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Analytical study of steel bolt profile and its influence on bolt load transfer
International Journal of Rock Mechanics & Mining Sciences 60 (2013) 188–195
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Analytical study of steel bolt profile and its influence on bolt load transfer Chen Cao n, Jan Nemcik, Naj Aziz, Ting Ren Engineering Faculty, University of Wollongong, NSW, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2011 Received in revised form 24 September 2012 Accepted 18 December 2012 Available online 15 February 2013
The load transfer capacity of fully grouted steel bolts has been the subject of research for the past three decades. Experimental studies have confirmed that the bolt surface profile plays an important role in load transfer of fully grouted rock bolting systems. This research work seeks to lay the foundation for the role of the bolt profile in rock bolting mechanisms. A shear failure surface which is parallel to the bolt surface has been assumed and investigated using stress analysis. The stress field within the resin introduced by the axial load of the bolt has been formulated based on the half space theory, and the influence of the bolt profile configuration on rock bolting failure is identified under Mohr–Coulomb’s failure criterion. Parametric studies show that bolts with smaller rib face angles or smaller profile height to length ratios are favourable to transfer load radially. Hence, they should be used in hard rock environments. On the contrary, bolts with large rib face angles and higher rib height to length ratios will transfer the major part of axial load into the resin in a direction parallel to the bolt surface. Thus, they should be used in soft rock conditions. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Fully grouted rockbolts Rebar profile Load transfer mechanics Stress analysis
1. Introduction The bolt load transfer capacity is governed by the shear strengths developed between the rock/grout and the grout/bolt interfaces. It is commonly accepted that the bonding strength has three components: cohesion, friction and mechanical interlock. Singer [1] demonstrated that there is no adhesion between the bolt–grout contact, but in most reported cases, there is a very small adhesion between grout and bolt [2]. The frictional components can be catalogued into dilation slip, shear failure of surrounding medium and torsional unscrewing of bolt [3]. Each of the components depends on the stress generated at the bolt– grout interface, which in turn depends on the internal reaction forces of the whole system. The mechanical interlock is created by the bolt profile configuration. However, there are only a few experimental studies to examine the influence of bolt profile in the rock bolting system, and there are no related theoretical models to evaluate the role of the bolt profile in the load transfer mechanisms. The profile configuration is defined by the rib profile shape, profile height, angle of wrap and spacing between the ribs, as shown in Fig. 1. In traditional rockbolting load transfer mechanism analysis, the effect of mechanical interlocking is often integrated into the analytical model using various manners without reference to the rib geometric configuration. Yazici and Kaiser [5] proposed a bond
n
Corresponding author. Tel.: þ61 0425 334 939. E-mail address: [email protected] (C. Cao).
strength model (BSM) to predict the ultimate load transfer capacity of fully-grouted cable bolt. In this model the mechanical interlock is simulated as zig-zag surface of the cable, which generates dilation or radial movement when debonding occurs. For the same problem, Hyett et al. [3] emphasised the ‘unscrewing’ effect during the deformation of the cable bolt and introduced an untwisting component into the axial force formula to quantify the interaction of the bolt and the grouting material. In the socalled ‘Interfacial Shear Stress’ (ISS) model, the deformation of surrounding materials is lumped into a zero thickness interface, which is assigned with specific shear behaviour to simulate the mechanical interlocking observed in pullout tests. For instance, Li and Stillborg [6] developed an ISS model for predicting the behaviour of rock bolts in pullout tests, in uniformly deformed rock mass and when subjected to opened joints. The effect of mechanical interlock is included into the shear load displacement behaviour of the bolt-resin interface. Ivanovic and Neilson [7] developed a lumped parameter model with varying shear load failure properties along the fixed anchor length to analyse the bolt behaviour under static or dynamic load. More recently, closedform solutions are obtained for the prediction of full range behaviour of fully grouted rockbolts under axial load [8,9]. A trilinear stress strain bond-slip model at the grout–bolt interface was adopted and consequently five consecutive deformation stages of the interface were identified in the rockbolting system. These are the most advanced achievements up-to-date in rockbolting mechanisms when a single bolt is subjected to axial loading. However, they described the effect of the interaction between bolt and grout but the profile configuration of the bolt
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Fig. 1. Steel bolt rib profile configuration [4].
Fig. 3. Laboratory studies of steel bolt pull-out tests showing the maximum load for various spacing of the bolt profile [18].
Fig. 2. FLAC axisymmetric model to compare the shear stress distribution for a rebar and a smooth bar. The rib geometry of the rebar is the same for T2 bolt shown in Fig. 1 and its interface is defined as the cylinder just above rib tips.
was ignored. Fig. 2 shows a comparison of shear stresses between a rebar and a conceptualised bolt (a smooth bar) used in ISS model when subjected to a same axial load within elastic range. It suggests that the ISS model is not a cause–effect based approach. In addition, the current theoretical models can rarely be used to achieve optimum bolt profile design or resin design, as they must be calibrated by pullout tests for each application. The literature review of research work which concerned with bolt profile shows that the bolt profile configuration has great influence on rock bolting performance. The smooth bar has a very low load transfer capacity compared with ribbed steel bar [2,10,11]. If a bolt has too many ribs, such as closed spaced thread bar, its load bearing capacity is also small [11,12]. In fact, a closely spaced rebar can be thought of as a smooth bar with a larger diameter. In studies of the reinforced concrete beams, civil engineers found that for bars with a rib face angle varying between 401 and 1051, it is likely to produce approximately identical behaviour during the pull-out tests. If the face angle is less than 301, then the bonding action is different [13]. Fabjanczyk and Tarrant [14] investigated the load transfer mechanism in push-out tests. They found that bolts with a lower profile height had smaller stiffness and concluded that the load transfer was a function of parameters such as hole geometry, resin properties, and bar surface configuration. Kilic et al. [11] studied cone shaped lugs of cement grouted steel bolt by pull-out tests. Single, double and triple conical lugged bars with different face angle were tested and the experimental results showed that the conical lugged rockbolt provide better anchorage strength. Ito et al. [12] used an X-ray CT scanner to visualise the patterns of failure in pull-out tests. The results show strong influence of bolt profile on the deformational behaviour of rock bolting. Moosavi et al. [15] studied the profile configurations in cementitious grout, leading to similar conclusions. Blumel [16] reported on the influence of the bolt profile spacing on load transfer capacity and found out
that widening of the spacing between the profiles enhanced the load transfer capacity of the bolting system. Later, finite element modelling of the bolts demonstrated that higher redial stress being developed in the bolt with wider spaced ribs as compared to the small rib distance [17]. Studies undertaken by Aziz et al. [18] indicated that, increased profile spacing contributed to improvement in bolt anchorage stiffness. The extent of the anchorage performance was found to be different with the bolt profile spacing (Fig. 3). Since these early works, there has been no further analytical or numerical work being undertaken to advance the load transfer capacity of the bolts with respect to profile configuration. Accordingly, this aspect of the topic is currently been further evaluated analytically, which is the subject of discussion in this paper.
