Mean deformation tensor and mean deformation ellipse of an excavated tunnel section

ARTICLE IN PRESS International Journal of Rock Mechanics & Mining Sciences 46 (2009) 1306–1314

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Mean deformation tensor and mean deformation ellipse of an excavated tunnel section S. Stiros b, V. Kontogianni a, a b

Geodesy Laboratory, Department of Civil Engineering, Patras University, Patras 26500, Greece Institute of Geology & Mineral Expl. of Greece (IGME), Acharnes 13677, Greece

a r t i c l e in f o

a b s t r a c t

Article history: Received 2 October 2008 Received in revised form 20 January 2009 Accepted 11 February 2009 Available online 15 April 2009

Terzaghi, based on graphical methods, introduced the concept of the deformation ellipse as a tool to understand and visualize the deformation of tunnel sections after their excavation, originally assumed as perfect circles. The deformation ellipse gave the opportunity to pass from a set of displacement data expressing the convergence of a tunnel section in a complicated and occasionally mutually inconsistent way, to the concept of deformation of a circle to an ellipse, and thereby to understand the pattern and the changes of the pattern of deformation of a tunnel during its excavation, and search for the causes of this change. Nevertheless, because of its graphic character and limitation in the available data, this approach seems to have rarely been used. In this paper we present analytical solutions for the mean tensor of deformation and of the mean deformation ellipse of a tunnel section, regarded as a deforming, pure mathematical curve. This transformation of a circle to an ellipse corresponds to the strain due to re-arrangement of stresses just after the section excavation, and hence describes the deformation of the surrounding rockmass and of support, independently of any stress-focusing models and measurements. This technique can be used for all types of geodetic and geotechnical monitoring data (tape, extensometer, total station, laser scanner etc. measurements) in tunnels and other underground excavations and its only requirements are (a) uniform, plane strain conditions in each control section during and shortly after the underground excavation, and (b) a nearly uniform distribution of observations along the tunnel section. Uncertainties in estimates permit to confirm whether the adopted model of a mean deformation ellipse is suitable to describe the deformed excavation section. The robustness and value of this approach are highlighted in two case-studies. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Convergence Deformation Ellipse Error Excavation Geodetic Strain tensor Tunnel

1. Introduction Rearrangement of geostatic stresses during underground excavations is followed by a deformation of the excavated area reflecting a tendency for closure of the ‘‘void’’. This deformation is usually described as convergence, i.e., as the displacement of control points, usually towards the void, and is measured using extensometers, tapes and wires and various surveying instruments. Measured displacements are either relative (for instance, roof subsidence or floor heave in a mine are only measured as a change of the net height of the panel using a common tape; [1]) or absolute (i.e., movement relative to an independent coordinate system using modern geodetic instrumentation and reference points in stable ground out of the excavation; [2,3]), and observations are usually summarized in diagrams showing the

 Corresponding author.

E-mail address: [email protected] (V. Kontogianni). 1365-1609/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2009.02.013

displacement history of the control points; in particular cumulative amounts and trends of displacement [4–6]. A basic characteristic of convergence diagrams is that for each section they display a usually complicated sequence of displacements of a number of points (or of changes in line lengths etc.), frequently not mutually consistent (Fig. 1). This obviously makes difficult any identification of the pattern of the deformation and of the changes in this pattern, and of course of the measures to be taken on time to optimize the excavation. A solution to this problem was first presented by Terzaghi [7]. During the excavation of the Chicago Subway in the 1930s, Terzaghi indeed did not only introduce the real-time monitoring of rock deformation as a tool for an efficient and safe way of excavation through soft rocks [7,8], but he also introduced the concept of the mean deformation ellipse based on convergence measurements. Terzaghi visualized the mean deformation of the tunnel section as a change from a circle to an ellipse and his approach was graphical, fitting an ellipse to the displaced points at the periphery of the excavation (Fig. 1; see below); an approach

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Fig. 1. Mean ellipse of deformation of the Chicago Subway tunnel (a) 47 days and (b) 180 days after support installation based on observed displacements across the section. Ellipse is drawn using graphic approach. After [7].

approach seems promising, and it only requires mathematical and computation documentation. In this article we try to respond to this challenge and try to document the theoretical and physical significance and limitations of this approach, and to present the necessary equations for practically all types of monitoring data in underground excavations (distance changes, coordinate changes, etc.). The application and advantages of this method are highlighted in two case studies.

