An analytical model for estimating the force and displacement of fully grouted rock bolts

Computers and Geotechnics 117 (2020) 103222

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Research Paper

An analytical model for estimating the force and displacement of fully grouted rock bolts

T

Weili Zhanga, Lei Huangb,c, , C. Hsein Juangc ⁎

a

Department of Civil Engineering and Mechanics, Faculty of Engineering, China University of Geosciences, Wuhan, Hubei 430074, China Three Gorges Research Center for Geohazards, Ministry of Education, China University of Geosciences, Wuhan, Hubei 430074, China c Department of Civil Engineering, Clemson University, Clemson, SC 29634, United States b

ARTICLE INFO

ABSTRACT

Keywords: Pull-out test Changing load Analytical model Interface Shear stress Fully grouted rock bolt

Previously proposed models for estimating the force and displacement of fully grouted rock bolts rarely describe the variations in shear stress and axial force in response to a changing load. In this work, an analytical model considering this variation pattern is developed. The model makes two assumptions: (I) failure occurs at the boltgrout interface, and (II) the shear stiffness is constant along this interface. First, according to the variation in interfacial stress, the complete rock bolt deformation process is divided into three stages: (1) the elastic stage, (2) the softening stage, and (3) the debonding stage. Equations for axial force and displacement during each stage are derived under the two assumptions. Using these equations, the developed model accommodates the three stages. The model estimates of force and displacement show good agreement with pull-out test observations, indicating that the model effectively describes the complete rock bolt deformation process during a pull-out test. However, one limitation of the proposed model is that, for simplification, as in most related models, the radial stress along the bolt is not considered.

1. Introduction Fully grouted rock bolts have widespread applications in civil and underground engineering. The mechanical behaviour of fully grouted rock bolts has long attracted extensive research. Significant studies include those by Farmer [1] and Freeman [2]. A complete system of fully grouted rock bolt is composed of three primary functional units: (1) the bolt, (2) the grout, and (3) the rock. Two interfaces are therefore present: (a) the bolt-grout interface and (b) the grout-rock interface. The boltgrout interface tends to be weaker than the grout-rock interface when the bolt is installed in hard rock and is more likely than the grout-rock interface to constitute the failure surface [3–6]. Previous models for estimating the axial force, shear stress, axial stress, and displacement at the bolt-grout interface fall into three main categories: (i) experimental models, (ii) numerical models and (iii) analytical models.

Experimental models are normally based on observations from pullout tests, including laboratory tests [3,5,7–11] and in situ tests [2,12,13]. Freeman [2] monitored for the first time the loading process and the stress distribution along fully grouted rock bolts in an experimental tunnel and introduced the concept of neutral points. Kilic et al. [8] revealed the relations between the grouting materials and fully grouted rock bolts via eighty laboratory rock bolt pull-out tests. Thenevin et al. [9] systematically investigated the influence of the confining pressure and the embedment length on the displacement. Benmokrane et al. [10] derived an empirical equation to estimate anchor pull-out resistance for a given embedment length according to the results of pull-out tests. Ivanović and Neilson [3] presented a model of the debonding of anchorage, which incorporates both the dynamic and static effects of debonding by considering load distribution and natural frequency. Recently, Martín et al. [5] proposed a semi-empirical

Abbreviations: τ, Shear stress at the bolt-grout interface; τr, Residual shear strength of the rock bolt; τu, Peak shear strength of the rock bolt; τ’, Shear stress of the rock bolt at the end point of the elastic-plastic part; τ0, Shear stress at the bolt-grout interface at the loading end; ω, Ratio of τr to τu; σ, Axial stress of the rock bolt; σ0, Axial stress of the rock bolt at the loading end; F, Axial force of the rock bolt; F0, Axial force of the rock bolt at the loading end; Fcr, Critical axial force during the debonding stage; Fu, Ultimate load of the rock bolt; s, Displacement of the rock bolt; s0, Displacement of the rock bolt at the loading end; Eb, Young’s modulus of the rock bolt; Em, Young’s modulus of the rock mass; Eg, Young’s modulus of the grout; Gm, Shear modulus of the rock mass; Gg, Shear modulus of the grout; μm, Poisson’s 2k ; l, ratio of the rock mass; μg, Poisson’s ratio of the grout; k, Shear stiffness; rb, Radius of the rock bolt; rg, Radius of the grout; β, An introduced parameter, = r bEb Length of the rock bolt; R, Radius of the rock bolt influenced; x, Distance from the loading end; x0, Coordinate of the end point of residual strength of the rock bolt; x1, Coordinate of the position of the shear stress equal to τu; a, Length of the bolt, a = x1 − x0 ⁎ Corresponding author at: Three Gorges Research Center for Geohazards, Ministry of Education, China University of Geosciences, Wuhan, Hubei 430074, China. E-mail addresses: [email protected] (W. Zhang), [email protected] (L. Huang), [email protected] (C.H. Juang). https://doi.org/10.1016/j.compgeo.2019.103222 Received 28 February 2019; Received in revised form 16 August 2019; Accepted 20 August 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.

