Models for analysis of shotcrete on rock exposed to blasting517588/FULLTEXT01.pdf · Models for analysis of shotcrete on rock exposed to blasting Lamis Ahmed Licentiate Thesis Stockholm,

Models for analysis of shotcrete on rock exposed to blasting

Lamis Ahmed

Licentiate Thesis Stockholm, May 2012

TRITA-BKN. Bulletin 114, 2012 ISSN 1103-4270 ISRN KTH/BKN/B--114--SE

Royal Institute of Technology (KTH) School of Architecture and the Built Environment SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungliga Teknisk högskolan framlägges till offentlig granskning för avläggande av teknologi licentiatexamen i byggvetenskap fredagen den 25 maj 2012 klockan 13:00 i sal B25, Kungliga Teknisk högskolan, Brinellvägen 23, Stockholm. ©Lamis Ahmed, May 2012

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Preface

The work presented in this thesis was carried out at the Royal Institute of Technology (KTH), Division of Concrete Structures. The study was made possible through financial support from Formas (The Swedish Research Council for Environment, Agricultural Science and Spatial Planning), with additional contribution from BeFo (Rock Engineering Research Foundation) that made the laboratory experiments possible. The support from Formas and BeFo is hereby gratefully acknowledged.

I would like to express my sincere thanks and gratitude to my supervisor associated professor Anders Ansell and assistant supervisor Ph.D. Richard Malm for their guidance, assistance, supervision, and encouragement during the term of this project.

I also wish to express my grateful thanks to the staff of the laboratory at Department of Civil and Architectural Engineering at KTH for providing all the necessities for executing the laboratory works.

Appreciation thanks to Professor Jonas Holmgren and Docent Anders Bodare for their support, and to all my colleagues and my brother for their help.

Stockholm, May 2012

Lamis Ahmed

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Abstract

In underground construction and tunnelling, the strive for a more time-efficient construction process naturally focuses on the possibilities of reducing the times of waiting between stages of construction. The ability to project shotcrete (sprayed concrete) on a rock surface at an early stage after blasting is vital to the safety during construction and function of e.g. a tunnel. A complication arises when the need for further blasting affects the hardening of newly applied shotcrete. If concrete, cast or sprayed, is exposed to vibrations at an early age while still in the process of hardening, damage that threatens the function of the hard concrete may occur. There is little, or no, established knowledge on the subject and there are no guidelines for practical use.

It is concluded from previous investigations that shotcrete can withstand high particle velocity vibrations without being seriously damaged. Shotcrete without reinforcement can survive vibration levels as high as 0.5−1 m/s while sections with loss of bond and ejected rock will occur for vibration velocities higher than 1 m/s. The performance of young and hardened shotcrete exposed to high magnitudes of vibration is here investigated to identify safe distances and shotcrete ages for underground and tunnelling construction, using numerical analyses and comparison with measurements and observations. The work focuses on finding correlations between numerical results, measurement results and observations obtained during tunnelling. The outcome will be guidelines for practical use.

The project involves development of sophisticated dynamic finite element models for which the collected information and data will be used as input, accomplished by using the finite ele-ment program Abaqus. The models were evaluated and refined through comparisons between calculated and measured data. First, existing simple engineering models were compared and evaluated through calculations and comparisons with existing data. The first model tested is a structural dynamic model that consists of masses and spring elements. The second is a model built up with finite beam elements interconnected with springs. The third is a one-dimensional elastic stress wave model. The stress response in the shotcrete closest to the rock when exposed to P-waves striking perpendicularly to the shotcrete-rock interface was simulated. Results from a non-destructive laboratory experiment were also used to provide test data for the models. The experiment studied P-wave propagation along a concrete bar, with properties similar to rock. Cement based mortar with properties that resembles shotcrete was applied on one end of the bar with a hammer impacting the other. The shape of the stress waves travelling towards the shotcrete was registered using accelerometers positioned along the bar.

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Due to the inhomogeneous nature of the rock, the stress waves from the blasting attenuate on the way from the point of explosion towards the shotcrete on the rock surface. Material damping for the rock mass is therefore accounted for, estimated from previous in-situ measurements. The vibration resistance of the shotcrete-rock support system depends on the material properties of the shotcrete and here were age-dependent properties varied to investigate the behaviour of young shotcrete subjected to blast loading. The numerical simulations require insertion of realistic material data for shotcrete and rock, such as density and modulus of elasticity.

The calculated results were in good correspondence with observations and measurements in-situ, and with the previous numerical modelling results. Compared to the engineering models, using a sophisticated finite element program facilitate modelling of more complex geometries and also provide more detailed results. 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. 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. It is acceptable to use elastic material formulations until the material strengths are exceeded, i.e. until the strains are outside the elastic range, which thus indicates material failure. The higher complexity of this type of model, compared to the engineering models, will make it possible to model more sophisticated geometries. Examples of preliminary recommendations for practical use are given and it is demonstrated how the developed models and suggested analytical technique can be used to obtain further detailed limit values.

Keywords: Shotcrete, Rock, Vibration, Stress waves, Numerical analysis.

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Sammanfattning

Inom undermarks- och tunnelbyggande leder strävan efter en mer tidseffektiv byggprocess till fokus på möjligheten att reducera väntetiderna mellan byggetapper. Möjligheten att projicera sprutbetong på bergytor i ett tidigt skede efter sprängning är avgörande för säkerheten under konstruktionen av t.ex. en tunnel. En komplikation uppstår när behovet av ytterligare sprängning kan påverka härdningen av nysprutad betong. Om betong, gjuten eller sprutad, utsätts för vibrationer i ett tidigt skede under härdningsprocessen kan skador som hotar funktionen hos den hårdnade betongen uppstå. Kunskapen i ämnet är knapphändig, eller obefintlig, och det finns inga etablerade riktlinjer för praktisk användning.

Slutsatsen från tidigare undersökningar visar att sprutbetong kan tåla höga vibrationer (partikelhastigheter) utan att allvarliga skador uppstår. Oarmerad sprutbetong kan vara oskadd efter att ha utsatts för så höga vibrationsnivåer som 0,5–1 m/s medan partier med förlorad vidhäftning till berget kan uppträda vid vibrationshastigheter högre än 1 m/s. Funktionen hos ung och hårdnande sprutbetong som utsätts för höga vibrationsnivåer undersöks här för att identifiera säkra avstånd och sprutbetongåldrar för undermarks- och tunnelbyggande, med hjälp av numeriska analyser och jämförelser med mätningar och observationer. Arbetet fokuserar på att finna samband mellan numeriska resultat, mätresultat och observationer från tunnelbyggande. Det slutliga resultatet kommer att vara riktlinjer för praktisk användning.

Projektet omfattar utveckling av sofistikerade dynamiska finita elementmodeller för vilka insamlad information och data kommer att användas som indata för det finita elementprogrammet Abaqus. Modellerna utvärderades och förfinats genom jämförelser mellan beräknade och uppmätta resultat. Först jämfördes befintliga enkla mekaniska, ingenjörsmässiga modeller vilka utvärderades genom beräkningar och jämförelser med befintliga data. Den första modellen är en strukturdynamisk modell bestående av massor och fjäderelement. Den andra är en modell uppbyggd av finita balkelementet sammankopplade med fjädrar. Den tredje är en endimensionell elastisk spänningsvågsmodell. Spänningstillståndet i sprutbetongen närmast berget, utsatt för vinkelrätt inkommande P-vågor simulerades. Resultat från icke-förstörande laborationsprovningar användes också som testdata för modellerna. Experimentellt studerades P-vågsutbredning i en betongbalk med egenskaper likvärdiga med berg. Cementbruk med egenskaper liknande sprutbetong applicerades på balkens ena ände medan en hammare användes i den andra. Formen hos den genererade spänningsvågen som propagerade mot sprutbetongänden registrerades med accelerometrar utplacerade längs balken.

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På grund av bergets inhomogena karaktär kommer spänningsvågorna från sprängningen att dämpas ut på vägen från detonationspunkten till sprutbetongen på bergytan. Materialdämpningen hos bergmassan måste därför beaktas och kan uppskattas utifrån resultat från fältmätningar. Vibrationståligheten hos förstärkningssystem av sprutbetong och berg beror av materialegenskaperna hos sprutbetongen och i den här studien varierades de åldersberoende egenskaperna för att undersöka beteendet hos nysprutad och hårdnande sprutbetong utsatt för sprängbelastning. De numeriska simuleringarna kräver realistiska materialdata för sprutbetong och berg, som t.ex. densitet och elasticitetsmodul.

De beräknade resultaten var i god överensstämmelse med observationer och mätningar i fält plats, och med de tidigare numeriska resultaten. Jämfört med de mekaniska modellerna kan ett sofistikerat finit elementprogram underlätta modellering av mer komplexa geometrier och också ge mer detaljerade resultat. Det visades att vågutbredning genom berg och fram emot sprutbetong kan beskrivas med tvådimensionella elastiska finita element i en dynamisk analys. Modellerna måste beskriva bergets egenskaper och noggrannheten i dessa parametrar kommer att ha stor inverkan på resultaten. Det kommer att vara möjligt att beskriva vågutbredningen genom bergmassan, från detonationspunkten till reflektionen vid bergytan, det vill säga gränssnittet mellan sprutbetong och berg. Det är acceptabelt att använda elastiska materialformuleringar tills materialet elasticitetsgräns överskrids tills töjningar utanför det elastiska området nås, vilket därmed indikerar materialbrott.

Den högre komplexiteten hos den här typen av modell, jämfört med de mekaniska modellerna, kommer att göra det möjligt att analysera mer komplexa tunnelgeometrier. Exempel på preliminära rekommendationer för praktiskt bruk ges och det visas hur de utvecklade modellerna och den föreslagna analysmetodiken kan användas för att fastställa ytterligare detaljerade gränsvärden.

Nyckelord: Sprutbetong, Berg, Vibrationer, Spänningsvågor, Numerisk analys.

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List of publications

This thesis is based on work contained in the following articles.

Paper I Ahmed, L. and Ansell, A. (2012). Structural dynamic and stress wave models for analysis of shotcrete on rock exposed to blasting. Engineering Structures, 35, 11-17.

Paper II Ahmed, L. and Ansell, A. (2012). Laboratory investigation of stress waves in young shotcrete on rock. Accepted for publication in Magazine of Concrete Research.

Paper III Ahmed, L., Malm, R and Ansell, A. (2012). Finite element simulation of shotcrete exposed to underground explosions. Submitted to Nordic Concrete Research.

In Paper I the calculations were performed by Lamis Ahmed while the analysis of the results and the writing was done by Lamis Ahmed with contributions from Anders Ansell.

Shared responsibility in planning of the experiments in Paper II with Anders Ansell. The writing and simulation was mainly done by Lamis Ahmed. The analysis of the results was done by Lamis Ahmed with contributions form Anders Ansell.

In Paper III the implementation of the models in the finite element code was carried out by Lamis Ahmed with contributions from Richard Malm. The finite element analysis was carried out by Lamis Ahmed. The writing and discussion of results was done by Lamis Ahmed with contributions from Anders Ansell.

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Contents

Preface ........................................................................................................................................ i

Abstract .................................................................................................................................... iii

Sammanfattning ....................................................................................................................... v

List of publications ................................................................................................................. vii

1 Introduction .................................................................................................................... 1

1.1 Background ............................................................................................................. 1

1.2 Rock support with shotcrete .................................................................................... 2

1.3 Previous research ..................................................................................................... 3

1.4 Aims and goals ........................................................................................................ 4

1.5 Contents of report .................................................................................................... 5

2 Materials properties ...................................................................................................... 7

2.1 Shotcrete .................................................................................................................. 7

2.2 Rock ......................................................................................................................... 9

2.3 Bond between shotcrete and rock ............................................................................ 9

3 Stress waves and structural dynamics ....................................................................... 13

3.1 Physical appearance of stress waves ..................................................................... 13

3.2 Principal frequency ................................................................................................ 15

3.3 Rock vibrations from detonations ......................................................................... 17

3.4 Attenuation of the velocity in rock ........................................................................ 19

4 Laboratory investigation ............................................................................................. 23

4.1 Test set-up ............................................................................................................. 23

4.2 Measurement results .............................................................................................. 25

4.3 Evaluation of results .............................................................................................. 27

x

4.4 Shotcrete bond stresses .......................................................................................... 31

5 Summary of engineering models ................................................................................ 33

5.1 Mass-spring model ................................................................................................ 33

5.2 Beam–spring model ............................................................................................... 35

5.3 Elastic stress wave model ...................................................................................... 35

5.4 Model comparison ................................................................................................. 36

6 Finite element analysis................................................................................................. 41

6.1 Explicit dynamic analysis ...................................................................................... 41

6.2 Solid infinite elements ........................................................................................... 42

6.3 Blast-induced vibration loads ................................................................................ 43

6.4 Finite element models ........................................................................................... 44

6.5 Examples ............................................................................................................... 49

6.5.1 Tunnel profiles ............................................................................................. 49

6.5.2 Tunnel plane ................................................................................................. 51

6.5.3 Young shotcrete ............................................................................................ 64

7 Conclusions ................................................................................................................... 67

7.1 Discussion and recommendations ......................................................................... 67

7.2 General conclusions .............................................................................................. 73

7.3 Further research ..................................................................................................... 73

Bibliography ........................................................................................................................... 75

Appended papers .................................................................................................................... 79

1

1 Introduction

1.1 Background

Over the last decades, shotcrete has become an important material for stabilising excavated tunnels and underground openings in hard rock. Shotcrete is concrete projected pneumatically onto a surface, using one of either the dry mix or the wet mix method. As shotcrete can be sprayed onto a horizontal or a vertical surface, it has been widely used for tunnelling works. Its unique flexibility in the choice of application thickness, material compositions (e.g., fibre content), output capacity and fast early strength development make shotcrete a material well suited for rock support. By the middle of the 20th century, the use of shotcrete as a support system became widespread, with an increasing need for reliable guidelines and calculation methods to control the mode of action of shotcrete in hard rock.

Most construction work in underground rock involves the use of explosives for excavation work. In tunnelling, the search for a more time-efficient construction process naturally focuses on the possibilities of reducing the periods of waiting between stages of construction. As an example, the driving of two parallel tunnels requires coordination between the two excavations so that blasting in one tunnel does not, through vibrations, damage temporary support systems in the other tunnel prior to installation of a robust, permanent support; see Figure 1.1. Similar problems also arise in mining. To be able to excavate as much ore volume as possible, the grid of drifts in a modern mine is dense. This means that supporting systems in one drift are likely to be affected by vibrations in a neighbouring drift. Thus, there is a need to know how close, in time and distance, to shotcrete blasting can be allowed to take place.