2. Methodology and governing equations To investigate the influence of bolt rib profile on the load transfer system of rock bolting, a single spacing between two bolt ribs is examined, as shown in Fig. 4. The rockbolt problem is often studied as an axisymmetric problem in three dimensions. When the bolt is subjected to an axial force, each bolt segment will experience a net axial load. This resultant force is then transferred into the resin via the rib profile. The stress distribution within the resin can be calculated according to half-space theory. In the resin, various weakness planes can be assumed as shown in Fig. 5. For each proposed weakness plane, critical load can be calculated according to a nominated failure criterion. Consequently, the influence of bolt rib geometry can be identified for each failure surface. It is essential to test a large number of weakness planes to establish the most probable failure surface. It is also necessary to confirm the theoretical calculation with experiments. However, to observe the initiation of resin failure in the laboratory is rather difficult. Past study by [19] used the flattened surface of a real bolt to examine the resin shear failure under constant normal stiffness conditions in the laboratory. The resin covered on the flattened plate contains all surface features of the bolt. This technique is introduced in this study, whereby the axisymmetric rockbolting problem is reduced to a plane stress problem and the theoretical solutions of half space are applicable. Stress analysis on the flattened plate is a half-space problem. Initiated from Kelvin’s problem, Boussinesq derived fundamental solutions for various loads on infinite or semi-infinite elastic media [20]. While loading an infinite strip on the surface of a
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Fig. 4. Rockbolting system and the bolt rib geometric parameters used in this study.
Fig. 5. A schematic drawing of a single spacing and proposed weakness planes.
semi-infinite mass (Fig. 6), the stress tensor anywhere within the media can be calculated as a function of the load, position and material properties. For a uniform normal load (Fig. 6(1)), the stress tensor within the medium can be calculated by integration of the solutions of Boussinesq’s problem, given as [20]:
sz ¼ p½a þ sina cosða þ 2dÞ=p
ð1Þ
sz ¼ p½asina cosða þ 2dÞ=p
ð2Þ
txz ¼ p sina sinða þ 2dÞ=p
ð3Þ
For a uniform shear load shown in Fig. 6(2), the stress distribution can be calculated via integration of the solution of Cerutti’s problem [20]:
sz ¼ q½sina sinða þ 2dÞ=p sx ¼
q
p
" ln
R21 R22
ð4Þ #
sina sinða þ 2dÞ
txz ¼ q½asina cosða þ2dÞ=p
ð5Þ
modelled and examined. When the bolt is loaded, the load is transferred to the resin as shown in Fig. 7. The direction of these loads depends on the bolt profile, while their magnitudes depend on both the bolt profile and the material properties. To investigate the failure of the resin between two bolt ribs, a plane of weakness spanning between the bolt profile tips is assumed as shown in Fig. 7. This failure mode is frequently observed in laboratory pullout tests whenever the confining material is stiff, as shown in Fig. 8 [21]. Both Boussinesq’s and Cerutti’s equations are for half-space problems. However, the contact of bolt and resin is not a half space, rather with irregularities due to rib profile. The sensitivity studies dealing with this issue were presented previously [22]. Another difference is that the resin and the rock are two materials with a common boundary. In practice the resin-rock boundary is invariably rifled due to drilling with wing bit [23]. The analysis can be simplified by extending the resin boundary to infinity as failure along the resin-rock interface is unlikely to occur. During bolt loading, the distributed loads within the bolt can be represented as shear forces and normal forces. Assume that the initial bonding shear forces S1, S3, S4 and S5 between the bolt and the resin contact are small and the tensile bond between the bolt and resin is also small when failure occurs, that is S1¼S3¼ S4¼S5E0 and N (tension)E0. Under these assumptions, the free body diagram of the bolt can be thus simplified as shown in Fig. 9, where only one normal force and one shear force to the inclined bolt profile remains. In most cases these stress components play a major role in the load transfer mechanism of rock bolting. Assuming the stresses along the bolt profile face are evenly distributed (Fig. 7), for static equilibrium of the bolt (Fig. 9), the normal and shear stresses at the rib face can be obtained as: p ¼ Fsin y=b
and
s2 ¼ q ¼ F cos y=b
ð7Þ
where F is the net axial force on one bolt profile, y is the bolt rib face angle, b is the bolt rib face length, p is the normal load on bolt face b, and s2 ¼q is the shear load on bolt face b. Mohr–Coulomb’s failure criterion is employed in this study as it is widely used for rock, soil and concrete studies. Accordingly, the normal and shear stresses to the chosen weakness plane must be calculated. Energy criterion is recommended as an alternative or in conjunction. Shear failure criterion is suitable for the bolt but not suitable for the resin material.
ð6Þ 4. Stresses on the proposed failure plane
3. Modelling of fully grouted bolt profiles To correlate the load transfer mechanism with the bolt rib configuration, a single spacing between two bolt profiles is
To calculate the normal and shear stress along the studied failure plane, the resin section is rotated as shown in Fig. 10(1) so that the load (p) becomes vertical. Line PQ represents the
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Fig. 6. Uniformly distributed (1) vertical load p and (2) horizontal load q on the surface of a semi-infinite mass.
stress parallel and perpendicular to the failure plane, to simplify the calculations, the plane stress vector needs to be transformed to a coordinate system parallel to the plane of failure. Accordingly, the normal and shear stress to the failure plane would be: 1 2
1 2
sn ¼ ðsx þ sz Þ ðsx sz Þ cos2ytxz sin2y 1 2
t ¼ ðsx sz Þ sin2y þ txz cos2y
ð14Þ
ð15Þ
The normal and shear stress, which is introduced by axial stress component p, to the assumed failure plane is calculated. Substituting Eqs. (1)–(3) and (8) into Eqs. (14) and (15) and simplifying, yields: Z Z p snp dh ¼ ð16Þ ½asina cosadh
p
Fig. 7. Load transfer between the steel bolt and the fully encapsulated resin.
