The use of probabilistic methods to analyse reliability of structures is being applied to a variety of engineering problems due to the possibility of establishing the failure probability on rational grounds. In this paper we present the application of classical reliability theory to analyse the safety of underground tunnels

Alarcón Álvarez, Enrique

Fraile de Lerma, Alberto

Gómez Lera, Sagrario

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Probabilistic Analysis of Tunnel Liners

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PROBABILISTIC ANALYSIS OF 
TUNNEL LINERS 
A. Fraile, M.S. Gomez Lera and E. Alarcon 
©1996 ,CJVJL-COMP Ltd., Edinhurgh, Scotland 
B.H. V. Topping, (Editor), 
Advances in Computational Methods for Simula! ion, 
Civil-Comp Press, Edinburgh, 171- 176 
Department of Structural Mechanics and Industrial Constructions, Technical 
University of Madrid, Madrid, Spain 
Abstract 
The use of probabilistic methods to analyse reliabiliy of 
structures is being applied to a variety of engineering 
problems due to the possibility of establishing the failure 
probability on rational grounds. In this paper we present the 
application ofclassical reliability theory to analyse the safety 
of underground tunnels. 
1 General 
The continuous development of transpmtation networks with 
stringent geometric conditions has induced an increasing pace 
of tunnel construction. Generally the techniques used to build 
them are based on a careful control of the structural response 
of the lining and this implies the use of sophisticated numerical 
techniques of analysis to predict and interpret the 
experimental results. In particular it would be interesting to 
have a measure of the liner reliability in relation with the 
di:fferent conditions that could be expected to limit the tunnel 
performance. 
In that sense the use of the Limit State philosophy, 
widely admitted in other areas of structural analysis, seems to 
be very promising. The same can be said in relation with the 
so-called Level II methods of structural reliability that allows 
the implementation of a general procedure as well as the 
possibility of calibration of Codes of Practice based on the 
factored load and resistance properties. 
In this paper we are going to show the way in which 
we have tried to establish a procedure to quantify the 
reliability of road tunnelliners. 
The plan of the paper is as follows: first of all we shall 
describe sorne details of the physical process, then a series of 
Limit States will be defined. Mter that, a simple mathematical 
model will be defined according to the very well known New 
Austrian Tunneling Method (NATM) procedure. After 
summarizing sorne results obtained with Level II methods, 
more general models will be presented that can be applied to a 
variety of situations including classical Terzaghi approach or 
the Panet procedure to simulate 3D conditions using 2D 
procedures. 
2 The physical model 
The problem that arises in tunnel piercing can be explained 
using figure l. In it four zones have been established in a 
sequence through which all sections of the tunnel have to 
pass. 
171 
LONGITUDINAL ARCHING 
1iner 
~-rrrr 1 -~._._1------r-
zone 1 1 Zone 11 Zone IV 
excavation front 
Figure 1 
Zones I and IV are stabilized situations. In the frrst, the soil is 
unaltered, because the effect of the perforation has not 
attained it, while in zone IV all the interaction effects between 
the soil and the liner have reached an equilibrium so the lined 
opening can be considered stable. 
In zone II the so-called convergence (that is, 
displacements directed towards the tunnel axis) of the soil not 
yet excavated, starts. The length of the front influence varies 
depending on the soil properties between one and two tunnel 
radü. The significance of this zone is that befare any opening 
has been materialized the soil has already suffered a 
decompression that alters the initial "at rest" stress-state. 
In zone III the excavation has been finished and two 
subzones can be identified according to whether or not the 
linning has been built. In zone (III a) the soil or rack is 
unsustained so a rapid convergence happens. 
Generally it is very difficult to measure those 
convergences due to the complicated operations that have to 
be done in arder to mantain the workers safety, but sorne of 
them can be obtained. As could be expected the length of the 
subzone is very dependent on the soil strength and is part of 
sorne empirical classifications (see for instance ref 2) 
In zone (III b) the liner has been input and the process 
of soil-structure interaction starts until the equilibrium of zone 
IV reached. It is generally in this area where most 
measurements are done: convergence is one of them, of 
course, but also total stresses in cells included in the liner or 
differential displacements measured with extensometers 
anchored to the liner. From those measurements it is possible 
to establish a data bank to be used in the calibration of 
