A Consideration on Deterioration Model for Cold
Region Tunnel Lining Based on Life-cycle
Concept
Atsushi SUTOH a, Osamu MARUYAMA a, Hiroaki T.-KANAKIYO b and Takashi SATO c a
Department of Urban and Civil Engineering, Tokyo City University, Tokyo, Japan b
Department of Civil, Environmental and System Engineering, Kansai University, Osaka, Japan c
Civil Engineering Research Institute for Cold Region, Sapporo, Japan
Abstract. This paper proposes the tunnel lining deterioration model which is based upon actual inspection data in order to carry out strategic maintenance and to rationalize life cycle cost analysis for tunnel structures. The evaluation value of deteriorating tunnel structures is a non-stationary stochastic process, and reliability problems of such structures are needed to the consideration of the future risk. In Japan, the probability that an earthquake occurs is high compared with other countries. A risk analysis of seismic motion and technical components as well as damages associated with cost variables have to be dealt with. The model of the tunnel management system presented in the paper is applied to the asset management of the cold region road tunnels. This research will be developed to the efficient tunnel maintenance system and a quantitative criterion from pictures of tunnel lining using the life cycle cost analysis. And also, in order to describe the deterioration model of the life cycle assessment is needed to consider the large-scale earthquake phenomena in Japan.
Keywords. Deterioration model, stochastic process, life-cycle cost, tunnel structure
1. Introduction
Optimum life-cycle-based engineering decisions must examine the influence of concepts related to both the target safety level of the initial system and the eventual repair and maintenance actions that may be undertaken during the life of the system. Reliability and expected performance functions of tunnel structures are sensitive to the process of damage accumulation associated with the random sequences of ground motion excitations that those systems may experience.
On the other hand, the deterioration model of tunnel structures becomes more and more an important with regard to life cycle assessment. The deterioration of conventional construction method road tunnel lining is non-stationary stochastic processes, and reliability problems of such structures are essentially different of time-independent reliability problems. While the forecasting model of deterioration is one of the central takes in infrastructure asset management, it is often the cases where are few data stocks available for estimating the deterioration forecasting model.
This presentation includes an overview of the general framework supporting these decisions, as well as some available results about a) the influence of damage accumulation of seismic risk for tunnel structures in Hokkaido and b) approximate estimates of accumulated damage is described by the Ito stochastic differential equation, c) a life-cycle cost analysis tool for cold region road tunnel structures containing tunnel lining damages influences has been developed.
Figure 1. Life cycle cost for tunnel performance
© 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
2. Concept of Tunnel Management System The model of tunnel management system presented in the paper is applied to the asset management of the road tunnels.
Also, this research project will be developed to the efficient tunnel maintenance system and a quantitative criterion from pictures of tunnel lining using the life cycle cost theory (See Figure 1).
2.1. Present of Tunnel Management System In this maintenance system is consist of the two following parts which shows in the Figure 2. Upper part is data store and estimated of the tunnel lining performance which based on inspection data.
Firstly, the database of the foundations of the basic specifications of the nation-al highway tunnel in Hokkaido, a repair history, checks data, etc. is created. Secondly, the degrading model based upon actual inspection data in order to carry out strategic maintenance and to rationalize life cycle cost analysis for tunnel structures.
The resistance of deteriorating tunnel structures is non-stationary stochastic processes, and reliability problems of such structures are essentially different of time independent reliability problems. Lower part is the tunnel maintenance planning of the suitable repair time, and the optimal repair construction method for tunnel lining is constituted in the developing system.
2.2. Quantification of Inspection Data
Inspected data from existing tunnels in Hokkaido are used for the quantification of tunnel lining (Sutoh A et al. 2008).
Here, six check damages or deteriorations of the tunnel lining shows as follow; 1) Crack, 2) Flaking off, 3) Water leak, 4) Joint and opening, 5) Faulty and Cave, 6) Efflorescence.
3. Stochastic Model for Degrading Process The present problem is the identification of the damage degree or performance function X(t) of
the tunnel concrete lining represented by the stochastic differential equation.
We generally suppose that it’s mean behavior can be described by the following differential equation. )) ( ( ) ( / ) (t dt 0 t g X t dX
P
(1) where g(x) is represents a shape function quantifying the growth rate,P
0is represents a parameter describing a mean growth resistance. Further, we assume the random behavior of the damage growth is driven by a noise incorporated in the growth resistance.^
( ) ()`
( ()) /)
(t dt 0 t W t g X t
dX
P
Z (2)where WZ(t) is a stochastic process with mean zero representing the driving noise. Equation (2) is then transformed into the following stochastic differential equation of Ito type.