2. Terzaghi’s graphical approach

Fig. 2. Cumulative displacements at a monitoring section (ch. 12+276m) at Sentvid’s tunnel measured at two periods after excavation: (a) first period limited by two circles in each point, (b) second period covering the whole measurement record. After [6].

common indeed in geotechnical engineering [9]. Mean ellipses of deformation permitted him to recognize a first period of nearly horizontal compression followed by a period of vertical compression in a segment of the Chicago tunnel, a change which was assigned to the consolidation of the ground due to water flow caused by tunneling. The basic merit of Terzaghi’s method for the computation of the mean deformation ellipse is that it permitted to recognize the pattern of the deformation and of its changes, tasks too difficult to complete based on simple convergence diagrams (see for example Fig. 2). Still, this technique was thereafter practically ignored because (a) of the limitations in detailed convergence measurements, and (b) because it was only a graphical approach, not compatible with the dominant approaches of tunnel analysis based on stress computations; this ellipse was not even explicitly associated with the deformation tensor. In recent years, however, with the advent and generalization of instrumentation and techniques permitting detailed records of convergence measurements, usually geodetic [2], Terzaghi’s

What Terzaghi [7] did during the excavation of the Chicago Subway in the 1930s was ingenious and very simple. He approximated the tunnel periphery by a scaled circle, and marked on it the stations of measuring tape lines for convergence measurements. The latter were subsequently plotted as radial vectors in an exaggerated scale and the deformed section of the tunnel in an exaggerated form was then produced connecting the convergence marks (vector edges; (see Fig. 3c); an approach conventionally used in various fields of engineering. In addition, although he did not specify it, Terzaghi implicitly assumed uniform strain (i.e. strain equal in all parallel gage lines) and plane-strain conditions and drew the ellipse best-fitting to the deformed shape of the circle. This was necessary because points defining the ellipse (convergence observations) were more numerous than the number necessary to define an ellipse. Because of the common scale in all convergence measurements, the orientation and the ratio of the axes of the ellipse are independent of the exaggeration in the scaling of displacements used and characteristic of the deformation of the circle. This deformation, obviously, does not refer to any medium (i.e., rockmass), but to the mathematical (geometric) substance (a circle) which approximates the tunnel periphery and describes its deformation.

3. Deformation tensor in a medium and deformation of a tunnel section Deformation analyses in various engineering structures are based on the estimation of elements of the deformation tensor, a very conventional approach for engineering mechanics [9]. However, in the case of a tunnel, the void is not a medium. Still, the deformation of the periphery of tunnel can be regarded as that of the internal surface of a cylinder with external diameter

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Fig. 3. Different types of deformation in tunnels (simplified from [16]).

Based on these assumptions in the general case of a circular hole in a homogeneous medium we may extend the use of deformation tensor to describe the deformation ellipse of a tunnel section. Alternatively, we examine the deformation of a mathematical curve (of a circle) which can be described by a tensor. In a general case, any 2-D deformation can be described by a 2  2 tensor: " # e11 e12 (1) T¼ e21 e22

Fig. 4. Principle of computation of the deformation ellipse from observations of distance changes. Diameter AB of a circle of unit radius at an azimuth y (angle measured clockwise from ‘‘north’’ (ordinate) to AB) is deformed to A0 B0 . Three other similar observations of distance changes, but at different azimuths, can be used to define the mean deformation ellipse. e11 and e22 are the points of intersection of the ellipse with axes x and y. Principal strain components correspond to the major axes of the ellipse, which are at an angle j (measured clockwise) from the ordinate axis u. Axes u, u shown in this graph correspond to x,y axes in a horizontal plane (for seismological studies) and y, z in a vertical plane normal to the x-tunnel axis.

