rheological properties of rock mass in tunnel construction …zbornik/doc/ns2019.056.pdf ·...

7th

INTERNATIONAL CONFERENCE

Contemporary achievements in civil engineering 23-24. April 2019. Subotica, SERBIA

| CONFERENCE PROCEEDINGS INTERNATIONAL CONFERENCE (2019) | 605

RHEOLOGICAL PROPERTIES OF ROCK MASS

IN TUNNEL CONSTRUCTION TECHNOLOGY

Dragan Lukić

1

Elefterija Zlatanović 2

UDK: 624.19:532.135

DOI: 10.14415/konferencijaGFS2019.056

Summary: The previous studies in the field of elasticity and elasto-plasticity did not take

into account the influence of time and changes caused by longer time-dependent effects,

only the final stress states were considered. However, observations and measurements

during construction, taking into account construction technology, confirmed that this

phenomenon is indeed happening. With the development of computers and software, the

possibility of analyzing the impact of the rheological properties of rock mass (the

influence of time) during the technological process by the construction stages is enabled.

This work has the task of showing a relationship between the rheological properties of

the rock mass and the technology of tunnel construction, as well as their

interdependence, based on the results of the most recent researches conducted in this

field.

Keywords: rock mass, tunnel, rheological properties, construction technology

1. INTRODUCTION

With regard to the tunnel objects, the greatest part of the hitherto published studies have

been aimed at a reliable evaluation of forces acting on the tunnel structure. The typical

dimensioning of the underground structure is carried out by the well-known principles of

the static of engineering structures, assuming that the pressure of the rock mass on the

structure is determined by one of the corresponding theories and hypotheses.

Nevertheless, the problem is much more complex if it is analysed having in mind that

under the term "construction" a tunnel lining and a rock mass in the area of secondary

stress effects are understood, thus implying the effect of the rock mass on the supporting

elements of the excavated contour, considering both the temporary insurance when it is

done in stages and the definite tunnel structure, and vice versa. This, in fact, represents

the phenomenon of interaction among the rock mass, the elements of the excavation of

the excavated contour, and the tunnel structure. In such way, the rock mass and

underground structure constitute a complex system, which must be taken into account as

a whole. When designing and constructing underground structures, it is of extreme

importance to determine variation of the stress field from the beginning of excavation of

1 Prof. dr Dragan Lukić, grad. Civ. Eng., University of Novi Sad, Faculty of Civil Engineering of Subotica,

Kozaraĉka 2a, Subotica, Serbia, tel: +381 24 554 300, e-mail: [email protected] 2 Assist. Prof. dr Elefterija Zlatanović, grad. Civ. Eng., University of Niš, Faculty of Civil Engineering and

Architecture of Niš, Aleksandra Medvedeva 14, Niš, Serbia, tel: +381 18 588 200, e-mail:

[email protected]

mailto:[email protected]

mailto:[email protected]

7. МЕЂУНАРОДНА КОНФЕРЕНЦИЈА

Савремена достигнућа у грађевинарству 23-24. април 2019. Суботица, СРБИЈА

606 | ЗБОРНИК РАДОВА МЕЂУНАРОДНЕ КОНФЕРЕНЦИЈЕ (2019) |

rock mass until the end of works, depending on the technological construction process.

Monitoring and measuring also demonstrated that there is a variation of secondary

stresses over time, as a result of rheological processes. Rheological processes and

materials are described with mathematical relations and corresponding boundary

conditions, so that they could be used within mathematical models for description of

excavation behaviour.

The advancement of computer softwares and the use of large-capacity computers have

led to the new opportunities for calculations and research. Taking advantage of the

method of finite elements, numerical simulations of the tunnel construction process are

enabled, and by that, analyses of strain development within the construction phases by

taking into consideration the rheological properties of the rock masses [1, 2]. The

acquaitance on changes of the stress state, deformation, and loads in the function of the

technological process of construction and time allows, before all, the proper choice of

the static system of tunnel, as well as a better fulfillment of the site conditions.

Accordingly, concerning the rheological properties of the rock mass, the main focus is

on the non-instantaneous deformations of the rock mass.

