Tunnelling and Underground Space Technology 42 (2014) 105–111
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Vibration vulnerability of shotcrete on tunnel walls during construction blasting Lamis Ahmed ⇑, Anders Ansell 1 Department of Civil and Architectural Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
a r t i c l e
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Article history: Received 6 September 2013 Received in revised form 7 February 2014 Accepted 9 February 2014 Available online 5 March 2014 Keywords: Shotcrete Tunnelling Vibrations FE model Young shotcrete
a b s t r a c t The effect on shotcrete from blasting operations during tunnelling is studied, with focus on young and hardening shotcrete. A ﬁnite element model specially adapted for analysis of the shotcrete behaviour is tested, it is able to describe stress wave propagation in two dimensions which is important for cases where shear stresses are dominant. The modelling results are compared with in situ measurements and observations, from construction blasting during tunnelling through hard rock. The comparison shows that the model gives realistic results and can be used to investigate the vulnerability of shotcrete, aiming at compiling recommendations and guidelines for practical use. The given recommendations emphasize that blasting should be avoided during the ﬁrst 12 h after shotcreting and that distance and shotcrete thickness are important factors for how much additional time of waiting is possibly needed. Ó 2014 Elsevier Ltd. All rights reserved.
1. Background For safe excavation and tunnelling through hard rock it is often important to apply shotcrete (sprayed concrete) on the rock surfaces already at an early stage to secure potentially loose blocks and the shape of tunnels or other subspace openings. For a time-efﬁcient construction process the time of waiting between stages of excavation needs to be minimized. Hardening concrete, and shotcrete, is vulnerable to disturbance during early age after casting or spraying, which can lead to failure or a reduction in strength. This is the case for the development of the bond between shotcrete and rock which is sensitive to vibrations, especially during the ﬁrst 12–24 h after shotcreting. For safe tunnelling and underground work it is thus necessary to know when e.g. the bond between rock and newly sprayed shotcrete has reached an age where it can withstand vibrations at certain levels. This is particularly important since most work in hard rock requires blasting operations resulting in stress waves that transport energy through the rock and which may cause severe damage on permanent installations and support systems within the rock, such as shotcrete. An important example is the driving of two parallel tunnels that requires coordination between the two excavations so that blasting in one tunnel does not damage temporary support systems in the other tunnel prior to
⇑ Corresponding author. Tel.: +46 8 790 68 86. E-mail addresses: [email protected]
(L. Ahmed), [email protected]
(A. Ansell). 1 Tel.: +46 8 790 80 41. http://dx.doi.org/10.1016/j.tust.2014.02.008 0886-7798/Ó 2014 Elsevier Ltd. All rights reserved.
placing of a sturdier, permanent support. Similar problems also arise in mining operation where the need to excavate as much ore volume as possible leads to that the grid of drifts in a modern mine is dense. Established, detailed guidelines for acceptable vibration levels from blasting close to young and newly sprayed shotcrete is missing. Therefore, unnecessarily strict limit values are often used, leading to longer production times and larger costs. The limits set up are often expressed as maximum allowed vibration velocities, or peak particle velocities (ppv). It is often difﬁcult to translate these into minimum distance and shotcrete age, allowed amount of explosives to be used, sequence of detonations, etc. For this to be possible information on geometry of e.g. the tunnel together with material properties of rock and shotcrete must be considered. Such detailed guidelines for how close, in time and distance, to young shotcrete blasting can take place would thus be an important tool in planning for safe and economical tunnelling projects. 2. Previous investigations 2.1. Numerical modelling As a step towards detailed guidelines for practical use, a series of projects with analytical and numerical modelling using in situ observations and measurements for veriﬁcation have been carried out. These projects have resulted in recommendations for fully hardened shotcrete exposed to small amounts of explosives at short distances (Ansell, 2005), large scale detonations at larger distance (Ansell, 2007a) and for young and hardening shotcrete
