Applied Mathematical Modelling 36 (2012) 4422–4438
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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Formation and accumulation of contact deficiencies in a tunnel
segmented lining
S.H.P. Cavalaro a,⇑, C.B.M. Blom b, J.C. Walraven b, A. Aguado a
a
Departamento de Ingeniería de la Construcción, E.T.S. de Caminos Canales y Puestos, Universidad Politécnica de Cataluña, BarcelonaTech, Calle Jordi Girona
Salgado 1-3, Módulo C1, Despacho 202, 08034 Barcelona, Spain
b
Department of Design and Construction, Structural and Building Engineering, Concrete Structures, TU Delft, Stevin II-North 2.02, Delft 2600, The Netherlands
a r t i c l e
i n f o
Article history:
Received 20 September 2010
Received in revised form 8 November 2011
Accepted 15 November 2011
Available online 23 November 2011
Keywords:
Contact deficiencies
Tolerance
Cracks
Segmented lining
Uneven support
a b s t r a c t
Damages observed in tunnels constructed with tunnel boring machines affect the overall
quality of the structure and the efficiency of the construction process. Most of these damages are caused by contact deficiencies between segments that are generated by the sum of
several tolerances on the shape and on the placement of the lining. Moreover, the imperfection of one ring affects the placement of the following ones, inducing an accumulation
mechanism that magnifies the imperfection expected due to the sum of tolerances in a single isolated ring. The overall consideration of these phenomena yields an intricate analysis
that must take into account some important probabilistic aspects. This paper explains how
the tolerances may evolve into the contact deficiencies found in practice. Initially the types
of tolerances and contact deficiencies more likely to affect the structural behavior of the
lining are analyzed. A mathematical model is proposed to explain the relation between tolerances and contact deficiencies. The predictions obtained with the model are then compared with the measurements performed in the tunnel of Line 9 in Barcelona. The
results obtained reinforce the importance of the model proposed in this study, which quantifies aspects that so far could only be studied qualitatively or on a trial and error basis.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
In the construction of tunnels with tunnel boring machines (TBM), the machine excavates the ground and installs a series
of concrete segments that form the final lining. In most cases, the lining serves as a reaction frame, receiving the load applied
by the TBM in order to generate the front pressure and perform the excavation. The number of tunnels constructed using this
technique has increased considerably on the last decades due to the technological advance observed in the machinery and in
the construction process [1–4]. It is now possible to reach high productivity levels keeping adequate safety standards even
under unfavorable work conditions and excavation depths [5,6].
However, the excavation on such extreme situations also affects the stress level of the lining during the construction. This
is one of the reasons for the increase on the severity and frequency of lining damages that compromise aesthetical aspects
and the durability of the tunnel [7–9]. Several works show that these damages are usually the result of contact deficiencies
between segments [8,10], which generate a partial support condition and critical internal stresses in the structure [9,11,12]
(see Fig. 1). Despite their harmful repercussion, just a few studies from the literature focus on the assessment of the contact
deficiencies that cause damages in tunnels. Some authors estimate that these deficiencies could be around 1 mm, 2 mm or
⇑ Corresponding author. Tel.: +34 934016507; fax: +34 934054135.
E-mail addresses: sergio.pialarissi@upc.edu, cavalaro-sergio.pialarissi@upc.edu (S.H.P. Cavalaro).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2011.11.068
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
4423
Fig. 1. Damages observed due to contact imperfections.
3 mm [13–16] – values big enough to cause cracks of the segments in some situations [17]. However, even if the contact
deficiencies were properly measured, a key question remains about their origin. The answer to such question is complex
given the abundant amount of factors to be considered.
It is known that the contact deficiencies are the result of a complex sum of several tolerances on the shape and on the
placement of the segmental lining [18]. In addition to that, the construction process of the tunnel magnifies the imperfection
expected of a single isolated ring. The overall consideration of this phenomenon yields an intricate analysis that must take
into account some important probabilistic aspects. Only that would make possible the assessment of the role played by the
tolerances both on the formation of the contact deficiencies and on the damages produced throughout the construction process. On a further step, this would also increase the knowledge about the behavior of the lining, thus contributing to the
proper definition of the tolerance during the design of the structure.
In this context, the objective of the present work is to explain and to predict how the tolerances may evolve into the contact deficiencies found in practice. Initially the types of tolerances and contact deficiencies more likely to affect the structural
behavior of the lining are analyzed. A mathematical model is proposed to explain the relation between tolerances and contact deficiencies. The results obtained with this method are then compared with some measurements performed in the tunnel of Line 9 in Barcelona.
2. Tolerances
The contact deficiencies are observed throughout the tunnel construction process mostly as unevenness on the space that
separates the segments. Also in cases of an apparent perfect contact, some small imperfections not visible to the unaided eye
may be present. The two places where contact deficiencies may occur with important structural repercussion are the longitudinal joints and the circumferential joints. The main cause of these contact deficiencies are the tolerances produced along
Fig. 2. Classification of tolerances and contact deficiencies.
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the width of the segments. Such tolerance may have three sources: the design of the lining, the production and the placement of the segment (see Fig. 2).
The design approximations are some simplifications assumed in the ideal geometrical shape of the lining that may compromise the perfect fit between segments, generating contact deficiencies. The production tolerances are mainly the result of
the dimensional variation of the molds used to cast the concrete segments. The placement tolerances are caused by variations induced by the jacks that lift and install the segments inside the TBM. Moreover, the summed tolerances from the previous rings limit the accurate positioning of the new rings, characterizing a mechanism of accumulation of imperfections.
The analysis of all these tolerances and the accumulation mechanism is presented in the following sections.