assumed plane of failure; angle y is the bolt rib face angle; point A represents an arbitrary point on the plane of weakness; and the length AP is variable (h) indicating its distance from point P. For convenience, line PQ is rotated around the x-axis and coordinate system is re-set as shown in Fig. 10(2). The location parameters in the governing equations (d, a, R1 and R2) have to be substituted by the variable h and the rib geometry, which is described by parameters b, c, and y. According to rib profile parameters of the steel bolt defined in Fig. 4, we get:
a þ d þ y ¼ p=2
ð8Þ
b siny sina ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þ h 2bh cosy
ð9Þ
hb cosy cosa ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þh 2bh cosy
ð10Þ
a ¼ tan1
b sin y hb cos y
R1 ¼ AP ¼ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 R2 ¼ AB ¼ b þh 2bh cosy
ð11Þ
Z
tp dh ¼
p
p
ð13Þ
where b, c, L and y are rib geometrical parameters shown in Fig. 4; d, a, R1 and R2 are positioning parameters used in the governing equations, and h is a variable. The stress tensor calculated using the governing Eqs. (1)–(6) are used to calculate the shear and normal stress along the plane of proposed weakness. Since the final solutions need to calculate
sin2 adh
ð17Þ
The stress calculations due to the shear load q (Fig. 6((2) are similar to the normal load calculations presented above. The results are: Z Z " q R2 snq dh ¼ sin2 y ln 12 a sin 2y þ cos 2y þsin2 2y sin2 a p R2 ð18Þ þsin 2yð1cos 2yÞsin a cos a dh Z
Z " sin2y R2 ln 12 þ a cos 2y þ cos2 2ysin tq dh ¼ p 2 R2 i a cos asin 2y cos 2y sin2 a dh q
ð19Þ
Through superposition, Eqs. (7) and (16)–(19) are combined and yield the final expression of the normal and shear stresses on the plane of weakness. The key procedure of integration and simplification is presented in the Appendix A. Closed form solutions can be obtained as: Z L G sn dh ¼ 1 F ð20Þ
p
0
ð12Þ
Z
Z
L
tdh ¼
0
G2
p
F
ð21Þ
where G1 ¼ 2sin2 yðkcosykmþ m lnm gsinyÞ þ ðsinysin2yÞ mðpygÞ þksiny þ gcosy 2
þ ðcos2y þ sin 2yÞgsiny þ ksinyðsin2ysinysin2ycos2yÞ
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Fig. 8. Post-test sheared bolt out of steel cylinder [21].
bolt load induced stress field. Thus, the failure criterion is expressed as net resistant force that can be summed together: Z L Z L f ¼ T 0 þ T ¼ ðcw þ tan f sn0 t0 ÞLþ tan f sn dh tdh 0
0
ð22Þ where f is the failure criterion, T0 is the pre-loading shear resistance at the failure plane, T¼bolt induced shear load at the failure plane, f is the internal frictional angle of the resin, sn0 ¼initial normal stress on the weakness plane, cw is the resin cohesion, t0 is the initial shear stress, sn is the normal stress introduced by axial load of the bolt, and t is the shear stress introduced by axial load of the bolt. Each component along the plane of weakness is shown in Fig. 11. Substituting Eqs. (20) and (21) into failure function (22), yields: f ¼ T0
1
p
ðG2 tan fG1 Þ F
ð23Þ
In thus equation, the expression (G2 G1tanj)/p can be referred to as the coefficient factor of the axial force. It is in a range of [ 1, 1], as an indicator of the rate of the axial load transferred to the shear load on the assumed weakness plane. In addition, the expression T0 can be thought as the initial shear resistance on the plane. Fig. 9. Free body diagram of the bolt after approximation.
G2 ¼ ½sin2ycosyðbkcosybkm þ bm lnm bgsinyÞ cos2y½bmðpygÞ þ bksiny þ bgcosy bksinycos2 2ybgsinyðsinysin2ycos2yÞ
g ¼ tan1
mcosy p þ y siny 2
L b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ln m2 2mcos y þ 1
m¼
5. Failure study along the plane of weakness Two combined stress fields are considered within the resin. The first one is the pre-loading stress field and the second is the
6. Parametric study of bolt profile geometry The T2 bolt (Fig. 1) is the most popular bolt in Australia and also studied in laboratory push/pull tests and double shear tests [2,24–26]. According to measurement, the rib geometric parameters of T2 bolt are estimated as (Fig. 12): rib spacing ¼12 mm, stem length c ¼7 mm, rib width¼ 1.6 mm, rib height bsiny ¼ 1.5 mm, length between the rib tips L¼8.8 mm, and rib face angle y ¼601. 6.1. Bolt rib face angle The rib face angle is investigated according to Eq. (23). The rib height is fixed to be 1.5 mm and the rib stem length L is to be constant at 8.8 mm. In addition, the grout material is assumed to be resin and accordingly tanj ¼0.7. Therefore, the coefficient factor of axial force (G2 G1tanj)/p with changing rib face angle y
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193
Fig. 10. Resin section under normal load p, and coordinate system after rotation.
Fig. 11. Forces on a small element along the plane of weakness.
Fig. 13. Coefficient factor of axial force at different rib face angle for T2 bolt.
Fig. 12. Geometric parameters of T2 bolt.
is plotted and shown in Fig. 13. From this diagram, it is found that: Case (1): rib face angle 01o y o191. It should be noted that there is a minimum rib face angle due to geometric constrains (grey dashed lines inFig. 12), for the investigated bolt: height ¼ 191 y Ztan1 rib L=2 Case (2): rib face angle 201o y o451. In this stage the influence of the axial load on the assumed weakness surface decreases with increasing rib face angle.
Case (3): rib face angle y E451. There is a stationary point around y ¼451, which is the most difficult case to cause parallel shear failure on the weakness plane. At this point, (G2 G1tanj)/ p E0.13, hence, to cause parallel shear failure on the supposed weakness plane: F ¼7.7T0. In other words, if the rib face angle is around 451, the axial load will be transferred to a direction perpendicular to the bolt axis and the rockbolting will have the greatest ability to resist parallel shear failure of the resin. Case (4): face angle 451 o y o901. With increasing the rib face angle above 451, increasing coefficient factor of the axial force indicates that parallel shear failure on the assumed weakness
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Fig. 14. The influence of rib face angles for different stem lengths of the bolt.
Fig. 16. Rate of load transfer with varying of length to height ratio.
small, say less than 8% of the stem length, parallel shear failure of the grout material will not occur. 6.4. Bolt rib height to length ratio Results shown in Figs. 14 and 15 suggest that the rib height and length should be studied together. Therefore, the coefficient of axial load is plotted for varying rib length to height ratios, and shown in Fig. 16. As expected, results show that if the rib length to height ratio is great than 13, parallel shear failure of the resin cannot occur due to insufficient load transfer.