tending to infinity. This is a problem first solved by Kirsch [10], in the case of 2-D stresses around a circular section. Later, Jaeger and Cook [11], Hoek and Brown [12], Brown and Bray [13] and many others introduced equations for non-homogeneous stress fields and various shapes, showing sections to take oval forms under the imposed stresses (Fig. 3b). In reality however, tunnel deformation is clearly a threedimensional effect and often follows a more complicated pattern, especially for large amounts of convergence (Fig. 3c; Fig. 4; [14–16]). However, in practice, in the vast majority of cases, planestrain conditions (exyEexzE0, with x corresponding to the tunnel axis and z to the vertical axis) at distances 42D (where D is the tunnel diameter) from the tunnel face (i.e. at the stabilized tunnel sections; [17–19]) can be assumed. Furthermore, initially circular sections are considered to deform gradually, taking the form of an oval.

with e11 and e22 indicating the tensor components of strain corresponding to the abscissa and ordinate (y-axis and z-axis in tunneling, u-axis and u-axis in a general case, see Fig. 4), respectively, and with elements e12 and e21, usually equal, indicating the shear components of strain. This tensor describes the transformation of the circle into an ellipse. The two semi-axes of this ellipse correspond to the principal strains emax and emin and their magnitude, as well as their orientation can be calculated based on the following simple equations (quoted from [11]) and are illustrated in Fig. 4. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e  e 2 e þ e22 11 22 þ (2) max ¼ 11 þ e212 2 2

min

e11 þ e22  ¼ 2

tan 2f ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e  e 2 11 22 þ e212 2

2e12 e22  e11

(3)

(4)

In tunneling and other underground excavations, the elements of matrix (1), as well as the principal strains, are unknown. What is usually known are certain observations of distance changes between two points (for instance change from L to L+DL for line AB in Fig. 5), or coordinate changes of certain points (for instance, Fig. 2). Such observations permit to form some equations which relate changes in the geometry of the circle with the elements of the deformation tensor. Practically this corresponds to the fitting of an ellipse to the deformed shape of the circle, i.e. an approach similar to that introduced by Terzaghi [7]. Definition of such an ellipse requires three independent observations, for instance three measurements of distance change between three points. In the case of additional measurements (for instance coordinates of five convergence stations), an ellipse best fitting to all available measurements is defined usually based on least square techniques. The basic assumption of the proposed method is that the deformation is uniform, i.e. that the strain only depends on the orientation of the line in which strain was measured relative to

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Fig. 5. Geological cross section along the western part of the Acheloos diversion tunnel, central Greece, examined in our study (after [25]). Numbers 1, 2 and 3 in circles indicate control sections of Table 2. Sections 4–7 are located outside (to the left) of this section, mostly in high-strength limestones.

the ordinate axis (angle y in Fig. 5), or in other words that the strain between points defining parallel lines is equal. As Terzaghi [7] has noticed, a change in the imposed stresses or the overall rock-strength conditions (for instance consolidation of ground due to tunneling-induced drainage of groundwater) will be reflected in a change of the pattern of the deformation ellipse (Fig. 1). In the case of no-change of the overall rock-strength conditions, the pattern of the deformation and of the corresponding ellipse will not change, and the only effect of cumulating deformation, mostly due to creep (time-dependent deformation) will be a gradual, quasi-symmetric change of the length of the ellipse axes. This is highlighted in one of the case studies presented below. The technique to form the necessary equations is derived from seismological studies, as shown below.