3. Design approximations
3.1. Shape of the ring
The universal ring is frequently used to construct segmented linings with TBM. This ring has a very particular geometry
given by the slicing of a hollow tube with two inclined planes that form an angle 2 b greater than 0 with each other (see
Fig. 3a). Consequently, the width of the segment (W) varies along its perimeter and thickness. Fig. 3b presents the coordinate
system and the main variables used to define the shape of the universal ring.
The ideal width of the ring at a certain internal angle h follows Eq. (1), where Wmed is the average between the maximum
(for h equal to 0°) and the minimum width (for h equal to 180°).
WðR; hÞ ¼ 2 R cos h tan b þ W med :
ð1Þ
However, the blueprints of several tunnels show the width of the ring (Wn) vary only in a finite number of fixed angles (hn)
instead of according to the continuous cosine function defined in Eq. (1). In the zone between fixed points, it is assumed that
the width of the ring follows Eq. (2), varying linearly with the internal angle h. In this function, Wn represents the fixed width
Fig. 3. Detail of (a) slicing planes and (b) width variation and coordinate system.
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
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corresponding to the closest smaller fixed internal angle hn whereas Wn+1 represents the fixed width corresponding to the
closest bigger fixed internal angle hn+1.
WðR; hÞ ¼ W n þ
W nþ1 W n
ðh hn Þ:
hnþ1 hn
ð2Þ
Fig. 4a presents the comparison between the ideal and the real outside width for the tunnel of Line 9 in Barcelona. The lining
in question presents an inside diameter, an outside diameter and an angle 2 b equal to 10,900 mm, 11,600 mm and to 0.52°,
respectively. The ring is composed by seven segments (A1, A2, A3, A4, A5, B and C) with an inside angle equal to 48° and a key
segment (K) with an inside angle equal to 24°, as shown in Fig. 4b. The outside width ranges from 1747.79 mm (at the
centerline of the segment K) to 1852.20 mm (at the centerline of the segment A1).
The comparison of the curves calculated with Eq. (1) and Eq. (2) shows that the real width approximates the ideal width
of the ring. The difference reaches a minimum of 1.12 mm at the segment A1 and a maximum equal to 1.01 mm at the
segments B and C. In general, the real width is greater than the ideal width from h equal to 72.12° (at segment A4) to
271.5° (at segment A3) whereas the opposite occurs in the rest of the ring.
The design approximations observed alter the contact surface of the ring, hence generating contact deficiencies after the
segments are installed in the tunnel. A model is proposed in the following section to estimate the magnitude of these contact
deficiencies after the segments are placed together.
3.2. Model for the contact between segments
In order to obtain a mathematical model for the calculation of the spaces between segments from adjacent rings it is considered that the design approximation and the contact deficiencies are small in comparison with the lining width and thickness. As any displacement produced due to the design approximation should also be small, the segments tends to move and
to achieve a stable support condition with the previous ring installed. In addition, it is assumed that the segments of the new
ring do not produce any movement on the ones of the previous rings, which have already reached a stable support condition.
The deduction of the model for the contact between segments is easily achieved after moving the coordinate system
origin from the center of the ring to the point of initial contact. In this case, the equations for the surfaces of the segments
in contact are obtained through a mathematical translation of Eq. (2). Such translation in represented in the first part of
Eq. (3), which is used to calculate the final contact deficiencies (S(R, h)) along the perimeter of the segments. The constants
k1 and k2 indicate the internal angles h of the segments in contact from each ring, whereas the constant K marks the internal
angle h corresponding to the point where the initial contact occurs.
SðR; hÞ ¼
WðR; h þ k1 þ KÞ WðR; k1 Þ WðR; h þ k2 þ KÞ þ WðR; k2 Þ
DSðhÞ:
2
ð3Þ
The second part of Eq. (3) represents the displacement DS(h) estimated in Eq. (4) and required to establish a stable contact
situation. The latter also depends on the spins x1 and x2 around the point K as well as on the angle x1,S1 that represents the
slope of the contact surface in the YZ plane. Fig. 5 shows the physical meaning of x1, x2 and x1,S1.
h
x1
x2 cos x1;S1 i
DSðhÞ ¼ 2 R ð1 cos hÞ tan
þ sin h tan
:
2
2
The spin x1 may be calculated in Eq. (5) depending on the constants k1, k2, K and the lining thickness (e).
Fig. 4. Comparison between ideal width and real width for the Line 9 in Barcelona.
ð4Þ
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Fig. 5. Physical meaning of the constants x1, x2 and x1,S1.
x1 ¼ arctan
WðR e; k1 þ KÞ
WðR e; k2 þ KÞ
arctan
:
2e
2e
ð5Þ
On the other hand, the angle x1,S1 is calculated in Eq. (6).
x1;S1 ¼ arctan
WðR e; k1 þ KÞ
:
2e
ð6Þ
Since the constants k1 and k2 are already known in advance, the definition of Eq. (3) for the final contact situation depends
solely on the estimation of the constants K and x2 that satisfy a set of conditions. The first condition is that the distance
between surfaces should always be greater than or equal to zero. Otherwise, there would be an unreal superposition of
the segments. The second condition is that the distribution of the contact points should balance the load applied to the segments by the TBM. The third condition is that the contact must happen in at least three points of the surface in order to
assure a stable plane. In other words, at least three sets of constants (K, x2) yield the final contact situation in Eq. (3).
The estimation of the final contact situation starts with the assumption of an initial contact point represented by a certain
value of K. Then, the x2 that gives the absolute minimum of Eq. (3) is calculated. The point of the surface where this minimum occurs is assumed as the new value of K, which is applied again to estimate x2. Repeating this iterative procedure
three or four times is usually enough to obtain the final contact situation in Eq. (3) that satisfies the set of conditions
described previously.