7. Discussion
Fig. 15. Coefficient of axial load with varying rib height at different stem length.
plane becomes easier. For example, if the face angle is designed to be 851, then (G2-G1tanj)/p E0.9, and consequently axial load of 1.1T0 will be sufficient to commence parallel shear failure of the rockbolting system. 6.2. Bolt rib spacing If the rib height is fixed at 1.5 mm, for different profile stem length, L, the influence of rib face angle on load transfer is plotted in Fig. 14. It can be seen that for large profile spacing the coefficient factor becomes negative. For example, a T2 bolt was modified to 25 mm rib spacing in some experiments. The influence of the axial loading of this modified bolt on the parallel shear failure of the resin was calculated and plotted as dashed line in Fig. 14. In case of rib face angle 351 o y o611, the failure criterion, f, is always positive. It indicates that shear failure of the resin on the assumed plane will never occur at this modified bolt profile configuration. Consequently, dilational slip failure of the resin will be inevitable as the majority of the axial load is transferred radially. 6.3. Bolt rib height If the rib face angle is to be constant at 601, the efficiency of axial load to commence parallel shear failure of the resin is demonstrated in Fig. 15. The results indicate that the increase in the rib height, increases the rate of load transferred to a direction parallel to the axis of the bolt. In addition, if the rib height is very
The novel idea of coupling the bolt profile geometry with the introduced stress field provides a tool to investigate the bolt profile configuration and its effects on the load transfer mechanism for the benefit of the mining industry and science in general. A new development in calculating load transfer capacity between two rib profiles of varying geometries is discussed. The derived mathematical equations calculate the stress distribution adjacent to the fully grouted bolt subjected to axial load. The parametric study suggests that the axial load can be transferred to surrounding material laterally and longitudinally. The transfer rate is sensitive to changes in bolt rib geometry. In general, lower rib face angle and/or small profile height to length ratios are favourable to transfer load radially. As a result, dilation is inevitable. On the contrary, high rib face angle with larger rib height to length ratio will transfer the majority of axial load to a direction parallel to the bolt surface. In this case, parallel shear failure would most likely occur. Therefore, this study suggests that the rock bolt with smaller rib face angle and larger spacing is suitable in high confinement such as hard rock, while the rock bolt with high profile height and larger rib face angle should be used in soft rock. There are limitations on the applications of derived mathematical expressions to rib profile design. The boundary effect becomes a problem for some rib geometry. For example, if the rib is closely spaced, then the grout material can no longer be represented by a half space. In addition, the calculation is inaccurate when the rib face angle is large. Dilational slip failure of the bolt is another very common failure mode in pullout tests. If dilational slip failure is investigated in a similar manner presented in this paper, calculation will be inaccurate as the failure surface is close to the bolt. The influence of bolt profile configuration in case of dilational slip failure is the subject of a paper in preparing.
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Appendix A Calculation of each component of Eqs. (16)–(19), substituting Eqs. (9)–(13) yields: !2 Z Z bsiny hbcosy 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ¼ bsiny tan1 sin a dh ¼ 2 2 bsiny b þ h 2bhcosy Z Z bsiny hbcosy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh sina cosa dh ¼ 2 2 2 2 b þ h 2bhcosy b þh 2bhcosy 1 2 2 ¼ bsinyln b þ h 2bhcosy 2 Z Z bsiny adh ¼ tan1 dh hbcosy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bsiny 2 2 þ bsinyln h 2bhcosy þ b ¼ htan1 hbcosy hbcosy þ bcosytan1 bsiny Z Z R1 h ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ln dh ¼ 2 2 R2 b þh 2bhcosy 2 2 ¼ ðhbcosyÞln h 2bhcosy þ b 2ðhbcosyÞ hbcosy þ 2bsinytan1 bsiny These solutions can be verified by reverse differentiation. For definite integral: RL 2 0 sin adh ¼ bgsiny RL 0 sinacosadh ¼ bksiny RL 0 adh ¼ bmðpygÞ þ bksiny þ bgcosy R L R1 0 ln R2 dh ¼ bkcosybkm þ bm lnmbgsiny In the procedure of simplification, it should be noted that: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bsiny L2 2bLcosy þb 1 ¼ klnm ¼ pyg; ln tan L Lbcosy References [1] Singer SP, Field verification of load transfer mechanics of fully grouted roof bolts. US Bureau of Mines, 1990, p. 9301. [2] Aziz NI, Webb B., Study of load transfer capacity of bolts using short encapsulation push test. In: Proceedings of the 4th underground coal operators conference, Wollongong, February 12–14 2003, p. 72–80. /http:// ro.uow.edu.au/coal/162/S. [3] 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. [4] Aziz NI, Jalalifar H, Concalves J., Bolt surface configurations and load transfer mechanism. In: Proceedings of the 7th underground coal operators conference, Wollongong, 5–7 July 2006, p. 236-44. /http://ro.uow.edu.au/coal/51/S. [5] Yazici S, Kaiser PK. Bond strength of grouted cable bolts. Int J Rock Mech Min Sci Geomech Abstr 1992;29(3):279–92.
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[6] Li C, Stillborg B. Analytical models for rock bolts. Int J Rock Mech Min Sci Geomech Abstr 1999;36(8):1013–29. [7] Ivanovic A, Neilson RD. Modelling of debonding along the fixed anchor length. Int J Rock Mech Min Sci 2009;46(4):699–707. [8] 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. Construct Build Mater 2010;24:361–70. [9] Martin LB, Tijani M, Hadj-Hassen F. A new analytical solution to the mechanical behaviour of fully grouted rockbolts subject to pull-out tests. Construct Build Mater 2011;25:749–55. [10] Aydan O., The stabilisation of rock engineering structures by rock bolts. Geotechnical Engineering, PhD Thesis, Nagoya University, Nagoya, 1989, 202p. [11] Kilic A, Yasar E, Atis CD. Effect of bar shape on the pull out capacity of fully grouted rockbolts. Tunnell Undergr Space Technol 2002;18:1–6. [12] Ito F, Nakahara F, Kawano R, Kang SS, Obara Y. Visualization of failure in a pull-out test of cable bolts using X-ray CT. Construct Build Mater 2001;15:263–70. [13] Tepfers R., A theory of bond applied to overlapped tensile reinforcement splices for deformed bar. Dissertation, Publication 73:2, Chalmers ¨ University of Technology, Division of Concrete Structures, Goteborg, Sweden, 1973, 328 p. [14] Fabjanczyk MW, Tarrant GC., Load transfer mechanisms in reinforcing tendons. In: Proceedings of the International conference on ground control in mining, Wollongong University, 1992, p. 212–18. [15] Moosavi M, Jafari A, Khosravi A. Bond of cement reinforcing bars under constant radial pressure. Cem Concr Compos 2005;27:103–9. [16] Blumel M., Performance of grouted rock bolts in squeezing rock. In: Proceedings EUROCK’96, predictions and performance in rock mechanics an rock engineering, Rotterdam, Balkema, 1996, p. 885–91. [17] Blumel M, Schweger HF, Golser H., Effect of rib geometry on the mechanical behaviour of grouted rock bolts. In: Proceedings of the world tunnelling congress, 23rd general assembly of the international tunnelling, Vienna, Austria, 1997, 6p. [18] Aziz NI, Jalalifar H, Remennikov A, Sinclair S, Green