4. Deformation tensor in tectonic and seismological studies

"

The problem of computation of the deformation tensor using geometric (mostly geodetic) data of a plane deformation was first solved in the framework of tectonic and seismological studies. In California for instance, it was realized that crustal deformation is accommodated by strike-slip faults, and hence vertical deformation is at least an order of magnitude smaller than the horizontal, and hence plane strain conditions can be assumed (ezxEezyE0; [20,21]). This permitted computation of some (in the case of angular observations; [22]) or of all of the elements of the deformation tensor (in case of distance measurements; [21]) of the ground surface. The technique used is the following. It is assumed that between two surveys the length of line i of initial length Li was increased by an amount DLi. Hence, strain along this line is defined by the equation ei ¼ DLi/Li. Assuming uniform strain, the observed strain ei along this line is related to e11, e22 and e12, i.e. the normal components of strain along the x and y axes and the shear component of strain on the x–y (horizontal) plane by the linear equation:

i ¼ e11 sin2 yi þ e12 sin 2yi þ e22 cos2 yi

uniform plane strain conditions are assumed, deformation along any line with azimuth yi is considered constant; still, measured values are contaminated by random errors and small-scale perturbations of the strain. Eq. (5) contains three independent unknowns, e11, e12 and e22, and hence three observations of this type are necessary to form a system of three linear equations from which the three unknowns can be computed. In the case of redundant observations, such equations are not compatible because of measurement errors (and possibly because of local, small-scale perturbations of strain), and the solution of the system of equations is made using standard least-square techniques. The latter also permit error estimates of obtained results. In the case of observations of coordinate changes along a horizontal plane, usually deduced in the last years from GPS data, the new coordinates x0i and y0i of point i are connected with the old ones xi and yi with the following linear equations:

(5)

where yi is the known, approximate azimuth (orientation) of line i (measured clockwise from north; [11,20–22]; Fig. 5). Since

x0i

#

y0i

" ¼

1 þ 12 g1 þ s 1 2

g2 þ o

1 2

g2  o 1  12 g1 þ s

#"

xi yi

# (6)

This system of equations is derived from Bibby [23] replacing rates of change with the corresponding values of g1, g2, s, o and assuming unit time between successive epochs of observations. This system of equations contains four unknowns, g1, g2, s, o; a solution is hence possible if coordinate changes of two points are available. If changes of coordinates of additional points are available, the system of equations is redundant and can be solved using standard least square techniques, which provide also estimates of errors (standard errors, but also the whole variance–covariance matrix [24]). In Eq. (6), g1 and g2 are measures of shear, while s measures a mean dilation and o a mean rotation, and are related to the elements of the deformation tensor matrix with the following relations (see [22]):

g1 ¼ e11  e22

(7)

g2 ¼ e12 þ e21

(8)

s¼

e11 þ e22 2

(9)

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o ¼ e12  e21

(10)

In the case of a symmetric tensor (i.e., if e12 ¼ e21), Eq. (8) yields

g2 ¼ 2e12

(11)

while Eq. (10) is eliminated. Based on Eqs. (7)–(11) e11, e22, e12 and e21, as well as their uncertainties, can be computed using the formulae for error propagation summarized in Stiros [24]. These last variables are related to the principal components of strain emax kai emin and the angle j relative to the x-axis, i.e. with the parameters of the strain ellipse with Eqs. (2)–(4). Application of the formulae for error propagation in these equations (see [24]) permits to estimate the uncertainties of emax kai emin and j. In the case of analysis of rates of strain changes and of the _ and _;o strain tensor, eij, g1, g2, s, o are replaced by e_ ij ; g_ 1 ; g_ 2 ; s Eqs. (5) and (6) yield:

DLi Li "

¼ i ¼ ðt  t 0 Þðe_ 11  sin2 yi þ e_ 12  sin 2yi þ e_ 22  cos2 yi Þ

# 0

xi

y0i

" ¼ ðt  t0 Þ

_ 1 þ 12 g_ 1 þ s 1 2

_ g_ 2 þ o

1 2

_ g_ 2  o 1_ _ 1  2 g1 þ s

#"

xi yi

coordinates is that length changes do not permit computation of the rotation, and hence the deformation tensor is a priori regarded symmetric and the overall solution is simpler. In the case of coordinate changes (based on geodetic, ‘‘optical’’ measurements), a more generalized case of deformation tensor can be examined, and for this reason the solution is more complicated, in steps. Drillhole extensometers can preferably be treated as reflecting coordinate changes along radii of the tunnel and used in the analysis outlines above. The above analysis is exemplified in two case studies, corresponding to the two types of observations. The case of the graphical approach of the Chicago Subway tunnel is also discussed.

