vibrations induced by blasting in rock a numerical approach

RIVISTA ITALIANA DI GEOTECNICA 2/2008

Vibrations induced by blasting in rock: a numerical approach

Cristina Jommi*, Anna Pandolfi*

SummaryBlasting is a powerful excavation method in terms of both production efficiency and economical costs, but its high en-

vironmental impact due to noise, vibrations, and potential damage to surrounding structures may limit its extensive appli-cation. The use of explosives as excavation tool is usually ruled by the national codes of practice, in which tolerable limitsfor induced motion are given for the different structure classes. In order to predict the vibrations induced in the groundat a given distance from the blast centre, attenuation laws, derived from either in situ measurements or analytical solutionsof simple elastic wave propagation problems, are adopted. As the attenuation laws usually refer to homogeneous continua,they may not be adequate as a predictive tool for complex geological sites. In such cases, numerical analysis provides a val-uable alternative, as the whole propagation history of stress waves could be simulated in principle, irrespective of the geo-logical complexity of the specific site. To describe the progressive effects of underground blasting on the surrounding site,a finite element approach is presented. The explosion energy is translated in a time history of pressure at the boundary ofthe blast hole. Cracking of the nearby rock mass is modelled according to a cohesive crack model, while elastic behaviouris assumed for the non cracked rock mass and soil deposits. Propagation of stress waves from the blast hole is simulated bya time domain 3-D finite element analysis, which is able to provide the time history of all the relevant quantities describingthe motion at any given distance. The numerical results can be post processed in order to derive attenuation laws for themost relevant quantities to which the codes of practice usually refer to, i.e., peak particle velocity and principal frequencyof the vibration. The model is energy-conserving, thus the energy supplied by the explosive is correctly partitioned intofracture energy of the rock mass close to the blast hole, elastic energy providing the stress wave propagation and kineticenergy of the fragmented rock blocks. Numerical simulations of two literature case-histories are presented, and the numer-ical results are compared to the available experimental data. Experimental peak particle velocity could be captured remark-ably. Principal frequencies for the rock mass could be reproduced as well. In layered sites, the ratio between the stiffness ofthe different media where stress waves propagate seems to play a key role in the determination of principal frequencies,while less influence is observed on peak particle velocities.

Keywords: blasting, dynamic wave propagation, induced vibrations, finite element analysis, cohesive fracture.

Introduction

The use of explosives is still a powerful excava-tion method, thanks to flexibility, robustness andhigh production rates achievable (see, e.g., HUDSON,1993). Blasting technology has been refined to agreat deal, although the practical methods havebeen optimised empirically. Many factors hindereda deep knowledge of all the dynamic effects pro-duced by explosive blasts, such as the debate onwhether the resulting rock breakage is due to thelarge amplitude stress waves travelling through themedium, or to the very high gas pressures. Severalresearch works published over the past 40 years sup-ported either one or the other hypothesis [FOURNEY,1993]. For very high charges detonated nearby anexposed plane free surface, BHANDARI [1979] sug-

gested that the most likely fragmentation mecha-nism could be the spalling induced by the reflectionof the compression wave generated by the explosionin the form of a tensile wave. On the contrary, gaspenetration into already opened fractures aroundblast holes seems to be more a symptom than acause of fracture [MC KENZIE, 1993].

Whatever fracture mechanism prevailed, part ofthe explosive energy is transmitted outward in thefar field through elastic stress waves. Vibrations areinduced inside the rock mass and on the freeboundaries, where surface waves and reflection mayamplify the incoming body stress wave. Commonly,peak particle velocity, i.e. the time derivative of thevibration displacement, is adopted as a convenientmeasure of disturbance level, although other alter-native parameters may describe the effects of blastinduced vibration on structures. DOWDING [1993]provides a comprehensive treatise of all the techni-cal aspects concerning the interaction between blastvibrations and structures.

* Department of Structural Engineering, Politecnico di Milano, Milano, Italy

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RIVISTA ITALIANA DI GEOTECNICA

The use of explosives is usually ruled by the na-tional codes of practice, in which tolerable limits forinduced motion are given for the different structureclasses (e.g. ANDERSON, 1993). As a consequence ofthe increasing environmental constraints on tolera-ble levels of disturbance induced by blasting opera-tions upon nearby residents, blast is designed so asto achieve levels of ground vibration and overpres-sure disturbance as close as possible to the permissi-ble levels. In order to predict the vibrations inducedin the ground at a given distance from the blast cen-tre, attenuation laws, coming from either in situmeasurements or analytical solutions of simple elas-tic wave propagation problems, are adopted. As theattenuation laws usually refer to homogeneous con-tinua, they may not be adequate as a predictive toolfor complex geological sites.

