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Arch. Min. Sci., Vol. 61 (2016), No 4, p. 823852 Electronic version (in col) of this paper is available: http://mining.archives.pl DOI 10.1515/amsc-2016-0056 MASOUD RANJBARNI1, AHMAD FAHIMIFAR, PIERPAOLO ESTE* NOWE ANALITYCZNE METODY OCENY SKUTECZNOCI DZIALANIA WSTPNIE NAPRANYCH ZACEMENTOWANYCH KOTEW PRZY STABILIZACJI TUNELI In this paper, two new analytical approaches are presented on the basis of convergence-confinement method to compute both the ultimate convergence of circular tunnel and its zone having been reinfced by systematically pre-tensioned fully grouted rockbolts. The models have two basic assumptions: (1) the grouted rockbolts increase the radial internal pressure within a broken rock mass by both the pre-tensioned fce and the probable following induced fce due to rock mass movement (2) tunnel convergence (specially sht-term) occurs only due to reducing and diminishing of the radial constrained stress on tunnel surface provided by the wking face. Hence, the values of both the pre-tensioned pressure and the mentioned radial constrained stress are specially taken into consideration in this paper. That is, accding to their magnitudes, two differenonditions occur: the magnitude of pre-tensioned pressure is greater than that of the constrained stress at bolt installation time and vice versa. The solutions are extended to each of conditions, and illustrative examples are solved. The proposed approaches predicting almost identical results show tha-tensioning of grouted rockbolts will increase the efficiency and effectiveness of rockbolts. Keywds: analytical approach; tunneling design; convergence-confinement method; pre-tensioned fully grouted rockbolt W pracy tej przedstawiono dwie analityczne metody oparte na metodzie badania konwergencji i napre wymuszonych wykzystane do obliczania zarówno granicznej konwergencji tunelu o przekroju kola az zachowania strefy plastycznej, po wzmocnieniu tunelu za pomoc wstpnie napranych i zacementowanych kotew. Model opiera si dwóch zaloeniach: (1) zacementowane kotwy prowadz do wzrostu cinienia wewntrznego w kierunku promieniowym w kruszonym materiale skalnym, spowodowanego sil wstpnego naprenia az sil spowodowan przez ruchy górotwu; (2) konwergencja * UNIVERSITY OF TABRIZ, FACULTY OF CIVIL ENGINEERING, DEPARTMENT OF GEOTECHNICAL ENGINEERING, EAST AZERBAIJAN, TABRIZ, 29 BAHMAN BLVD, IRAN. E-mail: m.ranjbarnia@tabrizu.ac.ir AMIRKABIR UNIVERSITY OF TECHNOLOGY (TEHRAN POLYTECHNIC), FACULTY OF CIVIL AND ENVIRONMENTAL ENGINEERING, DEPARTMENT OF GEOTECHNICAL ENGINEERING, TEHRAN, HAFEZ AVE, IRAN. E-mail: fahim@aut. ac.ir * POLITECNICO DI TINO, FACULTY OF ENVIRONMENTAL, LAND AND INFRASTRUCTURE ENGINEERING, TURIN, CSO DUCA DEGLI ABRUZZI, 24, 10129, ITALY. E-mail: pierpaolo.este@polito.it 1 CRESPONDING AUTH: E-mail: m.ranjbarnia@tabrizu.ac.ir tunelu (zwlaszcza w ujciu krótkoterminowym) pojawia si jedynie wskutek zmniejszenia wymuszonego naprenia promieniowego na powierzchni tunelu generowanego w rejonie przodka wydobywczego. W metodzie zwrócono szczególn uwag na wartoci cinienia wstpnego naprenia jak i naprenia wymuszonego. W zalenoci od wielkoci tych napre mamy do czynienia z dwiema zupelnie odmiennym sytuacjami: wielko cinienia wstpnego naprenia jest wiksza ni naprenia wymuszonego w trakcie mocowania kotew, lub odwrotnie. Podano rozwizania dla obydwu rozwaanych przypadków i zaprezentowano przyklady. Prawie identyczne wyniki otrzymane przy uyciu obydwu metod wskazuj, e wstpne naprenia cementowanych kotew poprawia ich skuteczno dzialania. Slowa kluczowe: metody analityczne, projektowanie tuneli, metoda obliczania konwergencji i napre wymuszonych, wstpnie naprane zacementowane kotwy 1. Introduction The use of systematic grouted rockbolts as a standard practice in design and construction of tunnels is widely increased due to their effectiveness e.g. in new technologies such as New Austrian Tunnelling Method (NATM) and their other advantageous e.g. speed, minimum installation space and cost. This stabilizing system can be installed as either passive active (pre-tensioned) types which the aim of pre-tensioning is to transfer initial compressive pressure to rock mass in der to increase its perfmance and its efficiency (Carranza-Tres, 2009). The study of behavi mechanism of grouted rockbolts as the systematic reinfcing suppt has been of considerable interest during the last three decades. A number of analytical methods of varying degrees of accuracy, efficiency, and sophistication have been developed. Among these wks, in a group of approaches, it has been attempted to obtain the engineering properties of reinfced rock mass as an improved composite material (Ranjbarnia et al., 2014a, 2015). However, in