Fig. 6 shows the contact deficiency profile calculated according to the iterative procedure when the segments A1 and A2
of the previous ring give support to the segment A3 from the new ring of the Line 9 of Barcelona. For this example, the stable
contact situation happens for K equal to 0° and 24° at the outer perimeter and equal to 48° at the inner perimeter. In the rest
of the points there is a space separating the segments that reach a maximum of 0.74 mm.
In Line 9 of Barcelona there are 93 possible combinations of segments with different contact deficiency profiles. Table 1
shows the maximum contact deficiencies calculated for every junction of three segments. As observed in the table, the contact deficiencies estimated in the circumferential joint reaches 2.16 mm. This value is 93% greater than the maximum design
approximation, which was clearly magnified due to the contact between segments. Notice that to eliminate these contact
deficiencies it is only necessary to specify the shape of the universal ring according to Eq. (1) rather than defining the width
in a reduced number of fixed internal angles.
Fig. 6. Contact between segments A3, A2 and A1 from the Line 9 in Barcelona.
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Table 1
Maximum contact deficiencies for the segments from the Line 9 in Barcelona (mm).
Segments that give support
Segments supported
B
A3
A2
A1
A5
A4
C
K–B
B–A3
A3–A2
A2–A1
A1–A5
A5–A4
A4–C
C–K
0.99
0.55
1.18
0.50
1.18
0.95
1.29
1.05
0.72
2.00
0.38
1.03
0.35
0.77
0.57
0.85
2.00
1.02
2.00
0.44
0.69
0.54
0.67
2.16
1.20
2.08
0.74
0.55
0.55
0.74
2.08
1.20
2.16
0.67
0.54
0.69
0.44
2.00
1.02
2.00
0.85
0.57
0.77
0.35
1.03
0.38
2.00
0.72
1.05
1.29
0.95
1.18
0.50
1.18
0.55
0.99
4. Production and placement tolerances
Even if the design of the ring were performed taking into account the perfect shape of the ring, some tolerances may arise
either from the production or from the placement of the segments. The tolerances from both sources observed along the
width of the segment (see Fig. 7) are the main responsible for generating the contact imperfections in circumferential joints
[17].
An initial assumption is made about the distribution curve that approximates the dimensional scattering (tolerance)
found in the width of the segments. Like in most industrial processes, the Gaussian distribution is used to describe the probability of finding a certain variation of this dimension [19–23]. Furthermore, a parametric definition of the normal distribution is adopted to guarantee a flexible application of this approach independently of the absolute characteristic value of the
dimension studied and the overall quality of the production process. In this context, any normal distribution is solely defined
in terms of the total tolerance (T), the quality of the construction process – assessed through the standard deviation (r) – and
the characteristic coefficient (k) – given by the ratio between T and r.
Fig. 8 shows the normal distribution curves of two construction processes with the same predefined tolerance value but
different standard deviations. In this figure the graphic definition of the characteristic coefficient k becomes clear, that is, the
number of times the standard deviation (r) is smaller than the tolerance (T). Physically, this parameter is an indicative of the
likelihood of finding in practice a tolerance bigger than the tolerance value (T) defined. Thus, the coefficient k of the wide
curve is smaller than that of the narrow curve meaning that the probability of finding an unfit segment in the former is
bigger.
The tolerance and the final distribution curve for the production of the segmented lining is the result of the summed
effect of two independent factors, all of them introduced during the production of the segments. Firstly, the most important
part of the dimensional variation comes from the tolerance (Tmold) and the standard deviation (rmold) of the molds used to
cast the concrete segments. Moreover, the packer sometimes placed on the joints (see Fig. 7) also introduces a characteristic
tolerance (Tpacker) and a standard deviation (rpacker).
If the distribution of the dimensional variation of the molds and of the packers is known in advance, the standard deviation of the overall production process (rpr) is calculated in Eq. (7). In this equation, kmold and kpacker represents the characteristic coefficient of the molds and the packers, respectively.
Fig. 7. Detail of position of packer and width tolerance.
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Fig. 8. Parametric normal distribution.
rpr
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ r2packer þ r2mold ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
2 2
T packer
T mold
þ
:
kpacker
kmold
ð7Þ
The overall tolerance of the whole production process (Tpr) is given by Eq. (8), which depends on the characteristic coefficient
kpr adopted for the overall production of the segment.
T pr ¼ kpr rpr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
2 2ffi
T packer
T mold
¼ kpr þ
:
kpacker
kmold
ð8Þ
Like in the case of the production tolerances, it is assumed that the placement tolerance also fits a normal distribution. Such
distribution is represented by a placement tolerance (Tpl) and the standard deviation (rpl) whose ratio gives the characteristic
coefficient kpl. The probabilistic combination of this curve with the one obtained previously for the production process gives
the final tolerance of a single ring after it is installed. Analogously to what was done in previous paragraphs, the final standard deviation for a single ring (r) is obtained according to Eq. (9).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r ¼ r2pr þ r2pl ¼
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2ffi
T packer
T pl
T mold
:
þ
þ
kpacker
kmold
kpl
ð9Þ
On the other hand, the final tolerance for a single ring (T) is obtained according to Eq. (9), which depends on the characteristic coefficient k adopted.
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2ffi
T packer
T pl
T mold
:
T ¼kr¼k
þ
þ
kpacker
kmold
kpl
ð10Þ
According to the classic Gauss distribution, Eqs. (9) and (10) may be used in Eq. (11) to obtain the final dispersion curve
(D(T)) for the width tolerance after the ring is installed in the tunnel.
DðTÞ ¼
T2
1
pffiffiffiffiffiffiffiffiffiffi e2r2 :
r 2p
ð11Þ
5. Accumulation mechanism
Imagine, for instance, a long tunnel stretch starting with the installation of the first rings against a perfectly plane reaction
structure. The imperfections in the position of the segments in the longitudinal direction should somehow approximate the
width tolerances. In this context, despite the perfect shape of the reaction structure, the surface of the first ring that gives
support to the ongoing construction should already present some imperfections.