Numerical analysis provides a valuable alterna-tive in these cases, as the whole propagation historyof stress waves could be simulated in principle, irre-spective of the geological complexity of the specificsite. GRADY and KIPP [1980] and ANG and VALIAPPAN

[1988] were among the first researchers tacklingwave propagation arising from detonation of explo-sives via finite element analysis. Recently, a few 3-Dnumerical analyses were proposed, including con-tinuum-based [LIU et al., 1997; HAO et al., 1998; JIA

et al., 1998; HAO et al., 2002; TORAÑO and RODRÍGUEZ,2003; WU and HAO, 2004; TORAÑO et al., 2006] anddiscontinuum-based approaches [HART, 1993; CHEN

and ZHAO, 1998].To describe the effects of underground blasting

on the surrounding site, a finite element approachis presented here. The explosion energy is trans-lated in a time history of pressure at the boundaryof the blast hole, based on Friedlander’s equation.Cracking of the inner portion of the rock mass ismodelled according to the cohesive crack modelproposed by Ortiz and Pandolfi [PANDOLFI and OR-TIZ, 1998; 2002; ORTIZ and PANDOLFI, 1999; 2004],

while elastic behaviour is assumed for the noncracked rock mass and soil deposits. Propagation ofstress waves from the blast hole is simulated by a 3-D finite element analysis, which is able to providethe time history of all the relevant quantities de-scribing the motion at any given distance. Numeri-cal simulation of two published case histories is pre-sented, allowing for capabilities and limitations ofthe proposed numerical model to be discussed.

2. Blasting and blast wave

The energy released by a conventional chemicalexplosive in air produces, with respect to atmos-pheric pressure, an overpressure wave, P, calledblast wave. The blast wave profile in air evolves inthe direction of travel (Fig. 1a), due to the differentvelocities of the overpressure components [KINNEY,1962]. Irrespective of the actual original overpres-sure shape, at a reasonable distance from the centreof the explosion, blast waves can be described by asharp overpressure front followed by a decaying tail,eventually dropping to zero or to negative values(Fig. 1b). At a fixed distance, r, from the explosioncentre, Friedlander’s equation may be adopted todescribe the homothetic decaying pressure p(t) intime:

(1)

where p0 is the atmospheric pressure, P is the peakoverpressure due to blasting, tr is the arrival time atdistance r, and Td is the duration of the blast in-duced positive overpressure. The parameter b gov-erns the decaying velocity of the overpressure.

Peak pressure, arrival time and duration of blastoverpressure at distance r, following the actual ex-plosion, may be calculated by scaling laws on the ba-

Fig. 1 – Evolution of blast pressure (a) nearby the explosion centre and (b) with distance.Fig. 1 – Evoluzione dell’onda di pressione (a) in prossimità del centro dell’esplosione e (b) a prefissate sufficienti distanze dall’esplosione.

79VIBRATIONS INDUCED BY BLASTING IN ROCK: A NUMERICAL APPROACH

APRILE - GIUGNO 2008

sis of experimental data relative to a standard ex-plosion of 1 ton TNT. Assuming that the explosionis characterised by the same temperature, referencepressure p0 and sound velocity as the standard con-ditions, the relevant parameters may be calculatedthrough a scaling parameter, λ, called yield, given bythe cube root of the ratio between the effective ex-plosion energy, W, and the standard reference ex-plosion energy of 1 t of TNT, Wref :

(2)

Distances may be simply scaled by the relation

(3)

and the relevant overpressure parameters by the re-lations

(4)

The non dimensional parameter b is an empiri-cal function of the scaled distance,

(5)

which can be derived from the tabular values report-ed by KINNEY [1962]:

The scaling laws indicate that a real explosion,characterised by a yield λ, generates the same over-pressure as the one generated by the standard ex-plosion at a distance r = λ rref, with arrival timetr = λ trref and duration Td = λ Td

ref.This approach may be adopted to describe the

effects of blasting in rocks, provided an air gap be-tween the explosive and the surrounding rock massallows for treating the pressure wave in the blasthole similarly to a blast wave in air.

3. Numerical modelling

To provide a description of the effects inducedby blasting in the surrounding environment, a finiteelement numerical model is proposed, based onprevious work by Ortiz and Pandolfi [PANDOLFI andORTIZ 1998, 2002; ORTIZ and PANDOLFI 1999, 2004],which will be briefly summarised in the following. Adetailed description of the relevant algorithms canbe found in the original papers.

3.1. Constitutive laws

A non linear kinematic approach is adopted inthe formulation of the model, due to the necessity ofdescribing processes involving fracture and possiblemotion of the fractured rock blocks.

The intact rock mass, in which blast holes aredrilled, may be modelled as an elastic brittle homo-geneous medium. This choice allows for capturingthe most relevant features governing the overall de-formational response to a blast load, i.e., intensefracturing of the inner zone near the blast hole andpropagation of elastic waves in the outer region. Forthe sake of simplicity, in the analyses presented herematerial damping in the intact zone is neglected.

An isotropic, non linear, hyper-elastic model isadopted [OGDEN, 1984; ORTIZ and PANDOLFI, 2004]for the rock mass. The model is characterised by theinitial tangent values of the two independent elasticconstants. Non linearity of the elastic law is a directconsequence of the finite kinematic approach.

Failure of the rock mass is described by crackformation and propagation and modelled by meansof the cohesive element proposed by ORTIZ and PAN-DOLFI [1999]. The cohesive model is based on thedefinition of an effective fracture opening displacement,δ, work-conjugate to an effective fracture, t. The twoquantities are defined based on the respective vec-tors, and t (Fig. 2), assigning different weights totheir normal and tangential components:

(6)

Fig. 2 – Relative displacements and traction over the co-hesive surface (CS).Fig. 2 – Spostamenti relativi e trazioni sulla superficie coesiva (CS).