another group of approaches, the grouted bolt have been considered as an individual element which its contribution to rock mass is in the fm of a radial load inducing the radial pressure within the influence domain of itself (Ranjbarnia et al., 2014a, 2015). In the first group, which the engineering properties of rock mass is assumed to be improved due to bolting effect, Indraratna and Kaiser (1990a, b) introduced a dimensionless parameter named "bolt density" which reflected the relative density of bolts with respect to the opening perimeter to obtain the reinfced rock mass properties. Osgoui and este (2007, 2010) improved Indraratna and Kaiser's solution (1990a, b) by applying "bolt density" parameter into all strength parameters of generalized Hoek-Brown criterion. Bobet (2006) obtained the properties of the rock- bolt material using the shear-lag method. Then, Bobet and Einstein (2011) discussed the imptance of few parameters on the perfmance of grouted rockbolts, and introduced a fmulation f mechanical contribution of the rockbolts based on shear interaction stress on the bolt surface. Carranza-Tres (2009) introduced a dimensionless coefficient named as "ground reinfcement- stiffness" (the contrast of stiffnesses of ground and rockbolt) which was a function of another coefficient named as reinfcement- density (the ratio of cross sectional area of rockbolt to the tributary area of rockbolt). These coefficients were used as the multipliers to obtain the confinement stress of composite material. Grasso et al. (1989) and Bernaud et al. (2009) wks are also the other attempts to model the composite material properties. All above mentioned researches have been perfmed f the passive grouted rockbolts except the wks by Carranza-Tres (2009) and Bobet and Einstein (2011) pre-tensioned grouted rockbolts have been also considered. In the other attempt, Fahimifar and Ranjbarnia (2009) presented an analytical approach f these types of rockbolts based on the extension of the wks by Stille et al. (1989) and Fahimifar and Soush (2005) having iginally been carried out f the passive types. In that study, it was assumed; due to applying compression pressure by the bolt around tunnel, the rock mass become stronger than the broken rock mass. Accdingly, the presence of pre-tensioned fce leads to development of greater confinement stress, and hence; the associated material would be much stronger. In the second group, which the reinfcemenontribution is in the fm of a radial load spread unifmly within the zone of influence of the rockbolt, a comprehensive series of studies have been also conducted (Aydan, 1989; Peila & este, 1995, 1996; Li & Stillbg, 1999; este, 2003, 2004, 2008, 2009; Cai et al., 2004a, b; Guan et al., 2007; Bobet & Einstein, 2011). However, like the first group methods, a few of them have been devoted to the pre-tensioned grouted rockbolts such as the wk by Bobet and Einstein (2011), Ranjbarnia, este and Fahimifar (2016) with great limitations which will be discussed in the following. In der to model the pre-tensioned grouted rockbolts as a systematic suppt of tunnels (at least in sht-time), the relation between the value of pre-tensioned pressure on the tunnel surface (produced by the pre-tensioned fce) and that of the fictitious constrained radial pressure by proximity of the face should be specially taken into consideration (Ranjbarnia et al., 2015). The advancement of tunnel face in front of bolted section leads to diminish the constrained radial pressure to zero and ultimately, the pre-tensioned pressure will only remain in that section. Providing the value of pre-tensioned pressure on the tunnel surface is greater than that of the constrained radial pressure, advancement of the tunnel face will nohange the stresses within the rock mass around tunnel, and the ultimate load will not be greater than the initial tensioning. Conversely, providing the value of pre-tensioned pressure is less than that of constrained radial pressure, the stresses around tunnel will redistribute, and tunnel convergence will occur immediately after the radial pressure becomes less than the initial value pri to bolt installation, and hence; the bolt fce will increase. The above-mentioned analytical approaches f the pre-tensioned grouted rockbolts are noomprehensive solution. Because, the relation between the pre-tensioned and the constrained pressures is neglected (Carranza-Tres, 2009; Fahimifar & Ranjbarnia, 2009; Bobet & Einstein, 2011). Furtherme, it would be almost impossible to extend and modify the approaches to use f pre-tensioned types, and to realistically model the influence of pre-tensioned load. On the other hand, the widely available commercial finite-element and finite-difference