A., Optimisation of the Bolt profile configuration for load transfer enhancement. In: Proceedings of the 8th underground coal operators conference, Wollongong, February 14–15 2008, p. 125–31. /http://ro.uow.edu.au/coal/11/S. [19] Aziz NI., A new technique to determine the load transfer capacity of resin anchored bolts. In: Proceedings of the coal operators’ conference, University of Wollongong & the Australasian Institute of Mining and Metallurgy, 2002, p. 176–84. /http://ro.uow.edu.au/coal/208/S. [20] Poulos H, Davis E. Elastic solutions for rock mechanics. New York: Wiley; 1974. [21] Aziz NI, Jalalifar H, Concalves J., Bolt surface configurations and load transfer mechanism. In: Proceedings of the 7th underground coal operators conference, Wollongong, 5–7 July 2006, p. 236–44. /http://ro.uow.edu.au/coal/51/S. [22] Cao C, Nemcik J, Aziz NI. Improvement of rock bolt profiles using analytical and numerical methods. In: Proceedings of the 12th underground coal operators conference, Wollongong, 5–7 July, 2011, p. 236–44. /http://ro. uow.edu.au/coal/350/S. [23] Aziz NI. An update of roof bolt research at the University of Wollongong, Aziz N. (editor). In: Proceedings of the Coal 2004: coal operators’ conference, University of Wollongong & the Australasian Institute of Mining and Metallurgy, 2004, p. 215–24. /http://ro.uow.edu.au/coal/147/S. [24] Aziz NI, Pratt D, Williams R., Double shear testing of bolts. In: Proceedings of the 4th underground coal operators conference, February 12–14 2003, Wollongong, p. 154–61. /http://ro.uow.edu.au/coal/172/S. [25] Jalalifar H, Aziz NI, Hadi M., Modelling of sheared behaviour bolts across joints, in Aziz N. (editor). In: Proceedings of the Coal 2004: coal operators’ conference, University of Wollongong & the Australasian Institute of Mining and Metallurgy, 2004, p. 225–32. /http://ro.uow.edu.au/coal/149/S. [26] Jalalifer H, Aziz NI, Hadi M., An assessment of load transfer mechanism using the instrumented bolts, Aziz N. (editor). In: Proceedings of the Coal 2006: coal operators’ conference, University of Wollongong & the Australasian Institute of Mining and Metallurgy, 2006, p. 255–65. /http://ro.uow.edu.au/ coal/53/S.
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Analytical study of steel bolt profile and its influence on bolt load transfer Chen Cao n, Jan Nemcik, Naj Aziz, Ting Ren Engineering Faculty, University of Wollongong, NSW, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2011 Received in revised form 24 September 2012 Accepted 18 December 2012 Available online 15 February 2013
The load transfer capacity of fully grouted steel bolts has been the subject of research for the past three decades. Experimental studies have confirmed that the bolt surface profile plays an important role in load transfer of fully grouted rock bolting systems. This research work seeks to lay the foundation for the role of the bolt profile in rock bolting mechanisms. A shear failure surface which is parallel to the bolt surface has been assumed and investigated using stress analysis. The stress field within the resin introduced by the axial load of the bolt has been formulated based on the half space theory, and the influence of the bolt profile configuration on rock bolting failure is identified under Mohr–Coulomb’s failure criterion. Parametric studies show that bolts with smaller rib face angles or smaller profile height to length ratios are favourable to transfer load radially. Hence, they should be used in hard rock environments. On the contrary, bolts with large rib face angles and higher rib height to length ratios will transfer the major part of axial load into the resin in a direction parallel to the bolt surface. Thus, they should be used in soft rock conditions. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Fully grouted rockbolts Rebar profile Load transfer mechanics Stress analysis
1. Introduction The bolt load transfer capacity is governed by the shear strengths developed between the rock/grout and the grout/bolt interfaces. It is commonly accepted that the bonding strength has three components: cohesion, friction and mechanical interlock. Singer [1] demonstrated that there is no adhesion between the bolt–grout contact, but in most reported cases, there is a very small adhesion between grout and bolt [2]. The frictional components can be catalogued into dilation slip, shear failure of surrounding medium and torsional unscrewing of bolt [3]. Each of the components depends on the stress generated at the bolt– grout interface, which in turn depends on the internal reaction forces of the whole system. The mechanical interlock is created by the bolt profile configuration. However, there are only a few experimental studies to examine the influence of bolt profile in the rock bolting system, and there are no related theoretical models to evaluate the role of the bolt profile in the load transfer mechanisms. The profile configuration is defined by the rib profile shape, profile height, angle of wrap and spacing between the ribs, as shown in Fig. 1. In traditional rockbolting load transfer mechanism analysis, the effect of mechanical interlocking is often integrated into the analytical model using various manners without reference to the rib geometric configuration. Yazici and Kaiser [5] proposed a bond
n
Corresponding author. Tel.: þ61 0425 334 939. E-mail address: [email protected] (C. Cao).
strength model (BSM) to predict the ultimate load transfer capacity of fully-grouted cable bolt. In this model the mechanical interlock is simulated as zig-zag surface of the cable, which generates dilation or radial movement when debonding occurs. For the same problem, Hyett et al. [3] emphasised the ‘unscrewing’ effect during the deformation of the cable bolt and introduced an untwisting component into the axial force formula to quantify the interaction of the bolt and the grouting material. In the socalled ‘Interfacial Shear Stress’ (ISS) model, the deformation of surrounding materials is lumped into a zero thickness interface, which is assigned with specific shear behaviour to simulate the mechanical interlocking observed in pullout tests. For instance, Li and Stillborg [6] developed an ISS model for predicting the behaviour of rock bolts in pullout tests, in uniformly deformed rock mass and when subjected to opened joints. The effect of mechanical interlock is included into the shear load displacement behaviour of the bolt-resin interface. Ivanovic and Neilson [7] developed a lumped parameter model with varying shear load failure properties along the fixed anchor length to analyse the bolt behaviour under static or dynamic load. More recently, closedform solutions are obtained for the prediction of full range behaviour of fully grouted rockbolts under axial load [8,9]. A trilinear stress strain bond-slip model at the grout–bolt interface was adopted and consequently five consecutive deformation stages of the interface were identified in the rockbolting system. These are the most advanced achievements up-to-date in rockbolting mechanisms when a single bolt is subjected to axial loading. However, they described the effect of the interaction between bolt and grout but the profile configuration of the bolt
1365-1609/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.12.013
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189
Fig. 1. Steel bolt rib profile configuration [4].
Fig. 3. Laboratory studies of steel bolt pull-out tests showing the maximum load for various spacing of the bolt profile [18].
