The Effect of Not Fully Grouted Rock Bolts on the Performance of Rock Mass

According to the characteristics of rock bolts in mining engineering, a mechanical model for not fully grouted rock bolt is presented to obtain the expression of shear displacement, the axial force and the shear stress along the anchor section. On these bases, the effect of the length of anchor section on the performance of rock mass and the effect of the axial force at the free end of the bolt on the performance of rock mass are analyzed by COMSOL Multiphysics. According to the results of numerical simulations, there are some conclusions as following: (1) a moderate length of anchor section, such as the length of 1.0 m~1.4 m, is favorable in mining supporting design, which can make sure grout and rock in coupled state and reduce the construction cost; (2) the favorable distance between the rock bolts is from 0.9 m to 1.3 m; (3) more rock bolts should be installed if the rock mass has larger deformation, which also can make sure grout and rock in coupled state. Conversely, less rock bolts should be installed if the rock mass has smaller deformation, which also can reduce the construction cost.


Introduction
Rock bolts have been widely used for rock reinforcement in underground engineering for a long time, especially in mining engineering in China.In order to improve the rock bolting effect, it is necessary to have a good understanding of the effect of rock bolts on the performance of rock mass.
Nevertheless, the characteristics of rock bolts in mining engineering are not the same as the characteristics of fully grouted rock bolts in civil engineering or other under-ground engineering in China.These differences are in two aspects: (1) the rock bolts in mining engineering are not fully grouted.A single rock bolt is divided into "anchor section" and "freedom section" as shown in Fig. 1.Here, the anchor section means the bolt section within grouted layer and the freedom section means the bolt section without grouted layer; (2) the faceplate is installed at the free end of the bolt, which can provide effective support to the rock mass and generate an axial force at the free end of rock bolt.These differences must be taken into account in developing analytical models for not fully grouted rock bolts.
The aim of this paper is to analyze the effect of not fully grouted rock bolts on the performance of rock mass.A mechanical model for not fully grouted rock bolt is presented first, and then the shear displacement, the axial force and the shear stress along the anchor section are obtained.On these bases, numerical simulations are carried out by COMSOL Multiphysics for analyzing the effect of the length of anchor section on the performance of rock mass and the effect of the axial force at the free end of the bolt on the performance of rock mass.Windsor (1997) proposed the concept that a reinforcement system comprises four principal components: the rock mass, the reinforcing element, the internal fixture and the external fixture.For reinforcement with a rock bolt in mining engineering, the reinforcing element refers to the rock bolt and the external fixture refers to the faceplate and nut.The internal fixture is the grouted layer, which usually uses epoxy resin in mining engineering in China.The internal fixture provides a coupling condition at the interface.When grouted bolts are subjected to an axial pull load, failure may occur at the bolt-grout interface, or at the groutrock interface.However, in this study we concentrate on the effect of rock bolts on the performance of rock mass, which is based on the hypothesis that there is no failure at the bolt-grout interface, or at the grout-rock interface.In other words, the decoupling at the interface is not considered in this paper.

Theoretical Analysis
It is well known that the faceplate restrains the deformation of the rock mass, and at the same time, there is an axial force in the rock bolt.For keeping balance, the anchor section of rock bolt is induced shear stress at the bolt-grout interface.And then the shear stress at the bolt-grout interface transfers into grouted layer, which also induces shear stress at the grout-rock interface.Finally, the shear stress transfers into rock mass.When the shear stress at the bolt-grout interface and the shear stress at the grout-rock interface are both lower than the shear strength, there is no slippage at the bolt-grout interface and the grout-rock interface, which is called coupling stage or compatible deformation stage.
The concept of the mechanical model of a single rock bolt and grouted layer is drawn in Fig. 2 and the equilibrium balance of the infinitesimal element in anchor section is shown in Fig. 3.According to the balance of the infinitesimal element of the rock bolt and grouted layer in anchor section, the equilibrium equation is written as, where, dx is the length of the infinitesimal element; N(x) and N(x+dx) are the axial force at left side and right side of the infinitesimal element, respectively; D is the diameter of the borehole and t(x) is the shear stress at the grout-rock interface.
According to Taylor expansion and ignoring highorder remainders, Eq. 1 can be expressed as below, (2) The geometrical equation of the infinitesimal element in anchor section is expressed as, where, Du is the extension of the infinitesimal element, which can be expressed as Eq.4; E a is the composite elastic modulus of rock bolt and grouted layer, which can be calculated as Eq. 5.