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(7)

where β is defined by means of the tensile strength,σt, and the uniaxial compressive strength, σc, of therock:

(8)

A simple model for cohesive fracture may be ob-tained by assuming a free energy density dependentonly on the effective opening displacement. Theuniaxial effective irreversible cohesive law imple-mented in the numerical code is visualised in Figure3. If the maximum attained effective opening dis-placement, q, is chosen as internal variable, a loadpath implying fracture propagation is described bythe conditions δ = q and δ· > 0. The critical energyrelease rate, Gc, is represented by the area boundedby the linear law depicted in Figure 2(b), i.e.:

(9)

The ratio between the energy dissipated in thecurrent fracture opening, Φ(q), and the critical en-ergy release rate defines an a posteriori damagemeasure, D, in the range between 0 and 1:

(10)

As depicted in Figure 3(a), irreversibility charac-terises an unloading process. Upon closure, the twosurfaces of the fracture are subjected to a unilateralcontact condition with friction. Contact and frictionare defined independently from the cohesive lawand implemented by a penalty approach [YU et al.,2002].

Besides the two parameters necessary to cali-brate the isotropic elastic law (shear modulus, G,and Poisson ratio, ν, are adopted here), three inde-pendent constants are needed to fully describe thecohesive law. A common choice is represented bythe tensile strength of the rock, σt, its uniaxial com-pressive strength, σt, and the critical energy releaserate Gc.

3.2. Boundary value problem formulation and discretisa-tion

The numerical code adopted for the analyses al-lows for a finite kinematics three-dimensional anal-ysis of dynamic boundary value problems in thetime domain. Explicit integration in time is per-formed by means of Newmark’s algorithm, byadopting βN = 0 e αN = 1 [HUGHES, 2000]. Thischoice implies algorithmic damping, which partiallycompensates the lack of material damping in theconstitutive law.

The finite element model is discretised into aninitially coherent mesh of 10-node solid tetrahedralelements. Fractures are allowed to propagate onlybetween the faces of the original elements, when theeffective traction reaches the critical value definedby the cohesive model. Fracture is then simulatedvia the local introduction of 12-node cohesive ele-

Fig. 3 – Graphical representation of the irreversible cohesive law.Fig. 3 – Rappresentaizone grafica della legge coesiva irreversibile.



ments (Fig. 4a) in between the faces of the originalelement mesh (Fig. 4b), through an auto-adaptiveremeshing procedure [PANDOLFI and ORTIZ, 1998;2002]. The cohesive law governs the subsequent ef-fective opening process along the new opened sur-face.

For the process zone of the cohesive fracture tobe correctly modelled without introducing spuriousscale effects, the characteristic size of the finite ele-ments in the numerical mesh should be smaller thanthe characteristic length, lc, introduced by the cohe-sive approach:

(11)

where E is the Young modulus of the intact rockmass. Besides, the integration time step should notbe larger than the critical time step for the stabilityof the numerical algorithm, ts:

(12)

where h is the minimum mesh size, ρ is the rockmass density and c is the rock longitudinal wavespeed [COURANT et al., 1967].

3.3. Boundary conditions

Different boundary conditions are imposed onthe different portions of the boundaries of the dis-crete domain.

Along the blast hole boundary, exposed to theexplosion, a time history of overpressure is im-

posed, based on the considerations summarised inSection 2. Given the explosion yield, a pressure timehistory is calculated through the scaling law as afunction of the distance between the centre of theexplosion and the surface element considered. Theresulting overpressure time history at any givennode of the blast hole surface follows the shape de-picted in Figure 5.

On the free boundaries of the domain, a zerotraction condition is imposed throughout the analy-sis. Fictitious outer boundaries, closing the numeri-cal domain, should in principle work as perfectly ad-sorbing boundaries. Among the different sugges-tions in the literature (e.g. ENGQUIST and MAJDA,1977; CLAYTON and ENGQUIST, 1977; KAUSEL, 1988;JIAO et al., 2007), in order to limit artificial reflectionof the stress waves, absorbing boundaries based onthe proposal by CLAYTON and ENGQUIST [1977] were

Fig. 4 – Geometrical definition of the 12-node cohesive element (a), introduced between the faces of the original coherentmesh (b).Fig. 4 – Definizione geometrica dell’elemento coesivo a 12 nodi (a), introdotto fra le superfici degli elementi del reticolo originario intatto (b).

Fig. 5 – Overpressure time history at a given distancefrom the explosion centre.Fig. 5 – Storia temporale di sovrapressione a distanza fissata dal centro dell’esplosione.

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introduced. At the nodes of the fictitious boundary,dashpots, calibrated on the longitudinal wave andthe shear wave velocities, are introduced. Thischoice allows for reducing, although not eliminat-ing, the artificial reflection of the transmitted wavein the problems analysed. Nevertheless, this limita-tion of the model does not affect the numerical re-sults presented in the following, where only the ar-rival waves are taken into consideration and dis-cussed.

4. Numerical simulations

The numerical model summarised in the previ-ous paragraph was adopted previously by ORTIZ andPANDOLFI [2004] to estimate damage induced onconcrete buried structures by a blast wave due to anexplosion over the soil surface. The present work isaimed at verifying whether the model was able tosimulate rock fragmentation due to blasting and in-duced vibrations in the outward rock mass and soildeposits. To this aim, two case-histories are consid-ered and the results of the numerical simulationsare compared with some published experimentaldata. The first case shows that the amount of energydissipated for rock fracturing is well caught by thenumerical model. The second example focuses onthe characteristics of the elastic waves propagationin the outer zone, and on the influence of the geom-etry and the material properties of the rock-soil de-posit on the overall vibration process.