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force and displacement and (2) clarify the variation patterns of the stress and axial force, this study (a) develops generalized equations of stress, axial force, and displacement and (b) conducts an analytical analysis of a rock bolt reinforcement system throughout the duration of an increasing load. First, according to the variation in the interfacial stress, the rock bolt is divided into several segments according to the deformation. Then, according to equilibrium and boundary conditions, the complete deformation of each segment is divided into several stages. Generalized equations for stress, axial force, and displacement are derived for each stage on the basis of Cai et al. [21], Xu et al. [23] and He et al. [26]. Third, the variation patterns of the stress and axial force curves are described, and a new model based on the generalized equations is developed. Finally, the strain and displacement observations from pull-out tests are used to verify the accuracy of the developed analytical model.

formula for describing the bolt-grout interface response based on pullout test data. Numerical models serve as a supplement to study and estimate the axial force, shear stress, axial stress, and displacement at the bolt-grout interface. Examples of these models are described in [4,14–16]. Deb and Das [4] analysed the bolt-grout interactions with finite element models and finite difference models. Ren et al. [14] developed a numerical simulation program for the prediction of the interfacial behaviour of fully grouted rock bolts. Li et al. [15] established a FLAC3Dbased numerical model of fully grouted rock bolts installed in concrete, where the material types and three interfaces of the rock bolts were considered. Recently, to investigate the working mechanism of anchor bolts and their anti-seismic support effect on surrounding rock in an underground cavern, Zhou et al. [16] presented a neutral-point-theorybased numerical model of joint action between fully grouted bolts and surrounding rock. These models require additional experimental tests for further validation, and in conjunction with the tests, these models can be used to optimize the parameters of grouted rock bolts [17–19]. Analytical models rely on mathematical and mechanical derivation. Farmer [1] provided an analytical solution to predict the behaviour of bolts under tension and found that the interfacial shear stress decreases exponentially from the loading point to the far end of the bolt before debonding occurs. Li and Stillborg [20] presented three analytical models for the distribution of the interfacial shear stress along the bolts and experimentally demonstrated that the interfacial shear stress attenuates exponentially along rock bolts with increasing distance from the point of loading. Cai et al. [21] proposed a tri-linear bond-slip model to predict the axial force of a grouted rock bolt during rock tunnelling, as shown in Fig. 1(a). This model is well suited to the cases of soft rocks, but it does not closely fit to hard rocks. For hard rocks, several models, e.g., [22–24], have been proposed. Ren et al. [22] presented an analytical solution for predicting the whole load-displacement behaviour and stress distributions along the bond length based on a tri-linear bond-slip model (shown in Fig. 1(b)). A similar model, as an improvement of the Cai model, was developed by Xu et al. [23], who derived the equations for shear stress, the axial force, and the displacement along the rock bolt. Recently, Showkati et al. [24] incorporated nonlinear shear stress into the tri-linear bond-slip model. Ma et al. [25] developed a nonlinear bond-slip model shown in Fig. 1(c). Additional factors influencing the interfacial bond strength of the fully grouted rock bolts, such as the surface profile of the bolt, the mechanical properties of the grout, the rock mass confinement and later changes in stress state, were not considered. These previous studies revealed different stress states along the length of rock bolts and noted that the debonding length gradually expands from the loading point towards the free end of the bolt as the load increases. However, the patterns of variations in the stress and axial force in different segments along the length of rock bolt were not discussed. To (1) simplify the analytical model for estimating the stress, axial

2. Division of a complete deformation process In this paper, the complete deformation process of a fully grouted rock bolt can be divided into three distinct stages (see also Fig. 2): (a) the elastic stage, (b) the softening stage, and (c) the debonding stage. The shear strength of a rock bolt is composed of adhesion, mechanical interlocking and friction in the axial direction at the coupling

Fig. 2. Examples showing the three stages in the complete deformation process of a fully grouted rock bolt under a pull-out load. Shear stress along the bolt is shown (a) during the elastic stage, (b) during the softening stage, and (c) during the debonding stage.

Fig. 1. Bond-slip models. (a) Model by Cai et al. [21]. (b) Model by Ren et al. [22]. (c) Model by Ma et al. [25]. τu and τr are the peak shear strength and residual shear strength, respectively, of a rock bolt. 2

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interface between the bolt and the grout. In the case of a relatively small applied load that is less than the shear strength of the three materials (grout, bolt and rock), the deformation in the grout is compatible with that in the bolt, and only elastic deformation occurs. Because of this compatible deformation, there is no interfacial friction; therefore, the shear stress is composed of adhesion and mechanical interlocking. As the load increases, the interfacial slippage and shear dilation appear, and in turn, the interfacial radial force and friction force, as well as the peak shear strength, are elevated [27]. This peak shear strength of a rock bolt, τu, is equal to the shear strength of the grout under confining pressure [20]. This stage, when τ ≤ τu, is referred to as the elastic stage, as shown in Fig. 2(a). Elastic and plastic deformations coexist during the elastic stage and provoke the anchoring force. If the interfacial shear stress is equal to the peak shear strength of the rock bolt, τu, debonding occurs when the grout separates from the rib of the bolt. The interfacial adhesion and interlock force disappear simultaneously in the debonding part, while the interfacial friction force gradually approximates the interfacial residual shear strength. This period is referred to as the softening stage and is shown in Fig. 2(b). If the interfacial shear stress is equal to the residual shear strength of the rock bolt, τr, then the frictional force depends on the friction surface of the grout. The period when the shear stress is equal to the residual shear strength of the rock bolt and the debonding appears is referred to as the debonding stage, as shown in Fig. 2(c). The debonding region occurs at the loading end and gradually expands to the free end as the load increases. Most rock bolts are in the elastic stage under normal working conditions, although a few are in the softening stage. If a rock bolt enters the debonding stage, reinforcement failure may occur because of large displacement.