The behaviour of rock mass over time is of exceptional significance for the analysis of

the stress–strain state around the underground structure, whether it is in the stage of an

unsupported or supported excavation. Monitoring and measurement of tunnel structure

displacements have shown that deformations are just initiated by the tunnel excavation

and that they may continue long after the tunnel construction is completed. The tunnel

contour movement (convergence) is the result of both the tunnel face advancement and

the time-dependent behaviour of the rock mass. Particularly interesting for the analysis is

the case of a supported excavation, from the aspect of determining the suppleness of the

supporting system. New methods for supporting the tunnel openings (as the case of the

NATM) involve the use of flexible supporting structures in order to engage the rock

mass in the process of receiving pressures. The investigations, which were carried out in

order to find out what is the impact of the load size on the rheological reduction of the

strength of rock mass, showed that for most types of rocks critical long-term load, which

causes material failure, is in the interval of 50-70% of the design strength (Fig. 1).

Figure 1. Reduction of the strength in the function of load and time [3]

The influence of rheological properties of rock mass in tunnelling has been a subject of a

number of studies [4-11]. Considering the interaction of rock mass and tunnel support,

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various methodologies have been proposed, which are commonly used as a preliminary

tool for quickly assessing the behaviour of the surrounding rock mass and tunnel lining

system considering both the design and construction process of underground structures.

The greatest part of these solutions is based on the well-known Convergence-

Confinement Method, which represents a two-dimensional simplified approach that can

be used to simulate three-dimensional problems such as the rock mass–tunnel lining

interaction in tunnelling, and is widely used for estimation of the required load capacity

for the proposed support system. The traditional approach of this methodology is based

on application of the Ground Reaction Curve, the Longitudinal Displacement, and the

Support Characteristic Curve, which provide information on the required support load in

regards to the tunnel face location as a percentage of the anticipated maximum tunnel

structure displacement. Recently, a modified Convergence-Confinement Method

approach has been proposed [12], considering the time-dependent material of the tunnel

support, by investigating different scenarios of the shotcrete and rock bolts supporting

system and how the behaviour of the supporting system changes over time. Simplified

formulations, on which the Convergence-Confinement Method is based, however, cannot

explicitly capture the time-dependent component of rock mass deformation (e.g., creep),

which can have a significant impact on loading of the tunnel support. Overlooking these

additional loads and deformations may lead to unexpected failures, causing safety issues

for the working personnel, as well as to cost overruns and project delivery delays [13]. In the subsequent part of this paper, the basic elements of rheological models of rock masses

are presented.

2. RHEOLOGICAL MODELS

The Bingham-Hooke model consists of a Bingham model and a spring in series.

Bingham’s material exhibits linear elasticity for stress values lower than the yield stress,

as in the Saint-Venant model, but flows linearly above that value, as in the Maxwell

model.

Figure 4 shows the outlined representation of common viscoelastic models with yield

stress and the response of these models before and after achievement of the yield stress

value σy.

Based on the relations of deformation, stress, and time scale, the materials can be divided

into the following four categories of rheological behaviour:

- Elastic materials: In a purely elastic material (such as steel at stresses below the

yield stress) all energy added is stored in the material;

- Viscous materials: In a purely viscous material (such as water) all energy added

is dissipated into heat;

- Viscoelastic materials: A viscoelastic material exhibits viscous as well as elastic

behaviour;

- Materials with yield stress: To such materials external stress must be applied to

make them begin to flow, indeed to yield.

2.1 Basic rheological elements




Classical continuum mechanics, in the past, acknowledged two types of material, i.e. two

rheological models: elastic solid bodies (Hooke’s body) and ideal fluids (Newton’s

fluid). The rheological behaviour of viscoelastic materials can be successfully described

by the combination of rheological models based on two fundamental elements: the linear

spring and the linear viscous dashpot.

Considering the linear spring, the relationship between stress and strain is given by

Hooke’s law (Eq. 1):

)()( tEt

(1)

where E stands for the modulus of elasticity (Young’s modulus).