6. Case studies

(12) 6.1. Chicago Metro

# (13)

where (tt0) is the lapse time between the two epochs of observations [20,23].

5. Implications for tunnelling The methodology presented above permits to compute the mean strain along a horizontal plane (x,y) fitting to the earth’s surface, assuming uniform plane-strain conditions using geodetic data describing changes of distances, angles or of coordinates of survey points. This technique can be used for tunnels as well, simply examining strain along vertical tunnel sections ((y,z) plane) instead of horizontal land surfaces (or in a more general case, sections normal to the main axis of an inclined tunnel). This technique is possible since in modern tunnels at least three survey points in each convergence control section are usually measured (e.g. Fig. 2). All types of data (tape, extensometer and coordinate measurements) or even their combination can be used. If distance changes between certain points (pins) are known, observations equations of type of Eq. (5) are formed, a system of equations is solved and the unknown values e11, e22, e12 (or their rates) along with their uncertainties are computed. Then, using Eqs. (2)–(4) emax kai emin and their angle j relative to the z-axis can be computed. Application of the law of error propagation [24] finally permits estimation of the uncertainties of emax, emin and of j. If coordinate changes are available, at a first step, using Eqs. (6) g1, g2, s, o (or their rates) as well as the corresponding errors are computed. However, instead of coordinates x and y, referring to a horizontal plane, coordinates y and z referring to a vertical plane were used. In this case, Eq. (6) becomes " 0# " #" # 1 yi 1 þ 12 g1 þ s yi 2 g2  o ¼ (14) 1 1 z0i zi g þ o 1  g þ s 2 2 2 1 At a second step, Eqs. (7)–(11) permit one to compute e11, e22, e12 (or their rates), while application of the law of error propagation [24] permits to compute the corresponding errors. At a third step using Eqs. (2)–(4) emax and emin and their angle j relative to the z-axis can be computed, while application of the law of error propagation finally permits estimation of the uncertainties of emax, emin and of j. The basic difference in computations of the deformation ellipse using length changes (mostly using tape extensometers) and

This tunnel was already discussed previously, as a graphical approach of the deformation ellipse, but is also repeated here as a pioneering and fundamental case study highlighting the significance of computation of the deformation tensor and of its variations in an underground excavation. The Chicago subway tunnels was an outstanding project in the 1930s due to innovative excavation techniques adopted; in particular, shield tunnelling at clays with variable strength parameters and at the vicinity of other structures and optimization of the excavation based on continuous modification of the excavation parameters (rate of advance, support etc.). The latter were dictated by the results of a real-time monitoring (levelling) system to control ground movements above the excavation as well as changes of the tunnel profile [7,8]. Distance changes between six control points were used for each monitoring section inside the tunnel. Terzaghi plotted changes of each of the measured lines and draw the ellipse bestfitting to the deformed section produced (see above). This is actually the graphical version of the analytical technique presented above. The main result of this study was an important change in the orientation of the deformation ellipse 47 and 180 days after the support installation (Fig. 1). This change in the orientation of the ellipse reflects a clear change in the deformation pattern of the rock/support system, most likely induced by the consolidation of clays following flow of water through the tunnel. This result could not be easily deduced from simple observations of displacements. 6.2. Sentvid tunnel, Slovenia The Sentvid tunnel in Ljubljana, Slovenia was recently excavated through various types of sheared rocks (sandstone, siltstone, etc.), locally faulted and of extremely poor quality. High (up to 37 cm) convergence was observed in certain weak sections, as is deduced from convergence diagrams (Fig. 2) derived by optical (geodetic) observations and published by Poschl et al. [6], who reported the problems faced during its excavation. 6.2.1. Monitoring data Our study was focused on the displacement history of monitoring station MS 1226 (left tube parking bay niche) of this tunnel, described by seven control stations, reported by Poschl et al. [6]. From this diagram, 2-D displacements of ix points for two periods were estimated. The first period is defined by two open circles and corresponds to the main deformation period, while the second covers the whole measurement period (Fig. 2).

