Theoretical analysis of anchorage mechanism for rock bolt including local stripping bolt

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

Contents lists available at ScienceDirect

International Journal of Rock Mechanics and Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

Theoretical analysis of anchorage mechanism for rock bolt including local stripping bolt

T

Hai-Chun Ma∗, Xiao-Hui Tan, Jia-Zhong Qian, Xiao-Liang Hou School of Geology and Environment, Hefei University of Technology, Hefei, 230009, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Bolt Axis stress Proportional coefficient Local stripping Experiment

We have proposed a theoretical model for the analysis of bolt anchorage effects through cross-section uniformity and elastic theory. When the surrounding rock and soil are fixed, then the slip resistance forces directly proportional to the displacement of the bolt. An equilibrium differential equation for the wholly bolt has been established. The axis stress expression has been presented from the combined boundary conditions. It is a monotonic function which consists of negative index and minus index function of the distance, away from the force. The effects of friction proportional coefficient and radius have been analyzed as well. For local stripping bolt, a different model having separate parts has also been analyzed. We have also deduced the axis stress distribution function. Moreover, the curves of different separate conditions were compared which conclude that there is a terrace for the separate part. The longer the separate part, the quicker the axis stress decreased. Finally, an experiment has been performed which show the distribution of axis stress has the same trend with the strain gauge record. The theory solution of axis stress is also proved to be reasonable.

1. Introduction Anchorage bolts play important role in the geotechnical engineering. Bolt anchoring mechanics has been studied for the wholly patterns.1 For the wholly grout bolt, many models are proposed by different researchers.2,3 Freeman et al.4 studied the behavior of fullybonded rock bolts in the Kielder experimental tunnel, through interaction between the rockbolt and the surrounding rocks. Stille et al.5 presented a theoretical solution (based upon a linear strength criterion), namely the Mohr–Coulomb criterion, and an elastic-brittle stress-strain model. Stille and Holmberg et al.6 modified the Stille's solution with the three-dimensional effect. On the other hand, Li and Stillborg7 developed three analytical models for the rock bolts. These models were based on some hypotheses such as intersecting surface force uniformity. Their material model was linear elastic. Laura et al.8 presented an analytical analysis of the rock bolt pullout behavior. Fahimifar9 reported the rock and rock-support interaction concepts and their corresponding relationships. They derived the two material behavior models from the non-linear strength criterion of rock. Farmer et al.10 have analytically shown the exponentially decreased axial load and the shear stress from the loading point along the anchored length of the bolt. Ren and his coworkers11 suggested a full-range analytical analysis for the pullout behavior of rockbolt grouted system. Their experimental results

∗

showed that the distribution of bolt shear stress was concentrated at some part of the bolt12–18 while the stress is mostly damped at the negative exponential distribution mode. For the laboratory measurement methods such as strain gauges, Skrzypkowski19 carried out the laboratory tests of point resin bolts and obtained load-displacement characteristic with determination of the elastic and plastic range of the bolt for Polish hard rock mines. Korzeniowski20 used force sensors, displacement sensors and strain sensors to discusses results of static loading test of the expansion shell rock bolts equipped with originally developed deformable component. Yu21 exploited a series of pullout tests on rock bolts with laser displacement transducer and strain gauge. The above test techniques can obtain the displacement and strain distributions of rock bolt. Actually, due to some factors such as rock strength and grout inhomogeneity, the contact between bolt and rock will also be different. Zhu22 presented a numerical study on the pullout behavior of the rock bolt grouted system having grout crack. It can cause the local stripping gap. Ghaboussi et al.23 employed the finite element simulations for the investigation of stress distribution and deformations in the rockbolt. So, the model for the local stripping anchorage bolt must be different from the integrated bolt, and the bolt force must follow other distribution regularity. A new model for local stripping bolt can be established to solve this issue. The theoretical model of local stripping bolt is suggested while the

Corresponding author. E-mail address: [email protected] (H.-C. Ma).

https://doi.org/10.1016/j.ijrmms.2019.104080 Received 18 December 2018; Received in revised form 28 April 2019; Accepted 13 August 2019 Available online 23 August 2019 1365-1609/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

H.-C. Ma, et al.