computer codes, while in principle are capable of modeling the pre-tensioning, are not yet treating a solution to include the above discussion (about the relation of the pre-tensioned and the constrained pressures). F this reason, this paper develops analytical approaches that have the following attributes · It quantitatively models the efficiency of the pre-tensioning of grouted rockbolts in terms of reduction in both the tunnel convergence and failure zone around tunnel in comparison with the passive systems. In other wds, it provides a solution f obtaining the optimum efficiency of the reinfced system in terms of the advancement of tunnel face (delay in installation), rockbolt arrangement, and the magnitude of pre-tensioning load e.g. the optimum pre-tensioning fce f a given delay installation time and the rockbolt arrangement. · It provides total length of the bolt in terms of the rockbolt arrangement and the magnitude of pre-tensioned load. The present paper develops two analytical approaches on the basis of two described groups of methods by the assumption of rigid connection between the bolt and the rock mass. The fmula- tions of both methods are derived on the basis of convergence- confinement method representing the response of a reinfced circular tunnel under unifm in-situ stresses by a systematically installed pre-tensioned grouted rockbolts. 2. Problem definition A circular tunnel of radius ri is driven in a homogeneous, isotropic, initially rock mass subjected to a hydrostatic stress field, p0. When the rockbolts are installed, it is assumed, a certain convergence of tunnel has already been occurred and an initial zone of radius re will develop around the tunnel (Fig. 1) (Ranjbarnia et al., 2014a). In this condition, there is a radial pressure on tunnel periphery supplied by proximity of the face, and its value is a percentage of field stress p0 ( pi = · p0), 0 < < 1). The magnitude of is mainly dependent upon the distance from the tunnel face within influence limit of tunnel face (which is about two tunnel diameters beyond wking face). The pre-tensioned grouted rockbolt installation consists of placing a grouted anch, tensioning the rockbolt and tying end of the bolt by nut and plate to the tunnel surface, and then grouting the reminder of the bolt length as (Fig. 2a) (Ranjbarnia et al., 2014a, 2015). The pre-tensioned fce applied by the bolt plate to tunnel surface develops the radial pressure extending within pi re Fig. 1. The axisymmetric tunnel problem a) re ri Sc b) Sl ri Fig. 2. Rockbolts arrangement (a) circular tunnel reinfced by systematic pre-tensioned grouted bolts; circumferential space between bolts (redrawn by Ranjbarnia et al. (2015)) (b) longitudinal spacing between bolts the rock mass in the radial direction. As the bolts are installed systematically (Figs 2a and 2b), it is assumed, each bolt increases radial pressure within the influence domain of itself. Therefe (Ranjbarnia et al., 2014a, 2015) p pre ten ten S l S (1) Tpreten and ppreten are the pre-tensioned fce and its associated radial pressure at tunnel surface, respectively. Sl and S are longitudinal and circumferential bolts' spacing at tunnel surface, respectively. It is assumed that the pre-tensioned fce provides equivalent radial stress in domain zone of each bolt (Fig. 3) (Ranjbarnia et al., 2014a, 2015). dr Fig. 3. Equivalent radial stress due to pre-tensioned load After installing the bolts, as tunnel face is again advancing within the influence limit of the wking face, the fictitious constrained radial stress will be furtheeduced and will be ultimately diminished. If the magnitude of constrained radial stress is less than that of the pre-tensioned pressure (Case I ), progressive advancement of tunnel face and then full diminishing of the constrained radial stress will not lead to furtheadial displacement. This is because; by applying the pre-tensioned pressure, the overall radial stress on tunnel surface after full diminishing of the constrained stress is greater than that of its initial value pri to the bolt installation moment. Hence, the final bolt fce is not greater than the initial applied tension i.e. the bolt fce will remain constant and will be equal to pre-tensioned fce, and grouting the reminder of bolt length has no influence on its behavi mechanism, but will only protect the bolt from crosion. In this case, the bolt behavi is similar to suppt systems e.g. un-grouted tensioned bolts the bolt and the rock mass act independently. However, if the magnitude of constrained radial stress is greater than that of pre-tensioned pressure (Case II), after somewhat diminishing of the constrained radial stress, the overall radial stress on tunnel surface will start to become less