Study on the Force Transfer Process of Bolt–Slurry Interface of Full-Length Bonding Anchor System at Earthen Sites

Abstract

The debonding and sliding of the bolt–slurry interface is the main failure form of the full-length bonding anchor system (FLBAS) of earthen sites, so it is urgent to carry out a quantitative study of the force transfer process of the anchorage interface. Based on field test results and existing research results, it was found that the bilinear bond–slip model is in line with the description of the constitutive relationship of the bolt–slurry interface. The whole process of debonding slip is discussed accordingly; the expressions for the slip, the axial strain of the bolt, and the load displacement at the bolt–slurry interface corresponding to the different loading stages are deduced; and the calculations of the ultimate load-carrying capacity and the effective anchorage length are given at the same time. On this basis, the bond–slip model parameters were calibrated by identifying the characteristic points of the bond–slip curve; a multi-parameter cross-comparison validation of the reasonableness of the theoretical analytical model was carried out on the basis of in situ pull-out tests; and the law of the influence of anchor bond length and axial stiffness on the anchorage performance was analyzed. The analytical model proposed in this study is widely applicable to the analysis of force transfer processes at the bolt–slurry interface in the presence of complete debonding phenomena and provides a useful reference for optimizing the design of anchors while minimizing interventions.

1. Introduction

Earthen sites are important immovable cultural heritage and direct evidence of the origins and evolution of human civilization [1]. Earthen sites of rich types are widely distributed in the main arid areas of Asia, North America, South America, and Africa. Some important earthen sites exist in the arid environments of China and along the Silk Road, such as the Great Wall and annex buildings from the Qin to Ming Dynasties, Buddhist temple sites, and ancient castles of different historical periods [2]. However, due to the influence of natural forces and human activities, there are common diseases that threaten structural stability at earthen sites, such as unloading deformation cracks, cracks caused by tectonic activities, and cracks caused by construction joints, which leave earthen ruins on the verge of instability and collapse. In the context of a large number of earthen sites in urgent need of protection and reinforcement, the situation is still in the “rescue reinforcement stage”, and the mechanical stability control of earthen sites is a top priority [3]. Forms of “full-length bonded anchor bolt anchoring technology” involve little construction disturbance and can effectively control the development of cracks, which is conducive to the self-stability of earthen sites, so that the earthen sites can be preserved for a long time, and have been widely used in earthen site reinforcement projects [4].

Since the 1990s, the Dunhuang Research Institute has introduced such technology into the reinforcement of earthen sites along the Silk Road [1]. The rescue and reinforcement of Chinese earthen sites have experienced a slow development process from the application of conventional anchor engineering experience to practical exploration, and then to the scientification of anchoring technology [5]. Earthen site anchoring is different from conventional rock–soil anchoring, because it has the dual attributes of rock–soil mass and cultural relics, which require its reinforcement to follow the principle of “minimum intervention, maximum compatibility, and no change to the original state”. In anchoring practice, it has been gradually recognized that conventional modern reinforcement materials such as metal anchors [6] and cement mortar [7] have poor compatibility with earthen sites in terms of physical, mechanical, and chemical properties, and are prone to fatal problems of inconsistent deformation and insufficient durability [3]. Therefore, many current studies mainly focus on the anchoring performance and anchoring process optimization of the full-length bonding anchor system (FLBAS) composed of multiple types of bolts [8,9,10] and grout [11,12,13], and the exploration of anchoring mechanisms is still shallow. Many studies have pointed out that the failure mode of the FLBAS at most earthen sites is debonding and sliding at the bolt–slurry interface [10,11,12,13]. The stress at the bolt–slurry interface is unevenly distributed, with single peak or double peak characteristics, and the stress peak gradually transfers to the anchoring tail end with the increase of load [11,12,13]. However, the current cultural relics protection industry standard «dry earthen sites protection and reinforcement engineering design specification (WW/T 0038-2012 [14])» is still based on the “interface average bonding stress” design method for earthen site anchoring designs, which involves an obvious overestimation of anchoring force. At the same time, in view of the limitations of directly obtaining data by the test method, the force transfer process at the bolt–slurry interface is numerically analyzed based on the trilinear interface constitutive model [15] and the improved trilinear [16], but the two constitutive models have the problems of high calculation results and cumbersome calculation due to excessive consideration of the strength of the friction section. Therefore, the existing practical experience, testing, and theoretical research only provide a qualitative understanding of the force transfer process at the bolt–slurry interface of earthen sites, or the optional interface constitutive model for quantitative analysis of the interface force transfer process is too singular. This can no longer meet the needs of anchoring of earthen sites that is being carried out on a large scale. To clarify the distribution of bond stress at the bolt–slurry interface along the bond length, accurately predict the ultimate bearing capacity, avoid “protective damage” to the earthen sites due to excessive reinforcement, and clarify the force transfer process at the bolt–slurry interface, it is urgent to carry out theoretical research on the force transfer mechanism of the bolt–slurry interface of the FLBAS at earthen sites.