Therefore, besides the natural shape deficiencies attributable to the tolerances of its segments, the second ring final position is also influenced by the imperfections provided by the first ring. As shown in Fig. 9, all these imperfections may sum up
so that the deficiencies on the position of the adjacent segments of the second ring are likely to be bigger than that observed
on the first ring installed. Likewise this is reflected on the surface of the second ring that gives support to the next one, which
could potentially show even worst imperfections.
The repetition of this chain of events creates a mechanism of accumulation of tolerances and imperfections from previous
rings, which, in theory, could increase indefinitely as the construction of the lining proceeds. Nevertheless, the observation of
real tunnels shows that the unevenness does not increase without a limit. As indicated by the measurement of the spaces
between adjacent rings conducted at stretches of the Metro Line 9 of Barcelona, the imperfection tends to remain below certain
limit values.
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
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Fig. 9. Mechanism of accumulation of tolerances and imperfections from previous rings.
Since the tolerance follows a statistical distribution, the accumulation mechanism must be treated from a probabilistic
point of view. It is theoretically possible to find segments whose dimensions are far above or far below the ideal width value.
In theory, it is even possible that both segments are placed in the same ring and that the same layout is repeated for long
series of rings, thus generating an every time bigger imperfection. However, this scenario shows an extremely unfavorable
combination of segments that, although feasible, has little probably of happening in practice. On the contrary, smaller imperfection could result from a higher number of combinations of segments, being more likely to happen in a real situation. The
probabilistic definition of the accumulation mechanism is performed according to two approaches.
5.1. Mathematical approach for ideal ring composed by infinite number of segments
The first approach considers the mathematical definition of the accumulation mechanism for an ideal ring composed by
an infinite number of segments. As shown in Fig. 9, the lining is initially placed against a perfectly shaped and rigid reaction
structure so that the imperfections introduced to the opposite surface of the first ring must equal the resultant tolerance of a
single ring. As a result, the function D1 that describes the frequency of imperfections found in this surface follows Eq. (11) for
the dispersion of the total tolerance of the width of the segments.
Considering that the majority of the segments of the second ring are put in contact with two other segments, it is clear
that not all imperfection of the first ring is passed along to the next. Rather than that, the segment of the second ring initially
makes contact with the element of the previous ring that present the maximum absolute value of contact deficiency. In other
words, only the maximum imperfection values are actually transmitted.
Therefore, the probability P2 of transmission of a certain imperfection T0 to the second ring is given by Eq. (12). In this
equation, T00 represents an auxiliary variable used in the integrals.
R T0
P2 ðT 0 Þ ¼ R 1
þ1
1
D1 ðT 0 Þ D1 ðT 00 Þ dT
00
D1 ðT 0 Þ D1 ðT 00 Þ dT
00
:
ð12Þ
As shown in Eq. (13), the distribution of imperfection (D2) in the opposite side of the second ring is the result of the probabilistic combination of the imperfection received from the first ring (estimated through P2) and the own production tolerance of the segments that form the new ring (represented by Eq. (11)).
D2 ðTÞ
Z
þ1
0
DðT T 0 P1 ðT 0 Þ dT Þ:
ð13Þ
1
The generalization of this expression for any ring (Dn) is given by Eq. (14), where n represents the number of the ring in
question.
Dn ðTÞ
Z
1
0
DðT T 0 Pn ðT 0 Þ dT Þ:
ð14Þ
1
It is important to consider that the accumulation mechanism is likely to produce bigger imperfections as more rings are installed. Therefore, as shown in Eq. (15), the worst possible distribution of imperfections happens once the number n of the
ring used in Eq. (14) tends to infinity.
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D1 ðTÞ ¼ lim ½Dn ðTÞ:
ð15Þ
n!1
Using the last function it is possible to determine the probability (DS) in the worst-case scenario of finding an imperfection
bigger than a reference value (smax) formed by two segments of the same ring. This is performed in Eq. (16) that stands as the
basis for the calculation of this probability. Notice that all the integrals used depend solely on the dispersion of the resultant
tolerance of a single ring, which was parametrically defined according to Eq. (11).
DS ¼ 1 Z
smax
smax
Z
1
2 D1 ðTÞ D1 ðT þ sÞ dT ds:
ð16Þ
1
In this case, the use of a normal distribution does not allow deriving an analytical solution for Eq. (16) once some of the integrals can only be solved numerically. According to the most intuitive numerical solution, the normal distribution is truncated
and divided into a sufficiently big finite number of sectors that are represented by an average value of tolerance. The integral
of the curve for each sector give the amount of elements with a certain tolerance. To obtain a relative distribution, the number of elements in each sector is divided by the total amount of elements of the whole curve. The relative distribution is then
used in Eq. (16) to obtain the probability (DS) of finding an imperfection bigger than a reference value (smax).
Throughout the analysis of parametric normal distributions described by a characteristic coefficient k between 1.5 and 4.0
(range found in practice), it was observed that acceptable results are obtained when the curve is truncated for a tolerance
equal to 1.5 T. Moreover, it was observed that the numerical integration up to the 300 rings is enough for the convergence
of the results. Any additional integration does not produce significant change in the results obtained.