Recommendations and guidelines for practical use are of great importance in e.g. civil engineering underground work, tunnelling and mining. A criterion for how close to the young shotcrete blasting can take place is therefore desired when planning for efficient underground and tunnelling projects. The work presented in this thesis contribute to the rock support research field by combining theory, in-situ observations and measurement with finite element modelling, covering concrete technology, rock mechanics and structural dynamics in a study of shotcrete on hard rock exposed to blasting.

Chapter

CHAPTER 1. INTRODUCTION

2

Figure 1.1: Construction of two parallel tunnels, redrawn from [20].

1.2 Rock support with shotcrete

The main design principle for rock support in underground construction is to help the rock carry its inherent loads. There are many elements, which are used in rock support systems; e.g., shotcrete, rock bolts, steel arches, mass concrete or pre-fabricated elements, used solely or in combination. In Sweden, shotcrete is widely used in rock support due to the quality of the rock, i.e. hard rock. Shotcrete is sprayed on the rock surface after the excavation of tunnel or a rock cavern to prevent fallout of rock blocks thereby securing the arch-shape of the tunnel profile. The most fundamental characteristic of shotcrete is its ability to adhere to a surface, forming a bond that depends on the bond strength between shotcrete and rock. The first type of shotcrete support consists of only a single un-reinforced shotcrete lining that bond to the rock surface. Other types are shotcrete linings anchored in fully grouted rock bolts or end-anchored rock bolts. For these, the shotcrete lining must be reinforced with steel mesh or fibres. Another type is shotcrete arches, built up by plain or alternatively fibre-reinforced shotcrete [3].

The rock support is generally designed for static loading conditions but in some cases, the openings are also subjected to dynamic loads. One example is the detonation of explosives during excavations of tunnels and underground spaces. These detonations lead to stress waves that propagate through the rock and may cause severe damage to installations and support such as shotcrete. In the past, not many attempts have made to understand the mechanisms behind shotcrete damage due to vibrations and there are no established guidelines for practical use. There are few results published on tests conducted in tunnels and mines where shotcrete has been subjected to vibrations from blasting.

1.3. PREVIOUS RESEARCH

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1.3 Previous research

Some attempts have been made to investigate the effect on shotcrete from blasting. In-situ tests, conducted underground in the Swedish mine at Kirunavaara [3, 4], was as a first step towards reliable guidelines for how close, in time and distance, to young and hardening shotcrete blasting can be allowed. The tests were conducted with sections of plain, un-reinforced shotcrete projected on tunnel walls and exposed to vibrations from explosive charges detonated inside the rock. The response of the rock was measured with accelerometers mounted on the rock surface and 0.5 m into the rock. The observed vibration levels showed that sections of shotcrete had withstood high particle velocity vibrations without being seriously damaged. It was concluded that shotcrete without reinforcement, also as young as a couple of hours, can withstand vibration levels as high as 0.5−1.0 m/s, while section with loss of bond and ejected rock where found for vibration velocities higher than 1.0 m/s. Similar measurements, based on in-situ experiments conducted in Japan [32], showed that vibration velocities of 0.7 m/s cracked the observed shotcrete lining. The response of steel fibre-reinforced and steel mesh-reinforced shotcrete linings subjected to blasts was investigated in a Canadian mine [30, 46]. It was observed that the shotcrete remained attached to the rock surface for vibration levels up to 1.5–2 m/s, with only partial cracking observed in the shotcrete. During other in-situ investigations [21] it was observed that the damage that can be expected during tunnelling is radial cracking across the shotcreted face nearest the blast and transverse cracking, from the face and into the tunnel. Maximum vibration levels are often defined in terms of peak particle velocity (PPV). Table 1.1 shows that bond between shotcrete and rock can withstand PPV levels of range 0.5–2 m/s before the bond damage occurs.

Table 1.1: Vibration velocities PPV when bond damage occurs. Based on in-situ measurements.

PPV Comments

Kiirunavaara, Sweden [3, 4] 0.5−1.0 m/s Young shotcrete

Tunnelling, Japan [32] up to 0.7 m/s

Gold mine, Canada [30, 46]. 1.5–2.0 m/s Steel fibre-reinforced and steel mesh-reinforced shotcrete

Numerical models have also been used to predict the bond damage on shotcrete from underground explosion, [3-5, 20, 33]. The advantage of these models, based on elastic stress wave theory and structural dynamics, is the efficient analytical procedure that, with small computational effort, makes it possible to compare a large number of calculations with various combinations of input data. The results from the in-situ measurements at Kirunavaara were evaluated [3, 4, 20] using numerical modelling with an approach based on elastic stress wave theory. This resulted in a prototype model for a one-dimensional analysis of shotcrete on rock through which elastic stress waves propagate towards the shotcrete. Another one-

CHAPTER 1. INTRODUCTION

4

dimensional model based on structural dynamics with lumped vibrating masses and stiffness contribution from discrete springs was evaluated by Nilsson [33]. The shotcrete on rock subjected to vibrations is in this case modelled in the same way as for a building on ground during an earthquake where the ground accelerations lead to inertial forces on the building. In order to interpret the dynamic behaviour of the vibrating shotcrete, parallel with and in right angels to rock walls, a two-dimensional model was tested [5]. The model consists of beam elements that represent the flexural stiffness and mass of the shotcrete and the fractured rock closest to the rock surface. Spring elements are used to obtain elastic coupling between shotcrete and rock. Based on results from this spring-beam model, recommendations for safe distances between shotcrete on rock and small amounts of detonating explosives in the rock are given [5], here in Table 1.2. The limit values are valid for rock with properties similar to Swedish type granite with 0.5 m spacing between discontinuities.

Table 1.2: Recommended minimum distance between shotcrete and a detonating explosive charge [5].

Thickness of shotcrete Weight of explosives

0.5 kg 1 kg 2 kg

25 mm 0.8 m 1.0 m 1.2 m

50 mm 1.2 m 1.5 m 1.9 m

100 mm 1.8 m 2.4 m 3.2 m

1.4 Aims and goals

This project aims at suggesting a numerical analysis method that can be used in the work of determining guidelines for practical use in tunnelling and underground work in hard rock involving blasting close to shotcrete. There are a number of important goals:

A comparison between existing prototype (engineering) models.

Development of a more detailed model, based on 2D solid finite elements.

A comparison between analysis results and results obtained in the laboratory and in-situ.

Identification of important factors and material properties that must be accounted for in the analysis.

A study of realistic and relevant tunnel geometries.

Practical recommendations will finally be given, adding further details and adjustment to existing, preliminary recommendations.

1.5. CONTENTS OF REPORT

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1.5 Contents of report

The thesis consists of six chapters, following the introduction. Chapter 2 provides material data used in the examples presented. Stress waves in rock are briefly reviewed in Chapter 3, also discussing the need for including the effect of rock material damping. The laboratory investigation is summarized in Chapter 4, also covered in the appended Paper II. The features of the first studied numerical prototype models, engineering models, are summarized in Chapter 5. A thorough description of these models is given in Paper I. The finite element analysis is briefly presented in Chapter 6, also covered in Paper III. Additional numerical examples are presented in this chapter. Conclusions and practical recommendations are given in Chapter 7, together with suggestions for further research. The appended papers contribute to the thesis as follows:

Paper I

Three different prototype models are compared and evaluated. The first model tested is a structural dynamic model that consists of masses and spring elements. The second is a model built up with finite element beam elements interconnected with springs. The third is a one-dimensional elastic stress wave model.

Paper II

Presents a non-destructive laboratory experiment that studies wave propagation along a concrete bar, with properties similar to rock. Cement based mortar with properties that resembles shotcrete was applied on one end of the bar with a hammer impacting the other.

Paper III

An elastic finite element model is presented to simulate the stress wave propagation from detonation through the rock towards the interface between the shotcrete and rock surface. The loads on the models are the peak particle velocity-time histories applied on the tunnel surface within the rock, at the damage zone boundary.

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2 Materials properties

The finite element (FE) numerical simulations presented in this study requires insertion of material data for shotcrete and rock, such as density and modulus of elasticity. This chapter contains a compilation of relevant material parameters. In the first and second sections selections for shotcrete and rock are given, followed by a section on bond strength, summarizing known observations and experimental tests results.

2.1 Shotcrete

As mention in section 1.1, shotcreting is a process of installing concrete pneumatically and it must therefore contain smaller aggregates than conventional, cast concrete. According to [26], shotcrete can be divided into three types with respect to the maximum aggregate size. The types are fine shotcrete, (normal) shotcrete and coarse shotcrete, with a maximum aggregate

size of 4, 8 and 15 mm, respectively. The density of shotcrete (shot varies, depending on type. In the following study, it is assumed to be 2100 kg/m3, equal to the standard shotcrete type used in the Kiirunavaara mine [3].

Based on tests of 150 mm cubes, performed at shotcrete ages of 6 and 18 hours as described in [Paper II], the age dependent compressive strength of shotcrete is here assumed to follow:

0.821.706, 24h5.506

stc cf f e ( 2.1)

where ts is the shotcrete age in days and fc,24h is the strength at 24 hours, equal to 25.3 MPa, see [Paper II]. An alternative formulation was used in the previous work, given in [2] as:

0.601.37 stc ccf f e ( 2.2)

where fcc = 33.6 MPa.

Chapter

CHAPTER 2. MATERIALS PROPERTIES

8

The development of the modulus of elasticity is here assumed to follow the average curve presented forth by Chang, [14]. The equation, derived from a large number of tests, gives the relation between Eshot and the age of shotcrete. The equation is

0.80.446

shot 28d1.062

stE E e ( 2.3)

where E28d is set to 27 GPa for this study. Alternatives to this relation are presented in [2], but give practically the same results for Eshot. The fastest development of the modulus of elasticity occurs around the first 14 days. The growth of the relative modulus of elasticity during the first 24 hours is shown in Figure 2.1, together with the corresponding curves for the compressive strength. It should be noted that the faster growing curve of Eq. (2.1) is representative for controlled laboratory environment while Eq. (2.2) represent in-situ conditions. At 24 hours of age, the first will reach 25.3 MPa while the latter here only reaches 8.5 MPa.

Figure 2.1: Compressive strength and modulus of elasticity vs. age of shotcrete. Relative values given in relation to the strength at 24 hours.

Theoretical values for Poisson’s ratio for concrete is presented by Boumiz et al. [12], based on tests performed on series of concrete cube specimens. It was concluded that Poisson’s ratio decreases from about 0.5 at early age of concrete to about 0.2 which is characteristic for hardened concrete materials. In addition, over the curing period it is seen that the propagation wave velocity increase rapidly during setting and more slowly afterwards. To evaluate the propagation of the stress waves in young and hardened concrete, series of tests are often made, see e.g. [24, 37]. From those tests, it was concluded that the wave propagation velocity is more sensitive to strength changes at early ages than later, when higher strength has been

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shotcrete age, hrs

Rel

ativ

e st

reng

th

Compressive strength, Eq.(2.1)

Compressive strength, Eq.(2.2)Modulus of elasticity, Eq. (2.3)

2.2 ROCK

9

reached. This can be attributed to the fact that at early ages the strength depends almost solely on the properties of the concrete matrix [24, 37].

2.2 Rock

It is difficult to define material properties for a fractured rock mass since its behaviour is site-specific. Its properties depend on geological discontinuities, e.g. fractures or zones containing weaker materials, that are the results of historical geological events [3]. Common values for

the density of granite are within rock = 2500 –2800 kg/m3. For comparison with in-situ results from the previous studies [3, 4] it should be noted that the iron ore in the Kiirunavaara mine has a higher density, 4800 kg/m3. However, the static Erock for intact rock in the mine is similar to that of granite and is in the range of 40–110 GPa [28]. Due to the heterogeneous nature of rock and discontinuities in the rock mass, a practical value for the static Erock in the Kiirunavaara mine can be estimated in the range of 10–30 GPa [7].

For this study a hard Scandinavian bedrock is assumed [36] with rock of 2500 kg/m3 and Poisson’s ratio (ν) equal to 0.15. The static Erock was assumed to be either 16 GPa or 40 GPa [7]. The higher value corresponds to intact rock, while the lower refers to fractured rock, according to Bieniawski [10]. A similar range, Erock= 6.7–59 GPa, was used by Song and Cho [41] to simulate wave propagation in granite.

2.3 Bond between shotcrete and rock

As previously mentioned, the most important feature of shotcrete is its ability to adhere to a rock surface. A number of bond test methods have been developed to estimate the bond strength between shotcrete and rock. There are for example non-destructive evaluation methods based on the impact-echo method that can be used to detect partial debonding or voids between shotcrete and rock, se e.g. [40, 41]. To obtain a measure on the bond strength must, however, destructive tests such as pull-out tests be performed. Holmgren [19] studied the modes of shotcrete failure using punching block tests to simulate load applied on a shotcrete lining. The test results indicate that for steel-fibre and mesh-reinforcement linings, direct shear failure tends to occur when the bond to the rock mass is good, while flexural and punching shear failure occur when the bond is poor, causing debonding. Other possible modes of shotcrete failure include bending, compressive and tensile failures, but these are not relevant for the cases studied here.

Hahn and Holmgren [16] suggested 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 [8]. Low bond strength will also occur for surfaces not initially cleaned, which means that the bond strength can be improved by first washing the rock surface using

CHAPTER 2. MATERIALS PROPERTIES

10

high pressure water. Malmgren et al. [27] investigated the influence of surface treatment of rock (water jet scaling), shrinkage and hardening of shotcrete on the bond strength. The bond tests were carried out in accordance with the Swedish standard [42]. It was concluded that the bond strength in that case was 0.33 MPa and 0.68 MPa for normal treatment and water-jet scaling, respectively. Saiang et al. [38] performed a series of laboratory tests on cemented shotcrete-rock joints, to investigate the strength and stiffness of the interfaces. The load at which two jointed pieces came apart was considered as the bond strength. Average bond strength of 0.56 MPa was obtained. A series of tests with a new approach for testing the bond strength between young shotcrete and a concrete plate, was performed by Bryne et al. [13]. The tests indicated a relatively rapid development of the bond strength which reached 1.0 MPa after 24 hours. This is in good correspondence to the range of 0.6–1.1 MPa reported from other investigations on 1–2 days old shotcrete and mortar [13]. The development of the bond strength thus follows

0.970.858

cb cb,24h2.345 e

stf f ( 2.4)

where the average strength at 24 hours is fcb,24h = 1.0 MPa, corresponding to laboratory conditions. The growth of the bond strength during the first 24 hours is shown in Figure 2.2, together with the lower values corresponding to in-situ conditions given in [2]. Note that the curve corresponding to Eq. (2.4) is obtained through curve-fit for tests with shotcrete older than 6 hours and that fcb in reality always is higher than zero. The above bond strength values obtained for young and hardened shotcrete are summarized in Table 2.2.