In recent decades, domestic and foreign scholars have carried out many studies on the constitutive relationship of the bond interface between two heterogeneous materials, which can be roughly divided into two categories: one involves the nonlinear continuous curve model [17,18,19,20,21,22], and the other the piecewise linear model [23,24,25,26,27,28]. Based on the constitutive relationship of a bond interface, the theoretical analysis of the force transfer mechanism of the anchor system is carried out using the load transfer method [16], the calculus method, and the transfer matrix method [29,30]. Yazicit et al. [31] explored the bearing performance of the anchor system based on the assumption that the bearing capacity mainly depends on the friction affected by the normal stress at the bolt–slurry interface, but this method ignores the interface bond strength, which is different from the actual situation. Gao et al. [20,32] used the bond–slip model of exponential attenuation of interfacial stress in the softening section to analyze the interfacial force transfer mechanism, but the relevant parameters need to be solved by iterative algorithm, which is not convenient for engineering application. Ma et al. [33,34] considered the slip at the loading end and the complete debonding phenomenon of the bolt–slurry interface, deduced the distribution of interface stress and strain, and enriched the debonding/decoupling mechanism of the bolt–rock interface, but the research on the whole process of force transfer at the anchoring interface is still insufficient. Referring to the algorithm of Yuan et al. [35], Ren et al. [36] used the trilinear bond–slip model to study the full range stress response behavior of the mortar–rock mass anchoring interface, but the model over considered the interface residual stress and did not consider the complete debonding phenomenon, which is different from the field pull-out test results of the FLBAS at earthen sites [11,12,13]. However, there are few reports on whether the mechanical properties of the bolt–slurry interface in the anchoring of earthen sites comply with the existing models.

Firstly, through previous field tests and existing research results, this study verifies the applicability of the bilinear bond–slip model in the bond–slip behavior of the bolt–slurry interface of the FLBAS of earthen sites. Then, based on the analysis methods of Ren et al. [15] and Lu et al. [16], a closed-form solution is derived to predict the full range mechanical behavior of the whole or part of the grouted bolt under tension. The whole process of interfacial bond sliding is divided into four stages, and the analytical solutions of a series of parameters such as the load–displacement relationship and interface slip corresponding to each stage are deduced. According to the drawing test results, the parameters of the model are calibrated, the test values are compared with the theoretical values, and the effects of different anchoring parameters are discussed.

2. Ideal Force Analysis Model for Bond–Slip at the Anchorage Interface

2.1. Ideal Force Analysis Model

A complete FLBAS at an earthen site includes not only the anchor bolt, the slurry, and the site soil, but also the bolt–slurry interface and the slurry–soil interface. The two heterogeneous materials form a bond transition zone near the interface, and the difference in the mechanical properties of the heterogeneous materials is most prominent at the interface. Existing research results show that the failure mode of the FLBAS at earthen sites is mostly the slip failure after the debonding of the bolt–slurry interface (Figure 1) [11,12,13]. This is mainly because the anchoring materials and techniques at earthen sites are different from conventional rock–soil anchoring engineering. The reasons are as follows: First, since the strength of the anchor slurry (such as modified glutinous rice mortar) of the earthen site is required to be close to or slightly higher than the strength of the site soil [4], and the site soil and slurry are granular, while the bolt is fiber material, there are differences in the elastic modulus and strength index between the bolt and the slurry, so the deformation coordination performance of the slurry–soil interface is stronger than that of the bolt–slurry interface [13,14,15,16]. Second, during the anchoring process, when the infiltration reinforcement of the anchor hole wall is carried out [3], the slurry penetrates into the soil, and the strength of the slurry–soil interface is increased after solidification. The soil is wrapped with slurry, and the slurry wraps around the bolt; that is, the action area of the slurry–soil interface is much larger than that of the bolt–slurry interface, and the shear resistance provided by the slurry–soil interface is much larger than that of the bolt–slurry interface.