Fig. 1. Bolt model.

then u x became as

theoretical solution of axis force is deduced; controlled by balance equations. And the parameters which influence the stress have been figured out. Furthermore, the development of local stripping scale was expounded. Finally, an experiment has been performed to prove the theory of distribution of bolts axis stress.

ux =

2. Bolt model

2k s ⎞ 2k s ⎞ x⎟ + C2 exp ⎜⎛− x⎟ σA = C1 exp ⎜⎛ Er ⎠ ⎝ Er ⎠ ⎝

∫l

x

σA dx E

(4)

(5)

where c1and c2 are undetermined coefficients. If is substituted by η which is dimensionless, then Eq. (5) can be written as σA = c1 exp(η x) + c2 exp(−η x) . At the loading position ( x = 0 ) and at the end ( x = l ) of the bolt σA are p and 0, respectively, A

⎧ σA

x=l

=0

⎨ σA ⎩

x=0

=

P A

(6)

so the c1 and c2 can be fixed as pexp(−lη)

⎧ c1 = − A (exp(lη) − exp(−lη)) pexp(lη) A (exp(lη) − exp(−lη))

⎨ c2 = ⎩

(7)

Bringing Eq. (7) into Eq. (5), it changes into

σA =

pexp(lη)exp(−η x) pexp(−lη)exp(η x) − A (exp(lη) − exp(−lη)) A (exp(lη) − exp(−lη))

(8)

From Eq. (8), the distribution of σA depends on ks E , A , l . However, σA is a linear superposition of p . So, the distribution state keeps the same as the p changes. The relevant conclusions from the above expression can be summarized as: The axial stress σA decreases with the increasing distance of force point which is a monotony decrease function. Combined with formula (8), it can be seen that the σA decreases with the increasing distance from the force point and decreases in negative exponential form, mixed with exponential decay, which is different from the expression in Refs. 4–7. The distribution must be concentrating near the force position. Example 1. The resistance coefficient ks and radius r are important parameters to influence the axial stress distribution. To illustrate the effects of Example 1, where the bolt is analyzed, and the bolt parameters are given in Table 1. The σA distribution is shown in Fig. 2, where ks is set as 1 × 109,

3.1. Whole bolt analyses The finite element of bolt mode is in stress balance state. So, the mechanical equilibrium equation can be written as (1)

As the loading force P increases, the tripping area extends to some extremity while the bolt will loss the effect. The key problem is to establish the relation between P and A1. To solve this problem, the cohesion strength C between bolt and surrounding rock or soil must be considered. As the P increases, σA will also changes. Equation (1) can be simplified as (2)

Table 1 Parameters of bolt model in Example 1.

where τ is the friction resistance and is the function of displacement (u x ) of the bolt. And the relation can be written as

τ = −ks u x A2

εdx =

2ks rE

3. Theoretical results

dσA·A + τ = 0

x

Simultaneous equations (2)–(4) can deduce

The geometry of bolt is fine and straight. To analyze the failure mechanism of local stripping anchorage bolt, theoretical model (longitudinal section) has been established as shown in Fig. 1. The finite element length is represented by dx . The cross-sectional area is A and the radius is r , the lateral area includes stripping area A1 and contact area A2 . Axis stresses at the ends of the element are σA and σA + dσA , respectively. The friction resistance is τ per unit area. Few assumed conditions are brought forward, based on the mode. (1) The stress of the section of anchor bolt is kept uniform. The stress value of the section in the same position is a function of the axial position. So, we can say that it is a one-dimensional problem. (2) The strain of Anchorage system is elastic and isotropic. The strain of surrounding rock mass is small compared to that of a strain of anchorage system, which can be neglected. (3) There is an elastic anti-sliding force between anchor system and the surrounding rock and soil, as the external force is added to anchor the system. It can be assumed that rock mass deformation is very small and the friction resistance τ is mainly caused by anchor strain as friction resistance. The proportional coefficient ks is the slip resistance force per unit area of displacement.

σA·A − (σA + dσA)·A − τ = 0

∫l

l [m]

ks [N/m]

(3) 8

For a whole bolt, A2 = 2πrdx and when surrounding rock is fixed 2

1 × 10

9

E [N/m2] 2.1 × 10

11

r [m]

A [m2]

p [N] −2

2 × 10

4 × 10

2

12.56 × 10

η −4

0.69

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

H.-C. Ma, et al.