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(Ansell, 2007b). Additional recommendations are given by Ahmed (2012). The ﬁrst models used and the cases studied were kept simple to facilitate the comparison between different modelling concepts and between calculated and in situ results, Ahmed and Ansell (2012a). With these simpliﬁed models, based on linear elastic material theory, the efﬁcient analytical procedure makes it possible to compare large numbers of calculations with various combinations of input data. Also, the interpretation of the results become straightforward but the possibility to study various geometrical conditions is heavily restricted. It is thus not possible to describe partially damaged structures, e.g. partial de-bonding of shotcrete, only to identify the limit for damage through series of calculations. One model tested was a one-dimensional elastic stress wave model, capable of describing wave propagation in one direction, including reﬂections and transmissions at material interfaces and free surfaces. The other two models were based on structural dynamic theories with vibrating masses and beams connected through elastic springs. The comparison and evaluation of the three numerical models showed that their results were comparable, although the deﬁnition of the dynamic loads was different. For the stress wave model the dynamic load was deﬁned as a time dependent velocity while a time dependent acceleration was used for the structural dynamic models. Based on the experience from these ﬁrst models the further work focused on developing a more advanced two-dimensional (plane strain) ﬁnite element (FE) modelling concept, Ahmed et al. (2012). The use of established FE programs facilitates the study of more complex geometries and it will also be possible to include the effect of rock bolts and partially damaged shotcrete, as the model is developed further. 2.2. In situ measurements The modelling results have been veriﬁed through comparison with in situ observations, measurements and laboratory tests. The latter were conducted as model tests using a shotcrete covered concrete beam with dynamic properties similar to good quality granite subjected to an impacting hammer that produced stress waves similar to that from blasting (Ahmed and Ansell, 2012b). Published, relevant in situ measurement data is scarce. In situ testing to determine how a shotcrete lining was affected by standard drift blasts at various distances from the lining was done by Kendorski et al. (1983), but no vibration levels were recorded during these tests. Measurements during the construction of parallel tunnels are presented by Nakano et al. (1993). Recorded accelerations or velocities are not available but particle velocities of 1450 mm/s are reported to have been reached with a smallest distance between blasting points and shotcrete that was 1.0 m. Results from tests carried out in a Canadian goldmine where steel ﬁbre-reinforced and steel mesh-reinforced shotcrete linings have been subjected to vibrations from explosions are presented by Wood and Tannant (1994) and McCreath et al. (1994), reporting vibration levels of 1500–2000 mm/s. As these results are either incompletely documented or not compatible with the numerical models tested, results from tests on young shotcrete performed during 1999 on site in the Kiirunavaara iron-ore mine, in the very northern of Sweden (Ansell, 2004) have been used for the veriﬁcation. The measurement set-up was similar to that used by Jinnerot and Nilsson (1998) who conducted tests aiming at determining damage zones in the rock due to blasting operations. Results from measurements during full scale production blasting in the mine has also been used, see Ansell (2007a). During these three measurement projects accelerometers were placed on a horizontal line along the length of the tunnel, two accelerometers at each measurement point to provide a two-axial description of the vibrations, parallel with and perpendicular to the length of the tunnel. Surface mounted accelerometers were placed on steel or aluminium plates bolted