4.1. Hydropower plant service tunnel excavation [BERTA, 1985]

The first case simulates one advancing step ofthe excavation of a service tunnel for a hydropowerplant in a granite rock mass [BERTA, 1985]. A sche-matic view of the tunnel geometry and of the exca-vation stages is given in Figure 6.

The tunnel section is excavated in three mainsteps. In the first one (bounded by the inner circlein Fig. 6) the central section is opened by first man-ually drilling the central relief hole, and then bycontrolled blasting in the inner blast holes. The sec-ond stage is the proper production stage, where arelevant part of the section (in between the innerand the outer circle) is excavated by controlledblasting, and most of the total blast energy is con-sumed. Finally, in the third stage, sequential blast-ing of multiple small explosions are used to refinethe external profile of the tunnel.

The geometry and the finite element meshadopted in the numerical analysis are described inFigure 7 (a-d). It is worth noting that only the sec-ond stage of excavation was simulated numerically.

To this aim, a single blasting episode was consid-ered, in which the total amount of explosive effec-tively used in the practice was concentrated at thecentre of the blast hole, irrespective of the real blast-ing sequence. The simplifying assumption is accept-able, as the aim of the first simulation was to verifythe global energy balance of the dynamic process.The numerical mesh is built with 13024 tetrahedralelements and 19585 nodes. Mesh refinement wasprovided close to the blast hole in order to satisfythe requirement on the characteristic length (Eq.11), although the typical size of the elements was in-creased outward from the front face.

The explosion yield was calculated on the basisof the design parameters of the production stage.The volume of rock to be excavated in the analysedstage was equal to 12.6 m3. Chosen the explosive,the equivalent mass of TNT to obtain the desiredvolume of excavation during the production stagewas estimated to be MTNT

eq =19 kg, whose equiva-lent explosive energy is W = 79 MJ, hence the yieldof the explosion is λ = 0.267. The rock mass param-eters were chosen on the basis of literature data andare summarised in Table I.

Relevant results of the numerical analysis areshown in Figures 8-9. The final damage level is rep-resented in Figure 8(a) on the cross section of theadvancing tunnel. The numerical simulation cor-rectly predicts the extension of cracking in the innertunnel section, which was the aim of the productionstage blasting. The outer part of the excavated sec-tion is yet damaged, but to a smaller level. It is worthnoting that, in practical applications, a third profil-ing stage is expected.

Figure 8(b) shows that the advancement depth iswell caught, although a slight mesh dependency canbe appreciated from the fractured zone beingbounded to a significant extent by the surfaces of

Fig. 6 – Schematic view of the hydropower plant servicetunnel and excavation stages.Fig. 6 – Disegno schematico della sezione della galleria di servizio della centrale idroelettrica e fasi di scavo.



the smaller elements. This mesh dependency couldbe eliminated by reducing the size of the elementsto values smaller than the characteristic length in abigger volume on the front of the tunnel. Neverthe-less, in this analysis the focus was on the global en-ergy balance rather than on following the actualfragmentation profile.

Figure 9 shows how the total blast energy is par-titioned during fragmentation and stress wave

propagation. The dots represent the total externalwork provided by the pressure wave on the bound-ary of the blast hole. Total energy is almost con-served by the numerical model, although the effectsof computational damping may be appreciatedstarting from about 15 ms after the explosion. Theanalysis predicts that most of the energy provided tothe system is transformed into kinetic energy duringthe whole time interval analysed. Fracture process is

Fig. 7 – Numerical model of the excavated tunnel: (a) solid view; (b) vertical section; (c) view of half FE mesh; (d) meshrefinement nearby the equivalent blast hole.Fig. 7 – Modello numerico dello scavo della galleria: (a) vista tridimensionale del modello solido; (b) sezione verticale; (c) reticolo di elementi finiti in metà del modello; (d) raffinamento del reticolo in corrispondenza del foro equivalente di esplosione.

Tab. I – Parameters adopted in the numerical analysis of tunnel excavation [BERTA, 1985].Tab. I – Parametri adottati nell’analisi numerica dello scavo della galleria [BERTA, 1985].

Rock mass density Young’s modulus Poisson’s ratio Tensile strengthUniaxial

compressive strengthCritical energy

release rate

(kg m-3) (GPa) (---) (MPa) (MPa) (N mm-1)

ρ E ν σt σc Gc

2720 51 0.26 8.3 157.7 0.33

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almost completed after the first 10 ms, and only asmall part of the total energy is transmitted to theouter rock mass as elastic propagating waves.Figure 9(b) shows the numerical prediction of thefracture area created by fragmentation of the innerrock mass. The ratio between the total volume of thefragmented rock and the fractured area may beused as a measure of the dimension of the blocksisolated by the excavation procedure. This informa-tion proves to be a relevant parameter in the design,for the removal, transport and storage of the exca-vated rock blocks.

Figure 10 shows how stress waves are transmit-ted to the surface of the rock mass, in correspond-

ence of the three sampling points (forward, centraland backward) indicated in Figure 7(b). The figurereports the vertical velocity component as a functionof time in the first 10 ms, before reflected wavesfrom the artificial lateral boundaries may interferewith the incoming stress waves. The different arrivaltimes of stress waves in the different sampling nodescan be clearly detected as a function of the distancefrom the blast hole. Stress waves propagating fromthe blast hole through the fractured zone reach firstthe central control point, located on the vertical tothe blast hole centre, where the ground vertical ve-locity is a result of the longitudinal stress wavesmainly. A longer time lapse is needed for the stresswaves to reach the rock mass surface forward and

Fig. 8 – Numerical simulation of tunnel excavation: (a) transverse section showing the fractured area; (b) longitudinal sec-tion showing excavation advancement.Fig. 8 – Simulazione numerica dello scavo della galleria: (a) sezione trasversale frammentata; (b) vista longitudinale dell’avanzamento dello scavo.