failure of a rock bolt may occur in the bolt, the rock, the bolt-grout interface, or the grout-rock interface or may involve a combination of multiple failure modes, while debonding failure most commonly occurs at the bolt-grout interface, as bolts are usually surrounded by hard rock [30]. The shear stiffness, k, depends on the surrounding rock type and the confining pressure. Although the surrounding rock and confining pressure may vary along a given bolt, the changes are small, so k is assumed to be equal to the most common value or an average value. 3.2. Parameters The parameters used in the analytical model are listed in Fig. 4. The subscripts m, g and b stand for rock mass, grout and bolt. These parameters are obtained through measurements or conventional laboratory tests, specifically: (1) Poisson’s ratio, μm and μg, can be determined from tensile experiments along the longitudinal and transverse directions, respectively. (2) Young’s modulus, Eb, Em, and Eg, can be determined from tensile experiments along the longitudinal direction. (3) The peak shear strength of the rock bolt, τu, can be determined from direct shear tests. (4) The residual strength of the rock bolt, τr, can be determined from direct shear tests. (5) The radii, rg and rb, and the length of the rock bolt, l, can be determined from the size of the rock bolt. 3.3. Derivation of equations for estimating the forces and displacements of the rock bolt To estimate the axial force and displacement of a fully grouted rock bolt, three aspects (i.e., the equilibrium condition, boundary condition, and deformation) under an increasing pull load were analysed. Here, the three stages, namely, the elastic stage, the softening stage, and the debonding stage, are considered separately.

3. The developed analytical model 3.1. Assumptions The developed analytical model requires the following two assumptions as demonstrated in Fig. 3: (I) failure occurs at the bolt-grout interface, and (II) the shear stiffness, k, is a constant along the boltgrout interface. If practical situations do not fulfil either of the two assumptions, then the proposed model is not applicable or would introduce error if applied. Previous engineering practices and experiments [4–6,28,29] have shown that most pull-out tests fulfil both assumptions, although occasional exceptions exist under special conditions. For example, the

3.3.1. The elastic stage Based on Cai et al. [21] and Xu et al. [23], the equations for estimating the forces and displacements of the rock bolt during the elastic stage are derived as follows. The equilibrium of the rock bolt system with the load F0 is shown in Fig. 5. Clearly,

dF (x ) = 2 r b (x )dx

(1)

Fig. 4. Parameters used in the analytical model. Eb, Em, and Eg are the Young’s modulus of the rock bolt, rock mass, and grout, respectively; μm and μg are the Poisson’s ratio of the rock mass and grout, respectively; τu is the peak shear strength of the rock bolt; τr is the residual strength of the rock bolt; rb and rg are the radius of the rock bolt and grout, respectively; and l is the length of the rock bolt.

Fig. 3. Two assumptions of the analytical model. (a) Assumption I: failure occurs at the bolt-grout interface. (b) Assumption II: the shear stiffness, k, is constant along the bolt-grout interface. 3

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Fig. 5. Sketch of the rock bolt system in equilibrium. (a) Coordinate system of a single rock bolt. (b) Stress distribution in a small-scale segment of a bolt. The variable x is axis of the bolt, dx is a small-scale segment of x, τ is the shear stress along x, F(x) is the axial force at x, s(x) is the displacement at x, dF(x) is the increment of F(x) corresponding to dx, and ds(x) is the increment of s(x) corresponding to dx.

where F(x) is the axial force, τ(x) is the shear stress, and rb is the radius of the bolt. According to Hooke’s law, the strain of the rock bolt can be expressed as

(x ) =

ds (x ) (x ) F (x ) = = dx Eb r b2Eb

d2s (x ) = d2x

2s (x )

(8)

Solving Eq. (8) using ds s (x )|x = 0 = s0, dx |x = l = 0 , we obtain

s (x ) =

(2)

the

boundary

s0 cosh[ (l x )] cosh( l)

conditions

(9)

where Eb is the Young’s modulus of the rock bolt, σ(x) is the axial stress of the rock bolt, and s(x) is the displacement of the rock bolt. Differentiating Eq. (2), we obtain

where l is the length of the rock bolt and s0 is the displacement of the rock bolt at the loading end. Substituting Eq. (9) into Eq. (2), we obtain

ds 2 (x ) = dx 2

F (x ) =

dF ( x ) r b2Eb dx

(3)

Then, substituting Eq. (1) into Eq. (3), we obtain

d2s (x ) d2x

=

2 (x ) r b Eb

(x ) = (4)

(x ) = ks0

Gg G m

k =

1 2

+ G m ln

rg rb

if R > > rb otherwise

rb

(6)

(x ) =

where k is the shear stiffness, Gm is the shear modulus of the rock mass E Gm = 2(1 +mµ ) , Gg is the shear modulus of the grout material

Gg =

Eg

, rb is the radius of the rock bolt, rg is the radius of the grout,

and R is the influence radius of the rock bolt R =

20E b r Eg + Em b

where Em is

the Young’s modulus of the rock mass, Eg is the Young’s modulus of the grout material, and Eb is the Young’s modulus of the rock bolt. To ease the simplification of Eq. (4), a parameter β [23] is introduced:

=

2k r b Eb

(11)

cosh[ (l x )] = cosh( l)