To evaluate how Eq. (16) changes as more rings are considered in the analysis, this probability is estimated depending on
the number of rings installed. The results of this study are presented in Fig. 10 for reference imperfections (smax) equal to
0.5 T and 2.0 T.
All curves show the same tendency with an initial phase in which the probability of finding an imperfection greater than
the smax increases significantly, characterizing a strong influence of the accumulation mechanism. As the installation of more
rings is considered, the growth rate of the probability reduces until an asymptotic behavior is observed. From this point on,
the accumulation mechanism reaches stability and the probability of finding an imperfection bigger than the reference value
remains practically constant, hence approaching a maximum limit. Such maximum value should be considered when predicting the accumulation mechanism of the contact deficiencies.
As expected, the comparison between the different graphs shows that an increase on the reference imperfection (smax)
from 0.5 T to 2.0 T causes a reduction not only at the beginning of each curve but also in the probability observed in
the asymptotic branch. In addition to that, it is clear that the parametric curves described by a higher characteristic coefficient k present a lower probability of exceeding the reference imperfection (smax). This result is reasonable since, in theory, a
higher characteristic coefficient k denotes a better production process and segments with a tolerance closer to zero.
The effect of a better production and placement processes becomes more noteworthy as the reference imperfection (smax)
adopted increases. Therefore, according to Fig. 8, the probability of exceeding an imperfection of 0.5 T is approximately 1.45
times bigger for the curve with characteristic coefficient k equal to 1.5 than for the curve with characteristic coefficient k
equal to 4.0. Still, the same comparison for the graph with a reference imperfection (smax) of 2.0 T shows a probability
30 times bigger for the curve with k equal to 1.5.
Although the previous analysis gives important insight about the stability and the evolution of Eq. (16), it does not allow a
clear visualization of its final shape after convergence for different reference imperfections (smax). For that, the abovementioned function is plot in Fig. 11 considering a parametric distribution of the tolerances with a characteristic coefficient k
equal to 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0. The figure illustrates the probability of exceeding an imperfection equal to n times
the initial parametric width tolerance T of a single isolated ring.
All probability functions show a considerable reduction for increasing values of imperfections. From a certain imperfection on, the curves present an asymptote, tending to zero. After this point, the probability of finding a bigger imperfection is
Fig. 10. Probability of finding a salience higher than: (a) 0.5T and (b) 2.0T.
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
4431
Fig. 11. Probability of exceeding a salience n times the initial tolerance.
almost inexistent. Such tendency reflects the stability of the accumulation mechanism, which has practically no probability
of generating bigger contact deficiencies. In fact, most of the imperfections found due to the accumulation mechanism are
concentrated in a range between s = 0 T and s = 4.0 T, meaning that that the final imperfection hardly ever would reach
four times the initial tolerance of a single isolated ring.
The probability curves obtained for distribution tolerances of a ring described by a characteristic coefficient k from 1.5 to
4.0 show clear differences. As expected, a distribution described by a lower characteristic coefficient k has higher probability
of generating bigger relative saliencies when compared with a distribution with a lower coefficient k.
Notice that the use of the parametric normal distribution represents a great advantage once the results obtained may be
generalized for an infinite number of proportional distributions. In other words, there is no need to determine again Eq. (16).
Rather than that, it is enough to multiply the total tolerance of a single ring (T) calculate in Eq. (10) and the value obtained for
the parametric curve with the same characteristic coefficient k.
In this context, to simplify the consideration of the accumulation mechanism, propagation coefficients (c) were estimated
in Eq. (16) for a reference probability of 2.5%. Table 2 shows the values obtained for the propagation coefficient (c) considering parametric distribution curves with different values of k.
Notice that the use of these accumulation coefficients (c) to calculate the final imperfection (d) is quite intuitive, being
performed in accordance with Eq. (17). Suppose, for instance, a construction process with a k of 1.5 and with a tolerance
T, calculated in Eq. (10) for a single isolated ring. In this case, according to Table 2, the propagation coefficients c equal to
4.01 must be used, leading to the conclusion that the maximum imperfection found in practice will not be greater than
4.01 T with a certainty of 97.5%.
d ¼ c T:
ð17Þ
5.2. Monte Carlo analysis for any ring with 6, 7 or 8 segments
Even though the analysis presented in the previous section gives a good idea about the accumulation mechanism and the
mathematical formulation that governs it, the initial hypothesis of a ring composed by infinite segments does not reflect the
reality. In practice, rings are usually formed by sets of 6, 7 or 8 segments, what may have some influence over the actual
value of the propagation coefficients (c) estimated. Therefore, further studies are required in order to obtain these coefficients depending on the number of segments per ring.
Since conceptually the formulation developed for the accumulation mechanism remains valid, the final curve of the probability of finding a certain imperfection shows the same general characteristics already outlined. Despite that, the reduction
Table 2
Propagation coefficient for ring with infinite elements.
Characteristic coefficient k
Propagation coefficient c
1.5
2.0
2.5
3.0
3.5
4.0
4.01
3.27
2.67
2.22
1.91
1.67
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S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
of the number of the segments affects the number of joints present in each ring. In this case, not all joints are subject to the
same random law definition. That, combined with the fact that only the maximum imperfection is transmitted to the next
ring, causes an additional concentration of the probability of a certain imperfection happening. Consequently, there is less
dispersion on the imperfection transmitted to the following rings, hence yielding smaller propagation coefficients.
Due to the complex description of this phenomenon, the mathematical approach from Section 5.1 is substituted by a
Monte Carlo analysis with an eye to the calculation time. In this approach, the installation of several rings is simulated.
The original production tolerance of each segment is determined randomly taking into account the frequency calculated
with the parametric dispersion for the resultant tolerance of a single ring, given by Eq. (11). After the whole sequence of rings
is defined considering the accumulation of imperfections, the assessment of the frequency of imperfections is performed.
This outcome is then translated into the probability of finding an imperfection bigger than a reference value. This probability
function should approach Eq. (16) if this analysis is repeated several times with a sufficient number of rings. Finally, the function obtained is used to estimate the propagation coefficients (c).