) ( )) ( ( )) ( ( ) ( / ) (t dt 0 t g X t dt g X t dZ t dX
P
(3)In which Z(t) an integrated process defined as ds s W t Z
³
t Z 0 ( ) ) ( (4) And X(t) represents a left-continuousversion of X(t) defined as
X
(
t
)
lim
X
(
s
)
t so
.Since actual data shows that the shape function g(X(t)) is proportional to X(t) (Maruyama et al. 2009), we assume that the Eq.(4) is reduced to the following linear equation. ) ( )) ( ( ) ( ) ( ) (t 0 t gX t dt g X t dZ t dX
P
(5)We propose a mathematical model by introducing a noise of a Poisson white noise (Tanaka et al. 2011).
3.1. Diffusive Model
The most widely used noise in constructing stochastic system is the well-known Gaussian white noise.
That is, the stochastic process Z(t) in Eq.(3) is given as Z(t) = BB(t) with positive constant B , where B(t) is a standardized Wiener process, which is a temporally homogeneous Gaussian process having independent increments with
E{B(t)} = 0, E{B(t)B(s)} = min{t,s} (6) The growth equation is then given as
) ( )) ( ) ( ) ( ) (t 0 t X t dt X t dB t dX
P
V
B (7)which has been widely used in many application fields, such as for describing random fatigue crack growth in the structural reliability analysis or temporary random variation of stock price in the mathematical finance (Black and Scholes, 1973).
Figure 3 show a fluctuation of the performance function and its log-normal distribution are indicated.
3.2. Improved Model using a Poisson White Noise
To remove, the fluctuation of the solution process in the diffusive model given by Eq. (7), we newly propose a mathematical model by introducing a noise of another type, i.e., a Poisson white noise (Maruyama et al. 2014).
The Poisson white noise is a formal derivative of a compound Poisson process C(t),
expressed as
¦
(1) ) ( t N k k Y t C (8)where N(t) is a temporally homogeneous Poisson
process with an intensity,
^ `
Yk k 1,2,is a family of i.i.d. (identically and indecently distributed) positive random variables, whose probability distribution function, is supposed to be given. ) ( ) (y PY y F k d (9) Further, it is also assumed that^ `
, 2 , 1 k k Y is
statistically independent of the Poisson process N(t). Setting the noise Z(t) as a compensated version of C(t) , i.e.,
t
q
t
C
t
Z
(
)
(
)
O
1 , q1 E^ `
yk (10) and substituting Eq.(10) into Eq.(5), we can obtain the following Ito stochastic differential equation which describe the damage accumulation.)
(
)
(
)
(
)
(
)
(
t
t
X
t
dt
X
t
dC
t
dX
P
(11) 1 0() ) (tP
tO
qP
(12)The first term of Eq.(11) represents continuous growth of the damage degree
according to the effect of rainfalls, salt or other usual effects.
On the other hand, the second term represents large-scale growth of the damage caused by unusual effects such as large-scale earthquake, serious frost damage etc. In this study we call the former growth large scale growth and the later small scale growth.
The solution of Eq. (11) can be obtained in analytical form, by applying the Ito formula (Ito, 1942), as
^ `
() 1 ) 1 ( exp ) 0 ( ) ( t N k k Y t X t XP
(13)The degrading process X(t) of the proposed model has following mean and variance.
>
X t@
X^ `
t E ( ) (0)expP
0 (14)>
( )@
(0) exp^
2 0` ^ `
(exp 2 1) 2 t q t X t X VarP
O
(15) which> @
2 0 EYkq is a second moment of i.i.d.
positive random variable Yk. In this research, we assume the exponential distribution for the Yk.
)
exp(
)
1
(
)
(
y
Q
y
Q
f
Y (16)> @
2 2 2 EYk 2Q
q (17) Comparing the mean and variance of the diffusive model and the proposed model, it leads,2 2
0
O
qV
(18)The accuracy and propriety of the proposed model will be examined in the numerical example.
4. Parameter Identification of Improved Model
4.1 Identification of Performance
Inspected data from existing tunnels in Hokkaido are used for the identification of parameter of improved model using a Poisson white noise. Inspection data of the conventional construction method road tunnel has been carried out last
several decades, such as crack width, crack length and crack expanse of about 240 tunnels.
It should be mentioned that most of the aged tunnels have repaired at certain period. It is quite difficult to identify based on the existing records, for example, when and how much repaired.