to the rock. For measurements inside the rock, accelerometers were mounted inside fully grouted pipes of PVC. The inner end of these pipes consists of a conical tip, of steel or aluminium that is in contact with the rock and contains holders for the accelerometers on the inside. The lengths of the pipes were in these cases adjusted so that the measurement points are about 300–500 mm behind the rock surface. The obtained acceleration measurements have provided a good set of data for the evaluation of the analytical and numerical models with respect to the risk for local shotcrete damage in the vicinity to the source of blasting. For further analysis of more advanced geometrical conditions the models need to be veriﬁed with respect to wave propagation along the length of the tunnel, caused by blasting at a remote point, e.g. at the tunnel front. This paper presents such an evaluation, through comparison with a series of measurements done in situ during tunnelling construction work. 2.3. Measurements during construction blasting An attempt to characterize the vibrations that occur along tunnel walls during excavation blasting has been performed by Reidarman and Nyberg (2000). The measurements were done during construction of the Southern Link (Södra länken) road tunnel system in Stockholm, Sweden. The accelerometers used were positioned following the same system as for the Kiirunavaara measurements, described above. The results from this investigation are well suited for evaluation of the FE models described above. No shotcrete damage was observed following the blasting, due to the very restrict guidelines used, and it can thus be assumed that the shotcrete-rock system behaves elastically throughout the passage of the stress waves. The FE model which is based on elastic material properties can therefore be used for a numerical study of the stress wave propagation along these tunnel walls. The measurement of vibrations from four blasting rounds was done using accelerometers located along an axis stretching approximately 5–50 m behind the tunnel front. Accelerations were measured in two directions, parallel with and perpendicular to the tunnel walls, recorded and later numerically recalculated into corresponding velocity–time records. All measurement points were situated 300 mm into the rock. The layout of the test tunnel with the position for the measurement points is shown in Fig. 1. It should be noted that the advancement of the tunnel front is towards the left in the ﬁgure and that each blasting round results in 5 m new tunnel length, except for the third round which gave a 10 m extension. The ﬁgure also shows how some measurement points were abandoned in favour of new points closer to the tunnel face, thus approximately giving equal spacing between the points for each of the four rounds. The maximum velocities for each point vs. the distances along the tunnel wall are shown in Fig. 2, with curves ﬁtted using the method of least squares, in the direction parallel with and perpendicular to the tunnel wall vmax,x and vmax,y, respectively, giving:
v max;x ¼ 74:81e0:065x ðmm=sÞ v max;y ¼ 18:48e0:026x ðmm=sÞ
with the distance x along the tunnel given in metres. It should be noted that the ppv from a single blast hole did not exceed 80 mm/s in any test point, in any direction. 3. Stress waves in rock 3.1. Wave types Detonations in rock give rise to stress waves that propagate outwards from the detonation, in all directions through the rock and
L. Ahmed, A. Ansell / Tunnelling and Underground Space Technology 42 (2014) 105–111 7
Round 1 9
Round 2 10
Round 3 11
Round 4 5
Fig. 1. Tunnel with advancing front during four excavation rounds. Test layout with position of measurement points.
coincides with the ﬁrst arriving peak at a point of observation, thus caused by the P-wave. In the following numerical analyses it is assumed that the stress waves and thereby the ppv follow a cosine shaped time history that would correspond to a sinusoidal shaped particle acceleration and displacement. The particle velocity of the propagating stress is here:
v ðtÞ ¼ 0:5v max cosð2pftÞ 0:5v max
where vmax is the ppv, t is time and f is the frequency of the incoming stress. This wave shape has previously (Ahmed and Ansell, 2012a) been tested and evaluated for use in these types of numerical calculations. 3.2. Wave propagation Fig. 2. Maximum particle velocity (ppv) vs. distance along the tunnel wall.