Therefore, the mechanical behavior of the bolt–slurry interface is the decisive factor affecting the performance of the anchoring system of earthen sites, and it is the main control surface of the anchoring system. The whole process of bond–slip at the bolt–slurry interface is affected by many factors.

The failure mode of the FLBAS of earthen sites is mostly slip failure after the debonding of the bolt–slurry interface (Figure 1). In order to facilitate the analysis of the whole bond–slip process behavior at the bolt–slurry interface, the following simplifications are made in this paper: (1) anchor rods are straight rods of equal cross-section with no defects in the body; (2) it is assumed that the grouting effect is good, there is no hollowing or leakage of the slurry, and both the slurry–soil interface and the bolt–slurry interface are well bonded; (3) to simplify the calculation, this study ignores the influence of Poisson’s ratio and assumes that the bolt–slurry interface only bears the bonding stress; (4) the volume and rigidity of the slurry and soil in the FLBAS at the soil site are large, so the deformation of the site soil or slurry can be ignored.

Based on the above assumptions, when there is a slip between the bolt and the slurry, the calculation diagram of the ideal model for the bond–slip analysis of the bolt–slurry interface as shown in Figure 2 will be generated.

2.2. Bilinear Bond–Slip Model

To analyze the force transfer process at the bolt–slurry interface of the FLBAS at earthen sites, it is necessary to clarify the constitutive relationship of the bolt–slurry interface. Figure 3a shows the bond–slip curve of the bolt–slurry interface in existing tests [37]. To show the whole process of the complete bond–slip curve, only the bond–slip curve of the interface at the midpoint (L = 90 cm) of two adjacent measuring points in the loading section is listed. Figure 3b refers to the bond–slip curve of the interface at the midpoint of two adjacent measuring points of the pull-out loading section of the bamboo bolt-modified slurry anchor system of Lu et al. [16]. According to the anchoring mechanism obtained from the existing tests [11,12,13] and the bond–slip curve between different measuring points in Figure 3, when the slip amount of the bolt–slurry interface changes to a small extent, the pull-out load between the bolt and the slurry presents a transition dominated by chemical bond force to mechanical interlocking force, and the bond stress gradually reaches its peak. Then the bolt–slurry interface enters the softening stage. At this time, the slip increases rapidly, the mechanical interlocking between bolt and slurry changes from loose to tight, and the cementation force gradually disappears and transitions to the state dominated by friction. With the continuous increase of slip, the bond stress at the bolt–slurry interface gradually decreases and tends to zero, and the bolt and slurry tend to become completely debonded.

Based on the above analysis of the actual play of chemical bond force, mechanical interlocking force, and friction force between bolt and slurry at different stages, and considering the form of the interface bond stress–slip curve in Figure 3 and the needs of subsequent numerical simulation calculations, a bilinear bond–slip model considering the form of complete interface debonding is selected to fit the bond–slip curve of the bolt–slurry interface of the FLBAS at earthen sites. Figure 3a shows that the bilinear bond–slip model considering the form of complete debonding of the interface is in good agreement with the bond–slip curve of the bolt–slurry interface obtained from the existing test [37]. Figure 3b shows that the curve of the model to the test value is in good agreement before the bond stress peak point, and the two also have good agreement within a certain slip after the bond stress peak point. When the slip exceeds about 12.5 mm, the test curve shows a slow downward trend, and the model underestimates the actual bond stress of the interface in this section, but the overall weight of the bond stress in this section is small. On the whole, the bilinear bond–slip model considering the complete debonding of the interface can well characterize the bond–slip relationship of the bolt–slurry interface of the FLBAS at earthen sites.

Through the above analysis, a bilinear bond–slip model of the bolt–slurry interface of the FLBAS at earthen sites considering complete debonding is proposed, as shown in Figure 4.