Table 2.2: Bond strength of the shotcrete-rock interface obtained from tests.

Bond strength Comments

Hahn and Holmgren [16] 0.3 – 1.7 MPa Granite

Malmgren et al. [27] 0.33 MPa 0.68 MPa

Normal treatment Water-jet scaling

Saiang et al. [38] 0.56 MPa Shotcrete-rock joints

Bryne et al. [13]* 0.06 – 0.07 MPa 0.36 MPa 0.35 – 0.48 MPa 0.67 – 0.83 MPa 0.74 – 1.15 MPa

Age: 6 hours 10 hours 12 hours 18 hours 24 hours

*Bond strength between shotcrete and concrete.

2.3 BOND BETWEEN SHOTCRETE AND ROCK

11

Figure 2.2: Bond strength vs. age of shotcrete.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shotcrete age, hrs

Bon

d st

reng

th ,

MP

a

Bond strength, Eq.(2.4)

Bond strength [2]

13

3 Stress waves and structural dynamics

The main difference in a structural dynamic analysis, compared to a static analysis, is the inclusion of inertia effects caused by motion of masses. In this chapter, the stress wave propagation in elastic media is described and the elastic stress wave theory reviewed. As a complement, a brief derivation of the natural frequencies for a longitudinally vibrating bar is given. The following section discusses the stress waves that occur in rock due to blasting and which may cause damage to the bond between shotcrete and rock. Finally, the need to account for damping in the rock mass is discussed, also explaining how a damping ratio can be estimated from in-situ measurements.

3.1 Physical appearance of stress waves

An elastic body with large extensions in all three dimensions can be approximated by a smaller volume restricted by boundaries given physically relevant properties i.e. boundary conditions. This approximation is accurate if the length of waves travelling in the body are small compared to its dimensions [3]. In light of this assumption, the stress wave propagation may be treated by means of a one-dimensional approximation. The following section briefly describes the approach to understand the underlying theory of one-dimensional wave propagation in elastic solids, which is based on [31]. To derive the equation of motion for longitudinal vibration, the axial deformation of a long, thin member can be considered, as shown in Figure 3.1(a). A freebody diagram for an element of length dx is shown in Figure 3.1(b).

The force acting on the cross section of a small element of the bar is given by P and P+dP with

uP A A E

x

(3.1)

Chapter

CHAPTER 3. STRESS WAVES AND STRUCTURAL DYNAMICS

14

Figure 3.1: Member undergoing axial deformation (a) and free body diagram (b).

where /u x is the axial strain, obtained by applying Newton's second law

( ) xF m u (3.2)

The summation of the forces in the x-direction gives the equation of motion (as shown in Figure 3.1) where ( , )f x t denotes the external force per unit length,

( ) AP dP f dx P xu (3.3)

where ρ is the density of the media. By using the relation ( / )dP P x dx and Eq. (3.1), the

equation of motion for the forced longitudinal vibration of a uniform bar, can be expressed as

2 2

2 2( , ) ( , ) ( , )

u uE A x t f x t A x t

x t

(3.4)

where 2 2( / )u u t . This is the equation of motion for axial vibration of a linearly elastic

bar. For the bar, the free vibration equation can be obtained from Eq. (3.4) by setting f = 0, as

2 22

2 2( , ) ( , )

u uc x t x t

x t (3.5)

where c is the propagation velocity. In a one-dimensional elastic body, e.g. a bar, the propagation velocities of longitudinal (cp) and transverse (cs) waves depends only on ρ and elastic constants, and are:

p

Ec (3.6)

and

(a) (b)

3.2 PRINCIPAL FREQUENCY

15

s1

2(1 )

E

c (3.7)

where E is the elastic modulus of the material and υ the Poisson’s ratio. When P-wave propagation is considered, which is the case for many construction and blast vibration problems [15], Eq. (3.5) should be employed.

The solution of Eq. (3.5), the wave equation, can be written as

1 2 3 4( , ) ( ) ( ) ( cos sin ) ( cos sin )

x x

u x t U x T t C C C t C tc c

(3.8)

where the function U(t) represents the normal mode, depending on position along the x-axis, and the function T(t), that depends on time t. To derive the stress wave propagation, f = 0 and ∂P = Δσ A is inserted in Eq. (3.3), giving:

A ( A )

v

xt

(3.9)

where σ is the stress and v is the particle velocity. For the constant cross section area, Eq. (3.9) can be written as:

x

vt

(3.10)

where Δx/Δt is equal to c. Then, Eq. (3.9) becomes:

c v (3.11)

Generalizing the above equation, the time-dependent relation between the stress (σ) and the two velocities, particle velocity (v) and propagation velocity (c), is

t c v t (3.12)

3.2 Principal frequency

To derive the natural frequencies for a longitudinally vibrating bar, the boundary conditions for the bar must be known. Common boundary conditions are shown in Table 3.1. In this study, two cases will be used together with the frequency equations for longitudinal vibration of uniform bars.


16

Table 3.1: Common boundary conditions for a bar in longitudinal vibration.

End condition of the bar Boundary conditions Fixed-free

u(0, t) = 0 ∂u/∂x (ℓ,t) = 0

Free-free

∂u/∂x (0,t) = 0 ∂u/∂x (ℓ,t) = 0

Fixed-fixed

u(0, t) = 0 u(ℓ, t) = 0

The first case will be used in the FE simulation and the second case is considered for the laboratory investigation of a suspended bar in axial motion, described in the following chapter. The frequencies are derived with Euler-Bernoulli beam theory; see e.g. [22, 44].

For the boundary conditions of a fixed-free bar of length ℓ the boundary conditions are

(0, ) 0u t (3.13)

and

( , ) 0

u

tx

(3.14)

Insertion of Eq. (3.13) in Eq. (3.8) gives C1=0, while Eq. (3.14) gives the circular frequency equation

cos 0

c (3.15)

The eigenvalues, or circular natural frequencies, are given by

(2 1), 1,2,3.....

2

nn c

n (3.16)

For n = 1 , the natural frequency is given by

11

(2 1) 1

2 2 2 4

n c c

f (3.17)

The boundary conditions for a free-free bar are:

3.3 ROCK VIBRATIONS FROM DETONATIONS

17

(0, ) 0

u

tx

(3.18)

and

( , ) 0

u

tx

(3.19)

Insertion of Eq. (3.18) in Eq. (3.8) gives C1=0, while Eq. (3.19) gives the circular frequency equation

sin 0

c (3.20)

The eigenvalues or circular natural frequencies are given by

, 0,1,2,...... n

ncn

l (3.21)

These natural frequencies include a zero-frequency for n = 0, i.e. a rigid-body mode. The behaviour of such bars has been investigated experimentally see Chapter 4. For n = 1 , the natural frequency is given by

11

1

2 2 2

c cf (3.22)

3.3 Rock vibrations from detonations

Detonations in rock give rise to stress waves that transport energy through the rock, towards possible free rock surfaces. Wave motion can be described as movement of energy through a material, transportation of energy achieved by particles translating and returning to equilibrium after the wave has passed [11]. This motion of particles in the rock can be described, as displacements, velocities or accelerations. The stress waves propagate outwards from the detonation in all directions. When the stress waves arrives at a free surface of the rock mass, such as that of a tunnel, they are reflected and transformed into tensile waves, which interact with the compressive waves and produce tensile cracking, if the material strength is reached. The particle accelerations, and velocity are doubled over the surface, while the stresses are zero [15]. In the case of exciting the shotcrete on a rock surface, the incident wave (aI) is transmitted through the interface between the shotcrete and the rock and reflected back through the rock, as shown in Figure 3.2(a). The transmitted wave reflects from the free surface of the shotcrete as a tensional wave. This reflected wave is then transmitted

CHAPTE

Figure 3

back thrreflectedthe refle

There ain this cThe mosubdividwaves, a

Figure 3

ER 3. STRESS

(a)

(b)

3.2: Tran

rough the shd wave willected wave

are three macase propagost importaded into coas shown in

3.3: Partishea

S WAVES AN

Negative

Positive

Negative

Positive

nsmission an

hotcrete untl be the alg(aRT), see F

ain wave typgate throughant surfaceompressive n Figure 3.3

icle motionar, secondary

Pm(a)

ND STRUCTUR

e

e

a

e

nd reflection

til it reachesebraic sum

Figure 3.2(b

pes, which ch the rock,

e wave is waves, den.

n for three wy (S-wave);

article motion

RAL DYNAM

18

aI

Ro

aR

aI

n of a wave

s the interfaof the trans

b).

can be dividand surfacthe Raylei

noted as P-

wave types; and (c) Ra

Propagation direction

MICS

ockInter

e in shotcret

ace. The netsmitted wav

ded into twoce waves traigh wave. -waves, and

: (a) comprayleigh wav

(b)

aT

aT

Shotcrrface

aRT

aR

te.

t acceleratiove (aTT), see

o varieties; ansmitted aBody wav

d shear wa

ressive, primes, redrawn

reteSurface

aR

R

on at the froe Figure 3.2

body wavealong surfacves can beaves, denote

mary (P-wan from [15].

(c)

ont of the 2(a), and

es, which ces [15]. e further ed as S-

ave); (b)

)

3.4 ATTENUATION OF THE VELOCITY IN ROCK

19

Many researchers, e.g. [17, 18, 23, 45, 47-50], have studied the possibility of predicting blast-induced damage in the surrounding rock mass. A thorough description of these types of problems is outside the scope of this thesis but of special interest is, however, the maximum allowed vibration level (PPV) that can occur in an intact rock mass. It is difficult to obtain a universally accepted damage criterion for rock since this involves many factors; e.g. loading density, explosive distribution patterns, chamber geometry, etc. For practical applications, PPV damage criteria are widely used because the PPV is less sensitive to change in geological conditions compared to acceleration or displacement [45]. For Swedish hard rock Persson [36] reported that the 1.0 m/s level of PPV is the limit for possible beginning of damage in hard rock after blasting. Zheng and Hong [50] carried out site investigation for internal explosion of a tunnel in granite rock mass. It was found that the threshold value of PPV was 0.7 m/s. Dowding [15] reported cracking observed for PPV at 1.0 m/s in a lined tunnel. Yang et al. [49] conducted experiment at a blast test site, concluding that the damage zone stretched from the blast hole to a distance of 8 m where the registered PPV value had been 0.8 m/s. Table 3.2 shows that the threshold PPV levels for rock damage in granite are within the range of 0.7–1.0 m/s.

Table 3.2: Threshold level of PPV for granite from in-situ measurements.

PPV at threshold damage Comments

Persson [36] 1.0 m/s 2.5 m/s

Damage rock Fragmentation

Zheng and Hong [50] 0.7 m/s Granite

Dowding [15] 1.0 m/s

Yang et al. [49] 0.8 m/s Moderately jointed granitic gneiss

3.4 Attenuation of the velocity in rock

The observed particle velocities in rock will show a decrease in magnitude with increasing distance R to the source of explosion. This decay is caused by geometrical spreading and hysteretic damping in the rock [15]. To predict peak particle velocity in the rock mass, functions of the scaled distance to the explosive charge are often used. The two most common approaches are square-root (R/Q1/2) and cube-root (R/Q1/3) scaling, where Q is the weight of the explosives. Square-root scaling is relevant for cases where the charge is distributed in a long cylinder (the blast hole) [15], while cube-root scaling corresponds to a case with a concentrated charge, see also [25]. In this study, the square-root scaling is employed to predict the attenuation or decay of PPV, which is also realistic since the geometry of the FE models are two-dimensional. Thus, the PPV is governed by a relation of the form


20

1max

Rv a

Q (3.23)

where vmax is the maximum PPV at a distance R from the point charge and a1 and β are constants. Eq. (3.23) is valid only for situations where R is large compared to the length of the explosive charge, thus assuming a concentration of the explosives charges. Based on regression analysis from in-situ results [3], the scaling relationship

1.5

700max

(mm / s)

Rv

Q (3.24)

gives the relation between PPV, R (in meter) and Q (in kg), for rock that corresponds to the cases studied in the following.

In rock, layers and cracks have a filtering effect on propagating vibration waves. Certain frequencies within a spectrum of a wave will be amplified while others will be damped out, depending on the characteristics of the medium, which can be rock with minor cracks. Therefore, a frequency spectrum of a measured acceleration signal will only contain a limited number of peaks, i.e. the characteristic frequencies. This can be used to forecast accelerations from known explosive charges. An amplified frequency f depends [15] on the layer thickness H and wave propagation velocity c as

4

cf

H (3.25)

For rock containing more than one characteristic distance between imperfections, H should be varied to calculate multiple characteristic frequencies.

The attenuation of peak particle velocity or decay of vibration is produced by, geometrical spreading and material damping, which is a function of the number of cycles of motion or wave length travelled. Higher frequencies have shorter wavelengths and require a greater number of cycles of motion to travel the same distance as those with low frequencies. The shorter the wavelength, the more material damping per unite distance travelled by high frequency waves [15]. In the present study, high frequency waves are studied and therefore the contribution from material damping has a noticeable effect on the results. For the following examples, the damping ratio is estimated on basis of measurement data from the Kiirunavaara mine [3]. The signals were recorded using accelerometers inside the rock, mounted through injection by cement grout. The frequency range of the waves was approximately 100–2400 Hz. The signals were sampled and digitized using the Matlab numeric software [29]. An example of a measured acceleration signal and a frequency spectrum, produced using the FFT routine of Matlab, is shown in Figure 3.4.

3.4 ATTENUATION OF THE VELOCITY IN ROCK

21

Figure 3.4: Example of in-situ measurements; (a) acceleration and (b) frequency spectrum.