0

cosh[ (l x )] cosh( l)

(12)

u

+

(1

) x1

u

x,

if 0

x

x1

(13)

where x1 is the position of the shear stress equal to τu. Combining Eqs. (1), (2), and (13) and applying the boundary condition F (x )|x = 0 = F0 , we obtain the axial force function and axial stress function as follows:

m

2(1 + µg )

x )]

3.3.2. The softening stage Based on He et al. [26], the equations for estimating the forces and displacements of the rock bolt during the softening stage are derived as follows. Hypothesizing that = r when the shear stress decreases to the u residual strength τr at the loading end, i.e., x0 = 0, as shown in Fig. 6(c), we can express the shear stress distribution function as

rb

Gg G m rg R Gg ln + G m ln rb rb

(10)

where τ0 is the interfacial shear stress at the loading end.

(5)

with R Gg ln rb

s0 Eb sinh[ (l cosh( l)

x )]

Substituting Eq. (9) into Eq. (5), we obtain

The shear stress during the elastic stage has been shown to be linearly correlated with the shear strain, which may be mathematically expressed as [21,23]

(x ) = ks (x )

r b 2s0 Eb sinh[ (l cosh( l)

F (x ) = F0

(x ) = (7)

0

2 rb

2

ux

rb

r b (1

ux

) x1

(1

) u 2 x , r b x1

u 2 x ,

if 0

if 0

x

x

x1

x1

(14) (15)

Similarly, combining Eqs. (3) and (14) and applying the boundary condition s (x )|x = 0 = s0 , we obtain the displacement function as follows:

Thus, Eq. (4) is rewritten as 4

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The displacement equation is F0 (x r b2 E b

s0 s (x ) =

(1 ) u (x 3r b E b x 1

+

u

x1 + a) +

r bEb

x1 + a) 2 if 0

(x

x

x1

if x1 < x

l

x1 + a)3,

cosh[ (l x )] , k cosh[ (l x1)]

(24) 3.3.3. The debonding stage When cracking extends along a rock bolt beyond the critical state, the debonding region is present and the debonding stage begins. We express the shear stress during the debonding stage as follows: Fig. 6. Examples showing shear stress along the rock bolt. The solid line is the effective region of the rock bolt in this stage, while the dashed line is outside of the rock bolt. (a) During the elastic stage. (b) During the softening stage. (c) At the critical state in which x0 = 0. (d) During the debonding stage. x0 is the coordinate of the end point of the residual strength of the rock bolt, and x1 is the coordinate of the position of the shear stress equal to τu.

F0 (1 ) u 3 u 2 x+ x + x , rb2 Eb r b Eb 3r b Eb x1

s (x ) = s 0

if 0

x

x1

r,

(x ) =

cosh[ (l cosh[ (l

x )] = x1)]

cosh[ (l cosh[ (l

x )] , x1)]

(16)

if x1 < x

l

F (x ) =

k

sinh[ (l cosh[ (l

cosh[ (l k cosh[ (l

if x1 < x

k

(17)

)u (x a cosh[ (l x )] , cosh[ (l x 1)]

(x ) =

(1

l

(18)

l

(x ) =

2 u (x

x1 + a) rb

(1

(19)

if x1 < x

l

(20)

x

x1

if x1 < x

l

) u (x rba

s (x ) =

F (x ) =

(x k

rb2 Eb

m (x

x1 + a) x )] , x1)]

cosh[ (l

2 rb

u (x

x0

x1 + a) if x 0 < x

x1

x1 + a) 2,

(x x )]

,

x 1)]

if x1 < x

l

0

2 rx , rb

0

2 r (x 1 a) rb

if 0 2 u (x

x 1 + a)

+

rb

(1

) u rba

x

x0

if x 0 < x

x1

if x1 < x

l

x1 + a) 2 ,

(x

(27)

x

x1

if x1 < x

l

s0 +

F0 x r b2 E b

s0

F0 (x r b2 E b

+

) u

if 0

x

x1

if x1 < x

l

x 2,

if 0 u

x1 + a) +

(1 ) u (x 3r b E b x 1

2 r b tanh( l) k

F0 = 2 r b

(22)

2 rb r b (1 x1

r

r bEb

r bEb

(x

x1 +

a) 2

x

x0

if x 0 < x

x1

if x1 < x

l

x1 + a)3,

As mentioned previously, in Eqs. (25)–(28), x1 is a variable that varies depending on the load. If x1 < 0, then τ′ = τ0. If x1 ≥ 0, then τ′ = τu. Substituting Eqs. (12), (21) and (25) into Eq. (1) and then integrating along the rock bolt, we obtain the axial force equations at the loading end as follows:

u u

{ {

s 0,

tanh[ (l tanh[ (l

if x1 x1)] x1)]

+ x1

1

x1 , if 0 < x1 < a

2a

+ x1 +

} a} ,

0

2

1 2

if x1

a

(29)

We refer to the critical axial force in the case x1 = a as Fcr:

x1 + a) 2 , sinh[ (l cosh[ (l

a

a)

x

(28)

The axial force equation is

2 rb

) u

cosh[ (l x )] , k cosh[ (l x1)]

(21)

x1 + a) 2, if 0

E b sinh[ (l x )] , k cosh[ (l x 1)]

F0

(25)

The generalized equation for displacement is

x1 + a), if 0

+

r b (1

E b sinh[ (l x )] , k cosh[ (l x 1)]