Du u x dx u x
where, u(x) and u(x+dx) are the axial displacement at left side and right side of the infinitesimal element, respectively.
where, E b and E g are elastic modulus of rock bolt and grouted layer, respectively; A is the area of the crosssection of the rock bolt (He & Li, 2006).
Combining Eq. 3 and Eq. 4, and ignoring high-order remainders, the following equation can be obtained.
After differential and substituting Eq. 2 into Eq.6, it turns to Eq. 7.
When the reinforcement system is in coupling stage, the constitutive equation for grouted layer can be expressed as, where, K is shear stiffness.
Substituting Eq. 8 into Eq.7, it can be written as, The solution for Eq. 9 can be expressed as, Substituting Eq. 10 into Eq.6 and Eq. 8, the expressions for N(x) and t(x) can be written as, According to Fig. 2, the boundary conditions are expressed as Eq. 13, where, l m is the length of anchor section, P is the axial force at the free end of rock bolt.
Combining Eq. 11-Eq.13, the expressions for u(x), N(x) and t(x) can be expressed as following:

Numerical Simulation Model
The deformation of rock mass before rock bolts supporting and after rock bolts supporting are shown in Fig. 4.An effective region, such as the red region in Fig. 4, is selected as the research object.Here, the range of effective region is wider than the influence range of a single rock bolt.Considering the unloading effect of excavation, the mechanical model of the effective region is proposed in Fig. 5.The left and right boundaries in Fig. 5    placement is zero and the vertical displacement is calculated according to Eq. 17; for freedom section boundary, the horizontal displacement is also zero but the vertical displacement is obtained according to Eq. 18.Thus, the numerical simulation model is built by COMSOL Multiphysics with free mesh, in which the height is 2.4 m and the width is 2 m.The boundary conditions of numerical simulation model and the coordinate system are shown in Fig. 7. where, l m is the length of anchor section, P is the axial force at the free end of rock bolt.

Analysis on the Influence of Not Fully Grouted Rock Bolt on the Performance of Rock Mass
According to the proposed model, the performance of rock mass mainly depends on the axial force P at the free end of rock bolt and the length of anchor section l m .The uniform load on the bottom boundary in Fig. 6 equals to 20´10 6 Pa and the other basic parameters are listed in Table 1.

Influence of the length of anchor section
For researching the effect of the length of anchor section l m on the performance of rock mass, the axial force P at the free end of rock bolt should be a constant.Here, the axial force P equals to 80´10 3 N and the length of anchor section l m changes from 0 m to 2.2 m in the proposed simulation model.The expression l m = 0 m means that there is no rock bolt supporting.
The numerical results including vertical displacement, vertical stress and vertical strain are shown in Fig. 8.The vertical displacement of bottom boundary in simulation model is shown in Fig. 9, which means the vertical displacement of rock mass at the roof of roadway.When the length of anchor section increases from 0 m to 0.6 m, the vertical displacement of rock mass at the faceplate decreases from 67.8´10 -3 m to 3.63´10 -3 m, which means that the vertical displacement of rock mass decreases significantly if the rock mass is supported by rock bolt.But on the other side, when the length of anchor section increases from 1.4 m to 2.2 m, the vertical displacement of rock mass at the faceplate decreases from 2.02´10 -3 m to 1.18´10 -3 m, which means that the supporting performance of rock bolt cannot improve significantly only by increasing the length of anchor section.
Compared with no rock bolt supporting, the relative reduction of vertical displacement of rock mass is shown in     If the range in which the relative reduction of vertical displacement of rock mass is more than 10% is defined as the influence range of a single rock bolt, the corresponding distance from the left boundary is defined as the influence radius of a single rock bolt.The influence radius of a single rock bolt with different length of anchor section is shown in Fig. 11.When the length of anchor section increases from 0.2 m to 2.2 m, the influence radius of a single rock bolt increases from 1.37 m to 1.49 m, which indicates that increasing the length of anchor section will increase the influence radius of a single rock bolt, however, the influence radius of a single rock bolt keeps almost constant if it exceeds certain length.
The shear stress at the grout-rock interface with different length of anchor section is also obtained in Fig. 12.
According to the curves in Fig. 12, the shear stress at the grout-rock interface has not uniform distribution along the axial direction of rock bolt.Taking l m = 1.8 m for example, the value of shear stress at the end of anchor section is 0.28´10 6 Pa while the value of shear stress at the point between anchor section and freedom section is 0.64´10 6 Pa, which shows that decoupling behavior occurs more easily at the point between anchor section and freedom section than at the end of anchor section.On the other hand, when the axial force P equals to 80´10 3 N and the length of anchor section increases from 0.2 m to 2.2 m, the minimum and the maximum shear stress decreases from 3.52´10 6 Pa to 0.20´10 6 Pa and from 3.57´10 6 Pa to 0.61´10 6 Pa, respectively, which means that increasing the length of anchor section will reduce the value of shear stress at the grout-rock interface.In other words, increasing the length of anchor section can make sure grout and rock in coupled state.
Soils and Rocks, São Paulo, 39(3): 317-324, September-December, 2016.321 The Effect of Not Fully Grouted Rock Bolts on the Performance of Rock Mass