Fig. 2. FLAC axisymmetric model to compare the shear stress distribution for a rebar and a smooth bar. The rib geometry of the rebar is the same for T2 bolt shown in Fig. 1 and its interface is defined as the cylinder just above rib tips.
was ignored. Fig. 2 shows a comparison of shear stresses between a rebar and a conceptualised bolt (a smooth bar) used in ISS model when subjected to a same axial load within elastic range. It suggests that the ISS model is not a cause–effect based approach. In addition, the current theoretical models can rarely be used to achieve optimum bolt profile design or resin design, as they must be calibrated by pullout tests for each application. The literature review of research work which concerned with bolt profile shows that the bolt profile configuration has great influence on rock bolting performance. The smooth bar has a very low load transfer capacity compared with ribbed steel bar [2,10,11]. If a bolt has too many ribs, such as closed spaced thread bar, its load bearing capacity is also small [11,12]. In fact, a closely spaced rebar can be thought of as a smooth bar with a larger diameter. In studies of the reinforced concrete beams, civil engineers found that for bars with a rib face angle varying between 401 and 1051, it is likely to produce approximately identical behaviour during the pull-out tests. If the face angle is less than 301, then the bonding action is different [13]. Fabjanczyk and Tarrant [14] investigated the load transfer mechanism in push-out tests. They found that bolts with a lower profile height had smaller stiffness and concluded that the load transfer was a function of parameters such as hole geometry, resin properties, and bar surface configuration. Kilic et al. [11] studied cone shaped lugs of cement grouted steel bolt by pull-out tests. Single, double and triple conical lugged bars with different face angle were tested and the experimental results showed that the conical lugged rockbolt provide better anchorage strength. Ito et al. [12] used an X-ray CT scanner to visualise the patterns of failure in pull-out tests. The results show strong influence of bolt profile on the deformational behaviour of rock bolting. Moosavi et al. [15] studied the profile configurations in cementitious grout, leading to similar conclusions. Blumel [16] reported on the influence of the bolt profile spacing on load transfer capacity and found out
that widening of the spacing between the profiles enhanced the load transfer capacity of the bolting system. Later, finite element modelling of the bolts demonstrated that higher redial stress being developed in the bolt with wider spaced ribs as compared to the small rib distance [17]. Studies undertaken by Aziz et al. [18] indicated that, increased profile spacing contributed to improvement in bolt anchorage stiffness. The extent of the anchorage performance was found to be different with the bolt profile spacing (Fig. 3). Since these early works, there has been no further analytical or numerical work being undertaken to advance the load transfer capacity of the bolts with respect to profile configuration. Accordingly, this aspect of the topic is currently been further evaluated analytically, which is the subject of discussion in this paper.
2. Methodology and governing equations To investigate the influence of bolt rib profile on the load transfer system of rock bolting, a single spacing between two bolt ribs is examined, as shown in Fig. 4. The rockbolt problem is often studied as an axisymmetric problem in three dimensions. When the bolt is subjected to an axial force, each bolt segment will experience a net axial load. This resultant force is then transferred into the resin via the rib profile. The stress distribution within the resin can be calculated according to half-space theory. In the resin, various weakness planes can be assumed as shown in Fig. 5. For each proposed weakness plane, critical load can be calculated according to a nominated failure criterion. Consequently, the influence of bolt rib geometry can be identified for each failure surface. It is essential to test a large number of weakness planes to establish the most probable failure surface. It is also necessary to confirm the theoretical calculation with experiments. However, to observe the initiation of resin failure in the laboratory is rather difficult. Past study by [19] used the flattened surface of a real bolt to examine the resin shear failure under constant normal stiffness conditions in the laboratory. The resin covered on the flattened plate contains all surface features of the bolt. This technique is introduced in this study, whereby the axisymmetric rockbolting problem is reduced to a plane stress problem and the theoretical solutions of half space are applicable. Stress analysis on the flattened plate is a half-space problem. Initiated from Kelvin’s problem, Boussinesq derived fundamental solutions for various loads on infinite or semi-infinite elastic media [20]. While loading an infinite strip on the surface of a
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Fig. 4. Rockbolting system and the bolt rib geometric parameters used in this study.
Fig. 5. A schematic drawing of a single spacing and proposed weakness planes.
semi-infinite mass (Fig. 6), the stress tensor anywhere within the media can be calculated as a function of the load, position and material properties. For a uniform normal load (Fig. 6(1)), the stress tensor within the medium can be calculated by integration of the solutions of Boussinesq’s problem, given as [20]:
sz ¼ p½a þ sina cosða þ 2dÞ=p
ð1Þ
sz ¼ p½asina cosða þ 2dÞ=p
ð2Þ
txz ¼ p sina sinða þ 2dÞ=p
ð3Þ
For a uniform shear load shown in Fig. 6(2), the stress distribution can be calculated via integration of the solution of Cerutti’s problem [20]:
sz ¼ q½sina sinða þ 2dÞ=p sx ¼
q
p
" ln
R21 R22
ð4Þ #
sina sinða þ 2dÞ
txz ¼ q½asina cosða þ2dÞ=p
ð5Þ
modelled and examined. When the bolt is loaded, the load is transferred to the resin as shown in Fig. 7. The direction of these loads depends on the bolt profile, while their magnitudes depend on both the bolt profile and the material properties. To investigate the failure of the resin between two bolt ribs, a plane of weakness spanning between the bolt profile tips is assumed as shown in Fig. 7. This failure mode is frequently observed in laboratory pullout tests whenever the confining material is stiff, as shown in Fig. 8 [21]. Both Boussinesq’s and Cerutti’s equations are for half-space problems. However, the contact of bolt and resin is not a half space, rather with irregularities due to rib profile. The sensitivity studies dealing with this issue were presented previously [22]. Another difference is that the resin and the rock are two materials with a common boundary. In practice the resin-rock boundary is invariably rifled due to drilling with wing bit [23]. The analysis can be simplified by extending the resin boundary to infinity as failure along the resin-rock interface is unlikely to occur. During bolt loading, the distributed loads within the bolt can be represented as shear forces and normal forces. Assume that the initial bonding shear forces S1, S3, S4 and S5 between the bolt and the resin contact are small and the tensile bond between the bolt and resin is also small when failure occurs, that is S1¼S3¼ S4¼S5E0 and N (tension)E0. Under these assumptions, the free body diagram of the bolt can be thus simplified as shown in Fig. 9, where only one normal force and one shear force to the inclined bolt profile remains. In most cases these stress components play a major role in the load transfer mechanism of rock bolting. Assuming the stresses along the bolt profile face are evenly distributed (Fig. 7), for static equilibrium of the bolt (Fig. 9), the normal and shear stresses at the rib face can be obtained as: p ¼ Fsin y=b
and
s2 ¼ q ¼ F cos y=b
ð7Þ
where F is the net axial force on one bolt profile, y is the bolt rib face angle, b is the bolt rib face length, p is the normal load on bolt face b, and s2 ¼q is the shear load on bolt face b. Mohr–Coulomb’s failure criterion is employed in this study as it is widely used for rock, soil and concrete studies. Accordingly, the normal and shear stresses to the chosen weakness plane must be calculated. Energy criterion is recommended as an alternative or in conjunction. Shear failure criterion is suitable for the bolt but not suitable for the resin material.