Fig. 9 – Numerical simulation of tunnel excavation: (a) energy balance; (b) evolution of fracture area.Fig. 9 – Modello numerico dello scavo della galleria: (a) bilancio energetico; (b) evoluzione dell’area di frattura.



backward sampling nodes. In these points, the ini-tial amplitude of the local velocity is smaller, due togeometric damping. The influence of surface stresswaves on these sampling points may be appreciatedby observing increasing amplitudes and loweringfrequencies with time. It is worth noting, anyway,that computational damping may contribute to fil-ter the highest frequencies with distance from theblast pressurised boundary.

Effectiveness of the dashpots introduced todamp the outgoing waves may be evaluated by thecomparison of the results of the previous analysis(Fig. 11a) with the results of an identical analysiswithout absorbing boundaries (Fig. 11b). The threevelocity components at the central control node aredepicted for the analysis time interval of 40 ms.Comparison between the time evolution of the threevelocity components demonstrates that the dash-

pots allow for reducing the effect of spurious re-flected waves although they are not able to eliminatethem completely. Differences between damped andnon-damped analysis may be fully appreciated afterabout 15 ms, while initially (until about 10 to 15 ms),before reflected waves begin to become significant,the numerical velocities are almost comparable.

The results presented in this section aim atdemonstrating the potential capabilities of the nu-merical model in the description of the differentprocesses involved in a rock mass when blasting isadopted as excavation technology. Energy is almostconserved, fracture area may be calculated, and inprinciple, the acceleration, velocity and displace-ment histories can all be described at any distancefrom the blast centre, at least before reflected wavesintroduce spurious effects in the result of the analy-sis. Usually, due to the characteristics of the stresswaves induced by blasting, which is an impulsiveload, the most relevant effects on ground accelera-tion and velocity are observed upon stress wave ar-rival, before spurious reflected waves may affect thenumerical results.

Reliability and limitations of the proposedmodel may be much better appreciated with refer-ence to the following literature example, where partof the numerical results could be compared to ex-perimental measurements.

4.2. Small-scale field blast test in a layered rock soil site [WU et al., 2003]

To investigate stress wave propagation fromdetonation of explosives in a quarry, a series of fieldblast test was designed and performed in a graniterock mass [WU, et al., 1998; HAO et al., 2001; WU etal., 2003]. Before the blast test, a 1.5 m thick com-pacted soil layer was backfilled onto the flat surface

Fig. 10 – Stress waves arrival at the top surface controlpoints.Fig. 10 – Arrivo delle onde elastiche nei punti di controllo della superficie superiore.

Fig. 11 – Velocity components at central control point from numerical analysis (a) with boundary dampers and (b) withoutboundary dampers.Fig. 11 – Componenti della velocità nel punto di controllo centrale, derivate dall’analisi numerica (a) con smorzatori ai bordi e (b) senza smorzatori ai bordi.

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of the test quarry site (Fig. 12a). The site was instru-mented with accelerometers, with a sampling rate of500 kHz, arranged inside the rock mass, on the rocksurface (section A-A in Fig. 12c), and on the soil sur-face (Fig. 12d). The accelerations following coupledand decoupled blast tests with different chargeweights were recorded.

A careful geological and geotechnical investiga-tion had been carried out before the blast tests, giv-ing a rather detailed characterisation of the rockmass and of the compacted soil layer. All model pa-rameters could be determined from the investiga-tion data. The sole critical energy release rate of un-weathered granite was calibrated, based on litera-ture data and on a preliminary numerical simula-tion of the fracture process in a granite disk [KUTTER

and FAIRHUST, 1971]. Although an elastic-plasticmodel could be equally adopted for the compactedupper layer [ORTIZ and PANDOLFI, 2004], the soil wasassumed to behave elastically, due to the small vi-

bration level expected. The rock and the compactedfill parameters adopted in the numerical analysisare summarised in Tables II-III.

The charge hole, of 11 m total depth, was lo-cated at the centre of the test field. The explosive(PETN) was buried at a depth of 8.5 m from therock mass surface, in a chamber 0.8 m wide. Differ-ent blast tests were performed with different chargeweights, ranging from 2.5 to 50 TNT equivalent kg.The 12.5 TNT equivalent kg test was simulated nu-merically, as most of the published data refer to thiscase.

Different views of the initial solid mesh adoptedin the analysis are provided in the Figures 12(b), 13-14. The finite element model has 7702 elementsand 11700 nodes. As in the previous example, theboundary of the charge hole was loaded with ascaled pressure time history, calculated from the ac-tual explosion yield λ = 0.232. Absorbing dashpots

Fig. 12 – Numerical model of the small blast test: (a) geometrical top view, (b) top view of the numerical mesh, (c) verticalsection A-A, (d) vertical section B-B. Numbers represent position of the accelerometers.Fig. 12 – Modello numerico dell’esplosione prova: (a) vista dall’alto della geometria, (b) vista dall’alto del reticolo numerico, (c) sezione verticale A-A, (d) sezione verticale B-B. I numeri rappresentano la posizione degli accelerometri.



were located at the fictitious outer boundary of thedomain, and no tractions were imposed elsewhere.