Based on the bilinear model, the bond–slip curve of the bolt–slurry interface is divided into an elastic phase, a softening phase, and a loosening phase. Therefore, the expression is:

$$\tau \left(\omega \right)=\left\{\begin{array}{l}{\displaystyle \frac{{\tau }_m}{{\omega }_l}}\omega \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{when} 0\le \omega \le {\omega }_l\hspace{1em}\hspace{1em} \left(\mathrm{a}\right)\\ {\displaystyle \frac{{\tau }_m}{{\omega }_m-{\omega }_l}}\left({\omega }_m-\omega \right)\hspace{1em} \mathrm{when} {\omega }_l\le \omega \le {\omega }_m\hspace{1em}\hspace{1em}\left(\mathrm{b}\right)\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{when} {\omega }_m\le \omega \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathrm{c}\right)\end{array}\right.$$

(1)

3. Interface Governing Equations

Based on the above analysis and basic assumptions, the stress balance equation of bolt can be established as follows:

$${\displaystyle \frac{d\sigma }{dz}}={\displaystyle \frac{2\tau }{r}}$$

(2)

where $\sigma$ is the axial stress of the anchor, $\tau$ is the bond stress at the interface, r is the radius of the anchor, and z is the axial length coordinate of the anchor.

The intrinsic equations of the interface and the anchor are given by earthen site anchoring systems as the bolt always remains linearly elastic during pull-out due to the relatively small load bearing:

$$\tau =\tau \left(\omega \right)$$

(3)

$$\epsilon ={\displaystyle \frac{\sigma }{E}}={\displaystyle \frac{du}{dz}}$$

(4)

where E is the modulus of the elasticity of the bolt, $\epsilon$ is the axial strain of the bolt, and u is the axial displacement of the bolt. According to the assumption, the slip ω is equal to the axial displacement u of the bolt.

Substituting Equations (3) and (4) into Equation (2), the control equation of the anchor bolt can be obtained as:

$${\displaystyle \frac{d^2\omega }{d{z}^2}}-{\displaystyle \frac{{\omega }_m}{{\tau }_m}}{\eta }^2\tau \left(\omega \right)=0$$

(5)

Thus:

$${\eta }^2={\displaystyle \frac{2{\tau }_m}{{\omega }_{m}Er}}$$

(6)

4. Behavior Analysis of the Whole Pull-Out Process and Derivation of Analytical Solutions

Based on the bilinear line bond–slip model, by solving the governing Equation (5) of each loading stage, the slip and bond stress distribution along the bolt–slurry interface, the axial stress of the bolt, and the load–displacement relationship can be obtained.

Figure 5 shows the evolution of the interface bond stress distribution when the bond length is significantly greater than the effective bond length. The ultimate bond stress transfers within the bond length, showing the complete bond–slip process of the interface, include four stages: elastic stage, elastic–softening stage, elastic–softening–loose stage, softening–loose stage. If the actual anchoring length of bolt is less than the effective anchoring length, the anchor system will be damaged earlier, and the development process of the bond stress of the bolt–slurry interface will not be as complete as the deduction process of this study. In engineering practice, considering the safety reserve of anchoring force, the actual anchoring length of a bolt is longer than the effective anchoring length. Therefore, the performance analysis and analytical solution derivation of the whole process of pull-out of the FLBAS at earthen sites in this study are of certain significance to engineering practice.

4.1. Elastic Stage

The bolt–slurry interface remains elastic throughout the bonding length under a small drawing force (Figure 5a,b). In this stage, the interface bonding stress decays exponentially along the anchoring length, and the interface bonding stress increases with the increase of the load, until the interface bonding stress at z = L increases to τ_m. The pure elasticity phase is also over. Substituting Equation (1a) into Equation (5), the governing equation of the elastic stage can be obtained:

$${\displaystyle \frac{d^2\omega }{d{z}^2}}-{\eta }_1^2\omega =0$$

(7)

Thus:

$${\eta }_1^2={\displaystyle \frac{{\omega }_m}{{\omega }_l}}{\eta }^2={\displaystyle \frac{{2\tau }_m}{{\omega }_{l}Er}}$$

(8)

The boundary conditions for this stage are: when $z=0$, $\epsilon =0$; and when $z=L$, $\epsilon ={\displaystyle \frac{F}{\pi r^{2}E}}$.