The most commonly used method for determination of the damping ratio is the Half-Power (Band-Width) method. In this method, the frequency spectrum of the signal is studied. Corresponding to each natural frequency, there is a peak in the frequency spectrum Rd; as seen in Figure 3.4(b). The Half-Power method calculates the damping by using the

relationship between the frequencies corresponding to Rd/ 2 , following:

2 1

2 1

f f

f f (3.26)

0 0.02 0.04 0.06 0.08 0.1 0.12-1.5

-1

-0.5

0

0.5

1x 10

4

Time,s

Acc

eler

atio

n,m

/s2

0 500 1000 1500 2000 25000

100

200

300

400

500

600

700

Frequency,Hz

|Y(f

)|

(a)

(b)


22

Around a frequency peak there are two points corresponding to Rd/ 2 , as seen in Figure 3.5. A wider distance between the two points gives a higher damping ratio. For example, from Figure 3.4 (b), the magnitude of the peak amplitude is 685|Y(f)| at the peak frequency (fp). Thus, the use of the half-power frequency points f1 and f2 on either side of fp gives the damping ratio ξ from Eq. (3.26), as illustrated in Figure 3.5. The two points f1 and f2 are here 168 and 198 Hz, respectively, giving ξ ≈ 0.08. Table 3.2 shows estimations of ξ from the measurements presented in [3]. For these limited data, two accelerometer readings from each test site, it can be concluded that the damping ratio ξ is approximately 8% within the frequency range of 100-2500 Hz.

Figure 3.5: Half-Power method to estimate damping.

Table 3.2: Estimated damping ratio from measurements presented in [3].

Site Damping ratio ξ Accelerometer 1 Accelerometer 2

1 0.08 0.16

2 0.08 0.08

3 0.10 0.08

4 0.09 0.13

Rd/

f1 f2

Rd

23

4 Laboratory investigation

This chapter describes laboratory tests performed to simulate stress waves travelling through the rock, striking at the shotcrete and rock interface. The tests, described in this chapter and [Paper II], give data to be used in the investigation and demonstration of how stress waves and structural vibrations are connected. The results are also used for evaluation of the elastic stress wave model, see [Paper I] and Chapter 5. In addition, the bond strength development is investigated. It was concluded that the bond strength growth is initially faster than that of the compressive strength during the first 24 hours, see [Paper II] for more details.

4.1 Test set-up

Simulation of stress wave propagation through good quality granite, from an explosive charge towards a shotcreted rock surface, has been performed in the laboratory. A non-destructive laboratory experiment was set up with P-wave propagation along a concrete bar, with properties similar to rock [Paper II]. Cement based mortar with properties that resembles shotcrete was applied on one end of the bar, with a hammer impacting the other. Due to practical reasons the rock was made of concrete with similar dynamic properties as rock. The shotcrete was here substituted with cement mortar cast onto one of the quadratic end-surfaces of the concrete bar. The mortar thus formed a slab with the same cross section as the bar, bonding to the end-surface of the bar that corresponds to the rock surface in a tunnel. The layout of a test-bar suspended in cables is shown in Figure 4.1. Stress waves, similar to those observed in-situ, were induced at the opposite end of the bar through the impact of a steel hammer. The tests simulated incoming stress waves, giving rise to inertia forces caused by the accelerations acting on the shotcrete. These will in turn yield stresses at the shotcrete-rock (slab-bar) interface, which may cause bond failure. It is also possible that shotcrete may fail due to low tensile strength, i.e. here a failure within the slab. The concrete beam with dimensions of 0.3×0.3×2.5 m3 is shown in Figure 4.2, with its material properties given in [Paper II].

Chapter

CHAPTER 4. LABORATORY INVESTIGATION

24

Hammer

LRigid steel frame

Concrete bar

Hinge

Manual hammer release

Accelerometer

Wood swing arm

Steel cable hanger

Shotcete(mortar)

Swing angle

Computer

p1

p2 3 4pp

Figure 4.1: Schematic view of the set-up for hammer impact tests.

0.3

Figure 4.2: Details of the test specimen.

The impacting hammer had a weight of 38.6 kg, a length (Limp) of 0.3 m and a quadratic cross-section of 0.1×0.1 m2. It was mounted on a swing arm, able to hit the bar end perpendicularly, as in Figure 4.1. When a stress pulse propagates along a rod or a bar, high frequency components propagate faster than low-frequency components resulting in dispersion of the pulse. This phenomenon is known as material dispersion, or attenuation in a two-dimensional case. Another type of dispersion, geometric dispersion, occurs when the wavelength of the stress wave is of the same order as the dimension of the cross section of the bar. Generally, to prevent the dispersion of the pulse, small dimensions are required so that the wavelength always is greater than the dimension of the bar [9]. The wavelength of the P-wave is twice the length of the impacting hammer [31]. Therefore, the wavelength of the P-wave transmitted in the test bar is 2Limp and equal to 0.6 m, wavelength which is greater than the dimension of the bar (0.3 m). According to the pendulum equation applied for the hammer, the impact velocity is:

h 2 (1 cos ) v g L ( 4.1)

where L is the length of the hammer arm, is the swing angle and g = 9.81 m/s2 is the gravity constant.

4.2 MEASUREMENT RESULTS

25

Three accelerometers were positioned along the bar with a forth mounted onto the shotcreted end, as shown in Figure 4.1, enabling recording of particle accelerations as stress waves travels along the bar. The recording time was 0.1 s with a sampling frequency of 9600 Hz, the highest possible. According to [43], the recorded signals of impact must be low-pass filtered before the data are sampled, as shown in Figure 4.3. In order to prevent out-of-band signals from being improperly interpreted within the analysis range, a phenomenon known as 'aliasing', the transducer signals were filtered using a low-pass (Bessel) filter with 1200 Hz sample rate, available in the data acquisition system.

Figure 4.3: Low-pass filtering of frequency response.

Frequency spectra were obtained using the fast Fourier transform (FFT) routines of the Matlab numeric software [29]. Using the FFT introduces limitations in the resolution of the spectrum so that it is only possible to obtain information on frequencies up to the Nyquist sampling frequency [44], i.e. here 4800 Hz. It should be noted that the number of longitudinal resonance frequencies measured up to this frequency depends on the material and the boundary condition. Also, in order not to exceed the accelerometer capacity, the length of the swing arm was limited to 1.5 m and the tests were carried out with swing angles of 5°–40°, corresponding to impact velocities between 0.33 and 2.6 m/s. A series of tests with 50 and 100 mm thick slabs of shotcrete (mortar) was carried out where the bar with shotcrete was subjected to different intensity of impacts until failure. The tests were performed at shotcrete ages of 6 and 18 hours, see [Paper II]. The growth of the relative concrete-mortar bond strength and compressive strength during the first 24 hours are determined from laboratory pull-out tests and compression tests [13].

4.2 Measurement results

The measurements were divided into two parts; verification of the properties of the test-bar and the search for failure limit loads for the shotcrete. The first part was done to verify that the behaviour of the suspended test-bar is close to that of a free-free bar. Further details are


26

Figure 4.4: Measured acceleration time history and frequency spectra, for an impact velocity of 1.85 m/s.

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Acc

ele

ratio

n, m

/s2

Time,s0 1000 2000 3000 4000 5000

0

50

100

150

200

250

|Y(f

)|

Frequency, Hz

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n, m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency, Hz

|Y(f

)|

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Acc

ele

ratio

n ,m

/s2

Time,s0 1000 2000 3000 4000 5000

0

50

100

150

200

250

|Y(f

)|

Frequency, Hz

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n, m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency, Hz

|Y(f

)|

p1

p2

p3

p4

Impa

ctin

g ha

mm

er

Con

cret

e ba

r G

rout

ed c

oncr

ete

BS1

00-1

8

4.3 EVALUATION OF RESULTS

27

presented in [Paper II]. The acceleration time history and acceleration–frequency spectra for the four points shown in Figure 4.1 of specimen BS100-18 are plotted in Figure 4.4. All acceleration time histories and spectra that can be drawn from the performed measurements are similar to these figures. The results from the measured acceleration show that two waves propagate in opposite directions in the bar due to wave reflections; i.e. the incident wave and the reflected wave overlaps. Thus, the acceleration at the middle point (p2) is the sum of the two opposing waves. When a propagating wave reaches the interface between shotcrete and rock, a part of the wave energy will be transmitted into the shotcrete while the other part reflects back into the rock. This proportion depends on the impedance ratio between rock and the shotcrete, see [Paper II]. The transmitted wave propagates in the shotcrete and reaches the free surface where it is reflected back while doubling the acceleration, e.g see Dowding [15].

The accelerometer measurements verified that the suspended test-bar has dynamic properties similar to a free-free bar. The results from the measurements of frequencies show that the longitudinal vibrations is dominated by the first vibration mode and that there is little contribution from the second mode, as shown in Figure 4.4. All results contain none or very small contributions from the third and higher modes and the used sampling frequency can therefore be accepted as accurate in this case. Figure 4.5 show the acceleration on the shotcrete surface prior to failure and at failure, respectively. It can be seen from Figure 4.5(a) that the shotcrete is rigidly tied to the concrete surface until failure occurs, as shown in Figure 4.5(b), i.e. fully bonded until a sudden failure. Similar bond behaviour has been observed during testing [13]. By applying Eq. (3.22), it can be found that the concrete beam has a zero-frequency or rigid-body mode at n = 0. The modes corresponding to the non-zero frequencies of the concrete beam is shown in Figure 4.6.

4.3 Evaluation of results

The engineering simulation program Abaqus [1] was used to create a three-dimensional FE model analyzed with the Abaqus/Explicit solver. A model of the test-bar is shown in Figure 4.7, the solid elements and features are presented in [Paper II]. The interaction between the hammer, the concrete bar and the shotcrete layer were constrained using displacement boundary conditions, restricting the model to primarily describe particle displacements in the wave propagation direction. Between the concrete bar and the hammer free translation was only allowed in the longitudinal direction. The shotcrete was rigidly tied to the concrete surface to simulate the bond behaviour observed during testing [13]. The incident disturbing stress wave caused by the hammer was applied as surface to a surface contact using the kinematic contact method with definition of the initial velocity in the longitudinal direction of the hammer, with the velocity assumed to follow Eq. (4.1). The acceleration-frequency spectra of the FE model were compared to the acceleration-spectra of the measured data to determine which element size and time steps that could be used in the explicit analysis. The model consists of about 17000 nodes and 14000 elements.


28

Figure 4.5: Frequency spectra from the measured accelerations, specimen BS100-18, (a) the impact velocity is 1.85 m/s and (b) the impact velocity is 1.92 m/s.

Figure 4.6: The mode shapes of the free-free bar.

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n, m

/s2

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n, m

/s2

0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

Am

plitu

tde

Mode 1, 800Hz

0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

Am

plitu

tde

Mode 2, 1600Hz

0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

Length of beam, m

Am

plitu

tde

Mode 3, 2400Hz

0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

Length of beam, m

Am

plitu

tde

Mode 4, 3200Hz

(a)

(b)

Figure 4

The accpoints ogood agof the pThe imprelease Figure 4accelerameasurethe FE mto Eq. (surface Note thincreasesimulateAbaqus correlat

4.7: Finit

celeration tiof specimengreement wiparticle accepact producwaves are

4.9. These ation and, ed acceleratmodel provi(4.1), and inis shown in

hat the straes, dependined and mea can predi

te well with

te element m

me history n BS100-18 ith the measeleration at ces compresgenerated release w

consequenttion at p1 dide more dentegrated pn Figure 4.

aight lines ng on age aasured resulct this typthe experim

model of the

and accelerare plotted sured accelethe first m

ssive wavesat the free aves interatly, of stresemonstrate etailed resulparticle velo10, togetherrepresentingand thicknelts demonste of impac

mental resul

29

e experimen

ration-frequin Figures 4

erations, shmeasurements which proside surfac

act continuoss and strathe same p

lts. The relaocities fromr with the cg the latteress of the strates that thcts and thalts.

nt bar.

uency spectr4.8. The resown in Figut point (p1)

opagate throce and trailously and ain at the phenomenonation betwee

m measured correspondir results di

shotcrete. Ahe kinemati

at the resul

4.3 EVALU

ra of simulasults from thure 4.4. Howcan be seen

ough the barl the main cause flucsurface of n, slightly den impact vaccelerationg results fiverge as th

A good agreic contact mlts obtained

UATION OF R

ated results he FE modewever, a flu

en in the FEr. At the sawave, as s

ctuation of the bar [3

different duvelocities, acons on the sfrom the FEthe impact eement betwmethod avad using thi

RESULTS

for four els are in uctuation E results. ame time hown in particle

31]. The ue to that ccording shotcrete E model. velocity

ween the ailable in s model


30

Figure 4.8: Calculated acceleration time history and frequency spectra, for an impact velocity of 1.85 m/s.

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n,m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency,Hz

|Y(f

)|

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n,m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency,Hz

|Y(f

)|

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n,m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency,Hz

|Y(f

)|

0 0.02 0.04 0.06 0.08 0.1-1500

-1000

-500

0

500

1000

1500

Time,s

Acc

ele

ratio

n,m

/s2

0 1000 2000 3000 4000 50000

50

100

150

200

250

Frequency,Hz

|Y(f

)|

p1

p2

p3

p4

Impa

ctin

g ha

mm

er

Con

cret

e ba

r G

rout

ed c

oncr

ete

BS1

00-1

8

4.4 SHOTCRETE BOND STRESSES

31

Figure 4.9: Release wave on the surface of the bar [31].

Figure 4.10: Measured maximum particle velocity on the surface of 50 and 100 mm shotcrete of 6 and 18 hours shotcrete age, compared to the corresponding FE results [Paper II].

4.4 Shotcrete bond stresses

In order to show the dynamic stress in the shotcrete, an example from a FE analysis is presented in Figure 4.11. In this figure, the average stresses over the shotcrete elements closest to the rock surface are shown as function of time. A summary of the average stresses from the FE analyses are plotted in Figure 4.12. The lines show the relations between impact velocity and maximum stresses in the shotcrete, at the shotcrete-rock interface.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3

Max. particle velocity on surface, m

/s

Impact velocity, m/s

50mm_6hr

100mm_6hr

100mm_18hr

50mm_18hr

100mm_18hr_FE

100mm_6hr_FE

50mm_18Hr_FE

50mm_6hr_FE


32

Figure 4.11: Simulated stress in the shotcrete element closest to the test-bar for 100 mm

shotcrete, 18 hours old with the impact velocity 1.85 m/s.