Figures 15-17 show time histories of horizontaland vertical velocity components at three controlpoints, at comparable distances from the chargehole, inside the rock mass, at the rock mass surfaceand at the upper soil surface, respectively. Compar-ison between the three figures confirms that theblast-induced stress waves differ substantially at thethree locations. In the rock mass and at the rock sur-face the vertical and horizontal components are ofthe same magnitude. The largest amplitudes are ob-

served at the rock surface (Fig. 16), due to wave re-flection and surface wave generation. On the con-trary, due to refraction at the rock-soil interface, onthe upper soil surface the vertical component is def-initely higher than the horizontal one. Although thecompacted fill soil is only 1.5 m thick, the attenua-tion properties of the backfill are rather evidentwhen the peak velocities on the soil surface are com-pared to those of the rock surface.

When environmental effects are of concern, aconvenient way to describe the effect of vibrationsinduced by blasting is to visualise the attenuation ofstress waves with the distance from the blast centre,choosing suitable parameters for the description ofthe stress wave. The most relevant parameter to in-fer potential damage on constructions is the peakparticle velocity, i.e., the maximum value of the dis-placement rate at a given location. Peak particle ve-locity is therefore adopted as a convenient measureof the strain level induced on the structure by thestriking wave.

Peak particle velocity can be defined in differentways, as the different velocity components are not inphase, in general. Here, it is defined as the modulusof the vector built with the independent peak valuesof the horizontal and vertical components.Although this definition has no physical counter-

Tab. II – Parameters adopted in the numerical analysis of small blast test [WU et al. 2003]: rock mass.Tab. II – Parametri adottati nell’analisi numerica dell’esplosione prova [WU et al. 2003]: ammasso roccioso.

Rock mass density Young’s modulus Poisson’s ratio Tensile strengthUniaxial

compressive strengthCritical energy

release rate

(kg m-3) (GPa) (---) (MPa) (MPa) (N mm-1)

ρ E ν σt σc Gc

2610 74 (or 25) 0.16 16.1 186 0.64

Tab. III – Parameters adopted in the numerical analysis ofsmall blast test [WU et al. 2003]: compacted soil fill.Tab. III – Parametri adottati nell’analisi numerica dell’esplosione prova [WU et al. 2003]: rilevato in terra.

Soil mass density Young’s modulus Poisson’s ratio

(kg m-3) (GPa) (---)

ρ E ν

2000 1.5 0.3

Fig. 13 – View of the whole numerical mesh adopted inthe analysis.Fig. 13 – Vista completa del reticolo adottato nelle analisi.

Fig. 14 – Inside view of the blast hole and some of the ac-celerometers positions.Fig. 14 – Vista dall’interno del foro di esplosione e della posizione di alcuni accelerometri.

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part and may overestimate the real peak value ofground motion (i.e. the modulus of the instanta-neous velocity), the choice allows for comparisonwith the published experimental data.

Figures 18-19 compare experimental peak par-ticle velocities with the results of the numerical sim-

ulation. On the horizontal axis, the scaled distance,defined as the ratio between the actual distance ofthe point under consideration from the blast centre,r in m, and the cube root of the equivalent TNTweight, W in kg, is reported. This definition doesnot seem totally appropriate, as the scaled distance

Fig. 15 – Velocity time history inside the rock mass, at a distance of 10.0 m from the blast centre: (a) horizontal componentand (b) vertical component.Fig. 15 – Storia temporale di velocità all’interno dell’ammasso roccioso, a distanza di 10.0 m dal centro dell’esplosione: (a) componente orizzontale e (b) componente verticale.

Fig. 16 – Velocity time history at the rock mass surface, 13.1 m from the blast centre: (a) horizontal component and (b)vertical component.Fig. 16 – Storia temporale di velocità sulla superficie dell’ammasso roccioso, a distanza di 13.1 m dal centro dell’esplosione: (a) componente orizzontale e (b) componente verticale.

Fig. 17 – Velocity time history at the soil surface, 14.1 m far from the blast centre: (a) horizontal component and (b) verticalcomponent.Fig. 17 – Storia temporale di velocità sulla superficie del terreno, a distanza di 14.1 m dal centro dell’esplosione: (a) componente orizzontale e (b) componente verticale.



Fig. 18 – Peak particle velocity at the rock surface: (a) comparison between numerical results and experimental data and(b) comparison between respective linear fits.Fig. 18 – Velocità di picco sulla superficie dell’ammasso: (a) confronto fra i risultati numerici e i dati sperimentali e (b) confronto fra le rispettive interpolazioni lineari.

Fig. 19 – Comparison between numerical and experimental peak particle velocity at the upper soil surface.Fig. 19 – Confronto fra le velocità di picco numerica e sperimentale sulla superficie del rilevato in terra.

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turns out to be a dimensional parameter, but it iswidely adopted as a measure of the actual distancescaled by the explosion energy (see, e.g., DOWDING,1996), and was adopted here to compare the nu-merical results with the experimental data [WU et al.,1998; WU et al., 2003].