Considering the above boundary conditions and solving the differential Equation (7), the slippage ω of the interface, the bonding stress τ, and the axial strain ε of the bolt are obtained:

$$\omega ={\displaystyle \frac{F{\omega }_l{\eta }_1\cosh \left({\eta }_{1}z\right)}{2\pi r{\tau }_m\sinh \left({\eta }_{1}L\right)}}$$

(9)

$$\tau ={\displaystyle \frac{F{\eta }_1\cosh \left({\eta }_{1}z\right)}{2\pi r\sinh \left({\eta }_{1}L\right)}}$$

(10)

$$\epsilon ={\displaystyle \frac{F\sinh \left({\eta }_{1}z\right)}{E\pi r^2\sinh \left({\eta }_{1}L\right)}}$$

(11)

Since the sliding amount at the loading section z = L of the bolt is the displacement (Δ) of the entire bolt, the load–displacement relationship of the bolt can be obtained by Equation (9):

$$F={\displaystyle \frac{2\pi r{\tau }_m\tanh \left({\eta }_{1}L\right)}{{\eta }_1{\omega }_l}}\mathsf{\Delta }$$

(12)

It is worth noting that the difference between the resistance value provided by considering tanh(2) ≈ 0.964, the bonding stress at the bolt–slurry interface, and the applied load in the elastic phase is less than 3.6%. In order to obtain the effective bonding length in the elastic phase, it is necessary to reach a consensus that the effective bonding length is the length of the bolt–pulp interface that is forced to participate in the resistance to the pull-out load, so the effective bonding length is approximately

$$l_{e,\mathrm{m}\mathrm{a}\mathrm{x}}=2/{\eta }_1$$

(13)

4.2. Elastic–Softening Stage

As the pull-out load continues to increase, the interface softens and gradually develops from the z = L interface to the z = 0 interface. At this time, the interface displays elastic and softening segments, as shown in Figure 5b–d. The axial coordinate corresponding to the peak value of the interfacial bond stress τ_m is also gradually shifted to the free end of the anchor, and the softening length α gradually increases until the interfacial bond stress at the loading end is reduced to 0. At this point, the critical state of transition from the elastic–softening stage to the elastic–softening–loosening stage is reached, and the anchoring force reaches its maximum F_e,s-max.

By substituting Equation (1b) into Equation (5), the governing equation of softening stage can be obtained:

$${\displaystyle \frac{d^2\omega }{d{z}^2}}+{\eta }_2^2\omega ={\eta }_2^2{\omega }_m\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}when {\omega }_l\le \omega \le {\omega }_m$$

(14)

Thus:

$${\eta }_1^2={\displaystyle \frac{{\omega }_m}{{\omega }_m-{\omega }_l}}{\eta }^2={\displaystyle \frac{2{\tau }_m}{\left({\omega }_m-{\omega }_l\right)Er}}$$

(15)

The previous boundary conditions still apply in this phase, while the following continuity conditions exist: when z = L − α, the value of ε is continuous and ω = ω_l.

Solving the control Equation (14) yields the interfacial softening zone $(0\le \omega \le {\omega }_l)$ or $(0\le z\le L-\alpha )$ ω, τ and ε:

$$\omega =({\omega }_m-{\omega }_l)\left\{{\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\alpha \right)\right]sin\left[{\eta }_2\left(z+\alpha -L\right)\right]-cos\left[{\eta }_2\left(L-\alpha -z\right)\right]+{\displaystyle \frac{{\omega }_m}{{\omega }_m-{\omega }_l}}\right\}$$

(16)

$$\tau ={-\tau }_m\left\{{\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\alpha \right)\right]sin\left[{\eta }_2\left(z+\alpha -L\right)\right]-cos\left[{\eta }_2\left(z+\alpha -L\right)\right]\right\}$$

(17)

$$\epsilon ={\displaystyle \frac{{2\tau }_m}{{\eta }_{2}rE}}\left\{{\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\alpha \right)\right]cos\left[{\eta }_2\left(z+\alpha -L\right)\right]+sin\left[{\eta }_2\left(z+\alpha -L\right)\right]\right\}$$

(18)

The total anchoring force at this stage can be obtained by Equation (18):

$$F={\displaystyle \frac{{2\pi r\tau }_m}{{\eta }_2}}\left\{{\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\alpha \right)\right]cos\left({\eta }_2\alpha \right)+sin\left({\eta }_2\alpha \right)\right\}$$