The response ε of such a spring to a stress σ is instantaneous, without any time

dependency, and the recovery after release of the stress is also instantaneous and

complete.

The viscous dashpot can be visualised as a piston displacing a Newtonian fluid in a

cylinder with a perforated bottom. The Newton’s law of viscosity is given by Eq. 2:

)()( tt

(2)

where

(t) = dε / dt is the strain rate and η is the viscosity coefficient. In this case, there is no instantaneous response, the deformation is directly proportional

to time, and no recovery takes place.

The rheological behaviour of viscoelastic materials with yield stress, however, cannot be

estimated accurately by rheological models based on only two fundamental elements. An

additional element is needed to represent the yield stress phenomenon. This is the so-

called Saint-Venant element. The deformation of this element is possible only after the

achievement of the critical stress value – the yield stress (Eq. 3):

)(

0

y

y

t

(3)

The representation of basic rheological elements is depicted in Figure 2.

Figure 2. Representation of basic rheological elements [14]

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2.2 Basic viscoelastic models

Complex rheological models are composed of several simple models. They can consist

of only two simple models such as, for instance, the Maxwell’s model and the Kelvin-

Voigt’s model, or of a combination of three or more basic models, which are often called

three-parametric or multi-parametric models of viscoelastic materials, such as e.g. the

Burgers’ model. The more a material is complex in its structure and behaviour, the more

basic models must be included in its behavior model [15].

The Maxwell viscoelastic model consists of a spring and a dashpot in series. As the

elements are connected in series, same stress acts on both elements, and so the total

strain is equal to the sum of the strain of the two elements.

The Kelvin viscoelastic model involves the parallel connection of a spring and a

dashpot. This model does not allow for instantaneous deformation since the stress on the

dashpot would be infinite. In this case, the elongation in each element remains the same.

Both models, Maxwell and Kelvin, are limited in their representation of actual

viscoelastic behaviour. The former is able to describe stress relaxation, but only

irreversible flow, whereas the latter can represent creep, but without instantaneous

deformation, and it cannot account for stress relaxation. A combination of both models,

however, enables a realistic description of the materials with more complex behaviour.

The Burgers viscoelastic model consists of the Maxwell (spring and dashpot in series)

and the Kelvin (spring and dashpot in parallel) section connected in series. Therefore, the

creep curve of Burgers model under the creep stress provides the superposition of the

Kelvin and the Maxwell models.

The standard Solid (Poynting) contact model consists of the Kelvin (spring and dashpot

in parallel) section and an additional spring connected in series. This contact model

shows the creep and relaxation behaviour and also the instantaneous elasticity.

The aforementioned viscoelastic models are presented in Figure 3.

Figure 3. Common viscoelastic models [14]




2.3 Basic models with yield stress

The Prandtl model consists of a Saint-Venant element and a spring element in series (as

shown in Fig. 4). The behaviour of that material is characterised by linear elasticity for

stress values below the yield strength. When yield stress is attained, the body exhibits

pure plasticity.

The serial connection of all the basic rheological elements can be presented as the

Maxwell model with St.-Venant element in series. Prior to attaining yield stress, the

system shows behaviour similar to the ordinary Maxwell model.

The Kelvin model with St.-Venant element in series gives another alternative in

representing plastic deformation.

The Bingham model consists of a Saint-Venant element and a dashpot in parallel (as

shown in Fig. 4). Deformation of the model is not possible before reaching yield stress

through the Saint-Venant element. When yield stress is achieved, the model exhibits

visco-plastic deformation.

Figure 4. Common viscoelastic models with yield stress [14]

A visco-elastoplastic model termed the Nishihara body (Fig. 5) was proposed in 1961

and is comprised of a Kelvin body and a Bingham body. The creep curve for the

Nishihara body can be used to describe decay, steady, and unstable creeps. The Nishihara model describes the variation in the attenuation period and steady period

fairly well. The parameters of the model are E1, E2, η2, η3, σs. The terms E1 and E2 are the

elasticity moduli, η2 and η3 are viscosity coefficients, and σs is long-term strength. E1 and

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σs can be obtained from the test creep curves directly, according to the results of the rock

creep test [16].

