Journal of Applied Mathematics and Physics, 2014, 2, 1073-1078
Published Online November 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.212123
How to cite this paper: Le, T.S., Huh, J. and Park, J.-H. (2014) Earthquake Fragility Assessment of the Underground Tunnel
Using an Efficient SSI Analysis Approach. Journal of Applied Mathematics and Physics, 2, 1073-1078.
http://dx.doi.org/10.4236/jamp.2014.212123
Earthquake Fragility Assessment of the
Underground Tunnel Using an Efficient SSI
Analysis Approach
Thai Son Le1, Jungwon Huh1*, Jae-Hyun Park2
1Department of Civil and Environmental Engineering, Chonnam National University, Yeosu, Ko rea
2Department of Geotechnical Research Engineering, Korea Institute of Civil Engineering and Building
Technology, Goyang, Korea
Received Sep te mber 2014
Abstract
This paper represents a simplified seismic fragility analysis approach of the underground tunnel
structure in consideration of the soil-structure interaction (SSI) effec t. SSI effect founds to be es-
sential in the estimation of dynamic analysis of underground structures like tunnels and thus
needs to be considered. The ground response acceleration method for buried structures (GRAMBS)
known to be a very efficient quasi-static method that can consider SSI effect is used in the pro-
posed approach to evaluate seismic structural responses without sacrificing much accuracy. Seis-
mic fragility curves are then developed by applying the maximum likelihood estimates (MLE) to
responses of a large set of artificial ground motion t ime histories generated for multiple different
levels of earthquake intensity. It is also assumed in this paper that the seismic fragility curve can
be represented by a two-parameter lognormal distribution function with median and log-standard
deviation that need to be defined using MLE.
Keywords
Soil-Stru ctu r e In ter ac tio n, Maximum Likelihood , Fragility Curves, Dynamic Ana lysis
1. Introduction
So far, soil-structure interaction (SSI) analysis is one of the main methods which are being used to observe the
struc tural behavior under seismic excitation considering interaction effect between soil and structure. Also it is
known that, due to unbounded nature of ground system, the computational size of this method is very large. For
this reason, the GRAMBS that can reduce the computational cost of analysis has been proposed [ 1] . In addition,
there is a necessity of a comprehensive methodology for risk assessment of tunnel structure, which has been
mainly based on the expert judgment or empirical fragility curves derived from actual damage record in past
earthquake. The study presents the development of fragility curves by integrating a simplified SSI analysis me-
*
Corresponding author.

T. S. Le et al.
1074
thod, RAM and maximum likelihood estimates (MLE).
2. A Simplif ied Seism ic Fragility Analysis Methodology
2.1. Response Acceleration Met hod—RAM
A simplified dynamic analysis approach GRAMBS is used for two dimensional seismic SSI analyses of tunnel
structures. First, the surrounding soil of the tunnel structure is subdivided into several layers and the free-field
soil response analysis without considering the tunnel is performed using the equivalent linear method-based
software such as SHAKE, M-SHAKE, etc. The time dependent displacement and acceleration response of each
layer are stored. Then, the time
m
t
when the maximum displacement difference occurs between the top and
bottom levels of the tunnel is sought. The response acceleration at
m
t
is taken over the full depth of the soil
column and converted into the body force in the static analysis. Finally general static structural analysis can be
executed to obtain resultant stress state of the tunnel at the time
m
t
and it can be considered as the approximate
maximum stress state.
2.2. Definition of Damage State
Seismic fragility curves are required to be developed according to the variation of damage index with increasing
of seismic intensity for different levels of damage states. Due to the lack of information available on damage
indexes and related parameters for the development of tunnel fragility curves, the damage index (DI) is assumed
to be the ratio between the actual bending moment
( )
M
and capacity bending moment
()
Rd
M
of the tunnel
cross section. A definition based on moments is compatible with the use of displacements, according to the
equal displacement approximation. It is assumed that the behavior of tunnel is approximated to that of an elastic
beam subjected to deformations imposed by the surrounding soil due to seismic waves propagating perpendicu-
lar to the tunnel axis. With help of the previous experience on damages of tunnels and engineering judgment, a
set of 4 different damage states is introduced in this study as shown in Table 1.
2.3. Fragility Curve Devel o p ment
It is generally assumed that the fragility curves can be expressed in the form of two-parameters of distribution
functions as shown below [2].
( )
lnx
P IMx
µ
β
 −
== Φ