Figure 4.12: Maximum stress in the shotcrete element closest to the test-bar. From FE

analyses for 50 and 100 mm shotcrete, 6 and 18 hours old [Paper II].

0 0.02 0.04 0.06 0.08 0.1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time,s

Str

ess

at i

nte

rfa

ce,M

Pa

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Impact velocity, vh (m/s)

Sho

tcre

te s

tres

s, σ

cb (

Mpa

)

18 hr., 100 mm6 hr., 100 mm18 hr., 50 mm6 hr., 50 mm

0 1 2 3 4 5

x 10-3

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

33

5 Summary of engineering models

Shotcrete support in hard rock tunnels is here first studied through numerical dynamic analysis using three different modelling approaches, engineering models with elastic material assumptions. The first model is a structural dynamic model that consists of lumped masses and spring elements. The second is a model built up with finite beam elements and connecting springs. The third is a one-dimensional elastic stress wave model. In this chapter, an outline is given of the principles of these three numerical models. The stress responses in the shotcrete closest to the rock when exposed to P-waves striking perpendicularly to the shotcrete-rock interface from these three models are compared in [Paper I]. The chapter ends with comparisons of the models with different load frequencies applied, for the modulus of elasticity set to 16 GPa and 40 GPa.

5.1 Mass-spring model

By using the theory of structural dynamics in analyzing the response of structures to ground acceleration caused by earthquake, the stress wave propagation within the shotcrete layer can be described. The considered system is shown schematically in Figure 5.1 and consists of lumped masses connected through elastic springs, i.e. a linear system, disregarding the damping. For the linear system, the relationship sf k u between the force fs and the resulting

displacement u depends on the stiffness k of the system.

Figure 5.1: Mass-spring systems; (a) single and (b) multi degrees of freedom.

Chapter

CHAPTER 5. SUMMARY OF ENGINEERING MODELS

34

The stiffness of a single spring is defined in terms of modulus of elasticity E, the cross section area A and the length L, as:

E Ak

L (5.1)

For un-damped single degree of freedom system subjected to a ground acceleration üg , the equation of motion governing the relative deformation u of the linear structure is

( ) mu k u p t (5.2)

where m is the mass and p the effective ground force, p = – m üg (t). Using Duhamel’s integral

to solve Eq. (5.2), the response to the ground acceleration is

0

1( ) ( ) sin[ ( )]

t

nn

u t p t dm

(5.3)

where /n k m . For the linear system, the constitutive material relationship is used to

calculate the stresses within the material where the strain is obtained by dividing the deformation by the original length. A similar idealization with lumped masses supported by a mass-less structure with stiffness k in the lateral direction can be used for multi degrees of freedom system. Therefore, the vibration of a unit area of the shotcrete layer connected through elastic springs can be described using lumped masses, as demonstrated in Figure 5.2. Details of the analysis is provided in [Paper I].

Figure 5.2: Mass-spring model with lumped masses and springs representing shotcrete exposed to vibrations from an explosive charge Q, [Paper I].

5.2 BEAM–SPRING MODEL

35

5.2 Beam–spring model

The fundamentals of this finite element based model are shown in Figure 5.3, where a section of rock with shotcrete is modelled using beam elements that represent flexural stiffness and mass. A finite beam element in two dimensions has six degrees of freedom describing the translations along the length of the beam and perpendicular to the axis, and the rotation. The model used for the numerical examples have 246 degrees of freedom. The beams are attached to the ground through elastic springs that also account for movement parallel to the rock surface. A beam element coupled to elastic springs at each end can be described by adding spring stiffness to the beam element stiffness. The formulae of this model is given in [Paper I] and [6]. The maximum allowed spring elongations are given by the tensile bending strength of rock and the bond strength between shotcrete and rock. The stresses that appear at the interface between rock and shotcrete are thus proportional to the elongation of the springs. In this two-dimensional model the load is represented by the time-dependent accelerations. The resulting response of shotcrete and rock is obtained through mode superposition analysis. For further details, see [Paper I].

Figure 5.3: Beam-spring model of shotcrete and rock exposed to vibrations from an explosive charge Q. Beam and spring elements interconnected at nodes with three degrees of freedom, [Paper I].

5.3 Elastic stress wave model

The elastic stress wave model is based on the theory of propagating waves, as is described in section 3.3. Since this model is one-dimensional, only an incident longitudinal P-wave (first arriving wave) is accounted for in the model. The transmitted and reflected waves are superposed to calculate the total stress in the shotcrete layer. A graphical description of the model is given in Figure 5.4 where a shotcrete strip with unity cross-section area is divided into n elements. The shotcrete-air interface has not been numbered as this interface is defined as a boundary condition. The thickness of each element, Δx, is chosen so that in each time


36

increment of the analysis the wave has propagated a distance of one element. A thorough description of the model is given in [Paper I] and [3, 20].

Figure 5.4: Elastic stress wave model of a shotcrete lining on a rock surface, exposed to a stress wave from an explosive charge Q, [Paper I].

5.4 Model comparison

The three models described in the previous sections are compared through a series of numerical examples. The material properties for shotcrete and rock are those given in chapter 2 and [Paper I]. A case with detonation of Q = 2 kg of explosives, corresponding to ANFO (ammonium nitrate and fuel oil), is considered. Series of calculations with 0 ≤ R ≤ 7 m have been performed, with corresponding particle velocities as given by Eq. (3.24). Note that for the elastic stress wave model these velocities are seen as impacting velocities and the particle velocities corresponding to stresses are half the maximum particle velocity, see [Paper I] for further details. For the other two models, the acceleration loads are considered. Results from the three models are compared and the maximum interface stresses that occur between shotcrete and rock for 100, 50 and 25 mm are shown in Figures 5.5–5.7, for E equal to 40 and 16 GPa, respectively. A crack distance of H = 0.5 m was assumed, with Eq. (3.25) giving a frequency of the incoming stress wave that is f = 2000 Hz and f = 1265 Hz and with a propagation velocity that is c = 4000 m/s and c = 2530 m/s, for E = 40 GPa and E = 16 GPa respectively. A full period for the incoming sinusoidal waves is thus 0.5×10-3 s.

5.4 MODEL COMPARISON

37

Figure 5.5: Maximum interface stresses for 100 mm thick shotcrete and (a) E=40 GPa and (b) E=16 GPa.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance from charge, m

Max

imum

inte

rfac

e st

ress

es,

MP

a

fcbfcbfcb

Elastic wave

Mass-springBeam-spring

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Max

imum

inte

rfac

e st

ress

es,

MP

a

fcb

Elastic wave

Mass-spring

Beam-spring

(a)

(b)


38


0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Max

imum

inte

rfac

e st

ress

es,

MP

a

fcb

Elastic wave

Mass-spring

Beam-spring

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Max

imum

inte

rfac

e st

ress

es,

MP

a

fcb

Elastic wave

Mass-spring

Beam-spring

(a)

(b)

5.4 MODEL COMPARISON

39


0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Max

imum

inte

rfac

e st

ress

es,

MP

a

fcb

Elastic wave

Mass-spring

Beam-spring

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Max

imum

inte

rfac

e st

ress

es,

MP

a

fcb

Elastic wave

Mass-spring

Beam-spring

(a)

(b)

41

6 Finite element analysis

The finite element (FE) method is a powerful tool for study of structures subjected to dynamic forces. In this chapter, two-dimensional (plane strain) dynamic FE models of shotcrete and rock subjected to stress waves are presented, where the models, load formulations and boundary conditions are described. The FE models are used to simulate the stress waves from blasting, propagating in rock towards shotcrete on a tunnel wall. Due to the inhomogeneous nature of the rock, the stress waves attenuate on its way from the point of explosion towards the shotcrete on the rock surface. Material damping for the rock-mass is estimated from in-situ measurements, as mention in section 3.4. The effect of material damping and the elastic modulus on the propagation of the stress waves are presented in [Paper III].

6.1 Explicit dynamic analysis

The models are here implemented using the FE program Abaqus [1]. For dynamic problems, Abaqus offers several numerical solution routines, implicit and explicit. For short dynamic response times, e.g. from blasts and impacts, the explicit dynamic analysis is computationally efficient. The explicit analysis procedure in Abaqus/Explicit is based on an explicit integration rule together with the use of diagonal or lumped element mass matrices. The equation governing the motion can be expressed as

( ) ( ) ( )i i iu M F I (6.1)

where M is the diagonal lumped mass matrix, F the applied load, and I the internal force vector. The equations of motion for the body are integrated using the explicit central difference integration rule, given by

( 1) ( )1 1( ) ( ) ( )2 2

2

i ii i it t

u u u

(6.2)

Chapter

CHAPTER 6. FINITE ELEMENT ANALYSIS

42

and

1( )( 1) ( ) ( 1) 2ii i iu u t u

(6.3)

The explicit procedure integrates through time by using small time increments Δt. The stability limit for the operator is given in terms of the highest element eigenvalue in the system as

max

2t

(6.4)

An amount of damping is introduced to control high frequency oscillations and therefore, the stable time increment is given by

2

max

2( 1 )t

(6.5)

where ξ is the damping ratio. Introducing damping reduces the stable time increment and a trial value is computed for each element in the mesh using Eq. (6.4). A conservative estimate of the stable time increment is given by the minimum value for all elements. For practical use Eq. (6.4) can be rewritten as

min e

p

Lt

c

(6.6)

where Le is the characteristic element length and cp the current effective P-wave propagation velocity in the material.

6.2 Solid infinite elements

Infinite elements are intended to be used for problems in which the region of interest is small in size compared to the surrounding medium, i.e. as here the tunnel cavity in the ground. Standard finite elements should be used to model the region of interest, with infinite elements for the far-field region. The node numbering for infinite elements must be defined such that the first face is connected to the finite element part of the mesh. These nodes are located away from the finite element mesh in the infinite direction [1]. An example is shown in Figure 6.1, where for a case with a load applied on a square hole of a quarter plate, the second node along each edge pointing in the infinite direction must be positioned so that it is twice as far from the applied load point. In addition to this length consideration, the second node must be specified in the infinite direction so the element edges in the infinite direction do not cross over, which would give non-unique mappings, see Figure 6.2.

6.3 BLAST-INDUCED VIBRATION LOADS

43

Figure 6.1: Quarter plate with square hole [1].

Figure 6.2: Examples of an acceptable and an unacceptable two-dimensional infinite element [1].

6.3 Blast-induced vibration loads

Dynamic excitation of an elastic system describing the rock results in displacements of particles that can be observed as a wave propagation. This motion of particles can be described either as displacements, velocities or accelerations. The time dependence can be used to describe as a sine-pulse acceleration with the period 1/f, as:

max( ) 2 sin (2 )a t f v f t (6.7)

where vmax is obtained from Eq. (3.24). The particle velocity is calculated by integrating the acceleration over time:


44

0

( ) ( )t

v t a t dt (6.8)

giving the particle velocity as a cosine wave with an initial boundary condition equal to vmax, as:

max max( ) cos(2 ) v t v f t v ( 6.9)

As mention in section 4.1, the length of the propagating wave will be twice that of an impacting body [31]. Therefore, the relation between the incident velocities v and the velocity vi in the rock will be

2iv v ( 6.10)

The particle velocity of the propagating stress wave is thus half of that of the body, or rock section, that impacts with the velocity vmax. The principle of this summation is described in [Paper I]. Substitution of Eq. (6.10) in Eq. (6.9), gives the particle velocity as

max max( ) cos(2 )2 2

v

v tv

f t ( 6.11)

Figure 6.3 depicts the quantities used to describe the vibratory motion. According to Eqs. (6.7) and (6.11), the particle acceleration and the integrated particle velocity can be drawn in Figure 6.3 (a) and (b), respectively. The vmax is here assumed to be 0.9 m/s and f is 2000 Hz.

6.4 Finite element models

A series of examples have been analyzed using the two-dimensional (2D) plane strain elements of the Abaqus/Explicit FE program [1]. The models used, describe two cases with respect to the geometry of the tunnel and position of the detonation point. The fundamentals of the models are shown in Figure 6.4. The first case is shown in Figure 6.4(a), where the wave propagates through the rock from the detonation point, the centre of the charge, towards the tunnel side. The second case is used for studying the effects of stress wave propagation along the tunnel sides as a result from different detonation points, as shown in Figure 6.4(b). The geometries of the FE models are created using the Abaqus/CAE interface, with the applied velocity loads written in input data files using the numeric software Matlab [29].

6.4 FINITE ELEMENT MODELS

45

Figure 6.3: The vibratory load as (a) particle acceleration and (b) particle velocity.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Time, s

Acc

eler

atio

n, m

/s2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-4

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Time, s

Vel

ocity

, m

/s

(a)

(b)


46

(a) (b)

Figure 6.4: Examples of stress waves in rock; (a) tunnel profile and (b) tunnel plane.

An example of meshing and use of finite elements is shown in Figure 6.5, describing one model of case (a) and one of case (b) in Figure 6.4. Note that only a short length of the tunnel is included for case (b). 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 rock and the shotcrete behave in a strictly elastic manner and possible failure thus occurs when the stresses at the shotcrete-rock interface exceeds the bond strength. That is, no plastic deformations or permanent failure, for example, crushing or cracking, was considered in the models and a linear elastic relationship between stress and strain was assumed.

6.4 FINITE ELEMENT MODELS

47

Y

X

Ground surface

Rock

Tunnel

RPPV

Infinite elements

Finite elements

R

20 m 20 m

Shotcrete

Figure 6.5: Configuration of the finite element model; (a) tunnel profile [Paper III] (b) tunnel plane.

(a)

(b)


48

The tunnel in case (a) is situated 11.5 m below the ground surface. The material surrounding the excavation is discretized with first-order 4-node plane strain elements of type “CPE4R”, recommended for simulations of impact and blast loading using Abaqus/Explicit [1]. The infinite extent of the rock is represented by a 20 m wide mesh that extends from the surface to a depth of 45 m below the surface. Far-field conditions on the bottom and right-hand side boundaries are modelled using infinite elements of type “CINPE4”. The interaction between rock and shotcrete was modelled using tie constraints, i.e. no relative displacement between the materials was assumed. The element size of the shotcrete part is 0.01×0.1 m2 for all models, with different element sizes used for the rock part. Depending on the accuracy and details of the solution, some regions of the rock are discretized with a refined mesh. More refinement adjacent to the tunnel opening and loading area was made, due to the significant deformations expected at these regions. The model consists of about 28000 nodes and 2500 elements.