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Table 1 Minimum and maximum strain and rotation angle of the ellipse of a highly deformed section of the Sentvid tunnel (Slovenia).

First period Second period

emax

emin

Angle (1)

0.0491 0.0532

0.0024 0.0022

115 75

6.2.2. Computation of the strain tensor For each of the two intervals of measurements and for each of the six stations, two equations of the type of Eq. (14) were formed (i.e., one for the y and one for the z coordinate), leading to two systems of twelve equations with four unknowns. These systems were solved using conventional least squares and the best estimates of the unknowns g1, g2, s and o were computed for each of the two periods. Subsequently, we computed the corresponding values of e11, e22 and e12 using Eqs. (7)–(10), as well as the minimum and maximum values of strain axes and the orientation of the deformation ellipse using Eqs. (2)–(4). These last three parameters for the two intervals of measurements are summarized in Table 1. It is evident that differences in the calculated tensors for the two intervals are small (mostly in the orientation of the deformation ellipse), and such differences may not be significant and to a major degree they may reflect input and computation errors, for displacements were read from Fig. 2.

6.3. The Acheloos Diversion Tunnel, Greece The Acheloos River Diversion Tunnel in central Greece, 17.4 km long, with an excavation diameter of 7.1 m and a net internal diameter 6 m, was under construction using full face open shield TBM during our study. During the first 3.5 km tunnelling advanced through high strength limestones, interchanges of cherts, marly cherts and cherty siltstones and flysch (silt-clay and sandstones, Fig. 5; [25]). However, along a distance of 30 m, when the tunnel entered to a zone of sheared flysch (siltstones and sandstones) probably corresponding to a thrust zone (ch.16+250–16+280 m, at 400 m overburden), observed displacements were about 110 mm, i.e. one order higher than in the adjacent limestone sections (maximum 10 mm), despite the relatively stiff support (steel ribs HEB140/ 75 cm, wire mess T140, 12 cm shotcrete shell). Deformation continued for more than 12 months after excavation and led to cracking and detachment of shotcrete, additional contraction of 6 cm, floor heave and lateral displacement of the preconstructed invert. Lining was subsequently repaired and deformation was finally stabilized without any significant deformation in the adjacent sections.

6.3.1. Monitoring data Deformation monitoring of Acheloos Diversion Tunnel was based on measurements of distances across tunnel sections with a tape extensometer of high accuracy (0.13 mm/10 m; [26]). Five pins were fixed at each control section and six different distances were measured (Fig. 6). The complete set of measured distances for seven control sections covering a distance of more than 1.5 km along the tunnel axis and a period of 400 days were available for our study. Emphasis is given to one of these sections (ch. 16+276 m), located in the weak zone of increased deformation. The corresponding displacement diagrams of this particular section are shown in Fig. 7.

Fig. 6. A typical monitoring section of Acheloos Diversion Tunnel. Six distances marked L1–L6 between five fixed survey points repeatedly measured with a convergence tape permitted to compile the monitoring record shown in Fig. 7.

6.3.2. Computation of the strain tensor For the available seven control stations we computed the deformation ellipse for the whole period of observations using cumulative displacements and the approximate ‘‘azimuth’’ of each line (i.e. of its orientation angle measured clockwise from ‘‘north’’, i.e. the z-axis). For each section examined using formula (5), a system of six equations relating observed length changes with the three unknowns e11, e22 and e12 was formed. This redundant equation system was solved on the basis of the least squares method and of the Mathematicas software, also permitting error estimates of obtained values of e11, e22 and e12. Subsequently, using formulae (2)–(4), minimum and maximum principal strains and rotation angle of the deformation ellipse were computed for seven sections and the results are given in Table 2. In order to shed more light to the deformation of Section 1, following the same approach we computed the deformation ellipses corresponding to six different intervals, i.e. 15, 40, 75, 120, 250 and 400 days since the excavation. Results are summarized in Table 3 and indicate a gradual increase of the maximum (horizontal) strain (emax) cumulating for an unusually long interval: emax ¼ 0.5 fifteen days after the excavation, and reaching emax ¼ 1.3 more than one year later. Using AUTOCADs, deformation for these intervals is visualized in Fig. 8. 6.3.3. Error analysis The obtained results include errors due to uncertainties in the input parameters e11, e22 and e12. Assuming very weakly correlated values, a reasonable assumption because most elements of the variance matrix were nearly zero, and applying the simplified law of error propagation [24] to Eqs. (8)–(10) the following formulae for the errors of emax, emin, and j are deduced:

s2max ¼

      @max 2 2 @max 2 2 @max 2 2 se11 þ se12 þ se22 @e11 @e12 @e22

(15)

s2min ¼

      @min 2 2 @min 2 2 @min 2 2 se11 þ se12 þ se22 @e11 @e12 @e22

(16)

s2f ¼



@f @e11

2

s2e11 þ



@f @e12

2

s2e12 þ



@f @e22

2

s2e22

(17)

For instance, for section ch. 16+276 m, the following results were obtained: e11 ¼ 0.010570.0023, e12 ¼ 0.005470.0034, e22 ¼ 4.1 10570.0032, emax ¼ 0.01370.0043, emin ¼ 0.0027 0.0039, and j ¼ 0.470.2558. These results indicate that the examined Section 1 was mainly affected by horizontal strain, and that emin and j are practically equal to 0. Similar results were obtained for the remaining Sections 2–7, in which the only difference found is that the corresponding deformation is one

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Fig. 7. Convergence diagrams for the critical cross Section 1 (ch. 16+276 m) of the Acheloos Diversion Tunnel in Greece.

Table 2 Minimum and maximum strain and rotation angle of the deformation ellipse of 7 tunnel sections at the Acheloos diversion tunnel (Greece). Control section no.

Chainage (km)

Geological setting

Overburden height (m)

emax

emin

Angle (1)

1 2 3 4 5 6 7

16+276 16+098 15+998 14+699 14+460 14+281 14+027

Sheared flysch (thrust zone) Flysch Flysch Alternations of limestones and flysch (transient zone) Alternations of limestones and flysch (transient zone) Flysch Flysch

400 420 430 650 o400 o400 o400

0.013 0.002 0.002 0.002 0.002 0 0.003

0.002 0 0 0 0 0 0

0.4 0.5 0.6 0.5 0.2 0.7 0.2

Geological setting is taken after Sfeikos and Marinos [25]. Control section 1 is further analyzed in Table 3.

Table 3 Minimum and maximum strain and rotation angle of the deformation ellipse of the highly deformed Section 1 (ch. 16+276 m) the Acheloos diversion tunnel (Greece). Time

emax

emin

Angle (1)

15 days 40 days 75 days 120 days 250 days 400 days

0.005 0.007 0.009 0.012 0.012 0.013

0.001 0.002 0.002 0.002 0.002 0.002

0.4 0.4 0.4 0.4 0.4 0.4

order of magnitude smaller than in Section 1 (Table 2). This indicates that the whole tunnel is primarily affected by horizontal compression, which is one order of magnitude higher in the weak Section 1.

conventional stress/strain-based approaches [10–12,27], and does not focus on the behavior and deformation of a specific rockmass. It focuses on the mean deformation of a closed planar curve which usually corresponds to the section of a tunnel or other underground excavation, and tends to ignore local perturbations of strain. The deformation of this curve is visualized as deformation of a circle of unit radius to an ellipse. However, since this curve defines the section of a tunnel (or of any other underground excavation), it describes the cumulative effect of deformation of the rockmass and the support system independently of any modelling; this provides a tool for judging the suitability and effectiveness of the excavation technique adopted. Furthermore, the shape of the ellipse (orientation and magnitude of strain) in conjunction with back analyses may give important information on the causative mechanisms of deformation and help in the decision making about the future of the project.