ð6Þ 4. Stresses on the proposed failure plane
3. Modelling of fully grouted bolt profiles To correlate the load transfer mechanism with the bolt rib configuration, a single spacing between two bolt profiles is
To calculate the normal and shear stress along the studied failure plane, the resin section is rotated as shown in Fig. 10(1) so that the load (p) becomes vertical. Line PQ represents the
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Fig. 6. Uniformly distributed (1) vertical load p and (2) horizontal load q on the surface of a semi-infinite mass.
stress parallel and perpendicular to the failure plane, to simplify the calculations, the plane stress vector needs to be transformed to a coordinate system parallel to the plane of failure. Accordingly, the normal and shear stress to the failure plane would be: 1 2
1 2
sn ¼ ðsx þ sz Þ ðsx sz Þ cos2ytxz sin2y 1 2
t ¼ ðsx sz Þ sin2y þ txz cos2y
ð14Þ
ð15Þ
The normal and shear stress, which is introduced by axial stress component p, to the assumed failure plane is calculated. Substituting Eqs. (1)–(3) and (8) into Eqs. (14) and (15) and simplifying, yields: Z Z p snp dh ¼ ð16Þ ½asina cosadh
p
Fig. 7. Load transfer between the steel bolt and the fully encapsulated resin.
assumed plane of failure; angle y is the bolt rib face angle; point A represents an arbitrary point on the plane of weakness; and the length AP is variable (h) indicating its distance from point P. For convenience, line PQ is rotated around the x-axis and coordinate system is re-set as shown in Fig. 10(2). The location parameters in the governing equations (d, a, R1 and R2) have to be substituted by the variable h and the rib geometry, which is described by parameters b, c, and y. According to rib profile parameters of the steel bolt defined in Fig. 4, we get:
a þ d þ y ¼ p=2
ð8Þ
b siny sina ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þ h 2bh cosy
ð9Þ
hb cosy cosa ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b þh 2bh cosy
ð10Þ
a ¼ tan1
b sin y hb cos y
R1 ¼ AP ¼ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 R2 ¼ AB ¼ b þh 2bh cosy
ð11Þ
Z
tp dh ¼
p
p
ð13Þ
where b, c, L and y are rib geometrical parameters shown in Fig. 4; d, a, R1 and R2 are positioning parameters used in the governing equations, and h is a variable. The stress tensor calculated using the governing Eqs. (1)–(6) are used to calculate the shear and normal stress along the plane of proposed weakness. Since the final solutions need to calculate
sin2 adh
ð17Þ
The stress calculations due to the shear load q (Fig. 6((2) are similar to the normal load calculations presented above. The results are: Z Z " q R2 snq dh ¼ sin2 y ln 12 a sin 2y þ cos 2y þsin2 2y sin2 a p R2 ð18Þ þsin 2yð1cos 2yÞsin a cos a dh Z
Z " sin2y R2 ln 12 þ a cos 2y þ cos2 2ysin tq dh ¼ p 2 R2 i a cos asin 2y cos 2y sin2 a dh q
ð19Þ
Through superposition, Eqs. (7) and (16)–(19) are combined and yield the final expression of the normal and shear stresses on the plane of weakness. The key procedure of integration and simplification is presented in the Appendix A. Closed form solutions can be obtained as: Z L G sn dh ¼ 1 F ð20Þ
p
0
ð12Þ
Z
Z
L
tdh ¼
0
G2
p
F
ð21Þ
where G1 ¼ 2sin2 yðkcosykmþ m lnm gsinyÞ þ ðsinysin2yÞ mðpygÞ þksiny þ gcosy 2
þ ðcos2y þ sin 2yÞgsiny þ ksinyðsin2ysinysin2ycos2yÞ
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Fig. 8. Post-test sheared bolt out of steel cylinder [21].
bolt load induced stress field. Thus, the failure criterion is expressed as net resistant force that can be summed together: Z L Z L f ¼ T 0 þ T ¼ ðcw þ tan f sn0 t0 ÞLþ tan f sn dh tdh 0
0
ð22Þ where f is the failure criterion, T0 is the pre-loading shear resistance at the failure plane, T¼bolt induced shear load at the failure plane, f is the internal frictional angle of the resin, sn0 ¼initial normal stress on the weakness plane, cw is the resin cohesion, t0 is the initial shear stress, sn is the normal stress introduced by axial load of the bolt, and t is the shear stress introduced by axial load of the bolt. Each component along the plane of weakness is shown in Fig. 11. Substituting Eqs. (20) and (21) into failure function (22), yields: f ¼ T0
1
p
ðG2 tan fG1 Þ F
ð23Þ
In thus equation, the expression (G2 G1tanj)/p can be referred to as the coefficient factor of the axial force. It is in a range of [ 1, 1], as an indicator of the rate of the axial load transferred to the shear load on the assumed weakness plane. In addition, the expression T0 can be thought as the initial shear resistance on the plane. Fig. 9. Free body diagram of the bolt after approximation.
G2 ¼ ½sin2ycosyðbkcosybkm þ bm lnm bgsinyÞ cos2y½bmðpygÞ þ bksiny þ bgcosy bksinycos2 2ybgsinyðsinysin2ycos2yÞ
g ¼ tan1
mcosy p þ y siny 2
L b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ ln m2 2mcos y þ 1
m¼
5. Failure study along the plane of weakness Two combined stress fields are considered within the resin. The first one is the pre-loading stress field and the second is the
6. Parametric study of bolt profile geometry The T2 bolt (Fig. 1) is the most popular bolt in Australia and also studied in laboratory push/pull tests and double shear tests [2,24–26]. According to measurement, the rib geometric parameters of T2 bolt are estimated as (Fig. 12): rib spacing ¼12 mm, stem length c ¼7 mm, rib width¼ 1.6 mm, rib height bsiny ¼ 1.5 mm, length between the rib tips L¼8.8 mm, and rib face angle y ¼601. 6.1. Bolt rib face angle The rib face angle is investigated according to Eq. (23). The rib height is fixed to be 1.5 mm and the rib stem length L is to be constant at 8.8 mm. In addition, the grout material is assumed to be resin and accordingly tanj ¼0.7. Therefore, the coefficient factor of axial force (G2 G1tanj)/p with changing rib face angle y
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Fig. 10. Resin section under normal load p, and coordinate system after rotation.
Fig. 11. Forces on a small element along the plane of weakness.
Fig. 13. Coefficient factor of axial force at different rib face angle for T2 bolt.
