Stress Analysis of Full-length Bond Bolt Under Shear Deformation of Anchor Interface

: Bolt anchoring force is closely related to the shear properties of the anchor interface. Considering the shear properties of anchoring agent and contact interface bonding behavior, the shear stress distribution of full-length bond bolt is analyzed based on the stress-strain relationship among bolt, anchoring agent, surrounding rock and bond interface. In this case, both the interface shear stress of the anchoring agent, surrounding rock and the bolt axial force is obtained respectively under drawing and actual working conditions. The results show that the peak shear stress of the interface, including the shear deformation of the contact interface, is significantly lower than that without it when the drawing force is applied. When designing the bolt parameters under the actual working conditions of grade IV and V surrounding rock, the relative deformation between surrounding rock and anchor should be considered, and the distribution of shear stress changes from “unimodal” to “bimodal ”. In the case of a lower elastic modulus of surrounding rock, both the shear stress concentration and distribution range are obvious, and the position of the neutral point is near the orifice. With the increase of elastic modulus, both the shear stress concentration and distribution range are reduced, and the position of the neutral point moves towards the depth of bolt. As a result, the optimum bolt length of full-length bond bolt can be determined by field test and decreases with the increase of elastic modulus of surrounding rock.


Introduction
Bolt support technology has achieved remarkable progress in the safety control of underground engineering, slope engineering in mining, water conservancy and transportation industries. Bolt support is achieved by combining the surrounding rock, anchoring agent and bolt, and the contact interface between bolt and anchoring agent --the first interface and the contact interface between anchoring agent and surrounding rock --the second interface. The stress transfer behavior between the two interfaces of these three materials is the key to studying the stress characteristics of the bolt. The mechanical characteristics of a full-length bond bolt are summarized below.
You [1][2]  interfaces of the anchoring system. Therefore, it is necessary to analyze the coordinated deformation relationship among bolt, anchoring agent and surrounding rock. Zhao [4] and He [5][6] obtained the stress distribution of the bolt based on the coordinated deformation relationship between the anchoring agent and the bolt. However, the shear properties of the interface between surrounding rock and the anchoring agent are also ignored, which fails to reflect the influence of the mechanical characteristics of the second interface. By comprehensively analyzing the stressstrain relationship between the bolt and anchoring agent, I. W. FARMER [7] obtained the distribution law of shear stress under different thicknesses of anchoring agent and proposed the calculation formula of pulling resistance based on indoor drawing tests. F. Delhomme [8] added the time variable in calculating the bolt drawing, which verified the significant creep behavior between the bolt and surrounding rock in the numerical simulation and laboratory experiments. However, the analysis process cannot reflect the relationship between the anchoring mechanism and the bonding interface. In addition, some scholars used mathematical models [9][10] to fit the experimental curve, which is highly accurate, with simple parameters, but not clear in the physical meaning, so further research is still needed.
However, there is a big difference in bolt stress between the actual working condition and the drawing test, mainly because the drawing test is difficult to reflect the influence of relative deformation between surrounding rock and anchor. Freeman [11] and Wang [18] put forward the neutral point theory, which divides the bolt length into anchor length and pick-up length under actual working conditions. The anchor length will gradually decrease due to the surrounding rock rheological properties. With the development of anchorage research, the neutral point theory has gradually become reasonable in reflecting the actual stress condition of the bolt. Based on the deformation theory of surrounding rock, Li [12] analyzed the interaction relationship between the full-length bond bolt and the surrounding rock. Under the condition of normal working and near failure, the analytical formula of axial force and shear stress distribution was obtained.
The results show that the stress distribution of the interface conforms to the neutral point theory. TAO [13] and Tetsuro ESAKI [14] gave the formula to determine the position of the neutral point on full-length bond bolt lately. Li [17] took the prestress as an influencing factor to analyze the relationship of coordinated deformation between bolts and surrounding rock, with the conclusion that with the increase of prestress, the position of the neutral point is gradually transferred to the deep surrounding rock, and the shear stress decreases in the range of pick-up length, and increases in the range of anchor length. It can be seen that considering the deformation of surrounding rock is of great significance to study the actual working state of the bolt. If the anchoring agent and surrounding rock are a whole, it is difficult to reflect the shear characteristics of the second interface. Therefore, properly considering the behavior of the contact interface can help us study the mechanical characteristics of the full-length bond bolt. Considering the plastic strength of the contact interface of the anchoring agent, Cai [15] proposed the prediction method of the axial force of the bolt in soft rock engineering and concluded that the debonding failure is most likely to occur near the neutral point. By using ABAQUS finite element method, Wu [16] concluded that the deformation of surrounding rock will increase the bolt stress and transfer the peak shear stress of the interface to the depth of the bolt. It can be seen that the shear properties of the contact interfaces (the first interface and the second interface) will have an important influence on the anchoring mechanism.
Establishing the shear transfer model of the two interfaces will promote the further study of the bolt stress relationship. In the actual working condition, the failure of the bolt usually occurs at the contact interface between the surrounding rock and the anchoring agent [18]. Therefore, analyzing the shear characteristics of the second interface is the key to studying the force of a full-length bond bolt.
Most previous studies found that the research of full-length bond anchorage mechanism focused on the "two materials with one interface". The bolts and the anchoring agent are assumed to be a complex whole when interface stress between the anchoring agent and surrounding rock is studied. Besides, the anchoring agent and the surrounding rock are assumed to a be complex whole when the interface stress between the bolts and the anchoring agent is studied. Nevertheless, neither of them can reflect the real mechanism of the two interfaces simultaneously. In addition, the shear stress of the bond interface is always regarded as a function of the relative rigid displacement of the two sides of the material. However, a large number of pull tests of bolts [19][20] found that a thin layer was left on the surface of the bolts or drill holes, which indicates that the anchoring bond damage is not a simple contact interface debonding failure, but a shear failure in the thin layer close to the contact interface.
In this paper, the bolt stress was analyzed with the consideration of the shear displacement caused by the contact interface. Based on the stress-strain relationship among bolts, anchoring agents, surrounding rock, and the two interfaces (the first interface and the second interface), the characteristics of the two interfaces were considered. The stress distribution of bolts was obtained under the actual working conditions and drawing conditions respectively, which can provide some reference for anchor design.