. (1)
where
( )
P IMx=
represents the probability of collapse, given a ground motion with
IM x=
,
( )
Φ
is the
normal cumulative distribution function,
µ
and
β
are the mean and standard deviation of ln
IM
, respec-
tively.
The Equation (1) can be expressed in terms of the median parameter as.
( )( )
ln x
P IMx
θ
β

== Φ

. (2)
where
θ
represents the median of the fragility function (the
IM
level with 50% probability of collapse)
The estimation of these two-parameters is done by the maximum likelihood estimates (MLE) treating each
IM
level of tunnel damage as a realization from a Bernoulli experiment. The likelihood is expressed as [4]:
( )
1
Likelihood 1
jj
jj
nz
mzz
j
jj
jj
npp
z
−
=

= −


∏
. (3)
where
j
p
represents the probability that a ground motion with
j
IM x=
will cause collapse,
j
z
is the num-
Table 1. Damage state provided by Pitilakis [5].
None
Minor
Moderate
Extensive
DI <= 1.0 1.0 < DI <= 1.5 1.5 < DI <= 2.5 2.5 < DI <= 3.5

T. S. Le et al.
1075
ber of collapses out of
j
n
ground motions,
m
represents the number of
IM
level.
The final goal is to choose the function that produces the highest probability of observing collapses. Therefore,
estimates of fragility function parameters can be obtained by maximizing the following likelihood function.
{ }
( )( )
1
ln ln
ˆ
ˆ, max1
jj
j
nz
z
mjj
j
ij
xx
n
z
θθ
µθ ββ
−
=


 

= Φ−Φ






∏
. (4)
The parameters which maximized likelihood function (4) will also maximize the log likelihood function be-
low:
{ }
() ( )( )
1
ln ln
ˆ
ˆ,maxlnlnln 1
mjj
j
j jj
ij
xx
nz nz
z
θθ
µθ ββ
=


 



 
=+Φ+ −−Φ
 
 



 


∑
. (5)
3. Numerical Example
An actual one-story two-cell reinforced concrete tunnel (cut and cover section) is considered in this numerical
example. As shown in Figure 1, the width of the tunnel is 29.3 m and the height is 7.8 m. The structure consists
of two reinforced concrete deck slabs (the thickness of the top and bottom slab are 1.2 m and 1.3 m, respectively)
and it is supported by 1.0 m thickness walls in both sides and a column in the middle (0.6 m × 0.74 m). The soil
profile around the tunnel consists of three layers namely sand, weathered soil, and rock, respectively. The prop-
erties of concrete tunnel and soil layers in this example are given in Table 2 and Ta bl e 3 in details.
In order to established the fragility curves for different damage states defined above, 200 artificial time histo-
ries (20 different time histories for each seismic level) based on the design spectrum [6] are randomly utilized as
input data for the application of free-field soil analysis approach. The acceleration profiles at
m
t
when dis-
placement value between the top and bottom of the tunnel is largest are obtained at each layers of ground system.
The process of RAM is illustrated in Figure 2.
The tunnel is analyzed with the MIDAS Civil finite element program [3], using the two-dimensional model.
According to this model, the structure is composed of vertical column and horizontal beam with specified sec-
tions in Table 4. Body forces converted from response acceleration results at each ground layers are applied
Figure 1. Dimension and ground condition of the example.
Table 2. Properties of reinforced concrete structure.
Material
Poisson Ratio
υ
Elastic Modulus Ec (kPa)
Unit mass
ρ
(t/m3)
Damping
Concrete 0.17 28693 2.5 0.5
Table 3. Properties of soils.
Soil name Shear wave velocity (m/s) Poisson Ratio Unit weight (t/m
3
)
Sand 275 0.35 1.8
Weathered soil 500 0.35 2.0
Rock 1500 0.25 2.3