In the present study, such analysis was conducted for rings composed by 6, 7 and 8 segments. For each case, 10 parametric curves for the dispersions of the tolerance of a single ring described by characteristic coefficients k ranging from 1.5 to 4.0
were studied separately. These curves were truncated for a tolerance equal to 1.5 T and divided into 30,000 equally spaced
sectors to guarantee an acceptable precision of the results. In order to assure a clear convergence of Eq. (16), a stretch compose by 200,000 adjacent rings was used. This was repeated 300 times for each parametric curve simulated. Since the maximum variation observed among the repetitions is practically insignificant (below 1%), the average of the results was used.
In Fig. 12, the resultant curve obtained for a ring composed by eight segments is compared with the probability function
calculated in the previous section for a ring with infinite segments. The data presented corresponds to segments with a characteristic coefficient k equal to 1.5 and 4.0.
There is a clear asymptotic behavior in all curves in spite of the difference in the calculation procedure applied. Moreover,
the curves obtained for a ring composed by eight segments show smaller probabilities of accumulating imperfections in
comparison with the curve obtained for the ring with infinite segments. This outcome corroborates the initial assumption
that the reduction on the number of segments per ring would cause a reduction on the probability of finding bigger imperfection in practice.
The same conclusion may be derived from Table 3, which gives the propagation coefficients estimated throughout the
Monte Carlos analysis with a probability of only 2.5% of being exceeded. Table 3 shows that the reduction on the number
of segments per ring leads to a decrease on the propagation coefficients (c).
5.3. Equations for the prediction of the accumulation mechanism
The propagation coefficient (c) deducted in the present work was obtained for a reference probability of 2.5%. In some
cases, however, it might be convenient to use other reference probabilities that obviously will yield different propagations
coefficients (c). Furthermore, in some situations, it might be necessary to estimate what is the probability that a certain
imperfection is exceeded.
Both tasks require the direct use of the probability curves that approximate Eq. (16), obtained through the Monte Carlos
analysis for rings composed by 6, 7 and 8 segments. The estimation using the graphical representation is not recommended
in this case since it would imply a significant imprecision for probabilities bigger than 90% due to the asymptotic behavior of
the curve. Rather than that, a mathematical regression of the results was performed to obtain Eq. (16) assuring the precision
required. The function for the probability (DS) of finding an imperfection bigger than a reference value (smax) is approximated with Eq. (18).
DS ¼
1
ðA þ B eCsmax ÞD
:
ð18Þ
Fig. 12. Probability of exceeding a certain salience depending on the number of segments per ring for characteristic coefficient k equal to (a) 1.5 and (b) 4.0.
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S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
Table 3
Propagation coefficient for ring composed by 6, 7 and 8 segments.
Characteristic coefficient k
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
Propagation coefficient c
6 seg./ring
7 seg./ring
8 seg./ring
2.62
2.44
2.25
2.05
1.87
1.71
1.57
1.45
1.35
1.26
1.18
2.69
2.50
2.30
2.11
1.92
1.75
1.61
1.49
1.38
1.29
1.21
2.74
2.55
2.35
2.15
1.96
1.79
1.64
1.52
1.41
1.31
1.23
Table 4
Constants of interpolated probability function for ring composed by six segments.
Characteristic
coefficient k
Parameters
A
B
C
D
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
0.0349
0.0336
0.0333
0.0326
0.0314
0.0306
0.0302
0.0296
0.0298
0.0297
0.0294
0.0219
0.0218
0.0222
0.0224
0.0219
0.0216
0.0214
0.0210
0.0211
0.0211
0.0209
1.2156
1.3041
1.4097
1.5405
1.6970
1.8637
2.0357
2.2122
2.3806
2.5520
2.7272
1.6050
1.5925
1.5929
1.5880
1.5711
1.5600
1.5532
1.5441
1.5469
1.5452
1.5412
Table 5
Constants of interpolated probability function for ring composed by seven segments.
Characteristic
coefficient k
Parameters
A
B
C
D
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
0.0249
0.0243
0.0236
0.0230
0.0227
0.0222
0.0216
0.0214
0.0213
0.0216
0.0214
0.0169
0.0170
0.0171
0.0171
0.0171
0.0169
0.0166
0.0164
0.0164
0.0166
0.0164
1.2637
1.3533
1.4671
1.6053
1.7600
1.9304
2.1126
2.2916
2.4713
2.6386
2.8206
1.4511
1.4455
1.4390
1.4319
1.4286
1.4217
1.4107
1.4070
1.4051
1.4105
1.4065
Table 6
Constants of interpolated probability function for ring composed by eight segments.
Characteristic
coefficient k
Parameters
A
B
C
D
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
0.0190
0.0183
0.0178
0.0177
0.0173
0.0170
0.0165
0.0164
0.0164
0.0164
0.0164
0.0137
0.0137
0.0137
0.0139
0.0138
0.0138
0.0134
0.0134
0.0133
0.0133
0.0133
1.3038
1.3983
1.5176
1.6535
1.8153
1.9880
2.1774
2.3587
2.5431
2.7225
2.9050
1.3460
1.3379
1.3314
1.3336
1.3274
1.3236
1.3124
1.3112
1.3091
1.3104
1.3108
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S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
On the contrary, in order to determine the propagation coefficient (c) for a certain reference probability (DS), Eq. (18) has to
be inverted, thus giving Eq. (19).
1
C
c ¼ smax ¼ ln
ðDSÞ1=D A
:
B
ð19Þ
Both equations consider the probability in percentage and the reference imperfection (smax) in number of times the width
tolerance for a single isolated ring (n T). On the other hand, the parameters A, B, C and D are constants obtained through
mathematical regression. These constants are presented according to the characteristic coefficient k in Tables 4–6 for rings
composed by 6, 7 and 8 segments, respectively. It is important to point out that interpolation is only valid for probabilities
(DS) bigger than 0.1%. For values below this limit, the function loses accuracy due to the asymptotic behavior of the probability curves.