(19)

It is clear that the anchorage force of the anchoring system reaches the limit value when the derivative of F with respect to α in Equation (19) is zero. Loosening starts at the loaded end when the interfacial bond stress τ at z = L in the loaded section decreases to 0. Substituting z = L and τ = 0 into Equation (17) gives:

$${\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\alpha \right)\right]sin\left({\eta }_2\alpha \right)-cos\left({\eta }_2\alpha \right)=0$$

(20)

From Equation (20), the corresponding softening length α_s occurs when the ultimate value of the pull-out load is reached:

$${\alpha }_s={\displaystyle \frac{1}{{\eta }_2}}arcsin\sqrt{{\displaystyle \frac{{\omega }_m-{\omega }_l}{{\omega }_m}}}$$

(21)

4.3. Elastic–Softening–Loosening Stage

With the increase of pull-out load, the bond stress at z = L decreases to 0, and the corresponding point of τ_m gradually transfers to the free end of the anchor, and the interface loosening starts to transfer from the loading end of the bolt to the free end. Therefore, within the bonding length, there are three stress states: elastic, softening, and loose. Using β to denote the length of interfacial loosening that occurs from z = L, Equations (16)–(19) are still valid when L is replaced with (L − β).

Substituting Equation (1c) into Equation (5) gives the control equation for the loosening phase:

$${\displaystyle \frac{d^2\omega }{d{z}^2}}=0 \hspace{1em}\hspace{1em}\hspace{1em}when \omega \ge {\omega }_m$$

(22)

The previous boundary conditions still apply in this phase, while the following continuity conditions exist: when z = L − β − α_s, the value of ε is continuous and ω = ω_m; when z = L − β, the value of ε is continuous and ω = ω_l.

The total slip of the interface at this stage is:

$$\mathsf{\Delta }={\displaystyle \frac{{2\tau }_m\beta }{{\eta }_{2}rE}}\left\{{\displaystyle \frac{{\eta }_2}{{\eta }_1}}tanh\left[{\eta }_1\left(L-\beta -{\alpha }_s\right)\right]cos\left({\eta }_2{\alpha }_s\right)+sin\left({\eta }_2{\alpha }_s\right)\right\}+{\omega }_m$$

(23)

At the end of this phase, when L − β = α_s, the softening–loosening phase is about to develop.

4.4. Softening–Loosening Phase

At this stage, the maximum interfacial bonding stress decreases with load at z = 0. In case ω ≥ ω_m, the displacement of the loaded section can be obtained by solving the control Equation (5). Alternatively, it is straightforward and simpler to derive the following load–displacement equation from the superposition of the displacement along the length of the bond:

$$\omega =F{\displaystyle \frac{\beta }{E\pi r^2}}+{\omega }_m$$

(24)

5. Analysis and Discussion of Results

The validity of the theoretical predictions was verified by applying the completed pull-out tests of the G1 anchorage system mentioned in Section 2.2. In this study, matlab was utilized to solve the implicit solutions in the equation, with two characteristic points $A(u_1,F_1)$ and $B\left(u_2,F_2\right)$ on the load–displacement curve obtained by pull-out test. The two characteristic points are substituted into the equations for the critical elastic phase and the elastic–softening phase at the loading end, respectively, which in turn leads to the model parameters.

5.1. Load–Displacement Curve Comparison

It can be seen from Figure 6 that the load–displacement curve predicted theoretically is in good agreement with that of the G1 anchorage system. In the elastic stage (before point A), it is almost the same; while in the elastic–softening stage (section AB), the slip of the G1 anchoring system is slightly larger. This may be due to the larger disturbance of the G1 anchoring system by the drawing method of cyclic loading, which makes the slip of the G1 anchoring system larger.

After point B, which corresponds to the end of the elastic–softening phase and the beginning of the elastic–softening–loosening phase, the data after this phase could not be accurately obtained due to the test conditions. Overall, the theoretical solutions for the elastic and elastic–softening phases achieved good predictions of the measured load–displacement curves.