The detonation is introduced in the model from a circular area within the rock where an incident 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 chosen so that the included rock remains undamaged outside this area. For Swedish hard rock, a PPV damage criterion is given by Persson [36], where the threshold damage (incipient damage) occurs at a PPV of 1.0 m/s, as shown in Table 2.1. 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 incipient damage according to Table 2.1. From Eq. (3.24), the depth of the damage zone RPPV, where the particles velocity reaches the threshold PPV, will thus be

0.846 (m)ppvR Q (6.12)

where Rppv then is the depth of the damage zone into the rock mass measured from centre of the charge to where the particles velocity reaches the threshold PPV. This distance thus corresponds to the limit for rock damage and therefore elastic properties can be assigned to the rock outside this area. In the real case, the rock area immediately around the hole 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. To allow the stress wave to disperse a relatively large volume of rock is modelled, compared to the distance to the centre of the explosives. The integration method used for calculating the response is exact for loads that vary piecewise linearly with time. Therefore, it is better to define the load-function of each node at the charge hole as tabular data. To reduce the effect of the numerical differentiation, a smooth curve of velocity time history is used with Δt = 0.01×10-3 s. To define the velocity at each node n, global Cartesian coordinate components are used as shown in Figure 6.6. The two components depend on the angle γ of the resultant velocity to the x-axis. Each component describes the velocity components in the given direction at one node.

6.5 EXAMPLES

49

Figure 6.6: Components of the incident velocity at each node.

6.5 Examples

Results from a series of different models with varying distance between shotcrete and explosive charge are presented in [Paper III]. The propagation of the waves in the rock was investigated by applying a particle velocity at the charge hole as a velocity boundary condition. The FE model consists of rock and infinite boundary conditions without the tunnel opening. In this model, the rock is assumed to be intact or fractured with the modulus of elasticity equal to 40 and 16 GPa, respectively. In the first examples of [Paper III], a damping ratio of 8 % is used, estimated from in-situ measurements. The attenuation of the particle velocities is compared with results from the empirical formula obtained from Eq. (3.24).

In this chapter, the two cases described in [Paper III] and here in section 6.4 are studied using a series of models with varying tunnel shapes and location of explosive charges. The shotcrete was assumed to have a density of 2100 kg/m3 and a modulus of elasticity of 27 GPa and the rock a density of 2500 kg/m3 and a modulus of elasticity of 40 GPa. Detonation of Q = 2 kg of explosives, corresponding to ANFO (ammonium nitrate and fuel oil), is considered for series of calculations for the two cases shown in Figure 6.4, with the corresponding PPV given by Eq. (6.11). The principle frequency of the incident particle velocity is assumed to be 2000 Hz for all models. Results for 100 mm thick shotcrete are studied. In addition, examples demonstrating the effect of shotcrete age are also presented, for detonation of Q equal to 0.5, 1.0 and 2.0 kg, respectively.

6.5.1 Tunnel profiles

As mention in section 6.3, the dynamic excitation can be described as a cosine-pulse velocity or a corresponding sine-pulse acceleration. Figure 6.7 shows the time history of acceleration and velocity, used for the case with a distance of 4 m from the centre of the charge, when the velocity boundary conditions are applied at the perimeter of the loading area.


50

Figure 6.7: Example of calculated acceleration (a) and corresponding particle velocity (b) at 4 m from the center of the explosive charge.

The tensile stresses that occur at the interface between shotcrete and rock are investigated. The results from six models of horseshoe shaped tunnels, as shown in Figure 6.4, with distances R of 2.2, 3.0, 4.0, 5.0, 6.0 and 7.0 m from the centre of the charge are presented in [Paper III] and compared with the results from the engineering models, described in chapter 5. The time histories of stresses at the shotcrete element closest to the rock surface, calculated for the distance R equal to 3, 5 and 7 m, are shown in Figure 6.8

0 1 2 3 4 5 6

x 10-3

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time, s

Vel

ocity

, m

/s

0 1 2 3 4 5 6

x 10-3

-2000

-1500

-1000

-500

0

500

1000

1500

Time, s

Acc

eler

atio

n, m

/s2

(a)

(b)

6.5 EXAMPLES

51

Figure 6.8: Stress in the interface between shotcrete and rock for various distances from the centre of the explosives (Q=2 kg) [Paper III].

In addition to the case of the horseshoe shaped tunnels, which is presented in [Paper III], the two other tunnel profiles corresponding to case (a) in Figure 6.4 are included in this section. The calculations were carried out with R = 3.0 m from the centre of the charge, with a damping ratio of 8 % and with Erock and Eshot equal to 40 GPa and 27 GPa, respectively. The same boundary conditions as used in [Paper III] is assumed. The stresses in the x- and y-directions are presented in Figures 6.9 – 6.14. The stresses are indicated with red areas showing tensile stresses (positive) over 0.2 MPa and blue showing compressive stresses (negative) lower than – 0.2 MPa. Far away from the charge hole and the tunnel, zero stress (green) can be assumed.

6.5.2 Tunnel plane

In this section, three different position of the explosive corresponding to case (b) in Figure 6.4 were simulated and presented in Figures 6.15 – 6.20. The calculations were carried out with R = 3.0 m from the centre of the charge, with properties identical to the models in section 6.5.1. The examples investigate the tensile stresses that occur at the interface between shotcrete and rock along the length of the tunnel when the explosion occurs in front of the tunnel face, as shown in Figure 6.4 (b). In this modelling approach, appropriate simulation of infinite boundary conditions must be considered, here applied around the finite element of the rock, representing a horizontal section of the ground, as shown in Figure 6.5 (b). Figures 6.15 – 6.20 show the tensile stresses propagating through the rock towards the shotcrete along the wall of the tunnel, from point A to point B (see Figure 6.4(b)).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time, s

Inte

rfac

e st

ress

es,

MP

a

R = 3 m

R = 5 mR = 7 m


52

Figure 6.9: Contours of stresses in x-direction (σx) of the horseshoe shaped tunnel. Deformation scale 1:1000.

t = 0.9×10-3s

σx, Pa

t = 1.1×10-3s

t = 2.3×10-3s

x

y

6.5 EXAMPLES

53

Figure 6.10: Contours of stresses in y-direction (σy) of the horseshoe shaped tunnel. Deformation scale 1:1000.

t = 0.9×10-3s

σy, Pa

t = 1.1 ×10-3s

t = 2.3 ×10-3s

x

y


54

Figure 6.11: Contours of stresses in x-direction (σx) of the square tunnel. Deformation scale 1:1000.

x

y

t = 0.9×10-3s

x, Pa

t = 1.8×10-3s

t = 3.5×10-3s

6.5 EXAMPLES

55

Figure 6.12: Contours of stresses in y-direction (σy) of the square tunnel. Deformation scale 1:1000.

t = 0.9×10-3s

σy, Pa

t = 1.8 ×10-3s

t = 3.5 ×10-3s

x

y


56

Figure 6.13: Contours of stresses in x-direction (σx) of the circular tunnel. Deformation scale 1:1000.

x

y

t = 0.9×10-3s

x, Pa

t = 2.1×10-3s

t = 2.7×10-3s

6.5 EXAMPLES

57

Figure 6.14: Contours of stresses in y-direction (σy) of the circular tunnel. Deformation scale 1:1000.

t = 0.9×10-3s

t = 2.1 ×10-3s

t = 2.7 ×10-3s

x

y

σy, Pa


58

Figure 6.15: Contours of stresses in z-direction (σz) of the side walls of the tunnel. Deformation scale 1:1000.

z

x

t = 1.8×10-3s

σz, Pa

t = 3.0×10-3s

t = 4.7×10-3s

6.5 EXAMPLES

59

Figure 6.16: Contours of stresses in x-direction (σx) of the side walls of the tunnel. Deformation scale 1:1000.

t = 1.8×10-3s

σx, Pa

t = 3.0 ×10-3s

t = 4.7 ×10-3s

z

x


60


z

x

t = 1.1×10-3s

σz, Pa

t = 2.1×10-3s

t = 4.0×10-3s

6.5 EXAMPLES

61


t = 1.1×10-3s

x, Pa

t = 2.1 ×10-3s

t = 4.0 ×10-3s

z

x


62


z

x

t = 2.0×10-3s

z, Pa

t = 2.3×10-3s

t = 4.3×10-3s

6.5 EXAMPLES

63


t = 2.0×10-3s

x, Pa

t = 2.3 ×10-3s

t = 4.3 ×10-3s

z

x


64

6.5.3 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. The model used is case (a), the horseshoe shape tunnel, but here Eshot is assumed to be a variable according to Eq. (2.1). Multiple calculations have been performed for shotcrete ages between 1 and 24 hours, for R = 2.2, 3.0 and 5.0 m with Q = 0.5, 1.0 and 2.0 kg, respectively. The results are presented in Figure 6.21 and compared to the bond strength curve according to Eq. (2.4) and results given in [2].

Figure 6.21: Tensile stress in hardening shotcrete, at the shotcrete-rock interface. Calculated for (a) Q = 0.5 kg, (b) Q = 1.0 kg and (c) Q = 2 kg.

0 3 6 9 12 15 18 21 240

0.5

1

1.5

2

2.5

3

Age of shotcrete, hrs

Max

imum

ten

sile

str

ess,

MP

a

R = 2.2 m

R = 3.0 m

R = 5.0 mBond strength Eq.(2.4)

Bond strength [2]

(a)

6.5 EXAMPLES

65

Figure 6.21(cont.).: Tensile stress in hardening shotcrete, at the shotcrete-rock interface. Calculated for (a) Q = 0.5 kg, (b) Q = 1.0 kg and (c) Q = 2 kg

0 3 6 9 12 15 18 21 240

0.5

1

1.5

2

2.5

3


Max

imum

ten

sile

str

ess,

MP

a

R = 2.2 m

R = 3.0 m


Bond strength [2]

0 3 6 9 12 15 18 21 240

0.5

1

1.5

2

2.5

3


Max

imum

ten

sile

str

ess,

MP

a

R = 2.2 m

R = 3.0 m


Bond strength [2]

(b)

(c)

67

7 Conclusions

7.1 Discussion and recommendations

The examples presented and the numerical models used are here summarized and commented. Recommendations for practical use are given based on the results from the engineering models and finite element models, also compared to some relevant previous results.

7.1.1 Laboratory investigation

The concrete bar used in the laboratory investigation had a wave propagation velocity close to 4000 m/s and can thus in this respect be accepted as a substitute for a bar of rock corresponding to good quality granite. In addition, the density of the concrete was close to that of granite with a corresponding modulus of elasticity of approximately 37 GPa, also accurate in this respect. The laboratory and numerical results given in Figures 4.4 and 4.8 show a good correspondence between measured and calculated shotcrete response, in both time and frequency domains. From Figures 4.4 and 4.8, it can be seen that the longitudinal vibrations are dominated by the first vibration mode and that there is very little contribution from the second mode. The frequencies of the first and the second mode are around 800 and 1600 Hz. Very small contributions from the third and higher modes can be seen. The used sampling frequency of 4800 Hz is therefore sufficient in this case.

The growth in bond strength of the mortar, used as a substitute for shotcrete, is initially faster than that of the compressive strength, when related to the levels reached during a relatively short time span of 24 hours. It should be noted that the tests were performed in laboratory environment under room temperature (approx. +20º C) which naturally is higher than in e.g. a tunnel environment, where also contact with the relatively cold rock affects the hardening of cementitious materials. The strength levels reached are however still comparable with previous observations, which are summarized in section 2.3, and the mortar can therefore be accepted as an accurate substitute for shotcrete in these tests.

Chapter

CHAPTER 7. CONCLUSIONS

68

The relation between impact velocities and particle velocities measured on the shotcrete surface is shown in Figure 4.10, together with the corresponding results from the presented FE model. Note that the straight lines representing the FE results diverge as the impact velocity increases, depending on age and thickness of the shotcrete. The highest surface vibration velocity for each case in Figure 4.10 agrees with previously observed maximum particle velocities (see section 1.3). For young (1–25 hours old), approximately 50 mm thick shotcrete the highest velocity measured for an undamaged lining is within 0.5–1.0 m/s [3]. The results marked in Figure 4.10 show that for 50 mm shotcrete values of around 0.2 m/s were measured at 6 hours of age and > 0.55 m/s at 18 hours. Numeric modelling has previously been used to show that these limits should be within 0.15–0.30 m/s [2]. For 100 mm thick shotcrete the corresponding limits are lower, approximately 0.10 and 0.40 m/s, respectively, which also are in good correspondence with the numerical results that gives a value just under 0.30 m/s for the thicker shotcrete. The results in Figure 4.12 show that the

limit values for 6 hours old shotcrete correspond to a bond stress (cb) of about 0.1 MPa. For 100 mm thickness, it is about 0.55 MPa for 18 hours. These results are in good agreement with the bond strength, e.g. 0.1 and 0.7 MPa, respectively, given in [Paper II]. If the line

representing 18 hours old, 50 mm thick shotcrete is extrapolated cb ≈ 0.55–0.70 MPa a relatively high impact velocity of about 5.5–7.5 m/s will be reached.

The effect of the aging shotcrete is studied in [Paper II]. When the two materials rock and shotcrete have the identical properties, the waves will pass through as if these were one material. For very young shotcrete with a low propagation velocity, the impedance ratio is low and therefore a relatively large fraction of the wave energy is reflected back into the rock. Young shotcrete will thus be exposed to lower stress levels compared to stiffer shotcrete from the same incoming wave.

7.1.2 Engineering models

For these models described in [Paper I] and chapter 5, it should be noted that two are based on structural dynamics and one on elastic stress wave theory and that the definition of the vibration load therefore is different. For the stress wave model the dynamic load is defined as a time dependent velocity while a time dependent acceleration is used for the other two model types. All models are based on linear elastic material theory and thus not able to describe partially damaged structures, e.g. partial debonding of shotcrete. The models can, however, be used to identify the limit for damage through series of calculations with increasing load level until either the tensile strength or the bond strength is exceeded

The three models have been tested with Q = 2 kg and three different shotcrete thickness; 100, 50 and 25 mm. The maximum bond stresses that occur between shotcrete and rock as a function of distance are presented in Figures 5.5–5.7. The limit values are compiled in Figure 7.1.

7.1. DISCUSSION AND RECOMMENDATIONS

69

Figure 7.1: Compiled minimum safe distances from detonation of Q= 2 kg. Dependence of load frequency f and rock modulus of elasticity E.