than that of its initial value pri to bolt installation. Thus, continuing excavation process will lead to furtheadial displacement of the surrounding rock mass and increase in the shear stresses between the bolt and the rock mass. That is, the bolt fce will increase till to full diminishing of the constrained radial stress, and the radius becomes greater (Fig. 4). In this case, the bolt interacts by its surrounding medium i.e. the bolt does not act independently of the rock mass, and hence; their defmations cannot be separated. F simplicity, further assumptions are made as follows: · The problem is studied under plane-strain conditions; thus the three-dimensional effect even near the tunnel face is disregarded. That is, the analysis considers a `slice' of tunnel of unit length i.e. 1 m along the axis of the tunnel else the bolt properties should be adapted f a unit length. · Rigid connection is assumed between the bolt and the rock. Therefe, the bolt fce (both of pre-tensioned and probable ultimate fce) provides a unifm radial suppssure in tunnel boundary and within the rock mass. · The bolt has a length that it is anched beyond the boundary of the broken zone in the iginal rock mass f both Cases of I and II (Figs 2a and 4). · The pre-tensioned fce applied to a rockbolt is typically a significant fraction of the bolt's yielding capacity. However, its magnitude is not a value leading the bolt to yield. re ri re Fig. 4. Increasing tunnel radius (redrawn by Ranjbarnia et al. (2015)) Also reminding again that · Tunnel closure is only assumed due to advancement of the wking face. In other wds, sht-term movements of rock mass are taken into account, and time-dependent properties of the rock mass are neglected. · The influence of the weight of the rock in the zone on tunnel displacements is disregarded. 3. The reinfcement mechanics of systematic pre-tensioned bolts The analytical solutions are proposed to model the pre-tensioned grouted rockbolts behavi in circular tunnels on the basis of both described methods. 3.1. The first method As mentioned in Introduction, in this method, the properties of the medium surrounding the tunnel is considered as an improved stronger material than the broken rock mass. To model the composite material properties, it will be wth figuring out the functional behavi of grouted rockbolt and its surrounding rock mass. In general, the grouted rockbolts assist the rock mass to fm a self suppting rock structure. They reinfce and mobilize the inherent strength of the rock mass by offering internal and confining pressure (Huang, 2002). Therefe, it is assumed in this paper, the grouted rockbolts reduce and control tunnel convergence through increasing the radial stress within the rock mass. In other wds, as the rockbolt restrains the defmation of rock mass, a tensioned fce in the rockbolt results in, and at the same time, it applies pressure to the rock mass. Therefe, the adjusted radial stress of the composite material is written by (2) 'r is the adjusted radial stress f composite material, T is the overall rockbolt tension fce, and C is the rockbolt effective area calculated by (Fig. 2) C = Sl · Sc C r i i L (3) (4) is the rockbolt effective area at tunnel surface, L is the bolt length located in the zone, and r is a variable showing the radial distance from tunnel center. Substituting Equation (4) into Equation (2) gives T ri r (5) In this paper, the Hoek- Brown strength criterion (Hoek & Brown, 1980) is adopted f iginal rock mass (6) and r are the circumferential and radial stresses, respectively. c is uniaxial compressive strength of the intact rock material, and parameters m and s are rock mass constants depending on the nature of the rock mass and its geotechnical conditions. Equation (6) can be used f the rock mass strength befe and after failure by using appropriate m and s. Substituting the adjusted radial stress of composite material into Equation (6) gives the strength criterion of composite material i.e. (7) Note that due to plane strain and the axial symmetry assumptions, the tangential and radial stresses, and r, will be principal stresses, 1 and 3, respectively. As well, due to the rigid connection assumption between bolt and its surrounding rock, presence of the bolts does nohange the principal stress directions. Substituting Equation (2) into Equation (7) gives m s (8) The axial bolt fce can be obtained by T = Ab · Es · b (9) Ab and Es are bolross section area and the modulus of ity of bolt, respectively, and b is axial bolt strain. As explained in section 2, f Case I, continuing excavation process will not impose furtheadial tunnel convergence and the bolt fce will remain constant and will be equal to pre-tensioned load, i.e. b = preten and