Figure 18(a) demonstrates that the calculatedpeak particle velocities at the rock surface fall en-tirely inside the range of experimental data, char-acterised by quite a large scatter. Both experimen-tal data and numerical results show that attenua-tion is non-linear in the log-log plane. The peakparticle velocity does not seem to decrease substan-tially, for scaled distances lower than 5, while an al-most linear decrease is observed at larger dis-tances. In Figure 18(b), replicating a commonpractice in the design, linear fits of both experi-mental and numerical data are drawn over thewhole range of analysed scaled distances. Compar-ison between the two linear fits shows that a linear-ization over the entire scaled distance range issomehow misleading, as the attenuation laws seemto differ much more than the direct comparisonpresented in Figure 18(b) shows.

In figure 19 the numerical results for the peakparticle velocity at the upper surface of the com-pacted soil fill are compared to experimental data.Again, the numerical results fall in between therange of measured data for scaled distance lowerthan 15, while for larger distances the numericalanalysis seems to underestimate the recorded peakparticle velocity.

Figure 20(a) presents the numerical peak parti-cle velocity of the rock mass. Although accelerome-ters were actually placed inside the rock mass, in thereference papers the experimental peak particle ve-locities of the rock mass were not reported. The nu-merical results confirm that a non linear attenuationlaw is expected inside the rock mass too, as damageinduced by blasting influences the overall stiffnessof the rock mass nearby the blast hole.

To analyse the sensitivity of the results of the nu-merical model on the assumed rock mass stiffness, asecond analysis was run, assigning a Young’s modu-lus for the rock mass three times lower than that ofthe unweathered granite. The comparison betweenthe results of the two numerical analyses shows thatthe rock Young’s modulus has little influence on thepeak particle velocity of the rock mass itself, exceptin the proximity of the explosion centre (Fig. 20b).A slightly higher influence may be appreciated atthe soil surface, although the two solutions tend toconverge as the distance from the blast hole in-creases.

The experimental data collected inside the rockmass were elaborated by WU et al. [1998], to give theattenuation law of the principal frequency of the ac-celeration. Experimental data, presented in

Figure 21, refer to three tests in which equivalentTNT charges of 10, 20 and 40 kg were exploded insequence. The results for the different explosivecharges were presented separately as a function ofthe actual distance, r, of the accelerometer from thecharge hole. The authors observed that the princi-pal frequencies of acceleration decrease dramati-cally in the near-field of detonation, and slowly forradial distances exceeding 10 m. This implies thatthe high frequency components are damped out inthe rock mass within a rather small zone surround-ing the explosion centre.

The blast tests were designed in order not to in-duce significant damage in the rock surroundingthe charge hole, so that multiple tests could be runat the same site [WU et al., 2003]. Nevertheless, ob-serving that smaller charge weights corresponded tohigher principal frequencies, the authors them-selves attributed the experimental evidence to thepossibility of damage around the blast chamber,which attenuated the high-frequency componentsof the shock wave nearby the charge chamber.

The results of the two numerical analyses, withtwo different Young’s moduli, seem to corroboratethis hypothesis. It is worth noting that the principalfrequencies are well caught by the analysis, althoughthe numerical attenuation is more gradual than theexperimental one. The analysis with a reducedYoung’s modulus, roughly simulating a weatheredrock mass, matches very well the experimental datafor the second and third explosions nearby the blasthole.

The extension of the damaged zone around thecharge hole, derived from the numerical analysis, isrepresented in Figure 22. The result confirms thatsome damage is likely to have occurred around theblast hole, even for an equivalent TNT weight of12.5 kg. The damaged zone has a limited extension,but it may justify the smaller dominant frequenciesobserved in the following experimental tests.

Finally, the principal frequencies of the acceler-ation at the backfilled soil surface calculated fromthe numerical analyses and from the experimentaldata are compared in Figure 23. In this case, WU etal. [2003] chose to represent the experimental databy means of the mean value of the frequency in therange bounded by the maximum spectral peak andhalf of the maximum spectral peak. Hence, a directcomparison between the experimental results andthe numerical values may be misleading to a certainextent. Nonetheless, it appears clearly that the prin-cipal frequencies calculated from the numerical re-sults are definitely higher than those recorded ex-perimentally. As the principal frequencies dependon the stiffness of the medium (see e.g. Fig. 21), thediscrepancy may possibly come from overestimatingthe compacted soil fill Young’s modulus. Further-more, decreasing the stiffness would increase



Fig. 20 – Sensitivity of peak particle velocity to the rock stiffness (a) inside the rock mass and (b) at the soil surface.Fig. 20 – Sensibilità della velocità di picco alla rigidezza della roccia (a) all’interno dell’ammasso e (b) sulla superficie del rilevato in terra.

Fig. 21 – Attenuation law for principal frequency of acceleration in the rock mass: comparison between experimental dataand numerical results.Fig. 21 – Legge di attenuazione per la frequenza principale dell’accelerazione nell’ammasso roccioso: confronto fra i dati sperimentali e i risultati numerici.

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slightly the peak particle velocity, apparently under-estimated in the present analysis, at least far fromthe blast hole (see Fig. 19).

5. Concluding remarks

The proposed finite element approach provedto be satisfactory to model the progressive effects ofunderground blasting on the surrounding site. Theexplosion energy was translated in a time history ofnormal pressure on the boundary of the blast hole.The pressure history, derived from theoretical con-

sideration on the blast wave propagation in air, canbe adopted in analysing the effects of blast in rockmasses, when an air gap is present between the ex-plosive and the blast hole surfaces.