The results are calculated for two different incoming stress waves with f = 2000 Hz and f = 1265 Hz and with propagation velocities of c = 4000 m/s and c = 2530 m/s, corresponding to E = 40 GPa and E = 16 GPa respectively. For 100 mm thick shotcrete it can be seen that the three models almost coincide and that the limiting, safe distance for fcb = 1.0 MPa is around 2.5 and 1.8 m from the centre of the charge, for Erock = 40 and 16 GPa, respectively. The corresponding results for 50 mm thick shotcrete give a safe distance of 1.5 and 1.0 m obtained with Erock = 40 and 16 GPa and for 25 mm thick shotcrete it corresponds to 0.8 and 0.7 m, respectively.

For 100 and 50 mm thick shotcrete Figures 5.5 – 5.7 show that the results from the three models almost coincide. For a shotcrete thickness of only 25 mm there is a scatter in the results. The mass-spring model gives higher stresses due to the relatively small mass that gives low inertia forces compared to the spring forces. It can also be seen that the elastic wave model gives stresses lower than 2.0 MPa for small values of R. This is caused by the superposition effect when the wave immediately reaches the interface and then, due to the thin shotcrete, directly reflects at the free surface and transmits back into the rock. It must, however, be pointed out that all three models are invalid when stresses over the limit fcb are reached since this indicates debonding of the shotcrete, e.g. corresponding to springs being uncoupled. For large values of R and 25 mm thickness, the elastic wave model and the beam-spring model give identical results while the mass-spring model still overestimates the stresses.

0 25 50 100

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Shotcrete thickness, mm

Dis

tanc

e fr

om t

he c

harg

e ce

ntre

, m

f = 1265 Hz, E = 16 GPa

f = 2000 Hz, E = 40 GPa


70

7.1.3 Finite element models

The tested finite element models provides detailed two-dimensional displacement fields not possible to obtain with the less sophisticated engineering models, that should be seen as prototype models. The results show how the explosion generate stress waves that propagates towards the free, shotcreted tunnel surface. Due to this motion, the stresses on the shotcrete vary in time, the stresses being the sum of the incident and reflected stress waves. An example of the variation of stresses at the shotcrete-rock interface is shown in Figure 6.8. It is clearly demonstrated that the peak stress occur later for longer distances while the stress levels decrease. The effect of damping can also be seen. The wave arrives at the point of observation as compressive stress (negative sign) directly followed by the tensile maximum, which is critical with respect to bond failure at the interface. As seen in Figure 6.8, the maximum response in the shotcrete occurs shortly after applying the blast load.

A feature of finite element stress analysis is the large amount of data generated, making the information suitable for presentation in graphical form. The stress distributions due to blast loads applied as boundary conditions at the perimeter of the circular area with the radius RPPV, are shown in Figures 6.9 – 6.20. The stresses are visualized as components in the x- and y- directions, or in the z- and x- directions. The stresses that occurs around three different tunnel profiles are depicted in Figures 6.9 – 6.14. Shortly after the explosion, the compressive stress waves in the x-direction (σx) reach the tunnel and is transmitted into the thin layer of the shotcrete tied to the rock surface. Then, the compressive stress waves starts to reflect to the surrounding rock, as shown in Figures 6.9 and 6.11. Reflected, tensile stress waves appear on the upper and lower sides of the shotcrete layer continuing towards the end of each side, being concentrated around the corners, especially for the square tunnel as shown in Figure 6.11. This phenomenon also appeared on the far side of the circular tunnel, as shown in Figure 6.13. In Figures 6.10 and 6.12, the contour of the stresses in the y-direction (σy) are depicted. It can be seen that the reflected tensile stresses appeared on the upper and lower sides of the square tunnel while there is no tensile stresses on the upper side of the horseshoe shaped tunnel. In the circular tunnel, the tensile stresses appeared on the far side from the location of the explosion, as shown in Figure 6.14. It should be pointed out that the time span of the analysis is 10 ×10-3 s in all analyses and absorbing boundaries were used to eliminate stress wave reflection from the outer edges of the models, beyond which the rock continues.

The models of case (b) demonstrate detonation ahead of a tunnel front and the stresses that occur along the tunnel walls, a geometry not previously studied. The centre of the explosive charge was here located 3 m from the front. Compared to the results in [Paper III], this is a safe distance with respect to damage at the front where the vibration velocity is vmax = 0.47 m/s, as given by Eq. (3.24), i.e. just below the damage limit of 0.5 m/s previously defined [4]. The results in Figures 6.15 – 6.20 show a dominance by stresses along the tunnel walls; i.e. in the z-direction (σz). Of the three previously used engineering models, shear stresses could only be described by the beam-spring model. In the study of large scale blasting during mining operations [2], where stress waves reached the shotcrete in an oblique angle, a domination of shear stresses was also observed. It can be seen that the only high normal

7.1. DISCUSSION AND RECOMMENDATIONS

71

stresses (x-direction) appear locally just behind the tunnel face while the maximum shear stresses (z-direction) are situated 2 m into the tunnel. In addition, note that the shear stresses are present more than 10 m into the tunnel. See [Paper III], for more details. Figures 6.16 shows that after the explosion the tensile stresses in x-direction (σx) appeared on both shotcreted sides of the tunnel. In Figures 6.18 and 6.20, it can be seen that relatively large tensile stresses also occurred at points on the far side of the tunnels.

Minimum ages at the time of blasting for 100 mm thick shotcrete are compiled and presented in Figure 7.2. The results are calculated for detonations of 0.5, 1.0 and 2.0 kg of explosive at 2.2, 3.0 and 5.0 m from shotcrete on a rock surface, obtained from comparison with the bond strength fcb of Eq. (2.4), as shown in Figure 6.21,

Figure 7.2: Compiled minimum ages for 100 mm thick shotcrete vs the distance from detonation.

The effect of introducing damping was investigated in [Paper III] and shown to give results closer to the empirical results of Eq. (6.11). The latter is based on in-situ measurements and therefore includes factors that influence the attenuation of vibration velocity with distance. In the presented examples a damping ratio of 8 % have been used, based on in-situ measurements. In Abaqus/Explicit the combined dynamic effect of damping and increased strain rate is accounted for, corresponding to a dynamic magnification of the modulus of elasticity. In the finite element models, the stress waves propagate from the edge of the loading area, making it difficult to calculate reliable results for very short distances R. As described in [Paper III] when calculating stresses it is important to use a representative modulus of elasticity since it governs the relation between displacement and stresses in the material. The assumption of a good quality rock mass with a high Erock will always give over-

0 2.2 3 5 66

8

10

12

14

16

18

20

22

Distance, m

Age

of

the

shot

cret

e, h

r

Q = 0.5 kg

Q = 1.0 kg

Q = 2.0 kg


72

estimated, safe distances. Lower, more realistic values for Erock can be used if the rock quality is well defined, allowing blasting closer to the shotcrete.

7.1.4 Recommendations for practical use

On basis of the results compiled in Figure 7.1, recommendations for minimum safe distances to a point of detonation of Q = 2 kg of explosives are given in Table 7.1. The results from [5] are also given for comparison. The table thus give a comparison between values for varying rock quality and load frequencies. Note that the safe distances for Erock = 16 GPa and f = 1265 Hz are lower than for Erock = 40 GPa and f = 2000 Hz, which is reasonable due to the different excitation values as well as to the lower values for the modulus of elasticity that represent the occurrence of cracks and imperfections. Note that an increase in load frequency leads to higher load levels and longer safe distances.

Table 7.1: Recommended minimum safe distance from the detonation of Q = 2 kg.

Rock and load characteristics Shotcrete thickness

100 mm 50 mm 25 mm

Erock= 16 GPa and f = 1265 Hz 1.8 m 1.0 m 0.7 m Erock= 40 GPa and f = 2000 Hz 2.5 m 1.5 m 0.8 m Erock= 40 GPa and f = 2500 Hz [5] 3.5 m 1.9 m 1.2 m

The results presented in Figure 7.2 are used as a basis for recommendations of allowed minimum shotcrete ages to be used in tunnelling and underground constructions. Recom-mended minimum ages for 100 mm thick shotcrete subjected to detonation of 0.5, 1.0 and 2.0 kg of explosive at 2.2, 3.0 and 5.0 m from shotcrete are given in Table 7.2. For comparison the table also contain the corresponding values obtained for the slower developing bond strength curve from [2], here given in Eq. (2.4).

Table 7.2: Recommended minimum ages for 100 mm thick shotcrete. The values within parentheses refer to the bond strength from [2].

Distance form explosive R Detonation Q

0.5 kg 1.0 kg 2.0 kg 2.2 m 12 hrs.

(>24 hrs.) 18 hrs.

(>48 hrs.) *

3.0 m 10 hrs. (24 hrs.)

13 hrs. (>24 hrs.)

21 hrs. (>48 hrs.)

5.0 m 7 hrs. (9 hrs.)

9 hrs. (21 hrs.)

12 hrs. (>24 hrs.)

*Not possible to obtain a sufficiently high bond strength.

7.2. GENERAL CONCLUSIONS

73

7.2 General conclusions

A type of finite element models suitable for dynamic analysis of shotcrete exposed to vibrations from blasting has been developed, implemented using the Abaqus/Explicit [1] program. The models based on 2D solid finite elements makes it possible to performed more accurate analyses than with the previously used prototype models, resulting in detailed results.

Three existing prototype (engineering) models have been compared and evaluated, also with respect to the results from the finite element models. The stress wave and structural dynamic properties have been compared with focus on the relation between applied loads and calculated resulting displacements and stresses. The analysis results have also been compared to results previously obtained in-situ and results from a laboratory study carried out within this project. The latter was set up in order to obtain further data to be used in the evaluation of the numerical results.

An important goal was to identify factors and material properties that greatly affect the analysis results and how these should be accounted for in the numerical models. It was seen that it is important to include rock damping in the models to include the effect of decreasing stress levels as the disturbing waves propagate through the rock, away from the source of blasting. It was also observed that the modulus of elasticity for the rock is important since it affects the wave propagation velocity and the frequency content of the stresses that reach the shotcrete-rock interface. The complex superposition of incident, reflected and transmitted stress waves at the interface can be illustrated by the results from the suggested analytical models.

It was demonstrated how the analysis method can be used to identify acceptable limits to be included in practical recommendations. A set of recommended minimum safe distances and minimum ages are given, adding further information to existing, preliminary recommenda-tions [2]. Thus, the suggested modelling approach is a suitable tool for the work of determining guidelines for practical use in tunnelling and underground work in hard rock involving blasting close to shotcrete.

7.3 Further research

A further development of the presented type of finite element models will be an important task for the future research. The goal is to find well-defined limits for safe blasting, with respect to both shotcrete age and distance to the vibration source. The models will be used to study more complex geometries than before, such as construction of multiple tunnels, parallel tunnels, crossing tunnels and also stress waves with oblique impacts. The effect of different shotcrete material properties should be investigated, with focus on young and hardening shotcrete, but also accounting for possible inclusion of fibres. The effect of additional reinforcing elements, such as rock bolts, cables and wire mesh, can also be included.


74

Cases with multiple explosive charges will also be studied, aiming at finding the least damaging sequence of detonation using different delay times. The models can be further developed to also consider the loading plan for the blasting rounds, i.e. initiation of explosives with a time interval. The initiation sequence of the explosives can e.g. be assumed according to the experiments performed at ASPE [34] and TASQ [35] tunnels in Sweden. In addition, the effect of loading density and rock mass properties along the charge hole will be investigated. The models can also be adjusted to include the effect of both intact rock and a zone of rock fractured due to blasting, situated around the tunnel perimeter, as shown in Figure 7.3. It will thus, by applying different modulus of elasticity to the two rock areas, be possible to study the protecting, filtering effect that the cracked rock volume has. Non-linear properties and behaviour of the rock and shotcrete can be introduced, making it possible to describe partial shotcrete failure.

Further in-situ measurements are required in order to establish a complete set of guidelines for efficient use of shotcrete in environments exposed to vibrations. Detailed information about tunnel geometry, rock types and plans for blasting operation must be used as input in the finite element analyses that will be used for evaluation of the measurement results. The material properties of shotcrete must also be investigated further, in laboratory tests that also include comparison with in-situ results and observations.

Explosive charages

Shotcrete lining

Fracture rock

(a) (b)

Figure 7.3: Schematic of tunnel surrounded by fractured rock, (a) tunnel profile and (b) tunnel plane.

75

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Appended papers

List of Bulletins from the Department of Structural Engineering, The Royal Institute of Technology, Stockholm TRITA-BKN. Bulletin Pacoste, C., On the Application of Catastrophe Theory to Stability Analyses of Elastic Structures. Doctoral Thesis, 1993. Bulletin 1. Stenmark, A-K., Dämpning av 13 m lång stålbalk "Ullevibalken". Utprovning av dämpmassor och fastsättning av motbalk samt experimentell bestämning av modformer och förlustfaktorer. Vibration tests of full-scale steel girder to determine optimum passive control. Licentiatavhandling, 1993. Bulletin 2. Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifierade avjämningsmassor. 1993. Bulletin 3. Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors and Geometry Tests and Theory. Doctoral Thesis, 1993. Bulletin 4. Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licentiate Thesis, 1993. Bulletin 5. Krus, J., Hållfasthet hos frostnedbruten betong. Licentiatavhandling, 1993. Bulletin 6. Wiberg, U., Material Characterization and Defect Detection by Quantitative Ultrasonics. Doctoral Thesis, 1993. Bulletin 7. Lidström, T., Finite Element Modelling Supported by Object Oriented Methods. Licentiate Thesis, 1993. Bulletin 8. Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Concrete Beams without Stirrups. Licentiate Thesis, 1994. Bulletin 9. Krus, J., Betongbalkars lastkapacitet efter miljöbelastning. 1994. Bulletin 10. Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Structures. Licentiate Thesis, 1994. Bulletin 11. Sanne, L., Information Transfer Analysis and Modelling of the Structural Steel Construction Process. Licentiate Thesis, 1994. Bulletin 12. Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin-Walled Columns. Doctoral Thesis, 1994. Bulletin 13. Kjörling, M., Dynamic response of railway track components. Measurements during train passage and dynamic laboratory loading. Licentiate Thesis, 1995. Bulletin 14. Yang, L., On Analysis Methods for Reinforced Concrete Structures. Doctoral Thesis, 1995. Bulletin 15. Petersson, Ö., Svensk metod för dimensionering av betongvägar. Licentiatavhandling, 1996. Bulletin 16. Lidström, T., Computational Methods for Finite Element Instability Analyses. Doctoral Thesis, 1996. Bulletin 17. Krus, J., Environment- and Function-induced Degradation of Concrete Structures.