Anchorage physical model
There is a great difference in the mechanical characteristics between the full-length bond anchorage and the end anchorage. The force transfer behavior of the anchoring agent has an important effect on the ultimate drawing force of the bolt, and its shear stress will change the shear strain of the contact interface.
Under the action of drawing force, the bolt and the surrounding rock will undergo axial deformation, and the anchoring agent will undergo shear deformation. On the interface region, the shear modulus will be affected where τ represents the shear stress at contact interface; r  represents the residual shear stress at contact interface; k represents the shear modulus at the bonded contact interface;  represents the shear strain at the contact interface; and m  represents the maximum shear strain at the contact interface.
Since the analysis in this paper is based on the elastic assumption, the residual interface strength after peak shear stress is not considered.

Bolt stress under drawing conditions
Under external drawing force, the elements of bolts element and anchoring agents were analyzed respectively, as shown in Fig. 2.
According to the stress balance relationship of bolt elements in Fig. 2 (a), the following equation can be obtained: where σ is the axial force of the bolt; 1 τ is the shear stress at the first interface; 1 D is the diameter of the bolt, and x is the distance from the bolt head.
where 2 D is the diameter of the borehole and 2 τ is the shear stress at the second interface.
The simplified equation can be obtained as follows: Since 2 D 1 D , 1 τ is larger than 2 τ , which is consistent with the common knowledge that the shear stress at the second interface is caused by the attenuation of the first interface. However, a simple linear attenuation was adopted in this paper. As the shear modulus of the interface is related to the properties of the contact bodies on both sides [19], the following equation can be obtained: where K is the shear modulus of interface; a K 、 b K are the shear modulus of the two contact bodies respectively. Since the shear modulus of the bolt is far greater than that of the surrounding rock, the shear strain generated at the first interface is relatively small. Therefore, the shear stress of the second interface (hereinafter referred to as the interface shear stress) plays an important role in maintaining the anchorage stability [18]. As shown in Fig 2 (b), the following equation can be obtained: where  is, the shear strain of anchoring agent; where F(x) is the axial force of the bolt at the x position; r E and r μ are the elastic modulus and Poisson's ratio of surrounding rock, respectively.
By recombining the axial direction of the bolt with the z-axis, and according to equation (3), the strain of surrounding rock generated by the drawing force can be described as: where Gr is the shear modulus of surrounding rock.
According to hypothesis (2), the relationship between shear stress and shear strain at the contact interface of the anchoring agent is: where 1 K and 2 K are the shear modulus of the first and second interface; 1 δ and 2 δ are the shear strain of the first and second interface, respectively.
According to the physical equation of the bolt, the following equation can be obtained: where a ε is the bolt strain; a E is the bolt elastic modulus.
As shown in Figure 3, the deformation relationship can be described as: Combining formula (6), (8), (9), (10) and (11), the following equation can be obtained: By introducing boundary conditions, we can get: where L is the length of the bolt, then the solution of interface shear stress and axial force under the drawing condition can be given as: where C1 is the coefficient related to the boundary, which can be described as:

Bolt stress under actual working conditions
The internal stress of the surrounding rock can be rebalanced after tunnel excavation, which will redistribute the field of stress and displacement.
Especially under the condition of soft rock, displacement can be substantial without efficient support. Therefore, the deformation of surrounding rock should be considered when designing anchorage support. The influence of surrounding rock deformation on the stress of anchor can be divided into two parts.
The first one is the shear stress caused by the relative deformation between surrounding rock and anchoring agent. The second one is the drawing force applied by the anchor plate directly, which is related to the deformation of the tunnel surface.  When the unit additional force dN, the following equation can be obtained: In this case, the shear strain 2  at the second interface, the relative shear strain  at the anchoring agent, the shear strain 1  at the first interface and the bolt strain a  caused by the additional force can be described as: Therefore, generated by additional unit force dN, the strain of surrounding rock   is: Assuming that r2 is the strain of surrounding rock caused by the bolt, then the additional force of the bolt can be described as: where Fr is the additional force of the bolt. The shear stress of the second interface can be described as: The axial force r F and shear stress 2  satisfy the relation: The general solution of the upper differentiation equation can be obtained as follows: Due to the lining and other constraints at the tunnel surface, the bolt head can be coordinated under the strain condition. Namely, when r equals r0, the rock strain at the tunnel surface can be expressed as where β is the lining influence coefficient, which can be obtained according to the elastic mechanics formula; r0 is the radius of the tunnel, and P is the in-situ stress.
Therefore, the strain of surrounding rock under the anchorage support can be described as: The deformation law of surrounding rock under the influence of anchor is consistent with the result of Song [19]. Therefore, considering the deformation of surrounding rock after tunnel excavation, the distribution of interface's shear stress   2 2  and bolt's can be described as: Substituting equation (8) Due to the restrain of the anchor plate, the relative displacement between the bolt and the surrounding rock will be limited. In this case, the surface of the tunnel and the bolt head should maintain the relationship of coordinate deformation, which means no relative slip between them. Therefore, the relative displacement between bolt and surrounding rock should be converted into the additional force of the bolt F. The following equation is obtained: According to the research of You [1], the distribution of the interface's shear stress and bolt's axial force caused by additional forces can be described as:  shear modulus of the bolt is bigger than that of the anchoring agent and surrounding rock, which will significantly enhance the shear stiffness of the first interface, resulting in that the value of K1 is much larger than that of K2. According to Equation (5) and reference [25], the interface parameters were set as K1= 83.6 GPa /m and K2= 9.95 GPa /m. The model was established by ABAQUS, as shown in Fig. 6 (a), and the interface was simulated by using the cohesive element, as shown in Fig. 6 (b). The comparison of results among numerical simulation, the model proposed in this paper and the solution of You [1] are shown in Fig. 6 and 7. was set to zero, and the β was equal to 1. Then the comparison results between the model proposed in this paper and the numerical simulation of Wang [22] are shown in Fig. 9.  Fig. 10 and Fig. 11.