so fth T = Tpreten (11) In this condition, tunnel convergence will be equal to its value pri to the bolt installation. It should be noted that, accding to the static equilibrium, the pre-tensioned load is constant along the bolt, and hence; its cresponding pressure applied along the bolt to its surrounding rock mass is unifm. However, this pressure within the rock mass is linearly decreased from the tunnel surface to the depth of rock mass due to the increasing the rockbolt effective area (Fig. 3). (10) p pre ten ten ri ri (12) ppreten is the pre-tensioned pressure. F Case II, furtheadial displacements of the rock mass after full diminishing of the constrained stress become less than that of the pre-tensioned pressure leading a further tension to be imposed to the rockbolt. The additional bolt strain will be the same as radial rock mass strain due to the rigid connection assumption between them. Therefe b = preten + 'r 'r is the radial strain within rock mass that takes place after installing the bolts. Pri to installing the bolts, on the other hand, a certain radial displacement of rock mass has occurred i.e. the displacement in the initial zone, e, and the defmar tions in the greater zone, re, have been already developed (Figure 4). Hence, Equation (13) can be rewritten as r b e r (13) e re re (14) (Note: re ri = L) and e are the radial strain within the r is total radial strain within rock mass, rock mass befe installing the bolts in the initial and developed zone, respectively. To obtain each of these radial strains, the rock mass stress-strain behavi is assumed strainsoftening which is me appropriate behavi of most rock masses in tunnelling design (Hoek & Brown, 1997). Different approaches and fmula have been presented to distribute stress and strain around tunnel f this behaviour (Brown et al., 1983; Alonso et al., 2003; Park et al., 2008; Lee & Pietruszczak, 2008; Wang et al., 2010; Ranjbarnia et al., 2014b) However, as the aim of this paper is only to assess the critical and the effective parameters of the pre-tensioned grouted rockbolts, Brown et al. (1983) wk (one of the simplest models) is used although it does not have very exacdictions (Alonso et al., 2003). The rockbolts parameters implemented into this approach is presented in Appendix 1 in the fm of calculation sequence to obtain Ground Response Curve of reinfced tunnel. Differential equation of equilibrium of iginal rock mass around tunnel at radius r from the tunnel centre in polar codinates is given by (15) Substituting the composite material criterion into Equation (15) results in the differential equation of equilibrium f the strongeock mass m r s (16) c r T r 2 c T (17) (18) F Case I, the ultimate tunnel convergence will be identical to that of the installing time. However, Equation (17) can be extended to this condition as follows m Ab Es r 2 c Ab Es (19) And f Case II Ab Es 2 c Ab Es and r ri re (20) Ab Es e r e r r 2 c Ab Es r re re (21) The differential Equation (19) has both boundary conditions of (1) and (2), while Equations (20) and (21) have boundary condition (1) and (2), respectively: (1) At r = ri, r = pi ri is the tunnel radius and pi is the magnitude of radial pressure in the tunnel surface. (2) At r = re, r = re, re is the radial stress at the outer boundary of zone and is obtained as follows (Brown et al., 1983) re = p0 M · c (22) m 4 p0 m 8 (23) To plot Ground Response Curve to calculate the ultimate radial convergence, the radial stress should be reduced to zero. This is because, in the first method, the material surrounding the tunnel is only considered as a stronger material and GRC of a stronger material is to be calculated. These differential equations can be solved by numerical method due to their algebraic complexity. An iterative finite difference solution on the basis of Brown et al. (1983) wk is used. That is, the zone is split into annulaings and calculations are starting from the unknown - interface towards the tunnel by incrementing the tangential strain value f each calculation step and then calculating the cresponding radial strain (Brown et al., 1983) If equation (19) is rewritten f a ring , ian be r ( ) r ( ) c Ab Es sa ( ) a 2 c 1) ) Ab Es ( ) 1) ) 2 1) ) (24) ) ri 2 1) 1) ma sa m( j s( j m( j ) s( j ) 1) Note that the average value of parameters in the right hand side of Equation (19) (i.e. m, s, r, r) were written in Equation (24) e.g. ma is the average value of m at two rings 1) and . different parameters are in 1) and radii. Defining some parameters and making simplifications results in the second der equation giving r(j) and solution is 2 b2 2a K pre K 4 ac (27) K pre ten ten r ( ) 2 2 sa 2 c r () 4K r ( ) 2 2 K pre 4 K 2 K pre ten (28) ) re ( ) ( ) (29) (30) ( ) ma 4 (31) (32) (33) K pre ten Ab Es 2 . a Perfming the same process f both Equations (20) and (21) changes the multiplier of second der as follows f ri r re zone 1 , 4K 2 r () 2 K pre ten r ( ) 2 ten 2 K pre r () 4K * 4 K1 K 2 K 2 2 K 2 K pre ten K pre sa 2 c (34) * r ( ) (35) (36) (37) (38) Ab Es 2 r ( ) r K1* and f re < r re zone Ab Es 2 1 , 4K 2 * K e(*) K1 K pre ten r ( ) 2 2 4 r ( ) 2 r () K 2 4K K e (* ) ten 2 sa 2 c K e (*) K pre K e (* ) K 2 K 2 K pre (39) Ke e r ( ) e Ab Es 2 re re e a (40) M C G (41) Note: superscript * refers to the first method. 