6. General overview of estimation of contact deficiencies
The formulation proposed is previous sections is completely flexible and could be used to estimate any of the parameters
involved in the transformation of the tolerances into the contact deficiencies. Nevertheless, on the design of the segmented
linings, the typical objective will be to estimate the contact deficiencies expected in reality. A general overview of the calculation procedure used in this case is illustrated in Fig. 13.
Initially the main input parameters required for the application of the model are defined and introduced in Eq. (10) to
estimate the final tolerance of a single isolated ring (T) after it has been installed in the tunnel. In parallel, a reference probability (DS) of exceeding an imperfection is defined and applied in Eq. (19) in order to obtain the corresponding propagation
coefficient (c). The latter and the tolerance of an isolated ring (T) are then combined in Eq. (17) to estimate the resulting contact deficiency (d).
7. Case studies
Some additional studies were conducted using the methodology developed for the sum of tolerances and the accumulation mechanism. The first study consists of a comparison between the predicted imperfection and the ones actually
measured in a real tunnel. The second study covers the influence of different types of packers on the imperfections most
likely to happen.
7.1. Comparison between prediction and real case measurements
The confirmation in a real case of the imperfections predicted by the methodology proposed in this work is a very complex task. For once, the reduced magnitude of the imperfections usually requires equipments with a minimal measuring
Fig. 13. Usual procedure to estimate the contact deficiencies.
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
4435
precision of 0.1 mm. Furthermore, the residual deformation of the packer and of the segment as well as the moment during
the construction process when the measurements are performed may have some influence over the results obtained.
In spite of that, the measurement of the imperfections produced after construction was performed at two stretches of the
Metro Line 9 of Barcelona from Bon Pastor to Cam Zam. The first stretch extends from ring 1108 to ring 1118 whereas the
second stretch includes the interval located between rings 1830 and 1848. All segments presented identical characteristics
except for the type of reinforcement applied. Elements from the first stretch were cast with conventional reinforcing steel
bars and 20 kg of steel fibers per cubic meter of concrete. In contrast, segments from the shorter stretch do not have conventional reinforcement that, for experimental reasons, was substituted by an extra amount of 40 kg of steel fibers per cubic
meter of concrete. Notice that the absence of conventional reinforcement led to cracking in most of the segments used in the
latter.
A set of metallic standard rules with different thicknesses were used to quantify the space separating adjacent rings. The
imperfections were then calculated by the difference between the spaces measured at each side of the longitudinal joints.
The accumulated frequency of these imperfections was calculated and subsequently translated into a percentage with respect to the total amount of segments assessed.
In parallel, a prediction about the accumulation mechanism was performed taking into account the formulation given in
previous sections and some data provided by the company responsible for manufacturing the segments. Due to the lack of
information regarding the standard deviation of the production tolerances of the lining and of the packer, it was assumed
they both respond to a productive process with k equal to 1.5. The width tolerance of the segments and the thickness
tolerance of the packer are 0.5 mm and 0.1 mm respectively. It was also considered that the constructive process is well executed, in a way that no considerable placement tolerance is introduced.
In this context, the final production tolerance is the result of the probabilistic combination of the individual curves for the
molds and the packer. The total standard deviation (r) of a ring after installation in the tunnel is estimated in 0.34 mm in
Eq. (9). The total tolerance (T) is calculated by using the characteristic coefficient k (equal to 1.5) in Eq. (10), which gives
approximately 0.52 mm.
A prior study on the universal ring of the Metro Line 9 of Barcelona show that the segments defined on the blueprints do
not follow the ideal shape, presenting some design approximations. This divergence should be reflected in all segments with
an absolute average value equal to 0.68 mm. Therefore, there is no need for the probabilistic consideration of such imperfection since it is always present. As a simplification, the average value was summed with the total tolerance (T) estimated
in the previous paragraph. This gives a final tolerance T for a single ring equal to 1.19 mm.
Eq. (18) was used to estimate the probability of finding a certain imperfection after the accumulation mechanism. The
result of that function along with the curves measured for the two stretches of Line 9 of Barcelona are presented in Fig. 14.
The estimated and the measured curve are very similar, tending to 0 as the magnitude of the imperfections approaches
4 mm. Nevertheless, the good fit is not the most important observation about the results since the own measuring procedure
may introduce a certain scattering. Besides, the accumulation process presents a natural variation that may affect the outcome due to the small number of rings assessed. In this case, the important outcome is that the estimated and the measured
curve clearly follow the same tendency situated within a similar range of values. It confirms that the methodology proposed
for the accumulation mechanism is suitable for estimating the imperfections if accurate input parameters are taken into
account.
7.2. Effect of different types of packers
According to the study performed by Cavalaro [17], the packer used as contact element at the circumferential joints may
be divided in two types. The most common of them consists of rubber plates while the second includes hard plywood plates.
Fig. 14. Comparison between estimated and measured probability curves.
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S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
Table 7
Tolerance measured for different types of rubber plates.
Tunnel
SD (mm)
Tolerance (mm)
Metro Line 9 of Barcelona
Pajares
La Cela
M30
Guadarrama
Average
0.065
0.057
0.054
0.053
0.055
0.130
0.113
0.108
0.105
0.111
0.114
Fig. 15. Probability of finding a certain salience in ring with rubber, plywood and no packer.
Aside from the mechanical behavior, there are clear differences regarding the production tolerance of both types. Based on
that, another study was conducted to assess the influence of the characteristics of different types of packers on the frequency
and the magnitude of the imperfections found in practice. Furthermore, the results obtained were compared with the case
with no packer at the joints.