Figure 5. Distortion characteristic of the Nishihara body: (a) model; (b) the first stage

creep curve; (c) the full creep curve [16]

3. TIME-DEPEDENT DEFORMATION IN TUNNELLING

The time-dependent deformation concerning tunnelling in a visco-elastic material has

been dealt in the literature as well. Analytical solutions that take into account the time-

dependent convergence have been proposed both for the case of supported circular

tunnels and for the case of unsupported circular tunnels in viscous medium (Fig. 6).

Figure 6: Visco-elastic models and analytical solutions for a circular unsupported

tunnel. (The analytical solutions for the Kelvin-Voigt and the Maxwell models are

adopted from Panet (1979) and for the Burgers model from Fahimifar et al. (2010)) [17]




Figure 6 summarises the visco-elastic models, their analytical solutions, and the radial

displacement–time relationships, considering their behaviour over time as well as creep

behaviour. In the above presented closed-form solutions, designations of the

corresponding physical values are as follows:

- σo is the in situ stress;

- σr stands for the radial stress;

- ur denotes the radial displacement of tunnel structure;

- r refers to the tunnel radius;

- t describes the time;

- T designates the retardation – relaxation time of each model;

- G is the shear modulus;

- η represents the viscosity;

- subscripts K, M and ∞ refer to the Kelvin-Voigt’s model, the Maxwell’s model

and the harmonic average, respectively.

It should be noted that in case when time is assumed to be infinite, the shear modulus

used in the Kelvin model is estimated with the harmonic average G∞ and is not equal

with the initial shear modulus of the rock mass G0.

Considering the time-dependent rheological rock mass properties in tunnel construction,

based on the Convergence-Confinement Method, the total observed displacement on

tunnel walls in an isotropic visco-elastic medium has been investigated recently [17],

taking into account both the tunnel advancement and the cumulated deformation induced

by the rheological behaviour of the material over time (Figs. 7 and 8).

Figure 7. Longitudinal Displacement Profile in an elasto-viscoelastic-plastic medium

[17]

Considering Figures 7 and 8:

- r is the tunnel radius;

- D is the tunnel diameter;

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- t is the time;

- x stands for the distance from the tunnel face which is a function of time;

- ur is the radial displacement of tunnel wall, which is the function of time and

distance from the tunnel face;

- G is the shear modulus;

- η is the viscosity;

- q denotes the deviatoric stress;

- σ designates the applied stress;

- subscripts M, K, y refer to the Kelvin’s model, the Maxwell’s model, and

yielding threshold, respectively;

- superscripts el, p, s, and tet denote the elastic response and primary, secondary,

and tertiary components of the creep behaviour, respectively.

Figure 7 illustrates the case of an elasto-viscoelastic-plastic material. In such case, the

material undergoes all three stages of creep up to the ultimate failure. This kind of

response is typical for severe squeezing rock masses, in which case the induced creep

behaviour leads the material to fracture and failure after exhibiting large deformations

and noticeable convergence.

The anticipated Longitudinal Displacement Profile of the tunnel displacement

considering an elasto-visco-elastic medium is presented in Figure 8, in which case the

phase of tertiary creep doesn’t exist. This behaviour is typical for more ductile materials

as in the case of rock salt.

As it is presented in both figures (Figs. 7 and 8), if time-effect is not taken into account,

the total displacements are underestimated. This may result in calculation errors at the

initial stages of the design process. Therefore, detailed investigation is recommended

when dealing with rock masses that exhibit time-dependent behaviour.