3.2. The second method As mentioned in Introduction, this method is on the basis of functional mechanism of grouted rockbolt reinfcemenontribution is in the fm of a radial load spread within its influencing zone. F tunnel with circular cross section, unifm in-situ stresses, and close spacing of the rockbolts, the differential equation of equilibrium is given as (este, 2008) dT 1 dr C dT ri 1 dr r (42a) (42b) Substituting failure criterion of iginal rack mass i.e. Equation (6) into Equation (42b) gives m c r s r 2 12 c dT ri 1 d (43) The above differential equations have the same boundary conditions as Equations (19)-(21). To solve differential equation f Case I, it is assumed that the radial pressure must be reduced from p0 to ppreten ppreten is greater than · p0 (the constrained stress at bolt installation time). That is, as the pre-tensioned pressure on tunnel surface ppreten is applied, it is added to · p0 and then an outward radial defmation occurs, and unloading takes place. Full diminishing of · p0 associated with progressive advancement of tunnel face leads again to loading of the rock mass around tunnel and therefe, accding to solid mechanics concepts, some inward radial defmation occurs which its magnitude is dependent upon ppreten value. However, in this paper, small defmations and variations of total defmations are disregarded in the fmula, and hence; the ultimate radial convergence will almost be the same as that of the bolt installation time. The differential equation f this condition will be (1) At r = ri, r = pi · p0 pi p0. (2) At r = re, r = re. From the mathematical point of view, on the other hand, as the load is constant along the bolt f Case I, dT/dr will be zero and Equation (44) will be obtained from Equation (43). F Case II, after dwindling of the radial pressure on tunnel surface from the remained in-situ radial pressure i.e. · p0 to pre-tensioned pressure i.e. ppreten, the radial defmations of rock mass will increase, and further tensioned is imposed to the bolt. This process is continued till to full diminishing of the constrained stress · p0, and till to decreasing the radial pressure on tunnel surface to ppreten. Thus, the differential equation f this condition will be (44) with the following boundary condition (Ranjbarnia et al., 2014a, 2015) with the following boundary conditions dT ri 1 dr r (45) (1) At r = ri, r = pi ppreten pi p0 (this is just f ri < r re zone). (2) At r = r , = (this is just f r < r r zone). e e e e Equation (45) can be rewritten as (Ranjbarnia et al., 2015) 12 c r r sa 2 c T T ri 2 2 1 (46) 12 c r r sa 2 c r ( ) r ( ) 2 Ab E s ( ) ri 2 (47) Processing Equation (47) in the similar way perfmed f Equations (19)-(21) gives the multiplier of second der, respectively as follows (Ranjbarnia et al., 2015) f ri < r re zone a 1 , 4K 2 r () 2 K1 K K1 K1 r ( ) 2 2 c r () 4K 2 K1 sa 2 c (48) r (49) (50) K1 Ab Es ri r r ( ) (51) (52) K1 Ab Es ri r and f re < r re zone a 1 , 4K 2 r () 2 * e K1 () K K1 r ( ) 2 2 c r () K1 4K K1e () K 2 K1e () sa 2 c (53) e () e e e r (54) (55) K1e () Ab Es ri r e () Note: superscript refers to the second method 4. Examples Ground Response Curve calculations, f a rock mass being reinfced by the pre-tensioned grouted rockbolts, are perfmed by the proposed methods. To quantify the effect of pre-tensioning, GRC calculations are also perfmed f the passive grouted rockbolts. Other parameters such as · the magnitude of pre-tensioned load and · bolt's spacing are investigated to identify the weight of each in tunnel stability. The examples were selected from (Ward et al., 1976; Brown et al., 1983) Example 1. Verification of the proposed models results with that of to the Kielder experimental tunnel Accding to Ward et al. (1976), total sht-term movement of tunnel surface in the unsuppted section of mudstone with weak engineering properties (Table 1) was about 8 mm less than 1 mm had occurred befe the face reached, and about 6 mm when the face had advanced 2 m beyond this position. If the reinfcement system was installed just in front of the face, ian be expected that tunnel closure was about 1-2 mm pri to bolt installation (assumed value is 1.5 mm in this