The anti-sliding ability of other parts of anchor bolt will be difficult to exert. The larger radius is responsible for the smaller axial stress of anchor bolt. So, the larger the radius, the higher will be the bearing resistance force.

3.2. Local stripping anchorage bolt analyses As shown in Fig. 4-a, the black filled part is called as local stripping anchorage bolt, separated from the surrounding rock and soil, having zero ks . So, the axis stress σA will remains the same. For the local stripping anchorage bolt, some part can be discarded for the computation of the resistance force. The σA of local stripping anchorage bolt can be solved as shown in Fig. 4-b. The 4-a graph is equivalent to the 4b computational model. The separate part length is Δl and its position is x1, away from p . To solve the model axis stress (σA ), the model is changed into wholly bolt as can be seen in Fig. 4. The calculation length should be changed as l − Δl . So, the formula becomes as

Fig. 2. The effect of ks on the σA distribution.

4 × 109, 9 × 109, and 16 × 109 N/m, severally. The curves are also shown in Fig. 2, which follow the same distribution trend. However, when the ks increases, σA decreases faster and the distribution becomes more concentrated near the p . It can be concluded that the ks has a direct relationship with that of the σA distribution. ks reflects the anchoring ability of the surrounding rock and soil. For example, the effective anchoring length of the bolt is in the range of 4.5 m when the ks is 1 × 109 N/m, while it reduced to 1.5 m when ks of 16 × 109 N/m is employed. Moreover, the radius r of the bolt has also a direct relationship with the σA . If r changes, the σA will also change the distribution of σA . So, r is set as 2 × 10−2, 3 × 10−2, 4 × 10−2, and 5 × 10−2 m simultaneously. Different curves of σA are shown in Fig. 3, where we can see that when the r increases the maximum value of σA decreases. For different values of r the decay rates of σA are different. The larger the r , the slower the decay rate of σA which is due to the area A of the bolt, but the effective anchoring lengths remained similar. So, the larger the proportional coefficient produces the faster the anti-slide force attenuates. Moreover, the force is more concentrated.

pexp((l − Δ l) η)exp(−η x)

⎧ σA = A (exp((l − Δ l) η) − exp(−(l − Δ l) η)) − ⎪ pexp((l − Δ l) η)exp(−η x1) ⎪ σA = − A (exp((l − Δ l) η) − exp(−(l − Δ l) η)) ⎪ ⎨ < x1 + Δ1 ⎪ σ = pexp((l − Δ l) η)exp(−η (x−Δ1)) − A (exp((l − Δ l) η) − exp(−(l − Δ l) η)) ⎪ A ⎪ < x< 1 1 ⎩

pexp(−(l − Δ l) η)exp(η x) A (exp((l − Δ l) η) − exp(−(l − Δ l) η))

0 < x< x1

pexp(−(l − Δ l) η)exp(η x1) x1 A (exp((l − Δ l) η) − exp(−(l − Δ l) η))


pexp(−(l − Δ l) η)exp(η (x−Δ1)) x1 A (exp((l − Δ l) η) − exp(−(l − Δ l) η))

+Δ

(9) For the same example a bolt has the same properties as that of the above. x1 is set as 1 and 2 m, respectively. On the other hand, the Δl can be set as 1 and 2 m, respectively. The σA curves are shown in Fig. 5. All these curves have similar trend (decrease) and a terrace which is a separate part of the bolt. The separate part works as transmitting axis stress and there is no anti bolt slide function. Δl will affect the σA decay speed. The longer the separate part the shorter the anchorage part which consequently increased the σA decay speed. In case of discontinuous separate parts, again the theory must be same but there may be several discontinuous terraces.

Fig. 3. σA distribution with respect to r 3

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

H.-C. Ma, et al.

Fig. 4. Local stripping anchorage bolt analysis.

⎧ τ = 0 u > u max τ ∝ u u ≤ u max ⎨ ⎩

(10)

Moreover, the curve of τ is shown in Fig. 6. So, according to this curve when the P increases to some extent, the u will approach the limit. Finally, the bolt will lose the anchoring effects. Combining with formula (5) the limit of the u can be deduced as:

umax = (c1(exp(ηl) − 1))/(ηE ) − (c 2(exp( −ηl) − 1))/(ηE )

(11)

So, when u max is brought into formula (11) the limit of P will become as

pmax =

umax ηEA (exp(lη) − exp(−lη)) exp(lη) + exp(−lη) − 2

(12)

While the friction resistance force gets the limit when u = 0 and can be shown as Fig. 5. σA for local stripping anchorage bolts.