7.1. Requirements and limitations of the method 7. Discussion Computation of the tensor of deformation and of the deformation ellipse of an underground excavation based on geometric (geodetic etc. data) is an approach different from the

The only requirements of the method are (1) the assumption of plane strain and of uniform strain conditions, a point already analyzed above, and (2) the existence of the necessary data, both concerning their number and their geometric pattern. Three

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Fig. 9. Buckling of a thin support lining due to bump in a Belgian Coal Mine (after [31]).

Fig. 8. Progressive deformation of the weak Section 1 (ch. 16+276 m) of the Acheloos tunnel, based on data of Fig. 7 and Table 3. An exaggerated scale for the ellipses is used.

observations (for example distance changes) are necessary to estimate the elements of the deformation tensor and of the parameters of the deformation ellipse, but redundant observations are necessary to estimate their errors. In addition, results are significant only if they are somewhat evenly distributed along the tunnel or any other excavation section. From this point of view, this technique is self-controlled, since uncertainties in the estimates of the tensor elements or the parameters defining the deformation ellipse indicate not only whether certain estimates are significant, but also whether the overall model (i.e. the assumption of planar, uniform strain) is consistent with the observations. A systematically changing pattern in the deformation ellipse (Table 3, Fig. 8) is also evidence of a successfully modelling. Additional tests of the stability of the modelling can be made gradually removing certain observations. As explained above, changes to the shape of a closed planar curve are used to visualize the rockmass mean deformation due to the main mechanisms of strain accumulation around a tunnel section. There are no a priori limitations concerning the geometry of the excavation void (circular, rectangular etc. void), its dimensions, or the magnitude of strain. Apparently, in the case of a void through different types of rock with very different mechanical characteristics, of unfavorable rock structure or minor geological imperfections, no uniform strain conditions can be assumed and thus this method should not be applied. Such an example is the case of the coal mine of Fig. 9: Deformed and undeformed parts of the tunnel correspond to rock masses with different mechanical characteristics, and hence no uniform strain conditions can be assumed. However, if the necessary observations exist, this analysis can be made for the two or more parts of an excavation separately. This is also the case of an excavation in two or more phases. Estimation of the mean deformation tensor and of the mean deformation ellipse is possible with data from repeated laser scanner surveys of underground openings, as well. In this case, instead of a number of individual points, a cloud of points is available, and the analytical processes outlined above can be used using representative points among them. The simplest, efficient process is to divide for example the tunnel in segments a few to a

few tens of cm wide using planes normal to its main axis, then divide the disks formed in short parts (for instance corresponding to angles of approximately 10201) and finally calculate mean values for each of these surfaces. If this process is repeated for the same areas for repeated laser scanner surveys, estimates of displacements similar to those obtained using conventional geodetic etc. data become available, and these data can be analyzed using the techniques described above.

8. Conclusions We have presented an analytical methodology to compute the mean deformation tensor and the mean deformation ellipse in a tunnel (or any other underground excavation) based on a wide range of displacement data: tape measurements, extensometers and various geodetic (total station or laser scanner). This approach is based on the algorithms developed for the estimation of the tectonic strain in tectonically active areas on the basis of geodetic data, and is in fact a generalized analytical approach of the method first introduced by Terzaghi [7]. The basic advantage of this method is that it can determine the average strain affecting a section of an underground excavation, if uniform, plane-strain conditions are assumed, ignoring local perturbations of the strain, in an approach independent of conventional stress-focusing approaches. This process permits to replace complicated convergence diagrams which permit no easy conclusions concerning the character and causes of the deformation by deformation ellipses. This approach permits an easy visualization of the pattern of the deformation and of its changes, and understanding of its real causes (see Fig. 1). The displacement and deformation-oriented analysis documented in this article should also be regarded in a more general trend for deformation and displacement oriented analysis in engineering (see [28–30]).

Acknowledgments We are indebted to A. Sfeikos for providing data from the Acheloos Diversion Tunnel, to undergraduate students M. Roumanis, D. Chiotis, K. Gavrilaki and E. Stirou for some of the calculations and drawings and to C. Panantonopoulos for fruitful discussions. Constructive comments by one anonymous reviewer are also acknowledged.

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