Figure 4 shows a degrading ratio of the conventional construction method type tunnels in terms of tunnel age and the reproduced degrading process identified by general solution of improved model using a Poisson white noise using the tunnel inspection data.
Above mentioned reason, these results indicate degrading process of a group of tunnel. Therefore, it is served that these results are satisfactory for the degrading process of the tunnel lining described the improved model using a Poisson white noise.
And in figure 4, shows the distribution of the conventional construction method type inspection data are divided 4 periods. Here, the using the degrading ratio of the conventional construction method type inspection data are divided 4 periods, in which divided the 10-20 years, 21-30 years, 31-40 years, 41-50 years. 4.2 I Distribution of Performance
Figure 4. Degrading and average value of tunnel lining
In the improved model using a Poisson white noise, the variance or distribution of considered repair effect is log normal distribution in Figure 5. The validity of these deterioration models of tunnel lining is verified through the actual inspection data in Hokkaido. And, the
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10 20 30 40 50 Years D e gr a di ng V a lu e Degrading Value Average
applicability of methodology presented is examined by the real data concerning the deterioration In addition, the average deterioration curves, variance and distribution of time history are obtained by the visual inspection data of tunnel lining, which was considered the repaired process of the each lining.
5 Durability Prediction Models for Life Cycle Assessment
5.1 Deterioration models from life cycle risk Any deterioration process reduces the durability as well as the integrity and reliability of a structure. Deterioration models are key components of life cycle assessment. They should predict the durability affecting deterioration processes sufficiently accurate risk in life cycle that is available from the structure.
Therein, the risk is described as a combination of the probability of any negative consequence and its resulting severity.
This definition is expressed in Eq. (19).
s
P
R
r
(19) where R is Risk, Pr is Probability, s is Severity.According to Sandoval-Wong and Schwartz (2009) it has to be added that risks inherit deviations that can result in positive or negative results. A positive deviation is regarded as an
opportunity, while a negative deviation represents a danger.
In Japan (especially Hokkaido), large-scale growth of the damage caused by natural disasters such as large-scale earthquake, serious frost damage etc. In this study we call the former growth large scale growth and the later small scale growth. And an event of large-scale earthquake is subject to uncertainty; the consequence is referred to as risk.
In order to describe the deterioration model of the life cycle assessment are needed to consider the large-scale earthquake phenomena in Japan. A probability of large-scale earthquake in the various part of Japan is opened from the Japan Seismic Hazard Information Station; J-SHIS.
To consider the risk, the large-scale earthquake probability Pr in life cycle model given by Eq. (20).
^
PT N`
Na ln1.0 ( d ) /
Q
(20)which N is years and
Q
a is annual average occurrence rate of the earthquake which makes shaking beyond optional seismic intensity a.Therefore, the Eq. (20) includes Eq. (11), gives us the new deterioration model to consider a life cycle.
5.2 Numerical results
First, numerical examples (Several samples of the Monte Carlo Simulations) are carried out to examine the effectiveness and accuracy of the proposed model by using the all inspected data obtained from existing the ordinary type tunnels (See Figure 6(a)).
Next, Figure 6(b) shows the several samples of the Monte Carlo Simulations from existing the NATM type tunnels.
The results indicated that a damage accumulation rapidly grow up the large-scale earthquake. The intense cold, the rough wave and wind with sea salt may accelerate the deterioration.
And, advance Monte Carlo sampling techniques, due to the small sample number necessary with respect to the required sample number of classical Monte Carlo simulation, are basic tools for efficient, highly accurate probabilistic reliability assessment methods in practical engineering.
6 Concluding Remarks
A method is developed to identify degrading process of tunnel lining concrete through inspected data of existing aged tunnels. In order to describe the deterioration model of the life cycle assessment is needed to consider the large-scale earthquake phenomena in Japan.
The stochastic model of degrading process is described by the Ito stochastic differential equation. Basic examples are demonstrated based on the inspected data from existing aged tunnels.
References
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(IALCEE2014㸧, 2012-2017
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der Baubetriebs und Immobilienwissenschaften,
Technische Universität München, 3.-4. Dezember, 71-94.
Figure 6(a). Monte Carlo Simulation (Ordinary Type)
Figure 6(b). Monte Carlo Simulation (NATM Type)
0 1 2 3 4 5 6 0 10 20 30 40 50 De g ra d ing Ra tio X(t)
Tunnel age (Year)
CL PW PE Observation (Mean) Montecalro 0 1 2 3 4 0 5 10 15 20 25 D e g ra d in g R a tio X (t)
Tunnel age (Year)
CL PW PE
Observation (Mean) Monte Calro
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