Fragmentation of the rock mass around the blasthole can be accounted for with a suitable cohesivemodel, defined by three constitutive rock parame-ters. The numerical simulations performed con-firmed that the percentage of explosive energytransmitted to the outer portion of the rock mass isrelatively low. This justifies the adoption of an elasticconstitutive law, characterised by the initial values ofthe elastic moduli, for the intact rock mass. Neglect-

Fig. 22 – Extension of the fragmented zone around the blast hole from the numerical analysis: (a) back view and (b) frontview.Fig. 22 – Estensione della zona frammentata attorno al foro di esplosione dall’analisi numerica: (a) vista da dietro e (b) vista di fronte.

Fig. 23 – Attenuation law for principal frequency of acceleration at the soil surface: comparison between experimental dataand numerical results.Fig. 23 – Legge di attenuazione per la frequenza principale dell’accelerazione sulla superficie del rilevato in terra: confronto fra i dati sperimentali e i risultati numerici.



ing material damping does not influence substan-tially the peak particle velocity of the rock mass, asdamping of the highest frequencies is accounted forby introducing a convenient algorithmic damping.

In the numerical analyses performed, the dash-pots, located at the fictitious outer boundary, weresufficient to avoid significant reflection of the outgo-ing waves as the comparison between numerical re-sults and experimental data demonstrated.

The numerical approach provides useful infor-mation for practical applications. Thanks to the en-ergy-conserving approach, partitioning of the ex-plosive energy into fracture energy, stress wavepropagation and kinetic energy can be estimated.The total fracture area, an important parameter inthe design for transport and storage of the fracturedrock volume, may be calculated. A full description ofacceleration, velocity and displacement histories isprovided at any location of the domain. The resultsmay be post-processed to obtain spectral densitiesand principal frequencies of the relevant quantities.

Following a common approach, the numericalresults were elaborated to give attenuation laws ofpeak particle velocity and principal frequency of theacceleration with scaled distance. The resulting atten-uation laws are not necessarily linear in non homoge-neous media, as wave reflection and refraction at thestrata interfaces modify the dominant characteristicsof the motion. Differently from empirical laws or sim-ple analytic solutions, the numerical approach allowsfor tracking the stress waves characteristics in generalnon homogeneous domains, hence providing a valu-able tool in the prediction of the environmental ef-fects of blast waves in practical applications.

Acknowledgements

The authors wish to thank Paolo Chiarati,former undergraduate student at the Politecnico diMilano, for helping in performing some of the nu-merical simulations here presented.

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Vibrazioni indotte da esplosivo in roccia: un approccio numerico

SommarioLo scavo mediante esplosivo è una metodologia versatile,

efficiente ed economica. Tuttavia, la sua applicazione può essere fortemente limitata dal forte impatto ambientale associato al disturbo, alle vibrazioni e ai potenziali danni indotti sulle costruzioni circostanti. L’uso di esplosivo come tecnica di scavo è regolato dalle normative nazionali, che definiscono i limiti di tollerabilità delle vibrazioni trasmesse alle strutture circostanti, in funzione della tipologia di struttura stessa. Per stimare l’entità delle vibrazioni indotte nel terreno a una data distanza dal centro dell’esplosione, vengono usualmente impiegate leggi di attenuazione, basate su misure effettuate in sito o su soluzioni analitiche semplificate di propagazione di onde. Le leggi di attenuazione sono riferite, in genere, a trasmissione in mezzi omogenei e, pertanto, possono non risultare del tutto adeguate come strumento predittivo in siti geologicamente complessi. In questi casi, l’analisi numerica può rappresentare una valida alternativa, poiché, in principio, consente di simulare l’intera storia di propagazione delle onde, qualunque sia la complessità del sito analizzato. In questo lavoro viene presentato un approccio agli elementi finiti per descrivere gli effetti provocati da un’esplosione sull’ammasso roccioso e sui terreni circostanti, mano a mano che ci si allontana dal suo centro. L’energia dell’esplosivo è tradotta in una storia temporale di pressione sul contorno esposto all’esplosione. Un modello di frattura coesiva consente di riprodurre la frammentazione della roccia in prossimità dei fori dove è posizionato l’esplosivo, mentre per l’ammasso roccioso non fratturato e per i terreni circostanti è assunto un comportamento elastico. La propagazione delle onde elastiche dal centro dell’esplosione è simulata mediante una analisi tridimensionale ad elementi finiti nel dominio del tempo, che fornisce tutte le variabili che caratterizzano il moto in qualsiasi posizione. I risultati dell’analisi numerica sono stati elaborati a posteriori, per derivare le leggi di attenuazione relative alle grandezze più importanti alle quali le normative fanno usualmente riferimento, ovvero la velocità di picco e la frequenza principale di vibrazione. Il modello numerico proposto è in grado di conservare l’energia. L’energia dell’esplosivo viene così correttamente ripartita fra la quota parte che va a frammentare la roccia attorno al foro di esplosione, l’energia di propagazione delle onde elastiche e l’energia cinetica dei frammenti di roccia. Vengono presentate le simulazioni di due casi di letteratura, confrontando i risultati dell’analisi con i dati sperimentali disponibili. Le velocità di picco ottenute dalla simulazione risultano del tutto paragonabili a quelle rilevate sperimentalmente, così come le frequenze principali in roccia. In sistemi stratificati, il rapporto fra le rigidezze degli strati è dominante sulle frequenze principali, mentre ha minore effetto sulle velocità di picco.

Parole chiave: esplosivi, propagazione dinamica di onde elastica, vibrazioni indotte, analisi ad elementi finite, frattura coesiva.

vibrations induced by blasting in rock a numerical approach

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