First, the production tolerance of five different types of rubber plates with nominal thickness between 1.9 mm and
2.9 mm was evaluated by direct measurement with a pachymeter. The samples tested correspond to the tunnels Metro Line
9 of Barcelona, Pajares, La Cela, M30 and Guadarrama. The standard deviation of the measurements and the tolerance calculated for a characteristic coefficient k equivalent to 2 (confidence interval of approximately 95%) is gathered in Table 7.
The reference value considered for this analysis is the average of the individual tolerances equal to 0.114 mm.
A similar assessment was not possible for the plywood plates since no samples of this material were available. In this
case, the tolerance was defined in accordance with the standard BS EN315:2000 [24] that set the limit dimensional variation
of plywood sheets. The maximum production tolerance of 0.3 mm recommended for a 3 mm thick plate was considered.
For this simulation, the same ring composed by eight segments with 0.5 mm production tolerance of the molds and a
characteristic coefficient k equal to 2 was considered. Applying the provided data to Eq. (11) gives a total tolerance of a single
ring equal to 0.51, 0.58 and 0.50 mm for the cases with rubber, plywood and no packer, respectively. Eq. (18) was then used
to obtain the probability of finding a certain imperfection after the accumulation mechanism has stabilized. The results
obtained are shown in Fig. 15.
The graph shows that the curve obtained for the plywood is slightly higher than the equivalent one calculated for the
rubber plate and for the case with no packer. In this context, the use of plywood cause a sensible increase in the imperfections expected in practice.
To better understand the practical repercussion of that, imagine the case of a tunnel where the production tolerances
were originally designed for the use of rubber plates as the packing material. Besides, consider that this design was performed based on the propagation coefficient (c) from Table 3, which would be 2.35 for the lining. Therefore, the maximum
reference imperfection exceeded only in 2.5% of the cases is calculated by multiplying this propagation coefficient (c = 2.35)
and the overall tolerance of the segment with rubber plates (0.51 mm), what gives approximately 1.20 mm. In this case,
damage will only happen in 2.5% of the segments installed.
Suppose now that for economical reasons the constructor decides to change the rubber plate for plywood with equivalent
mechanical characteristics. The change is performed without modifying the tolerance of the segments. Consequently, the
overall tolerance of a single ring will increase in about 37.2%, passing from 0.51 mm to 0.58 mm. In this new situation,
the incidence of imperfection bigger than 1.20 mm also increases. According to Eq. (18), such imperfection will be exceeded
in approximately 4.3% of the joints and the percentage of damaged segments should reach the same value.
S.H.P. Cavalaro et al. / Applied Mathematical Modelling 36 (2012) 4422–4438
4437
It is obvious that the small variation on the overall tolerance (only of 37.2%) caused by changing the packing material
would generate an increase of 72% on the number of damaged segments. To obtain the same frequency of damage as in
the original situation with rubber plates (2.5%), it is necessary reduce either the production tolerance of the plywood to
0.11 mm or the tolerance of molds to 0.41 mm.
8. Conclusions
The definition of the width of the universal ring in a reduced numbers of fixed points represents a design approximation
that generates contact deficiencies between adjacent segments. The analytical model proposed to estimate the contact deficiencies generated show that in the Línea 9 de Barcelona the spaces between segments of adjacent rings may reach up to
2.16 mm. To eliminate these contact deficiencies, the universal ring must be designed according to Eq. (3) rather than
through the definition of the ring width in a reduced number of fixed points.
On the contrary, the elimination of the production and the placement tolerances is not feasible. The probabilistic combination of these tolerances should be performed with Eq. (10) that allows the estimation of the resultant tolerance of a single
isolated ring.
The transformation of this tolerance into the contact deficiencies must be considered from a probabilistic point of view,
using a formulation that describes how the tolerances of several rings interact and accumulate during the construction process. The mathematical formulation developed allows the assessment of the probability of finding a certain imperfections for
the hypothetical case of a ring composed by an infinite number of segments. Alternatively, Eq. (18) obtained in the Monte
Carlo analysis serves to estimate such probability for rings with finite number of segments.
To simplify the prediction of the accumulation mechanism, propagation coefficients (c) ranging from 1.18 to 2.74 are estimated for a reference probability of 2.5%. The multiplication of this coefficient by the tolerance of a single assembled ring
gives directly the maximum contact deficiency produced by the accumulations mechanism with a 97.5% of certainty.
Furthermore, the formulation proposed allows the estimation and the comparison of aspects that so far could not be
directly linked. One example of that is the assessment of the repercussions related with the use of different types of packers
with dissimilar tolerances. In this analysis, although the substitution of a rubber plate by plywood plates produces an increment of only 37.2% on the tolerance of a single isolated ring, it leads to an increase of 72% on the frequency of damaged
segments.
The spaces between rings measured in the Metro Line 9 of Barcelona shows the same tendency and similar absolute values when compared with the values predicted with the model proposed here. This indicates that the latter is suitable for the
prediction of the contact deficiencies found in practice.
For the proper application of the model, further experimental data should be used to calibrate the parameters taking into
account the particularities of each precast industry and work site. Despite that, it is important to remark that the model represents a considerable advance towards the prediction of the contact imperfections and the reduction on the frequency of
damaged segments in tunnels.
Acknowledgments
The corresponding author thanks the EDUCATIONAL MINISTRY OF SPAIN for the FPI Scholarship linked with the national
funding project BES-2006-13592. The authors thank the company FOMENTO DE CONSTRUCCIONES Y CONTRATAS, S.A. for
the dedication to several research projects (especially HATCONS) throughout the years. The authors also thank Enrique Bofill
and Francisco Capilla for their faith on this work.
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