paper). TABLE 1 Mechanical properties of mudstone in the Kielder experimental tunnel (Hoek & Brown, 1980) Prarameter Value Axial compressive strength c (MPa) Radius of tunnel, ri (m) In-situ stress, p0 (MPa) Defmation modulus, E (MPa) Poisson's ratio of rock mass Strength parameter, m peak s peak m residual S residual Dilation angle (degree) Of the eight sections with different suppt systems in mudstone, one of them is only reinfced by passive grouted rockbolts (Table 2). TABLE 2 Geometrical parameters of passive grouted rockbolts in the Kielder experimental tunnel (Hoek & Brown, 1980) The parameter The value Fully grouted Rockbolt Length, L (m) Young's modulus of rock bolt, Es (GPa) Bolt diameter, db (mm) Distance between rockbolt, Sl * Sc (m2) 1.8 210 25 0.9 * 0.9 The cresponding ground response curves (Fig. 5) and output results (Table 3) show the calculated and measured defmations data at tunnel surface f suppted and unsuppted rock mass. As observed, the proposed methods can almosdict the identical results f the reinfced section, and agree, in a satisfacty way, with the in-situ measurements. The First method 0,4 The Second method Un-reinfced tunnel 0,1 Fig. 5. Ground response curves f the rock mass around Kielder experimental tunnel TABLE 3 The measured defmations by (Ward et al., 1976) and the calculated defmations by the proposed methods at the rock surface f suppted and unsuppted rock mass Parameter Measured Calculated Un-reinfced tunnel Passive grouted bolt section 8 mm 4-5 mm 8.05 mm 4.27 mm (the first method) 3.96 mm (the second method) In continue; the parameters associated to pre-tensioned grouted rock bolts are investigated. Note: the curves of GRC associated to the passive and the pre-tensioned reinfcements in Examples 2-5 are only calculated by the second method, meanwhile; the output results of both methods f these examples are available in Table 6. Example 2. Evaluating the perfmance of grouted bolts f Case I circumstance A highway tunnel with 10.7 m in diameter is driven in a fair to good quality limestone at a depth of 122 m below the surface (Brown et al., 1983) (Table 4). TABLE 4 Mechanical properties of the rock mass (Brown et al., 1983) in Example 2-5 Prarameter Axial compressive strength c (MPa) Radius of tunnel, ri (m) In-situ stress, p0 (MPa) Defmation modulus, E (MPa) Poisson's ratio of rock mass Strength parameter, m peak s peak m residual S residual Dilation coefficients, f h Value 27.6 5.35 3.31 1380 0.25 0.5 0.001 0.1 0 1.2 2 3.5 The pre-tensioned grouted rockbolts are installed f Tpreten =17 ton. As can be calculated (Table 5), the pre-tensioned pressure is greater than the fictitious constrained pressure of tunnel face. Consequently, the circumstance of Case I will take place i.e. continuing excavation process will not induce any further tunnel convergence. TABLE 5 Geometrical parameters of reinfcement systems in Examples 2-5 Example Rockbolt Density (m2) pre-tensioned fce (ton) constrained pressure due to wking face 2 2P* 3A 3P 4A 4P 5AI 5AII 5AIII = 1 = 1 = 0.5 = 0.5 = 0.5 = 1 = 0.75 = 1 16.5 ton/m2 (0.165 MPa) 16.5 ton/m2 (0.165 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) 33.1 ton/m2 (0.331 MPa) * Letters A and P in the examples denote the employing of active (pre-tensioned) and passive grouted rockbolt, respectively. Note: The diameter of bolts in all examples is 25 mm. The output results (Fig. 6 and Table 6) show the efficiency of pre-tensioning. The convergence of tunnel by pre-tensioning of bolts is reduced considerably. Reinfced by Pre-tensioned grouted bolt, Tpre = 17 ton, C = 1 Reinfced by Passive grouted bolt, Tpre = 0 ton, C = 1 Un-reinfced tunnel Fig. 6. Ground response curve f the rock mass around tunnel in Examples 2 TABLE 6 The output results of Example 2-4 Ultimate convergence of tunnel reinfced by .... * Pre-tensioned Passive bolt bolt radius of tunnel reinfced by .... Pre-tensioned Passive Example Approach The first method The second method The first method 3 The second method The first method 4 The second method Unreinfced tunnel * unit is millimetre unit is metre Example 3. Evaluating the perfmance of grouted bolts f Case II circumstance If the pre-tensioned grouted rockbolts in Example 2 are installed sooner i.e. at section pi = 0.1 p0 = 33.1 ton/m2, the circumstance of Case I will take place. Comparing the results of 3A and 3P (Table 6 and Fig. 7), the result shows that employing the pre-tensioned grouted rockbolts will not be as efficiency as the previous circumstances in Example 2A (Case I). Hence, the less delay to install the bolt, the less need to apply the pre-tensioned load. On the other hand, if the bolts are installed with great delay i.e. the tunnel advancement is so long that insignificant and leasonstrained