Figure 8. Longitudinal Displacement Profile in an elasto-viscoelastic medium [17]




4. CONCLUSION

When designing and constructing underground structures, it is of paramount importance

to assess variation of the stress field from the beginning of excavation of rock mass until

the end of works, depending on the technological construction process. Monitoring and

measuring also demonstrated that there is a variation of secondary stresses over time, as

a result of rheological processes. As rheological properties of rock mass may have a

significant influence on the tunnel support in regard to additional loads and

deformations, rock mass and underground structure must be considered as a complex

system in analyses, in particular when tunnelling in rock masses with time-dependent

behaviour takes place. Overlooking of these additional loads and deformations may lead

to support yielding, abrupt rock mass instabilities, unexpected failures, safety issues, as

well as cost overruns and project delivery delays. In order to overcome these issues, it is

recommended to perform the project optimisation by comparing different excavation

methods with an aim of prediction and estimation of the rock mass responses due to

excavation. This will result in the appropriate selection of the excavation method and

the support system that would allow the rock mass to further deform over time avoiding

overstressing. In addition, it would be of huge importance if a complete tunnel dataset is

utilised with monitoring data and laboratory data in a numerical back-analysis.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of the Ministry of Education, Science,

and Technological Development of the Republic of Serbia within the Projects OI

174027 and TR 36028.

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Dissertation, Faculty of Civil Engineering of Podgorica, Podgorica, Montenegro,

2004, p. 141 (in Serbian).

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Serbian).

[6] Roylance, D.: Engineering viscoelasticity, Cambridge, 2001.

[7] Jiang, Q., Cui, J., Chen, J.: Time-Dependent Damage investigation of Rock Mass in

an In Situ Experimental Tunnel, Materials, 2012, No.5, p.p. 1389-1403.

[8] Effinger, V., Bois, P.D.: Modeling of viscoelastic materials, Dyna, 2012.

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[9] Khoshboresh, A.R.: A study on Deformation of Тunnel Excavation in Fractured

Rocks, Quebec, 2013.

[10] Grosh, R.M., Rao, K.S.: Analysis of creep behavior of soft rocks in tunnnelling,

Indian Institute of Technology Delhi - Conference paper, 2015.

[11] Wang, H.N., Chen, X.P.., Jiang, M.J., Song, F., Wu, L.: The analytical predictions

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Tunnelling and Underground Space Technology, 2018, vol.71, p.p. 403-427.

[12] Gschwandtner, G.G., Galler, R.: Input to the application of the convergence

confinement method with time-dependent material behavior of the support,

Tunnelling and Underground Space Technology, 2012, vol. 27, No. 1, p.p. 13-22.

[13] Paraskevopoulou,C., Benardos,A.:Assessing the construction cost of tunnel projects,

Tunnelling and Underground Space Technology, 2013, vol. 38, p.p. 497-505.

[14] Shyshko, S.: Numerical simulation of the rheological behavior of fresh concrete,

PhD Dissertation, Technical University of Dresden, Dresden, Germany, 2013.

[15] Fritz, P.: An Analytical Solution for Axisymmetric Tunnel Problems in Elasto-

Viscoplastic Media, Int. J. for Numerical and Analytical Methods in Geomechanics,

1984, Vol. 8, pp. 325-342.

[16] Song, Z-P., Yang, T-T., Jiang, A-N., Zhang, D-F., Jiang, Z-B.: Experimental

investigation and numerical simulation of surrounding rock creep for deep mining

tunnels, The Journal of the Southern African Institute of Mining and Metallurgy,

2016, Vol. 116, pp. 1181-1188.

[17] Paraskevopoulou, C., Diederich, M.: Analysis of time-dependent deformation in

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Space Technology, 2018, Vol. 71, p.p. 62-80

РЕОЛОШКА СВОЈСТВА СТЕНСКИХ МАСА

У ТЕХНОЛОГИЈИ ГРАЂЕЊА ТУНЕЛА

Резиме: Досадашња изучавања у области еластичности и еласто-пластичности

нису узимала у обзир утицај времена и промене изазване дужим временским

ефектима, већ су разматрана коначна напонска стања. Међутим, опажања и

мерења у току грађења узимајући у обзир технологију грађења су потврдила да се

ова појава заиста догађа. Са развојем рачунара и софтвера омогућена је анализa

утицаја реолошких својстава стенске масе (утицај времена) у току технолошког

процеса по фазама изградње.

Овај рад има задатак да прикаже међусобни однос реолошких својстава стенске

масе и технологије грађења тунела, као и њихову међузависност, a на бази

резултата најновијих истраживања у овој области.

Кључне речи: стенска маса, тунел, реолошка својства, технологија грађења

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