Note that Eq. (12) is an extreme case of Eq. (21) at x1 = 0. Similar to Eq. (21), the equations for axial stress, axial force, and displacement can be derived. The equation for axial stress is 0

l

(26)

Note that in Eqs. (17)–(20), x1 is a variable dependent on the load. If x1 < 0, then τ′ = τ0. If x1 ≥ 0, then τ′ = τu. The curve of Eq. (13) is shown in Fig. 6(b). Suppose that a = |x1 − x0|; by replacing x in Eq. (13) with (x − x1 + a), we can express the equation for shear stress during the softening stage as follows:

+

x1

if 0

r b2 E b sinh[ (l

and the axial stress is formulated as

u

x0

if x1 < x

2 r b r (x1 +

(x ) =

if x1 < x

Eb sinh[ (l x )] (x ) = , k cosh[ (l x1)]

x

x1 + a), if x 0 < x

The generalized equation for the axial stress is

x )] , x1)]

x )] , x1)]

(x

2 r b r x,

F0 F (x ) =

the displacement is formulated as

s (x ) =

)u

a cosh[ (l x )] , cosh[ (l x 1)]

F0

the axial force is formulated as

rb2 Eb

(1

Interestingly, Eq. (21) is an extreme case of Eq. (25) when x0 = 0 and x1 = a. Eq. (25) is a generalized equation applicable to the complete deformation process. Similarly, the generalized equations for the axial force, axial stress, and displacement can be derived. The generalized equation for the axial force is

When x > x1, the region of interest undergoes elastic stage, and the equation is similar to the one describing the elastic deformation. The only difference is that the length of the rock bolt in this equation is l-x1 rather than l as in the case for the elastic deformation. The shear stress here can be formulated as

(x ) = ks x1

if 0 +

u

Fcr = 2 r b

(23) 5

u

tanh (l

a)

+

a (1 + ) 2

(30)

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When x1 > a and

x1 = l

1 1+ ln 2 1

dF0 dx 1

= 0 in Eq. (29), the following is obtained:

1 1

(31) 1 2

1+ 1 1 1

ln We refer to the ultimate load in the case x1 = l as Fu. Similarly, by integrating Eqs. (10), (23), and (26) and then substituting the integrations into Eq. (2), with s(l) = 0, we obtain the displacement equation at the loading end as follows: F0 , 2 r b tanh( l) k

s0 =

F0 x1 r 2 Eb b F0 x1 r 2Eb b

-

if x1

2 (a x )2] u [a 1 r bEb 2 u x1 r bEb

+

+

(1 ) u a2 3r b E b

(1

+

) u [a3 (a 3x1 r b E b u , k

x1)3]

+

u , k

0

if 0 < x1 < a if x1

a

(32) Fig. 7. Examples showing the axial force along the rock bolt. (a) During the elastic stage. (b) During the softening stage. (c) At the critical state in which x0 = 0. (d) During the debonding stage. F0 is the axial force of the rock bolt at the loading end, and Fcr is the critical axial force during the debonding stage.

4. Properties of the shear stress and axial force An example of the interfacial shear stress of a rock bolt during the elastic stage is shown in Fig. 6(a). The shear stress in the elastic-plastic part decreases towards the free end along the rock bolt. The shear stress during the elastic stage changes exponentially along the bolt [1,7]. The stressed length increases when the load increases, and the shear stress curve appears to move towards the free end as the load increases. As the pull-out load increases, the shear stress at the loading end reaches the ultimate shear strength of the grouted material. Eventually, shear failure and debonding occur at the loading end, and the shear stress rapidly decreases to a stable value in the debonding part. The shear stress curve includes two parts: (1) the bonding part and (2) the debonding part. The resistance force in the bonding part is constituted by the interfacial adhesion and interlocking, whereas the resistance force in the debonding part is only constituted by friction force. When the interfacial adhesion and interlocking disappear and only friction force exists, the shear stress is equal to the residual strength, τr, which is provided by the interfacial friction, as shown in Fig. 6(c). In the case of x0 = 0, the load reaches its critical value, signifying that the critical state appears. Subsequently, the shear stress drops sharply from the peak shear stress to the residual shear strength [19,22]. For simplification, it was assumed the shear stress changes linearly along the rock bolt from x0 to x1, thereby producing a tri-linear bond-slip model for the whole bolt. This model is similar to that by Xu et al. [23] (shown in Fig. 1(b)). The points x0 and x1 move towards the free end of the rock bolt as the load increases. The debonding part constantly expands as the load increases. The interfacial interlocking is known to contribute to the shear stress in the rock bolt. However, this contribution is not significant, and the rib of a rock bolt can cut the grouted material easily and repeatedly. The interlocking at the debonding interface continues to decrease until it disappears. As a result, the shear stress becomes approximately equal to the residual shear strength, τr, which is close to the friction between the bolt and the grout. The part from the loading end to the point x0 is referred to here as the debonding part in the debonding stage, as shown in Fig. 6(d). The end point of the debonding part is x0, which moves from the loading end to the free end as the load increases. Cracking occurs in the debonding part, and it extends to the free end of the rock bolt until the rock bolt is broken. The axial force of a rock bolt according to Eq. (26) is shown in Fig. 7. The debonding part continuously expands from the loading end to the free end along the rock bolt. In the case of x0 = 0, the load has reached the critical value, as shown in Fig. 7(c). We refer to the critical axial force in the case of x0 = 0 as Fcr. This force is not the ultimate force of the rock bolt, indicating that the cracking likely appears at the loading end and that the rock bolt is expected to enter the debonding stage. With the mechanical properties of the rock mass varying along the rock bolt, the curve shape will have minor local changes during this