radial pressure of tunnel head is remained pri to bolt installation, the bolt may be much pre-tensioned to apply greater initial pressure to the tunnel surface (trying to alter close the condition from Case II to Case I). However, it may lead to bolts with a final load too close to yield. Reducing spacing of bolts may be an appropriate alternative f pre-tensioning of bolts to confine tunnel convergence in case increasing pre-tensioned fce is impossible and is insufficient else the bolt will yield. Example 4 evaluates this issue. Reinfced by Pre-tensioned grouted bolt, Tpre = 17 ton, C = 1 Reinfced by Passive grouted bolt, Tpre = 0 ton, C = 1 Un-reinfced tunnel Fig. 7. Ground response curve f the rock mass around tunnel in Examples 3 Example 4. Evaluating the influence of bolts density in the perfmance of pre-tensioned grouted bolts in tunnel stabilization In this example, the effect of rockbolt density is evaluated. Pre-tensioned grouted rockbolts in Example 3 are installed with = 0.5 m2. In this case, the circumstance of Case I is produced, and the convergence will be significantly reduced in comparison to employing of passive grouted bolts (Table 6 and Fig. 8). Example 5. Comparing the increase of bolt density parameter with the increase of pre-tensioned fce in tunnel stabilization The weighting of spacing of bolts and increasing the pre-tensioned fce which both increase the radial pressure are quantified and compared. Both parameters variations have the identical influence on confining of the convergence and stability of tunnel provided they lead Case I to occur. However, if the pre-tension fce is increased by percent, the ratio of new cresponding pressure to the radial constrained pressure on tunnel surface will be pi ,new pi 1 ten ten (58a) Reinfced by Pre-tensioned grouted bolt, Tpre= 17 ton, C = 0.5 Reinfced by Passive grouted bolt, Tpre = 0 ton, C = 0.5 Un-reinfced tunnel Fig. 8. Ground response curve f the rock mass around tunnel in Examples 4 pi ,new 1 pi (58b) On the other hand, reducing bolts' spacing as percent gives the following result f tunnel surface pressure Tpre pi ,new pi ten 1 T (59a) pi ,new pi 1 (59b) Comparing the Equations (58b) and (59b) gives pi ,new pi ,new (60) Therefe, ian be concluded that reducing spacing of bolts is me effective than increasing of the pre-tensioned fce. On the other hand, this parameter has great influence on confining convergence in the case of passive grouted rockbolts. This is because; it reflects the reinfcement density in the rock mass. In other wds, reducing spacing of bolts not only efficiently increases tunnel surface pressure but also effectively improves broken rock mass strength. In this Example, the pressure on tunnel surface is increased up to 25% either of the pretensioned fce the rockbolt density. As output results show (Table 7 and Fig. 9), the rockbolt density has me effect in tunnel stability than the pre-tensioned fce. This is because; the radial pressure will be greater while increasing rockbolt density. However, if the pressure on tunnel wall is improved by increasing the pre-tensioned fce to the magnitude achieved by increasing rockbolt density i.e. ppreten = 22.67 ton/m2, the suppting perfmance of bolts will not be as effective as the case of increase in the bolt density (Example 5AII and 5AIII). Reinfced by Pre-tensioned grouted bolt, Tpre = 21.25 ton, C = 1 Reinfced by Pre-tensioned grouted bolt, Tpre = 17 ton, C = 0.75 Reinfced by Pre-tensioned grouted bolt, Tpre = 22.67 ton, C = 1 Reinfced by Pre-tensioned grouted bolt, Tpre = 17 ton, C = 1 0,1 Fig. 9. Ground response curve f the rock mass around tunnel in Example 5 TABLE 7 The output results of Example 5 Example Ultimate convergence (mm) radius (m) 3 5AI 5AII 5AIII 5. Conclusions Two new analytical approaches were proposed f the computation of the Ground Response Curve of a circular tunnel reinfced by the pre-tensioned grouted rockbolts, and also computation of ultimate zone around tunnel. These models were developed on the basis of shtterm convergence of tunnel i.e. tunnel convergence would only occur in term of reducing the constrained stress on surface due to tunnel face advancement. Consequently, the value of the constrained radial stress at bolt installation time and applied pre-tensioned pressure on tunnel were focused on in process of modeling. Different examples were solved by these approaches. The results obtained through these two models were almost identical and comparable. The results showed pre-tensioning of grouted rockbolts will be very appropriate option when installation of rockbolts is carried out with delay.
Archives of Mining Sciences – de Gruyter
Published: Dec 1, 2016
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