towards possible free rock surfaces. Wave motion can be described as movement of energy through a material, i.e. transportation of energy achieved by particles translating and returning to equilibrium after the wave has passed. This motion of particles in the rock can be described as displacements, velocities or accelerations. When a wave-front reﬂects at a free surface, for instance a tunnel, the particle velocities are doubled and the stresses are zero over the surface. Also, a compressive wave reﬂects backwards as a tensile wave, etc. The velocity of propagation through elastic materials depends on the type of wave. The most important types for analysis of stress waves in rock are longitudinal waves (P-waves) and shear waves (S-waves), see e.g. Dowding (1996). Of these two types P-waves are the fastest propagating which means that the ﬁrst arriving stress peak for most cases is that of a longitudinal wave. The propagation velocity cp of P-waves in an elastic material is given by its stiffness and the density q of the material, following:
The particle velocities that can be measured remote from detonating explosives in rock will show a decrease in magnitude with increasing distance to the source of explosion. This decay is caused by geometrical spreading and hysteretic damping in the rock (Dowding, 1996). In this paper, the examples have been performed based on the properties of the hard quality granite at the location of the Southern Link tunnels, giving (Holmberg and Persson, 1979):
v max ¼ A
R ðf1 Q Þ
where vmax is the ppv in m/s, Q is the charge weight in kg and R the distance from the point of observation to the charge, in m. Based on regression analysis of in situ measurement data, the constants A, a, and b are in this case equal to 0.7 m/s, 0.7, and 1.5, respectively. Here f1 is a factor that includes the effect from long explosive charges, formulated as:
R arctanðLe =2RÞ ¼ Le ðLe =2RÞ
where E is the elastic modulus. The corresponding velocity for Swaves is:
where Le is the charge length. It should be noted that Eq. (6) is valid only for situations where R is larger than Le. An alternative expression to predict the blast vibration is given by e.g. Arora and Dey (2010), as:
v max ¼ 0:61
where G is the shear modulus. As a wave passes the particles of the material are displaced with a velocity that depends on the magnitude of the disturbing source and in the direction given by the wave type. The maximum velocity is usually given as ppv and this often
Ph /h nR e qc R
qe v 2det 8
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with n, qe and vdet, denoting the damping ratio, density of explosives and velocity of detonation, respectively. The diameters of the explosive charge and the pre-drilled charge hole are /e and /h, respectively. The constant c can vary between 2.4 and 2.6 while the constant 0.61 is a geometrical scale factor. For the conditions at the Southern Link tunnels the ppv vs. distance relations given by Eqs. (5) and (7) are plotted in Fig. 3, together with some measurement points which are located approximately 11 m behind the tunnel front. Note that the measurements normal to the tunnel wall registered at points 9 and 10 are signiﬁcantly lower than the theoretical curves. These points were closest to the tunnel front during round 2 and 3. They are spaced 6 m apart which may indicate an inﬂuence from local rock, such as wide cracks. The explosives used were a type of Site Sensitise Emulsion, deﬁned by qe = 993 kg/m3 and vdet = 4500 m/s. The charge hole diameters are here /e = 21 mm inserted in holes /h = 48 mm, with the length Le = 3.7 m, giving Q = 1.27 kg per explosive charge. 4. Material properties The numerical simulations require insertion of realistic material data for shotcrete and rock. To evaluate the propagation of the stress waves in rock and young and hardening shotcrete focus was put on the development by age of shotcrete properties such as the elastic modulus, tensile strength and the bond strength between shotcrete and rock. No rock bolts or mesh reinforcements are included as supporting elements and the material properties are chosen as typical representative average values to exclude the effect of extreme material strengths, etc. To facilitate comparisons the same properties of the rock as documented by Reidarman and Nyberg (2000) were chosen. 4.1. Shotcrete The material properties chosen for the examples resembled that of the shotcrete used in the Kiirunavaara tests (Ansell, 2004) resulting in a density qs of 2100 kg/m3 and a modulus of elasticity E28d of 27 GPa after 28 days. The development of the Es for hardening shotcrete is herein assumed to follow an average function, derived from a large number of tests (Chang, 1994). The relation between Es and the age of shotcrete ts (in hours) is here:
Es ðt s Þ ¼ 1:06E28d e5:7ts
The tensile strength fct in this study is approximated using the relation between tensile and compressive strength of concrete given in Eurocode 2 (2004). Based on the compressive strength of shotcrete fcc, experimentally found by Bryne et al. (2013), the following equation emerges:
fct ðt s Þ ¼ 0:3½fcc ðt s Þ
ðMPaÞ for 0 6 ts 6 72 h
with the shotcrete age ts given in hours. 4.2. Rock The rock properties are here chosen to represent the hard granite at the Southern Link tunnel in Sweden. The density of the rock qr is 2500 kg/m3 and c = 5300 m/s, with a damping ratio of n = 10%. It is difﬁcult to deﬁne the modulus of the elasticity for a large rock mass since its behaviour is site-speciﬁc with geological discontinuities and fractures that affect the characteristics of propagating stress waves. A representative value can be identiﬁed as Er = 70 GPa obtained through insertion of density and wave propagation velocity in Eq. (2), corresponding to high quality intact rock. For this study a section of 0.5 m thick slightly fractured rock was assumed along the tunnel walls, in the numerical model represented by a low modulus of elasticity, here chosen as 16 GPa which is in accordance with Bieniawski (1978). The characteristic frequency extracted from the acceleration measurements was about 1000 Hz. 4.3. Bond between rock and shotcrete As previously mentioned, the most important feature of shotcrete is its ability to adhere to a rock surface. Hahn and Holmgren (1979) suggest that the bond strength should be set to 0.5–1.0 MPa at 28 days. However, in cases where signiﬁcant fractures or other planes of weakness exist parallel to the shotcrete rock interface, it must always be assumed that the effective bond strength will be low (Barrett and McCreath, 1995). A series of tests with a new approach for testing the bond strength fcb between young shotcrete and rock is presented by Bryne et al. (2013). The tests were performed during laboratory conditions at +20 °C, resulting in a strength growth that follows: 1:49
fcb ðt s Þ ¼ 1:55e43:9ts
ðMPaÞ for 0 6 t s 6 72 h
Also in this case ts should be given in hours. The bond strength curve is shown in Fig. 4 where it can be compared to the corresponding tensile strength. 5. Numerical examples
Fig. 3. Maximum particle velocity vs. distances, from detonation of one contour hole in the Southern Link tunnels (Reidarman and Nyberg, 2000). The rock properties are in this case n = 10%, q = 2500 kg/m3 and c = 5300 m/s. The numbers refer to the measurement points given in Fig. 1.
For this study, a dynamic ﬁnite element model of rock and shotcrete subjected to stress waves has been developed using the ABAQUS/Explicit ﬁnite element program (Simulia, 2013). The simulations were performed using two-dimensional (2D) plane strain elements. The fundamentals of the models, where the wave propagates through the rock from the detonation point at the centre of the charge towards the front of the tunnel and along the tunnel sides, and an example of meshing and use of ﬁnite elements is shown in Fig. 4. The detonation is introduced in the model from a circular area within the rock where an impulsive particle velocity is applied. An incident ppv wave caused by an explosion is applied as a boundary condition at the perimeter of the circular area, with the radius Rppv as the distance that corresponds to the limit for rock damage and outside which elastic properties can be assigned to the rock. In the real case, the rock area immediately around the hole
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explosion occurs in front of the tunnel face, were calculated with the purpose to demonstrate the variation of the stresses along the tunnel wall of the tunnel. The resulting curves have been added in Fig. 5 for comparison with the measurement results. 5.2. The effect from higher amounts of explosive
Fig. 4. The detonation in rock ahead of the tunnel face (horizontal plane). Conﬁguration of ﬁnite element model.