process. For simplification, the shape of the curves is considered to be unchanged in this study. As mentioned previously, the shear stress curve and axial force curve move from the loading end to the free end as the load increases. In practice, most rock bolts are in the elastic stage, in which the shear stress of the rock bolt is less than the peak shear strength of the grouted material. When the pull-out load reaches Fcr, the rock bolt is in the critical state. The critical state is the boundary between the softening stage and the debonding stage. Debonding occurs at the loading end in the critical state. Beyond this point, the rock bolt is in the debonding stage, the axial force increases gradually, and the displacement increases markedly. After the ultimate force is exceeded, the displacement increases suddenly, and the axial force starts to decrease, signifying possible failure. 5. Experimental verification Pull-out tests allow the examination of the anchoring capacity of rock bolts [28,29]. Here, pull-out tests were conducted on specimens comprising rock bolts, cement mortar, and concrete. 5.1. Rock bolt specimens The concrete block was cast in a mould as shown in Fig. 8, and a hole was made with a PVC pipe in the middle of the block. A bolt was then inserted into the prepared hole and grouted with cement mortar. Concrete was used instead of a rock mass to avoid joints and heterogeneity, which can result in the different mechanical properties among

Fig. 8. Cubical mould for casting the pull-out test specimens. 6

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Fig. 9. Specimen and hanging basket for the pull-out tests. (a) Overall view. (b) Details of hanging basket setup for the pull-out tests. (c) Details of the specimen and placement of the strain gauges A1, A2, and A3 (cross section).

the different specimens being compared. The dimensions of the concrete specimens are 200 mm on each side. The specimen dimensions and the internal rebar layout are shown in Fig. 9(a–c). The diameter of the hole in the centre of the block was 40 mm, which was 20 mm greater than the diameter of the bolt. The material of the bolt was 20MnSi steel. The concrete mixture proportion by weight was 1.0 (cement):1.42 (sand):3.15 (gravel):0.5 (water). Simple Portland cement 425 was used to manufacture the blocks, and the collapsed slump of the concrete was 40 mm. The bolt was grouted with a 325 slag Portland cement mortar. The water-cement ratio was 0.45, and the cement-sand ratio was 1.0. The 28-day compressive strengths of standard-cube concrete samples composed of 325 and 425 grade cements were 19.8 MPa and 37.8 MPa, respectively. A picture of the specimens is shown in Fig. 10.

Fig. 10. Specimens for the pull-out tests.

same cross section of the bolt symmetrically to produce an average value to eliminate error. The specimens were tested using an electrohydraulic servo universal testing machine. A hanging basket setup was used as the pull-out experimental apparatus, as shown in Fig. 9(b). The top of the hanging basket was linked to the universal junction of the testing machine, from which it hung. The specimen was put on the bottom board of the hanging basket, and a hinge support was fixed around the hole in the bottom board. The loading end of the bolt was connected to the bottom beam of the testing machine. The hinge support and the universal junction of the machine were designed to ensure the bolt was axially loaded. The diameter of the hole in the bottom board of the hanging basket was 50 mm, which is 10 mm larger than the diameter of the prepared hole in the block. Two displacement sensors were used to determine the displacement of the loading end and the free end of the bolt. Strain and displacement data were collected with a

5.2. Experimental methods Concrete paste was poured into an aluminium box to cast each block. The mould was removed 24 h after casting. The bolt was inserted into the centre of the pre-set hole, and cement mortar was poured into the hole and cured for 28 days. Eight specimens were prepared and stored in the laboratory environment at an ambient temperature of 20 °C and a relative humidity (RH) of 50% until testing. Strain gauges were glued to the bolt and concrete surface (Fig. 9(c)) to measure the strain along the bolt and concrete during the pull-out tests. Two narrow slots, 2 mm deep, 5 mm wide, and 200 mm long, were milled symmetrically on the anchoring cross section of each bolt; strain gauges were glued in each slot. Two strain gauges were glued in the 7

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Fig. 12. Relationship between axial stress and load for the rock bolts.

Fig. 11. Examples of failed specimens after the pull-out test to show that failure occurs at the bolt-grout interface.

DH3815N static strain testing system. 5.3. Experimental results The failure occurring at the cement-rock interface or the bolt-cement interface was influenced by many factors, such as the strength of the surrounding rock. Debonding at the cement-rock interface may occur in soft and fractured rock. If the strength of the surrounding rock is high, failure occurs at the bolt-cement interface, as detected in previous pull-out tests [8,26]. Therefore, only the failure occurring at the bolt-grout interface is discussed in this study. Most rock bolts pulled out at the bolt-cement interface in the pullout tests, as shown in Fig. 11. Only one specimen failed at the cementconcrete interface (concrete took the place of rock in these tests), and the concrete mass split because of internal defects in the concrete mass. The ultimate force of the cracked sample was obviously lower. This case was excluded as an exception. Obviously, bolt yielding could not be determined from the load-displacement curve of the pull-out test. The failure occurred at the bolt-cement interface, so the destructive phenomena of pull-out tests are consistent with the assumptions of the analytical model.