τ=−

pη exp(lη)exp(−η x) exp(lη) − exp(−lη)

τmax = −

−

pη exp(−lη)exp(η x) exp(lη) − exp(−lη)

pη exp(lη) + exp(−lη) exp(lη) − exp(−lη)

(13)

Moreover, the limit of P can also be expressed as

pmax = −τmax

exp(lη) − exp(−lη) η (exp(lη) + exp(−lη))

(14)

When the value of P is larger than pmax , then the striping part will appear and propagate until the whole bolt lose the anchorage effects. 5. Experiment To experimentally analyze the axis stress of the bolt, we have carried out an experiment where the strain gauges (SG1 to SG6) were set at six positions of the bolt as shown in Fig. 7. The type of strain gauge is BX120-3CA and the resistance is 120Ω. The dimension of strain gauge is 3 mm × 2 mm. After that, the pullout force was applied at the top by the oil pressure jack (see Fig. 8). The strain gauges recorded the strain and its distribution is shown in Fig. 9. These curves show the distributions of axis strain at different forces, ranging from 320 to 500 kN. All these curves have a similar trend. The strain is attenuated continuously until a certain length is reached, called effective length. The effective lengths of the rock bolt were kept similar. Moreover, the strain became nearly zero. So, the strain is at the same distribution with the curves of Figs. 2 and 3. Results and discussion of these curves led us to conclude that this model (stress) of rock bolt is reasonable and appropriate.

Fig. 6. The τ curve.

4. The limit of P For the rock bolt pullout tests, the P must have a limit. In case of the theoretical model, the common model can be set by the values of τ and can be called as elastic and completely soften model. Mathematically, in view of paper24 it can be represented as

4

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

H.-C. Ma, et al.

Fig. 7. Strain gauges positions (mm).

Fig. 8. Experimental model.

analyzed and the conclusions can be summarized as follows. The axis stress of bolt can be solved from the wholly bolt model which is a monotonic function and consist of negative index function and minus index function. When the proportional coefficient ks increases, the distribution of σA concentrates near the force position and the effective length of the bolt becomes shorter. Moreover, when the radius r increases the maximum of σA decreases while the effective length of bolt remains constant. In case of local stripping bolt, the curves have a terrace for the stripping part. And the stripping part is just as transmitting axis stress. The limit of P has been analyzed from the elastic and completely soften models. Finally, our experimental results have a strong correlation with that of theory solution of axis stress. The above results are consistent with previous results on similar rock bolts and cable bolts, and provide a theoretical method for analyzing the anchorage mechanism. The results can be used to be a theory to predict the pullout force by monitoring the displacement of rock bolt. The parameters of rock bolt can be designed on the basis of the conclusions.

Fig. 9. Strain distribution of bolt.

6. Conclusions

Acknowledgement

It is found that the effect of the external force on the anchor system is passive. Initially, the stress and strain response of the anchor system occurs and then it is transferred to the interface of the rock mass. Which further play the role of friction force based on the properties of the material. Therefore, it is important to analyze the interface friction for the characteristics of stress and strain of anchor bolt. For the proper analysis of stress characteristics of the anchor bolt, a reasonable model is required. Starting with the interface of an elastic model, the governing equations of the axial stress of the anchor bolt under the framework of the elastic theory are established through strict equilibrium conditions. The influence of some of the physical parameters are

The authors would like to deeply appreciate the support by the National Natural Science Foundation of China (Nos. 41831289 and 41772250).

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijrmms.2019.104080.

5

International Journal of Rock Mechanics and Mining Sciences 122 (2019) 104080

H.-C. Ma, et al.

References 13.