Influence of the axial force at the free end of rock bolt
For researching the effect of the axial force P at the free end of rock bolt on the performance of rock mass, the length of anchor section l m should be a constant.Here, the length of anchor section l m equals to 1.4 m and the axial force P at the free end of rock bolt changes from 20´10 3 N to 160´10 3 N in the proposed simulation model.
The vertical displacement of rock mass at the roof of roadway is shown in Fig. 13.When the axial force at the free end of rock bolt increases from 20´10 3 N to 160´10 3 N, the vertical displacement of rock mass at the faceplate increases from 0.50´10 -3 m to 4.03´10 -3 m, which indicates that increasing the axial force at the free end of rock bolt will increase the vertical displacement of rock mass.However, the vertical displacement of rock mass cannot be significantly increased.On the other hand, the increasing axial force at the free end of rock bolt only influences the performance of rock mass in a small region, in which the distance from left boundary is less than 1 m.
The axial force at the free end of rock bolt is very important for mining engineering.According to the axial force at the free end of rock bolt, researchers can optimize the design of rock bolt supporting.The vertical displacement of rock mass at the faceplate with different axial force is shown in Fig. 14.Obviously, the vertical displacement of rock mass at the faceplate linearly increases with the increasing of axial force at the free end of rock bolt, which agrees well with the theoretical analysis.Therefore, more rock bolts should be installed if the rock mass has larger deformation, which also can make sure grout and rock in coupled state.Conversely, less rock bolts should be installed if the rock mass has smaller deformation, which also can reduce the construction cost.
The shear stress at the grout-rock interface with different axial force is also calculated in Fig. 15.When the axial force at the free end of rock bolt increases from       3 N to 160´10 3 N, the minimum and the maximum shear stress increases linearly from 0.103´10 6 Pa to 0.824´10 6 Pa and from 0.176´10 6 Pa to 1.41´10 6 Pa, respectively, which also agrees well with the theoretical analysis.It indicates that decoupling behavior can more easily happen with bigger axial force.Thus, more rock bolts are installed in this case for reducing the axial force of rock bolts and preventing the decoupling behavior.

Conclusions
A mechanical model is proposed for not fully grouted rock bolts in rock mass and the effect of not fully grouted rock bolts on the performance of rock mass is discussed based on a numerical simulation model.It supplies a sufficient way to analyze the influence of not fully grouted rock bolts on the performance of rock mass for mining engineering.According to the model, the performance of rock mass mainly depends on the length of anchor section and the axial force at the free end of rock bolt, and the following findings are obtained.1) Increasing the length of anchor section will reduce the vertical displacement of rock mass and reduce the value of shear stress at the grout-rock interface.However, the influence radius of a single rock bolt keeps almost constant if the length of anchor section exceeds certain length.Thus, a moderate length of anchor section, such as the length of 1.0 m~1.4 m, is favorable in mining supporting design, which can make sure grout and rock in coupled state and reduce the construction cost.
2) For improving the effectiveness of rock bolts supporting, the distance between the rock bolts should be less than the influence radius of a single rock bolt.Therefore, the favorable distance between the rock bolts is from 0.9 m to 1.3 m, which can explain the phenomenon that the distance between the rock bolts is about 1.1 m in the roadway supporting of mining engineering.
3) The increasing axial force at the free end of rock bolt only influences the performance of rock mass in a small region, moreover, it will increase the value of shear stress at the grout-rock interface.Thus, more rock bolts should be installed if the rock mass has larger deformation, which also can make sure grout and rock in coupled state.Conversely, less rock bolts should be installed if the rock mass has smaller deformation, which also can reduce the construction cost.

Figure 1 -
Figure 1 -A not fully grouted rock bolt in surrounding rock mass.

Figure 2 -
Figure 2 -Mechanical model of a single rock bolt and grouted layer.

Figure 3 -
Figure 3 -Equilibrium element of the rock bolt and grouted layer in anchor section.

Figure 4 -
Figure 4 -Deformation of rock mass in coal roadway.

Figure 5 -
Figure5-Mechanical model of the effective region.

Fig. 10 .
Fig.10.The distance from left boundary increases from 0 m to 1.4 m while the relative reduction of vertical displacement of rock mass decreases from about 90% to about 10%.If the range in which the relative reduction of vertical displacement of rock mass is more than 10% is defined as the influence range of a single rock bolt, the corresponding distance from the left boundary is defined as the influence radius of a single rock bolt.The influence radius of a single rock bolt with different length of anchor section is shown in Fig.11.When the length of anchor section increases from 0.2 m to 2.2 m, the influence radius of a single rock bolt increases from 1.37 m to 1.49 m, which indicates that increasing the length of anchor section will increase the influence radius of a single rock bolt, however, the influence radius of a single rock bolt keeps almost constant if it exceeds certain length.The shear stress at the grout-rock interface with different length of anchor section is also obtained in Fig.12.

Figure 9 -
Figure 9 -Vertical displacement of rock mass at the roof of roadway.

Figure 10 -
Figure 10 -Relative reduction of vertical displacement of rock mass at the roof of roadway.

Figure 11 -
Figure 11 -Influence radius of a single rock bolt.

Figure 12 -
Figure 12 -Shear stress at the grout-rock interface.

Figure 13 -
Figure 13 -Vertical displacement of rock mass at the roof of roadway.

Figure 14 -
Figure 14 -Vertical displacement of rock mass at the faceplate.

Table 1 -
Basic parameters of rock mass and rock bolt.