Fig. 13. Axial stress along the rock bolts.

Section A1. This pattern is due to the movement of the stress curve along the rock bolt as the load increased. The curves are consistent with previous conclusions [14,22]. The axial stress can be calculated using Eq. (4), and the results are shown in Fig. 13. The developed analytical model estimates of axial stress are almost the same as the experimental observations. The axial stress decreases from the loading end to the free end. The estimates are slightly higher than the experimental observations when the load is low and increase slower than the experimental observations as the load increases. The curve is flatter than before and tends to be a straight line when the load is sufficiently large. The axial stress of the interior of the rock bolt clearly increases faster than the stress of the loading end, which means that the load is transferred from the loading end to the free end. The splitting occurs at the loading end and extends to the free end, which is consistent with the change in the axial stress along the rock bolt in Fig. 13.

5.3.1. The developed analytical model estimates versus experimental observations of axial stress For a fully grouted rock bolt, the Young’s modulus of the rock bolt Eb = 210 GPa, the Young’s modulus of the mortar Eg = 35 GPa, the Poisson’s ratio of the mortar μg = 0.25, the Young’s modulus of the concrete Em = 45 GPa, and the Poisson’s ratio of the concrete μm = 0.25. According to Li and Stillborg [20], ω = τr/τu = 0.1. The constants k = 379.05 GPa/m and β = 19 are obtained from Eqs. (6) and (7), respectively. The experimental observations and the developed analytical model estimates are compared in Figs. 12 and 13. The model estimates of axial stress are slightly higher than the experimental observations during the elastic stage, whereas the former are slightly lower than the latter during the elastic stage. The developed analytical model estimates are slightly less than the experimental observations near the loading end, whereas the opposite pattern is found in the middle part of the rock bolt. The axial stress in the three different cross sections (A1, A2, and A3) increased with the load, and the stress at the loading end was clearly larger than that at the free end. At the beginning of the loading, the load is relatively low, and stress is present only in the part of the bolt near the loading end; thus, the axial stress could be tested only at Cross Section A1. The deformation extended to the free end as the load increased. The axial stresses in Cross Sections A2 and A3 were tested successively, but the stresses were notably less than that in Cross

5.3.2. The developed analytical model estimates versus experimental observations of the load-displacement relationship The axial force of the rock bolt at the loading end can be calculated with Eq. (29), and the displacement of the rock bolt at the loading end can be calculated with Eq. (32). When x1 = 0, F0 = 39.62 kN, x1 = a = 0.05 mm, Fcr = 60.12 kN, x1 = 0.105 mm, and Fu = 62.44 kN, the developed analytical model estimates are shown in Fig. 14. Comparison of the two sets of curves reveals that the developed analytical model estimates of displacement during the elastic stage and softening stage are close to the displacement values observed in the experiment. The analytical solution of the developed analytical model is 8

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Fig. 14. Relationship between load and displacement for a 20-cm-long fully grouted rock bolt.

Fig. 15. Relationship between load and displacement for a 40-cm-long fully grouted rock bolt.

close to that of the five-consecutive-stage model by Ren et al. [22]. However, the debonding stage is not apparent in the curve of the experimental observations because the bolt used for the experiment was very short and failure occurred soon after the debonding at the loading end. Consequently, the debonding stage was transient in the short specimen and difficult to detect with the available devices. An example excluding the debonding stage is shown in Fig. 15. Using the same parameters as He et al. [26], we show that the results of the proposed analytical model are similar to those of He et al. [26]. This result demonstrates that the developed analytical model has the ability to estimate the displacement variation pattern for fully grouted rock bolts.

that in actual engineering. The length of rock bolts in actual engineering is usually over 3 m, making it difficult or impossible to test in laboratory if the full length is considered due to laboratory space or pull-out instrument height limits. A simple way to relax this limit and obtain the needed information is by selecting the section of interest for the pull-out test rather than considering the full length, as commonly adopted by many scholars, such as Benmokrane et al. [10], Ma et al. [25], Martín et al. [5] and Thenevin et al. [9]. Among these researchers, Benmokrane et al. [10] concluded that the test results of bond stress and the bolt slip by selecting a section of interest for the pull-out test have sufficient accuracy for defining their constitutive relationship. Ma et al. [25] used bolts with lengths of 36 and 75 mm to demonstrate the bond-slip relationship. Martín et al. [5] performed forty-seven pull-out tests at laboratory scale, in which the tested bolts used to represent the actual bolts were also short, ranging between 90 and 150 mm. Recently, Thenevin et al. [9] used bolts with a length of 130 mm to investigate the relationship between load and displacement. Longer bolts used in laboratory pull-out tests are 1–3 m, e.g., those in Li and Stillborg [20], Rong et al. [31], Liu et al. [32], Liang [33] and Nie et al. [17]. The lengths of the tested bolts used in these studies to represent the actual bolts, however, are still shorter than the length of rock bolts in actual engineering. It should be noted that this simplification (i.e., using shortened bolts for tests) does result in differences (e.g., in shear stress and axial force) with actual rock bolts, but the differences have been shown to normally remain within an acceptable level [30]. Generally, short-length laboratory tests are preferred over fulllength tests in the laboratory or in situ. One advantage of short-length laboratory tests is that they are cheaper and easier to perform than fulllength tests. A second advantage is that the former are well suitable for the investigation of the influence of different parameters (e.g., the confining pressure and the grout quality) on the bond strength. In comparison, the latter are well suitable to explore whether the bolt and the grout in use are compatible with the mechanical properties of the surrounding rock mass and its degree of damage. The surrounding rock mass radius in this study is smaller than that in actual engineering, while the bolt diameters (20 mm) and cement used in the test are the same as those used in actual engineering. The influence of the surrounding rock radius has been taken into account in Eq. (6).