containing the explosives will be severely cracked. However, there is no need to include this effect in the present model since the load here is applied at a long enough distance from this point. In the following examples a damage limit at 0.9 m/s for granite is assumed, which is on the safe side with respect to damage according to Persson (1997). From Eq. (5) is given the depth of the damage zone Rppv = 1.2 m that corresponds to Q = 3 kg. A ﬁne mesh was used around the loading area and tunnel opening with coarser mesh used further away from the tunnel opening. Inﬁnite elements were utilized to represent the non-reﬂecting boundaries and prevent the wave reﬂections. The presence of cracks is very important for the propagation and reﬂection of waves from blasting. The models included the effect of both intact rock and a zone of rock fractured due to blasting, situated around the tunnel perimeter, as shown in Fig. 4. The excavation damage zone is represented by rock with a low elastic modulus, as previously discussed in Section 4.2. The rock and the shotcrete behave in a strictly elastic manner and it is assumed that failure occurs when the stresses at the shotcreterock interface exceeds the bond strength or tensile strength of the shotcrete. The material surrounding the excavation is discretized with ﬁrst-order 4-node plane strain elements of type CPE4R, which is recommended for simulations of impact and blast loading using ABAQUS/Explicit (Simulia, 2013). Far-ﬁeld conditions are modelled using inﬁnite elements of type CINPE4 and the interaction between rock and shotcrete was modelled using tie constraints, i.e. no relative displacement between the materials was assumed. The element size for the shotcrete part is 0.01 0.1 m2, with various element sizes used for the rock part. Depending on the accuracy and details of the solution, some regions of the rock were discretized with a reﬁned mesh, resulting in a model that consists of about 27,000 nodes and 26,000 elements. The examples in the following sections are based on the material properties given in Section 4 and are valid for cases with 100 mm shotcrete. 5.1. Construction blasting For the case studied by Reidarman and Nyberg (2000) the maximum particle velocity registered at each measurement point, 300 mm below the tunnel surface, is shown in Fig. 5. The diagram covers the distance between points B and C shown in Fig. 4, in the directions perpendicular and parallel to the tunnel walls. From Fig. 3 it can be seen that R = 1.2 m corresponds to a ppv that is vmax = 340 mm/s, as given by Eq. (5). For numerical calculations, this wave velocity from detonation of Q = 1.27 kg was applied at the edge of the loading area Rppv = 1.2 m, as seen in Fig. 5, which includes the measurement points up to 20 m distance, also given in Fig. 3. The maximum tensile stresses that occur at the interface between the shotcrete and the rock along the tunnel, when the
The response of the shotcrete in the tunnel from the previous example is here investigated for higher blast loads, using 2 and 3 kg of explosives. The response of 100 mm shotcrete for various explosive charge weights Q is shown in Fig. 6. From Eq. (5) it is given that the investigated amounts of explosives, Q = 1.27, 2, and 3 kg, correspond to ppvs that are 340, 550 and 840 mm/s, respectively. Under the higher blast loads, the shotcrete stress along the tunnel length increased, giving stress concentrations of which an example can be seen in Fig. 7 where the stress distributions in the x- and y-directions at about 1.9 and 2.7 ms after the detonation are shown. The high stresses are indicated with black areas showing tensile stresses over 1 MPa and bright grey showing compressive stresses lower than 1 MPa. Far away from the loading area, sections with zero stress (grey) can be seen. At 1.9 ms after the detonation, the ﬁrst peak of shear stress is concentrated mainly at the tunnel front while at 2.7 ms, peaks of high shotcrete stress can be seen forming close to the corners between front and wall. Such stress concentrations also appeared propagated along the tunnel walls, but with decreasing stress levels. It should be pointed out that the analysis covered 100 ms duration. 5.3. Young shotcrete This last example demonstrates how the development of the shotcrete modulus of elasticity affects the stresses at the shotcrete-rock interface, i.e. the bond stress at point A, see Fig. 4. The model used is identical to those in the previous examples, but here Es is assumed to be a variable according to Eq. (9). Multiple calculations have been done within the age range 1–24 h, for R equal to 3.0 m. The results are presented in Fig. 8 and compared to the bond and tensile strength curves given by Eqs. (10) and (11). The intersections between the stress load curves and the strength curves give the critical, minimum shotcrete age for each case. 6. Discussion The presented numerical example demonstrated that the numerical FE model used can give results that are in good agreement with in situ measurements and observations, in this case the vibrations recorded during construction blasting by Reidarman and Nyberg (2000). Compared to the previously tested models (Ahmed et al., 2012) the current, updated model showed higher stresses. This was due to the inclusion of a 0.5 m section of fractured rock that contributed to the reﬂection, transmission and superposition of stress waves. For the rock mass, representative average values for the modulus of elasticity must be used accounting for the occurrence of cracks and imperfections, also in combination with a certain amount of damping that should be based on in situ measurements. This is necessary so the attenuation of the ppv with increasing distance follows the empirical relations that often can be established by compiling numerous in situ measurements. The application of the dynamic load, i.e. the stress wave, must be done in a correct manner. The presented examples demonstrate the method used for the presented model, with a circular front of ppv applied along the perimeter of a circular area surrounding the centre of the explosives. Empirical relations are used to give the correct level of ppv. The displacement ﬁeld is two-dimensional and therefore the effect from both P and S-waves
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Fig. 5. Maximum ppv from Q = 1.27 kg along the tunnel wall of the Southern Link, (a) parallel with and (b) perpendicular to the tunnel wall. Measurement results from Reidarman and Nyberg (2000) and numerical results for 100 mm shotcrete.