Fig. 12. Geometric parameters of T2 bolt.
is plotted and shown in Fig. 13. From this diagram, it is found that: Case (1): rib face angle 01o y o191. It should be noted that there is a minimum rib face angle due to geometric constrains (grey dashed lines inFig. 12), for the investigated bolt: height ¼ 191 y Ztan1 rib L=2 Case (2): rib face angle 201o y o451. In this stage the influence of the axial load on the assumed weakness surface decreases with increasing rib face angle.
Case (3): rib face angle y E451. There is a stationary point around y ¼451, which is the most difficult case to cause parallel shear failure on the weakness plane. At this point, (G2 G1tanj)/ p E0.13, hence, to cause parallel shear failure on the supposed weakness plane: F ¼7.7T0. In other words, if the rib face angle is around 451, the axial load will be transferred to a direction perpendicular to the bolt axis and the rockbolting will have the greatest ability to resist parallel shear failure of the resin. Case (4): face angle 451 o y o901. With increasing the rib face angle above 451, increasing coefficient factor of the axial force indicates that parallel shear failure on the assumed weakness
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Fig. 14. The influence of rib face angles for different stem lengths of the bolt.
Fig. 16. Rate of load transfer with varying of length to height ratio.
small, say less than 8% of the stem length, parallel shear failure of the grout material will not occur. 6.4. Bolt rib height to length ratio Results shown in Figs. 14 and 15 suggest that the rib height and length should be studied together. Therefore, the coefficient of axial load is plotted for varying rib length to height ratios, and shown in Fig. 16. As expected, results show that if the rib length to height ratio is great than 13, parallel shear failure of the resin cannot occur due to insufficient load transfer.
7. Discussion
Fig. 15. Coefficient of axial load with varying rib height at different stem length.
plane becomes easier. For example, if the face angle is designed to be 851, then (G2-G1tanj)/p E0.9, and consequently axial load of 1.1T0 will be sufficient to commence parallel shear failure of the rockbolting system. 6.2. Bolt rib spacing If the rib height is fixed at 1.5 mm, for different profile stem length, L, the influence of rib face angle on load transfer is plotted in Fig. 14. It can be seen that for large profile spacing the coefficient factor becomes negative. For example, a T2 bolt was modified to 25 mm rib spacing in some experiments. The influence of the axial loading of this modified bolt on the parallel shear failure of the resin was calculated and plotted as dashed line in Fig. 14. In case of rib face angle 351 o y o611, the failure criterion, f, is always positive. It indicates that shear failure of the resin on the assumed plane will never occur at this modified bolt profile configuration. Consequently, dilational slip failure of the resin will be inevitable as the majority of the axial load is transferred radially. 6.3. Bolt rib height If the rib face angle is to be constant at 601, the efficiency of axial load to commence parallel shear failure of the resin is demonstrated in Fig. 15. The results indicate that the increase in the rib height, increases the rate of load transferred to a direction parallel to the axis of the bolt. In addition, if the rib height is very
The novel idea of coupling the bolt profile geometry with the introduced stress field provides a tool to investigate the bolt profile configuration and its effects on the load transfer mechanism for the benefit of the mining industry and science in general. A new development in calculating load transfer capacity between two rib profiles of varying geometries is discussed. The derived mathematical equations calculate the stress distribution adjacent to the fully grouted bolt subjected to axial load. The parametric study suggests that the axial load can be transferred to surrounding material laterally and longitudinally. The transfer rate is sensitive to changes in bolt rib geometry. In general, lower rib face angle and/or small profile height to length ratios are favourable to transfer load radially. As a result, dilation is inevitable. On the contrary, high rib face angle with larger rib height to length ratio will transfer the majority of axial load to a direction parallel to the bolt surface. In this case, parallel shear failure would most likely occur. Therefore, this study suggests that the rock bolt with smaller rib face angle and larger spacing is suitable in high confinement such as hard rock, while the rock bolt with high profile height and larger rib face angle should be used in soft rock. There are limitations on the applications of derived mathematical expressions to rib profile design. The boundary effect becomes a problem for some rib geometry. For example, if the rib is closely spaced, then the grout material can no longer be represented by a half space. In addition, the calculation is inaccurate when the rib face angle is large. Dilational slip failure of the bolt is another very common failure mode in pullout tests. If dilational slip failure is investigated in a similar manner presented in this paper, calculation will be inaccurate as the failure surface is close to the bolt. The influence of bolt profile configuration in case of dilational slip failure is the subject of a paper in preparing.
C. Cao et al. / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 188–195
Appendix A Calculation of each component of Eqs. (16)–(19), substituting Eqs. (9)–(13) yields: !2 Z Z bsiny hbcosy 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ¼ bsiny tan1 sin a dh ¼ 2 2 bsiny b þ h 2bhcosy Z Z bsiny hbcosy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh sina cosa dh ¼ 2 2 2 2 b þ h 2bhcosy b þh 2bhcosy 1 2 2 ¼ bsinyln b þ h 2bhcosy 2 Z Z bsiny adh ¼ tan1 dh hbcosy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bsiny 2 2 þ bsinyln h 2bhcosy þ b ¼ htan1 hbcosy hbcosy þ bcosytan1 bsiny Z Z R1 h ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ln dh ¼ 2 2 R2 b þh 2bhcosy 2 2 ¼ ðhbcosyÞln h 2bhcosy þ b 2ðhbcosyÞ hbcosy þ 2bsinytan1 bsiny These solutions can be verified by reverse differentiation. For definite integral: RL 2 0 sin adh ¼ bgsiny RL 0 sinacosadh ¼ bksiny RL 0 adh ¼ bmðpygÞ þ bksiny þ bgcosy R L R1 0 ln R2 dh ¼ bkcosybkm þ bm lnmbgsiny In the procedure of simplification, it should be noted that: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bsiny L2 2bLcosy þb 1 ¼ klnm ¼ pyg; ln tan L Lbcosy References [1] Singer SP, Field verification of load transfer mechanics of fully grouted roof bolts. US Bureau of Mines, 1990, p. 9301. [2] Aziz NI, Webb B., Study of load transfer capacity of bolts using short encapsulation push test. In: Proceedings of the 4th underground coal operators conference, Wollongong, February 12–14 2003, p. 72–80. /http:// ro.uow.edu.au/coal/162/S. [3] 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. [4] Aziz NI, Jalalifar H, Concalves J., Bolt surface configurations and load transfer mechanism. In: Proceedings of the 7th underground coal operators conference, Wollongong, 5–7 July 2006, p. 236-44. /http://ro.uow.edu.au/coal/51/S. [5] Yazici S, Kaiser PK. Bond strength of grouted cable bolts. Int J Rock Mech Min Sci Geomech Abstr 1992;29(3):279–92.
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