6. Discussion This paper developed an analytical model to estimate the axial force, shear stress, axial stress and displacement of fully grouted rock bolts. The model reveals the changes in a rock bolt subjected to an increasing load and helps reveal the mechanical characteristics of a fully grouted rock bolt. Stillborg [7] tested the Swellex bolt in his pull-out tests. Li and Stillborg [20], Cai et al. [21] and Xu et al. [23] gave analytical models for estimating the axial force, shear stress, axial stress and displacement of rock bolts, but they did not consider the variation of the shear stress with load. In comparison, the model developed here considers such variation. Another improvement of the model developed here in comparison to that developed by He et al. [26] is that their model is relatively complicated in comparison to the model developed here, which provides an easier alternative because of the use of generalized equations for axial force, shear stress, axial stress and displacement. Certain factors may contribute to the error of the developed analytical model. First, the developed analytical model, like most related models, was based on the assumption that the rock masses are homogeneous and did not consider the confining pressure. In reality, however, natural rock masses have spatially variable mechanical properties and confining pressures, resulting in variation in the radial stress in the rock bolt. For simplification, the radial stress along the bolt was not considered in the proposed analytical model, as in most related models, although shear stress increases with radial stress. Consideration of the variability in rock masses and confining pressure will result in a change in the stress, and thus, the analytical model will be more complex if radial stress is considered. Second, the transmission of the load in the pull-out test is different from that in an actual rock bolt. Therefore, further field verification of the patterns of changes in rock bolts is required in the future. The length of the rock bolt used in our pull-out tests is shorter than

7. Conclusions In this study, the mechanical behaviour of a fully grouted rock bolt with an axial load was analysed and the generalized equations relating the directly observed quantities to four parameters (i.e., axial force, 9

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shear stress, axial stress and displacement) were given. Based on these equations, an analytical model for the four parameters was developed. After a long elastic stage and softening stage, debonding occurs in the plastic part. When the shear stress becomes equal to the residual strength at the loading end, shear stress extends to the free end with the expansion of plastic deformation. Moreover, three different deformations co-exist in a rock bolt. As the load increases, the curve moves from the loading end to the free end. The estimates produced by the developed analytical model are close to the experimental observations from pull-out tests. This similarity suggests that the analytical model provides accurate estimates of the axial force, shear stress, axial stress and displacement for the fully grouted rock bolts under a pull-out load.

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Funding This work was supported by the National Natural Science Foundation of China [grant numbers 41302232 and 41572281]. Declaration of Competing Interest None. References [1] Farmer IW. Stress distribution along a resin grouted rock anchor. Int J Rock Mech Mining Sci Geomech Abstr 1975;12:347–51. https://doi.org/10.1016/01489062(75)90168-0. [2] Freeman TJ. The behaviour of fully-bonded rock bolts in the Kielder experimental tunnel. Tunn Tunn Int 1978;10:37–40. https://doi.org/10.1016/0148-9062(78) 91073-2. [3] Ivanović A, Neilson RD. Modelling of debonding along the fixed anchor length. Int J Rock Mech Min Sci 2009;46:699–707. https://doi.org/10.1016/j.ijrmms.2008.09. 008. [4] Deb D, Das KC. Bolt-grout interactions in elastoplastic rock mass using coupled FEM-FDM techniques. Adv Civ Eng 2010;7:1–13. https://doi.org/10.1155/2010/ 149810. [5] Martín LB, Tijani M, Hadj-Hassen F, Noiret A. Assessment of the bolt-grout interface behaviour of fully grouted rock bolts from laboratory experiments under axial loads. Int J Rock Mech Min Sci 2013;63:50–61. https://doi.org/10.1016/j.ijrmms.2013. 06.007. [6] Li CC, Stjern G, Myrvang A. A review on the performance of conventional and energy-absorbing rockbolts. J Rock Mech Geotech Eng 2014;6:315–27. https://doi. org/10.1016/j.jrmge.2013.12.008. [7] Stillborg B. Experimental investigation of steel cables for rock reinforcement in hard rock. Sweden: Lulea University of Technology; 1984. [8] Kilic A, Yasar E, Celik AG. Effect of grout properties on the pull-out load capacity of fully grouted rock bolt. Tunn Undergr Space Technol 2002;17:355–62. https://doi. org/10.1016/S0886-7798(02)00038-X. [9] Thenevin I, Martín LB, Hassen FH, Schleifer J, Lubosik Z, Wrana A. Laboratory pullout tests on fully grouted rock bolts and cable bolts: Results and Lessons learned. J Rock Mech Geotech Eng 2017;9:843–55. https://doi.org/10.1016/j.jrmge.2017.04. 005. [10] Benmokrane B, Chennouf A, Mitri HS. Laboratory evaluation of cement-based grouts and grouted rock anchors. Int J Rock Mech Mining Sci Geomech Abstr 1995;32:633–42. https://doi.org/10.1016/0148-9062(95)00021-8.

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