5.2. Comparison of Interface Stress Distribution Curves

In Figure 7a, experimental axial stress distribution curves and theoretical predicted axial stress distribution curves under different load conditions of 2~12 kN are presented. When the load is small, the coincidence degree between theoretical value and test value is better than that of a large load; that is, the test value fluctuates greatly with the increase of load. On the whole, the axial stress distribution curves predicted by theory agree well with the experimental values. In Figure 7b, the experimental bolt–slurry interface bonding stress distribution curve and the theoretical predicted bolt–slurry interface bonding stress distribution curve under different loading conditions of 2~12 kN of the G1 anchoring system are presented. The closer the loading end is to the bond stress, the greater the fluctuation; on the contrary, the test value of the free end is closer to the theoretical value. The reasons are as follows: on the one hand, the stress concentration at the loading end is close to the free surface, and the change in the soil–slurry–bolt anchorage system is increased; on the other hand, the test value in Figure 7b is not directly measured, but calculated.

5.3. Analysis of Anchorage Parameters

This section focuses on the analysis of the influence of bolt bonding length and axial stiffness on the performance of the anchoring system. In order to clarify the axial stiffness of the anchors, the product $E_s{A}_s$ of the modulus of elasticity $E_s$ of the anchors and the cross-sectional area $A_s$ $\left(\pi r^2\right)$ of the anchors was selected as the variable. The values were taken as 2.46 MN, 3.69 MN, and 4.93 MN, respectively, and the computational parameters were the same as those in

Table 1. Anchorage parameters and characteristic point parameters for in situ tests of the anchorage system [37].

r (mm)	E (GPa)	L (m)	τm (m)	Characteristic Point A		Characteristic Point B
				μ1 (mm)	F1 (kN)	μ2 (mm)	F2 (kN)
14	4	1	0.3528	0.51	6.25	2.25	13.21

. From Equations (13) and (21), the effective bonding length L_e,max = 0.402 m of the elastic section and softening length α_s,max = 0.406 m can be obtained, so the total effective bonding length is 0.808 m. As can be seen from Figure 8, when the bonding length of the anchoring system is less than 0.808 m, the ultimate anchoring force increases significantly with the increase of the bonding length. When the bonding length of the anchoring system is greater than 0.808 m, the load–displacement curves of the anchoring system with different bonding lengths completely coincide from the origin to the peak point (elastic stage, elastic–softening stage and partial elastic–softening–loosening stage). When the bonding length exceeds the effective bonding length, increasing the bonding length can only maintain the ultimate anchoring force, increase the limit slip of the bolt, and enhance the ductility of the anchoring system. When the bonding length of the anchoring system is fixed, the axial stiffness of the bolt is increased, the ultimate anchoring force is increased, and the length of the stable section of the load–displacement curve is shortened. Figure 9 shows that when the axial stiffness remains unchanged and the bonding length is greater than 0.808 m, the distribution of bonding stress and axial stress under the same drawing load remains unchanged as the bonding length continues to increase. When the bonding length is equal to 1 m and the axial stiffness of the bolt is increased, the area of the bonding stress distribution curve and the axial stress distribution curve increases with the bonding length axis, which reveals the reason why the axial stiffness of the bolt increases and the ultimate anchoring force increases.

To sum up, in the anchorage design of earthen sites, the bonding length of the bolt is designed to be slightly larger than the effective bonding length according to the comprehensive consideration of the safety reserve of the anchoring force, the limit slip amount of bolt, and the minimum intervention degree of earthen sites. Based on practical engineering experience, it is suggested that the bonding length of the bolt should be 1~2 m. Although a higher ultimate anchorage force can be obtained by increasing the axial stiffness of the bolt in actual engineering, in order to avoid the poor compatibility of mechanical strength and deformation of the designed bolt with earthen sites, that is, the brittle failure phenomenon of “chopsticks inserted into tofu” often mentioned in the industry, the diameter of the bolt can only be appropriately increased or the elastic modulus of the bolt can be appropriately increased under conditions similar to the elastic modulus of earthen sites.