Fig. 6. Maximum tensile stresses in the shotcrete element closest to the rock along the wall of the tunnel; (a) parallel to and (b) perpendicular to the tunnel wall.
Fig. 8. Maximum stresses in hardening shotcrete, compared with shotcrete strength. Fig. 7. Examples of stress distribution in the rock and tunnel, at 1.9 and 2.7 ms for the x- and y-directions, respectively.
6.1. Recommendations originating from the same event can be studied. This is highly important for cases where shear stresses will be dominant.
The results from the FE model, in the range of 4–17 m distance, are in agreement with the measured ppv levels shown in Fig. 5. The
L. Ahmed, A. Ansell / Tunnelling and Underground Space Technology 42 (2014) 105–111
model can thus be used to identify limits for damage through series of calculations with increasing load levels, as shown Fig. 6. The last example demonstrates the variation of maximum stress at the shotcrete-rock interface as the shotcrete hardens. The stress peaks that occur in Fig. 8, within 1–5 h of age, are characteristic and this phenomenon has also been observed for concrete. Previous investigations (Ansell and Silfwerbrand, 2003) show that hardening, cast concrete goes through a phase during a period of 3–11 h after casting where it has low resistance to vibrations and it is therefore recommended that blasting before 12 h of shotcrete age should be avoided. The intersection between stress curves and strength curves in Fig. 8 shows that for the studied example with 100 mm shotcrete the minimum required shotcrete ages are 12, 15 and 23 h, for detonation of Q = 1, 2 and 3 kg of explosives at 3.0 m distance. These limits are in good correspondence with the previous recommendations given by Ahmed (2012). It should be noted that the present comparison is based on realistic material strength data from laboratory investigations of young and hardening shotcrete (Bryne et al., 2013) which give credibility to the results. It should be remembered that a decrease in tunnel air temperature will reduce the rate of shotcrete strength increase while thinner shotcrete layers will reduce the stress load levels (Ansell, 2007a,b). 6.2. Conclusions It was demonstrated that wave propagation through rock towards shotcrete can be modelled using two-dimensional elastic ﬁnite elements in a dynamic analysis. The models must include the properties of the rock and the accuracy of the material parameters used will greatly affect the results. However, this also means that it will be possible to describe the propagation of the waves through the rock mass, from the centre of the explosion to the reﬂection at the shotcrete-rock interface, also after this has occurred. Using a sophisticated ﬁnite element program with elastic models in dynamic analyses gives a relatively efﬁcient and fast analysis process. It is acceptable to use elastic material formulations until the material strengths are exceeded, i.e. until the stresses are outside the elastic range which thus indicates material failure. It is in this case of little interest to continue the analysis for a partially damaged supporting structure. For the future research, this type of ﬁnite element models will be used to study more detailed cases, with respect to varying shotcrete material properties and complex geometry. Cases that need focus are construction of large diameter tunnels, parallel tunnels, crossing tunnels and stress waves with oblique impacts. The use of multiple explosive charges will also be considered, aiming at ﬁnding the least damaging sequence of detonation by setting different delay times. The effect of additional reinforcing elements, such as rock bolts, cables and wire mesh, can also be included. Of special interest is to further study how vulnerable young and hardening shotcrete is when exposed to impacting vibrations. The ultimate goal is here to establish guidelines for safe blasting, with respect to both shotcrete age and distance to the vibration source. Acknowledgements The presented investigations constitute part of a project that studies the material properties of young and hardening shotcrete. The project is supported by SBUF, the Development Fund of the
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