strength; \(\:{f}_{y}^{{\prime\:}}\) is standard value of compressive strength of reinforcement; \(\:{h}_{0}\) is effective height of section; \(\:x\) is height of concrete compression zone; \(\:{A}_{s\eta\:}\) is the cross-sectional area of the corroded reinforcement; \(\:{f}_{y\eta\:}\) is the yield strength of corroded reinforcement; \(\:{k}_{m}\) is correction coefficient of flexural bearing capacity of anti-slip pile considering bond slip effect;

By simulating the reduction factor \(\:{k}_{m}\) of the flexural capacity of the normal section of the corroded reinforced anti-slip pile when the corrosion rate is 0 −15%, the calculation model of the flexural capacity of the normal section of the corroded reinforced anti-slip pile considering the bond effect of reinforced concrete is obtained as follows:

$$\:{k}_{m}=\left(1-0.679\eta\:\right)$$
(10)
$$\:{M}_{y}^{{\prime\:}}=\left(1-0.679\eta\:\right)\left[{\alpha\:}_{1}{f}_{c}bx\left({h}_{0}-\frac{x}{2}\right){+f}_{y}^{{\prime\:}}{A}_{s}^{{\prime\:}}\left({h}_{0}-{a}_{s}^{{\prime\:}}\right)\right]$$
(11)

In order to verify the accuracy of the formula, the test data of 25 corroded reinforced concrete demolition members in actual projects were selected for verification and analysis52. The \(\:{k}_{m}\) fitting results and calculation results are shown in Fig. 19:

Fig. 19
figure 19

The relationship between corrosion rate and degradation coefficient of bearing capacity and the verification calculation results.

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It can be seen that the average value of the ratio between the calculated value and the test value is 0.975, and the standard deviation is 0.031. The calculated value is in good agreement with the test value as a whole. Most of the calculated values are lower than the test values, which is safe. When the corrosion rate is low, the calculated results are in good agreement, but for the reinforced concrete members with serious corrosion, the deviation between the calculated results and the test values is relatively large.

Conclusions

This study aims to investigate the indoor testing and simulation methods for the bearing performance of cantilever anti-slip piles under the combined effects of acid rain corrosion and thrust. A loading test was conducted on a cantilever anti-slip pile, and a numerical model of the cantilever anti-slip pile under the combined action of acid rain corrosion and thrust was developed based on bilinear cohesive elements. Then, the simulation results were used to verify the experiment, and the following conclusions were drawn:

  1. (1)

    The crack distribution is relatively uniform, with a total of five main cracks appearing between 110 mm above and 150 mm below the slip surface, and the crack distribution presents an “olive-like” pattern when unfolded. The bending moment of the anti-sliding pile is non-linearly distributed along the direction of the pile, with a bending moment peak near the slip surface.

  1. (2)

    The load-bearing capacity of the anti-sliding pile is positively correlated with concrete strength. When the concrete strength increases by 11.14%, the load-bearing capacity of the anti-sliding pile increases by 0.85%. When the concrete strength decreases by 23.51%, the load-bearing capacity of the anti-sliding pile decreases by 1.53%. Considering both the reinforcement mechanical properties and the damage to the reinforcement-concrete interface, when the reinforcement corrosion rate reaches 2.68%, the load-bearing capacity of the anti-sliding pile decreases by 3.98%, and when the corrosion rate reaches 6.79%, the load-bearing capacity of the anti-sliding pile decreases by 8.01%. With the increase of the corrosion rate, the peak stress of the reinforcement-concrete bond interface decreases, and the maximum displacement increases, leading to a decrease in load-bearing capacity.

  1. (3)

    The reinforcement strength has the greatest impact on the load-bearing performance of anti-sliding piles under acid rain corrosion, followed by bond strength, and concrete strength has the least impact. To prevent the failure of cantilever anti-slip piles due to corrosion, the most critical measures are to apply anti-corrosive coatings to the reinforcement and to strictly control the quality of reinforcement connections.Carry out a long-term testing of cantilever anti-slip pile engineering and establishment of a more accurate corrosion model combined with time-varying reliability, and understand the structural degradation situation through monitoring data in order to timely discover and solve problems.