6. Discussion

This paper confirms that the bilinear model conforms to the bolt–slurry interface constitutive relationship through experiments and existing studies, which enriches the descriptive form of the stress–strain relationship at the bolt–slurry interface at earthen sites. This study lays the foundation for the mechanical behavior of the bolt–slurry interface, which is the key to controlling the performance of soil site propriety anchoring systems. This paper analyzes the force transfer process at the bolt–slurry interface of a full-length bonded anchorage system at earthen sites based on a bilinear model, which theoretically confirms the phenomenon of peak distribution of interfacial bond stresses found in existing studies, confirming that the code-suggested design method of “interfacial average bond stress” overestimates the anchorage force. This paper uses the theoretically predicted bilinear model, load–displacement curves, axial force distribution, and bond stress distribution, and the experimental bilinear model, load–displacement curves, axial force distribution, and bond stress distribution are cross-corroborated with each other. On the basis of ensuring the correct theoretical analysis, the interfacial slip, interfacial bonding stress, and axial stress of the bolt–slurry interface mechanical behavior at different loading stages are revealed in a quantitative way, and the interfacial force transfer mechanism is clarified. The maximum anchorage force during the elastic phase, ultimate anchorage force, and effective bond length, as well as the law of influence of anchor bond length and axial stiffness on the performance of the anchorage system derived in this study, are all conducive to the realization of earthen site protection based on the imposition of a minimum of external intervention. This paper strives to be compatible with earthen sites in terms of material, structure, parameters, etc., and at the same time to avoid “protective damage” to earthen sites due to over-reinforcement, so that the optimal design of earthen site anchorage can be based on the evidence.

Of course, even a model that fits the stress–strain relationship at the bolt–slurry interface does not reveal the full constitutive relationship at that interface, and the bilinear bond–slip model used in this paper also has its own limitation. The limitation of this study is specifically the presence of a horizontal plateau on the theoretically predicted load–displacement curves, a phenomenon that occurs after the interface exhibits elasticity/softening/debonding phenomena. However, in practice, the slurry acts as a force transfer medium between the anchor and the sites soil, which causes the anchor to be constrained in all directions. Although the mechanical strength of the slurry is close to that of the site soil due to the material compatibility type, after the rod and the slurry are debonded and slipped, the slurry will inevitably exert pressure on the anchor, which in turn will generate more or less frictional resistance on the surface of the anchor. Therefore, the load–displacement curves of pull-out tests of anchoring systems usually do not have an absolute horizontal stabilization phase. Therefore, if the results of this paper are used in the elastic stage and elastic–softening stage before reaching the ultimate bearing capacity, the results will be more concise and accurate. Whereas, the predicted bearing capacity values will be small if the results of this paper are used in the elastic–softening–debonding stage to the complete debonding stage.

7. Conclusions

In this paper, based on a bilinear model considering the phenomenon of complete debonding, the force transfer process at the bolt–slurry interface of a full-length bonded anchorage system at the earthen sites is investigated, the theoretical prediction results are compared with the results of the pull-out test of the anchorage system based on the anchorage parameters determined in Table 1, and the following conclusions are obtained:

(1) The bond–slip bilinear model considering the phenomenon of complete debonding satisfies the need to characterize the constitutive relationship at the bolt–slurry interface of earthen site anchoring systems. The whole bond–slip process at the bolt–slurry interface was divided into four stages: elastic, elastic–softening, elastic–softening–loosening, and softening–loosening stages.

(2) In the process of reaching the ultimate bearing capacity of the anchoring system, the theoretically predicted values of the load–displacement relationship, the axial force distribution of the rods, and the interfacial bond stress distribution at each stage are highly consistent with the test values, which verifies the correctness of the theoretical solution.

(3) The theoretically predicted effective bond length of the elastic stage, softening length, total effective bond length, and ultimate bearing capacity were 0.402 m, 0.406 m, 0.808 m, and 13.21 kN, respectively.

(4) When the bonding length of the anchor is less than the effective bonding length of 0.808 m, increasing the bonding length can effectively improve the ultimate bearing capacity. If the bonding length is more than the effective bonding length of 0.808 m, when the bonding length of the anchor is increased, the ultimate anchoring force will remain unchanged, but the ultimate slip of the anchor will be improved, which means that the ductility of the anchoring system is enhanced.

(5) Increasing the axial stiffness of the anchor by increasing the diameter of the anchor or the modulus of elasticity of the anchor will increase the ultimate load carrying capacity of the anchoring system, but it will also reduce the ductility of the anchoring system. Therefore, the appropriate stiffness of anchors should be selected according to the needs of mechanical compatibility and deformation coordination between the bolt and the earthen body of the sites.