1. Aziz N, Jalalifar H. Optimization of the bolt profile configuration for load transfer enhancement. 8th Underground Coal Operators' Conference. University of Wollongong & the Australasian Institute of Mining and Metallurgy; 2008:125–131. 2. Nguyen T, Ghabraie K, Tran-Cong T. Applying bi-directional evolutionary structural optimisation method for tunnel reinforcement design considering nonlinear material behaviour. Comput Geotech. 2014;55:57–66. 3. Ghadimi M, Shahriar K, Jalalifar H. Analysis profile of the fully grouted rock bolt in jointed rock using analytical and numerical methods. Int J Rock Mech Min Sci. 2014;24(5):609–615. 4. Freeman TJ. The behaviour of fully-bonded rock bolts in the Kielder experimental tunnel. Tunnels Tunnelling. 1978;10:37–40. 5. Stille H. Theoretical aspects on the difference between prestresses anchor bolt and grouted bolt in squeezing rock. Proceeding of the International Symposium on Rock Bolting, Abisko. 1983; 1983:65–73. 6. Stille H, Holmberg M, Nord G. Support of weak rock with grouted bolts and shotcrete. Int J Rock Mech Min Sci. 1989;26(1):99–113. 7. Li C, Stillborg B. Analytical models for rockbolts. Int J Rock Mech Min Sci. 1999;36:1013–1029. 8. Laura BM, Michel T, Faouzi HH. A new analytical solution to the mechanical behaviour of fully grouted rockbolts subjected to pull-out tests. Constr Build Mater. 2011;25:749–755. 9. Ahmad F, Hamed S. A theoretical approach for analysis of the interaction between grouted rockbolts and rock masses. Tunn Undergr Space Technol. 2005;20(4):333–343. 10. Farmer IW. Stress distribution along a resin grouted rock anchor. Int J Rock Mech Min Sci Geomech Abstr. 1975;12:347–351. 11. Ren FF, Yang ZJ, Chen JF, Chen WW. An analytical analysis of the full-range behaviour of grouted rockbolts based on a tri-linear bond-slip model. Constr Build Mater. 2010;24:361–370. 12. Hyett J, Bawden WF, Reichert RD. The effect of rock mass confinement on the bond

14. 15. 16. 17.

18. 19. 20.

21.

22. 23. 24.

6

strength of fully grouted cable bolts. Int J Rock Mech Min Sci Geomech Abstr. 1992;29(5):503–524. Hyett AJ, Bawden WF, Macsporran GR, Moosavi M. A constitutive law for bond failure of fully-grouted cable bolts using a modified Hoek cell. Int J Rock Mech Min Sci Geomech Abstr. 1995;32(1):11–36. Jalalifar H, Aziz N. Analytical behaviour of bolt-joint intersection under lateral loading conditions. Rock Mech Rock Eng. 2010;43:89–94. Massarsch KR, OikawaI K. Design and practical application of soil anchors. Proc. Ground Anchorages and Anchored Structures. London: Thomas Telford; 1997:153–158. Josef M. Analysis of Grouted Soil anchors.Proceeding of International Symposium on Anchors in Theory and Practice. 1995; 1995. Fuller PG, Cox RHT. Mechanics load transfer from steel tendons of cement based grouted. Fifth Australasian Conference on the Mechanics of Structures and Materials. Melbourne: Australasian Institute of Mining and Metallurgy; 1995 Published by. Stillborg B. Experimental Investigation of Steel Bolts for Rock Reinforcement in Hard Rock. Sweden: Lulea University of Technology; 1984. Skrzypkowski K. Evaluation of rock bolt support for polish hard rock mines. E3S Web Conf. 2018;35. Korzeniowski W, Skrzypkowski K, Krzysztof Z. Reinforcement of underground excavation with expansion shell rock bolt equipped with deformable component. Studia Geotechnica Mech. 2017;39(1):39–51. Yu SS, Zhu WC, Niu LL, Zhou SC, Kang PH. Experimental and numerical analysis of fully grouted long rockbolt loadtransfer behavior. Tunn Undergr Space Technol. 2019;85:55–66. Zhu CX, Chang X, Men YD, Luo L. Modeling of grout crack of rockbolt grouted system. Int J Mining Sci Technol. 2015;25:73–77. Ghaboussi J, Wilson EL, Isenberg J. Finite element for rock joint interfaces. J. Soil Mech. Found Div., ASCE. 1973;99(SM10):833–848. Thenevin I, Blanco-Martin L, et al. Laboratory pull-out tests on fully grouted rock bolts and cable bolts: results and lessons learned. J Rock Mech Geotech Eng. 2017;05:65–77.