Doctoral Thesis, 1996. Bulletin 18. Editor, Silfwerbrand, J., Structural Loadings in the 21st Century. Sven Sahlin Workshop, June 1996. Proceedings. Bulletin 19. Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of Frame Type Structures. Licentiate Thesis, 1996. Bulletin 20. Troive, S., Optimering av åtgärder för ökad livslängd hos infrastrukturkonstruktioner. Licentiatavhandling, 1996. Bulletin 21. Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles. Licentiate Thesis, 1996. Bulletin 22. Hallgren, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs. Doctoral Thesis, 1996. Bulletin 23. Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Aluminium Extrusions. Licentiate Thesis, 1996. Bulletin 24. Yagi, T., Wind-induced Instabilities of Structures. Doctoral Thesis, 1997. Bulletin 25. Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Extreme Events proceedings from a symposium, May 1997. Bulletin 26. Paulsson, J., Effects of Repairs on the Remaining Life of Concrete Bridge Decks. Licentiate Thesis, 1997. Bulletin 27. Olsson, A., Object-oriented finite element algorithms. Licentiate Thesis, 1997. Bulletin 28. Yunhua, L., On Shear Locking in Finite Elements. Licentiate Thesis, 1997. Bulletin 29. Ekman, M., Sprickor i betongkonstruktioner och dess inverkan på beständigheten. Licentiate Thesis, 1997. Bulletin 30. Karawajczyk, E., Finite Element Approach to the Mechanics of Track-Deck Systems. Licentiate Thesis, 1997. Bulletin 31. Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams. Licentiate Thesis, 1997. Bulletin 32. Edlund, S., Arbitrary Thin-Walled Cross Sections. Theory and Computer Implementation. Licentiate Thesis, 1997. Bulletin 33. Forsell, K., Dynamic analyses of static instability phenomena. Licentiate Thesis, 1997. Bulletin 34. Ikäheimonen, J., Construction Loads on Shores and Stability of Horizontal Formworks. Doctoral Thesis, 1997. Bulletin 35. Racutanu, G., Konstbyggnaders reella livslängd. Licentiatavhandling, 1997. Bulletin 36. Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stöd för byggprocessen. Licentiatavhandling, 1997. Bulletin 37. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete Beams Subjected to Combined Mechanical and Thermal Loading. Licentiate Thesis, 1998. Bulletin 38.

Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: Nordic Seminar on Computational Mechanics, October 1998. Bulletin 39. Luo, Y., On some Finite Element Formulations in Structural Mechanics. Doctoral Thesis, 1998. Bulletin 40. Troive, S., Structural LCC Design of Concrete Bridges. Doctoral Thesis, 1998. Bulletin 41. Tärno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hyparform Suspended Roofs. Licentiate Thesis, 1998. Bulletin 42. Hassanzadeh, G., Betongplattor på pelare. Förstärkningsmetoder och dimensioneringsmetoder för plattor med icke vidhäftande spännarmering. Licentiatavhandling, 1998. Bulletin 43. Karoumi, R., Response of Cable-Stayed and Suspension Bridges to Moving Vehicles. Analysis methods and practical modeling techniques. Doctoral Thesis, 1998. Bulletin 44. Johnson, R., Progression of the Dynamic Properties of Large Suspension Bridges during Construction A Case Study of the Höga Kusten Bridge. Licentiate Thesis, 1999. Bulletin 45. Tibert, G., Numerical Analyses of Cable Roof Structures. Licentiate Thesis, 1999. Bulletin 46. Ahlenius, E., Explosionslaster och infrastrukturkonstruktioner - Risker, värderingar och kostnader. Licentiatavhandling, 1999. Bulletin 47. Battini, J-M., Plastic instability of plane frames using a co-rotational approach. Licentiate Thesis, 1999. Bulletin 48. Ay, L., Using Steel Fiber Reinforced High Performance Concrete in the Industrialization of Bridge Structures. Licentiate Thesis, 1999. Bulletin 49. Paulsson-Tralla, J., Service Life of Repaired Concrete Bridge Decks. Doctoral Thesis, 1999. Bulletin 50. Billberg, P., Some rheology aspects on fine mortar part of concrete. Licentiate Thesis, 1999. Bulletin 51. Ansell, A., Dynamically Loaded Rock Reinforcement. Doctoral Thesis, 1999. Bulletin 52. Forsell, K., Instability analyses of structures under dynamic loads. Doctoral Thesis, 2000. Bulletin 53. Edlund, S., Buckling of T-Section Beam-Columns in Aluminium with or without Transverse Welds. Doctoral Thesis, 2000. Bulletin 54. Löfsjögård, M., Functional Properties of Concrete Roads − General Interrelationships and Studies on Pavement Brightness and Sawcutting Times for Joints. Licentiate Thesis, 2000. Bulletin 55. Nilsson, U., Load bearing capacity of steel fibree reinforced shotcrete linings. Licentiate Thesis, 2000. Bulletin 56. Silfwerbrand, J. and Hassanzadeh, G., (editors), International Workshop on Punching Shear Capacity of RC Slabs − Proceedings. Dedicated to Professor Sven Kinnunen. Stockholm June 7-9, 2000. Bulletin 57.

Wiberg, A., Strengthening and repair of structural concrete with advanced, cementitious composites. Licentiate Thesis, 2000. Bulletin 58. Racutanu, G., The Real Service Life of Swedish Road Bridges - A case study. Doctoral Thesis, 2000. Bulletin 59. Alavizadeh-Farhang, A., Concrete Structures Subjected to Combined Mechanical and Thermal Loading. Doctoral Thesis, 2000. Bulletin 60. Wäppling, M., Behaviour of Concrete Block Pavements - Field Tests and Surveys. Licentiate Thesis, 2000. Bulletin 61. Getachew, A., Trafiklaster på broar. Analys av insamlade och Monte Carlo genererade fordonsdata. Licentiatavhandling, 2000. Bulletin 62. James, G., Raising Allowable Axle Loads on Railway Bridges using Simulation and Field Data. Licentiate Thesis, 2001. Bulletin 63. Karawajczyk, E., Finite Elements Simulations of Integral Bridge Behaviour. Doctoral Thesis, 2001. Bulletin 64. Thöyrä, T., Strength of Slotted Steel Studs. Licentiate Thesis, 2001. Bulletin 65. Tranvik, P., Dynamic Behaviour under Wind Loading of a 90 m Steel Chimney. Licentiate Thesis, 2001. Bulletin 66. Ullman, R., Buckling of Aluminium Girders with Corrugated Webs. Licentiate Thesis, 2002. Bulletin 67. Getachew, A., Traffic Load Effects on Bridges. Statistical Analysis of Collected and Monte Carlo Simulated Vehicle Data. Doctoral Thesis, 2003. Bulletin 68. Quilligan, M., Bridge Weigh-in-Motion. Development of a 2-D Multi-Vehicle Algorithm. Licentiate Thesis, 2003. Bulletin 69. James, G., Analysis of Traffic Load Effects on Railway Bridges. Doctoral Thesis 2003. Bulletin 70. Nilsson, U., Structural behaviour of fibre reinforced sprayed concrete anchored in rock. Doctoral Thesis 2003. Bulletin 71. Wiberg, A., Strengthening of Concrete Beams Using Cementitious Carbon Fibre Composites. Doctoral Thesis 2003. Bulletin 72. Löfsjögård, M., Functional Properties of Concrete Roads – Development of an Optimisation Model and Studies on Road Lighting Design and Joint Performance. Doctoral Thesis 2003. Bulletin 73. Bayoglu-Flener, E., Soil-Structure Interaction for Integral Bridges and Culverts. Licentiate Thesis 2004. Bulletin 74. Lutfi, A., Steel Fibrous Cement Based Composites. Part one: Material and mechanical properties. Part two: Behaviour in the anchorage zones of prestressed bridges. Doctoral Thesis 2004. Bulletin 75. Johansson, U., Fatigue Tests and Analysis of Reinforced Concrete Bridge Deck Models. Licentiate Thesis 2004. Bulletin 76.

Roth, T., Langzeitverhalten von Spannstählen in Betonkonstruktionen. Licentitate Thesis 2004. Bulletin 77. Hedebratt, J., Integrerad projektering och produktion av industrigolv – Metoder för att förbättra kvaliteten. Licentiatavhandling, 2004. Bulletin 78. Österberg, E., Revealing of age-related deterioration of prestressed reinforced concrete containments in nuclear power plants – Requirements and NDT methods. Licentiate Thesis 2004. Bulletin 79. Broms, C.E., Concrete flat slabs and footings New design method for punching and detailing for ductility. Doctoral Thesis 2005. Bulletin 80. Wiberg, J., Bridge Monitoring to Allow for Reliable Dynamic FE Modelling - A Case Study of the New Årsta Railway Bridge. Licentiate Thesis 2006. Bulletin 81. Mattsson, H-Å., Funktionsentreprenad Brounderhåll – En pilotstudie i Uppsala län. Licentiate Thesis 2006. Bulletin 82. Masanja, D. P, Foam concrete as a structural material. Doctoral Thesis 2006. Bulletin 83. Johansson, A., Impregnation of Concrete Structures – Transportation and Fixation of Moisture in Water Repellent Treated Concrete. Licentiate Thesis 2006. Bulletin 84. Billberg, P., Form Pressure Generated by Self-Compacting Concrete – Influence of Thixotropy and Structural Behaviour at Rest. Doctoral Thesis 2006. Bulletin 85. Enckell, M., Structural Health Monitoring using Modern Sensor Technology – Long-term Monitoring of the New Årsta Railway Bridge. Licentiate Thesis 2006. Bulletin 86. Söderqvist, J., Design of Concrete Pavements – Design Criteria for Plain and Lean Concrete. Licentiate Thesis 2006. Bulletin 87. Malm, R., Shear cracks in concrete structures subjected to in-plane stresses. Licentiate Thesis 2006. Bulletin 88. Skoglund, P., Chloride Transport and Reinforcement Corrosion in the Vicinity of the Transition Zone between Substrate and Repair Concrete. Licentiate Thesis 2006. Bulletin 89. Liljencrantz, A., Monitoring railway traffic loads using Bridge Weight-in-Motion. Licentiate Thesis 2007. Bulletin 90. Stenbeck, T., Promoting Innovation in Transportation Infrastructure Maintenance – Incentives, Contracting and Performance-Based Specifications. Doctoral Thesis 2007. Bulletin 91 Magnusson, J., Structural Concrete Elements Subjected to Air Blast Loading. Licentiate Thesis 2007. Bulletin 92.

Pettersson, L., G., Full Scale Tests and Structural Evaluation of Soil Steel Flexible Culverts with low Height of Cover. Doctoral Thesis 2007. Bulletin 93 Westerberg, B., Time-dependent effects in the analysis and design of slender concrete compression members. Doctoral Thesis 2008. Bulletin 94 Mattsson, H-Å, Integrated Bridge Maintenance. Evaluation of a pilot project and future perspectives. Doctoral Thesis 2008. Bulletin 95 Andersson, A., Utmattningsanalys av järnvägsbroar. En fallstudie av stålbroarna mellan Stockholm Central och Söder Mälarstrand, baserat på teoretiska analyser och töjningsmätningar. Licentiate Thesis 2009. Bulletin 96 Malm, R., Predicting shear type crack initiation and growth in concrete with non-linear finite element method. Doctoral Thesis 2009. Bulletin 97 Bayoglu Flener, E., Static and dynamic behavior of soil-steel composite bridges obtained by field testing. Doctoral Thesis 2009. Bulletin 98 Gram, A., Numerical Modelling of Self-Compacting Concrete Flow Licentiate Thesis 2009. Bulletin 99 Wiberg, J., Railway bridge response to passing trains. Measurements and FE model updating. Doctoral Thesis 2009. Bulletin 100 Athuman M.K. Ngoma, Characterisation and Consolidation of Historical Lime Mortars in Cultural Heritage Buildings and Associated Structures in East Africa. Doctoral Thesis 2009. Bulletin 101 Ülker-Kaustell, M., Some aspects of the dynamic soil-structure interaction of a portal frame railway bridge. Licentiate Thesis 2009. Bulletin 102 Vogt, C., Ultrafine particles in concrete Doctoral Thesis 2010. Bulletin 103 Selander, A., Hydrophobic Impregnation of Concrete Structures Doctoral Thesis 2010. Bulletin 104 Ilina, E., Understanding the application of knowledge management to safety critical facilities Doctoral Thesis 2010. Bulletin 105 Leander, J., Improving a bridge fatigue life prediction by monitoring Licentiate Thesis 2010. Bulletin 106 Andersson, A., Capacity assessment of arch bridges with backfill – Case of the old Årsta railway bridge Doctoral Thesis 2011. Bulletin 107 Enckell, M., Lessons Learned in Structural Health Monitoring of Bridges Using Advanced Sensor Technology Doctoral Thesis 2011. Bulletin 108 Hansson, H., Warhead penetration in concrete protective structures Licentiate Thesis 2011. Bulletin 109 Gonzalez Silva, I., Study and Application of Modern Bridge Monitoring Techniques Licentiate Thesis 2011. Bulletin 110

Safi, M., LCC Applications for Bridges and Integration with BMS Licentiate Thesis 2012. Bulletin 111 Guangli, D., Towards sustainable construction, life cycle assessment of railway bridges Licentiate Thesis 2012. Bulletin 112 Hedebratt, J., Industrial Fibre Concrete Slabs – Experiences and Tests on Pile-Supported Slab Doctoral Thesis 2012. Bulletin 113 The bulletins enumerated above, with the exception for those which are out of print, may be purchased from the Department of Civil and Architectural Engineering, The Royal Institute of Technology, SE-100 44 Stockholm, Sweden. The department also publishes other series. For full information see our homepage http://www.byv.kth.se

www.kth.se

TRITA-BKN. Bulletin 114, 2012

ISSN 1103-4270

ISRN KTH/BKN/B--114--SE

Documents

Models for analysis of shotcrete on rock exposed to blasting517588/FULLTEXT01.pdf · Models for analysis of shotcrete on rock exposed to blasting Lamis Ahmed Licentiate Thesis Stockholm,