T. S. Le et al.
1076
Figure 2. Response acceleration method process.
Table 4. Specification section of the structure model.
Section name Longitudinal (m) b Thickness (m) h Area (m2) A = bh
Upper slab 2.0 1.2 2.4
Lower slab 2.0 1.3 2.6
Walls 2.0 1.0 2.0
Column (CTC = 4m) 0.6 0.74 0.44
to the tunnel. Static analysis is performed and bending moment results of the structure which illustrated in Figure 3
are obtained.
From the results of structure analysis, moment data at weakest point (the intersection point between vertical
column and horizontal beam) are collected. Then, the failure probabilities for three damage states according to
Table 1 are evaluated at each increment of seismic intensity in the Table 5 where (1), (2), and (3) are minor
damage, moderate damage and extensive damage, respectively and IM is the intensity measure of the ground
acceleration in terms of standard gravity.
Two statistical parameters for the fragility curve of the tunnel are obtained using proposed method and fragil-
ity curves are derived from it. The Maximum Likelihood method applying to obtain two-parameters is clarified
in Figur e 4. The median peak ground acceleration at each damage state is acquired with its corresponding stan-
dard deviation value summarized in Table 6.
Figure 5 displays the computed fragility curves which associated with damage state definition provided in
Table 1. This computation can be performed using EXCEL program in the study.
The figure shows that the fragility curves for various damage indexes which provided in Ta ble 1 have similar
shape. The number of failures dramatically increases with the slightly rise of seismic intensity which is satisfac-
tory agreement compared to empirical fragility curves provided by [5]. The definition of damage states for tun-
nel structure needs to be investigated further since there is no universal guideline for it in the profession.
4. Conclusion
A simple yet comprehensive numerical methodology is proposed for fragility assessment of an underground
tunnel subjected to seismic loading by integrating responses acceleration method (RAM) and Maximum Like-
lihood approach. It is elaborated with an illustrative numerical example and found to be computationally effi-
cient. However, there is a necessity for further study on various types of structures and adequate-rational dam-
age states need to be established in the development of seismic fragility curves for tunnel structure.

T. S. Le et al.
1077
Figure 3. Model example and bending moment result.
Figure 4. Two parameters obtained using Likelihood approach.
Figure 5. Fragility analysis results.
Table 5. Obtained failure probability data.
IM (g)
Number of failure
Probability of failure
(1) (2) (3) (1) (2) (3)
0.060 0 0 0 0.00 0.00 0.00
0.085 7 0 0 0.35 0.05 0.00
0.100 15 1 0 0.75 0.25 0.00
0.125 20 5 0 1.00 0.65 0.00
0.135 20 13 0 1.00 1.00 0.00
0.150 20 20 0 1.00 1.00 0.00
0.200 20 20 2 1.00 1.00 0.10
0.220 20 20 7 1.00 1.00 0.35
0.240 20 20 16 1.00 1.00 0.80
0.250 20 20 20 1.00 1.00 1.00

T. S. Le et al.
1078
Table 6. Summary static parameter value.
Damage state Minor Moderate Extensive
Median
θ
0.09 0.13 0.22
Standard Deviation
β
0.13 0.11 0.07
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A4A01015343).
References
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[2] Shinouka, M., Fen g , M.Q., Kim, H., Uzaw a, T. and Ueda, T. (2001) Statistical Analysis of Fragility Curves.
[3] Mi das-IT (2011) MIDAS Civil On-Line Manual. http://manual.midasuser.com/EN_TW/civil/791/index.htm
[4] Baker, J.W. (2011) Fitting Fragility Functions to Structural Analysis Data Using Maximum Likelihood Estimation.
http://web.stanford.edu/~bakerjw/fragility/archived_versions/Baker_(2011)_fragility_fitting.pdf
[5] Pitilakis, K. (1995) Fragility Function for Roadway System Elements. SYNER -G, No. 244061 .
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