numerical models that can be utilized to compute the failure 
probability of the system as indicated in figure 2. 
MATHEMATICAL MODELING FINITE 
ELEMENTS 
BOUNDARY ELEMENTS 
BLOCK THEORY 
CONVERGENCE 
CURVE 
FITTING 
Figure 2 
3 Limit States 
The limit states are related, as usual, to conditions that have 
to be fulfilled in order to mantain the serviceability or to 
guarantee the resistance of the tunnel. We propase to selectas 
a first limit state a value related to the convergences. 
On the one hand convergences measure the degree of 
stabilization of the soil-liner system but on the other they also 
reflect the accuracy of the geometric dimensions of the 
structure. In case of symmetric deeply embedded structures 
the convergence is twice the radial component of the 
displacements. 
The admissible value is usually taken as a proportion 
a, R of the liner radius, so the first condition is 
where ft (Ko, Ks, H, etc) is a function of the at rest coefficient 
Ko, soil density "(s, depth of embedment H, and other 
parameter influencing displacements. If conditions are non 
symmetric the limit-state could be based on the absolute 
maximum of the displacements or on a weighted mean value 
that can take into consideration the different importance of 
displacements in relation with the use of space (i.e.: a 
ventilation duct, or a traffic space or the part outside the 
circulation area, etc). 
The second and third limit states are related to the 
liner resistance and soil resistance respectively. The liner 
resistance can be expressed in terms of hoop stresses for the 
case of symmetric scenarios as a function 
where t, R, f~k , etc are respectively the liner thickness, liner 
radius, characteristic value of concrete strength, and any other 
parameter influencing the resistance. 
In less simple cases it would be better to work with 
actions effects and to establish the admissibility on the basis of 
an interaction diagram of bending moments, and axial and 
shear forces. 
172 
The soil resistance is another factor to control because 
the key idea in the New Austrian ·Tunneling Method 
(N.A.T.M.) (3,4) it just to create a mixed structure in which 
the liner supplies radial confmement to the soil in the degree 
neccesary to obtain a ductile behavior of the rock mass. In 
that way a soil arch or plastified zone is created around the 
tunnel. This arching effect is fundamental to obtain the 
stability and the economy, so the third limit state is related 
to the soil resistance. A way to control the soil arch in a 
symmetric situation is to compare the radius of the plastified 
zone with the liner radius, i.e. 
f3 (K o , y s , H , t, R, <¡>, e, etc) ~ a 3 R 
where the parameters are those of the previous states plus 
sorne others like the interna! soil friction cp, cohesion e, etc. 
In more complicated cases the plastified zone is not 
concentric with the liner and then it is better to establish a 
maximum distance or a weighted mean distance to the liner as 
a criterium. 
Other limit states can be envisaged similary depending 
on the use of the tunnel, the soil conditions, the embedment 
depth, etc. For instance in shallow depths it is possible that the 
formation of a key block due to the rock fissuration could be 
more critica! because the confinement pressure could be not 
high enough to induce ductile behavior. In those cases a 
different behavioral model would be used as well as a different 
criterium. In this paper only the above mentioned criterium 
will be used in order to clarify the application of the reliability 
philosophy. 
4 Mathematical model 
T~e classical interpretation of the N.A.T.M. can be explained 
usmg figure 3 that can be compared with figure l. 
The idea is to plot for a symmetric situation the 
pressures that should be applied to the boundary of a circular 
hole (figure 3.a) to obtain a fixed radial displacement o. The 
curve AB (figure 3.b) shows after an elastic part AA' a curved 
branch A 'B, reflecting the plastic behaviour of the continuous 
media. 
(a) (b) (~) 
Figure 3 
Similarly it is possible to plot the same relation for a ring 
representing the liner (figure 3.c). in this case the curve is Po 
p* and starts at a point (Oo,O) because before the liner is built 
(zones 1, 11 and 111 a of figure 1) the soil has contracted. The 
equilibrium is reached at point P1 except if the soil is viscous 
in which case a family of AB curves will develop for different 
time intervals t¡. In a viscoplastic soil the fmal curve will be 
reached for t = oo where all the plastic deformations are 
developed Point p* will be then the design point or check-
point because there all the three above mentioned limit states 
are in the worst conditions. 
P. Fritz (ref 5) using a Mohr Coulomb criterium has 
developed the formula to quantify the soil model. If a linear 
behaviour of the liner is assumed it is possible to compose 
Table 1 
1+sencp 
m=--__;_ 
1- sencp 
2Po -as 
a = 
P 1+m 
SOIL 
2c coscp 
1- sencp 
k1 = (a + ~) [(1 - v ) 1 + m2 - v ] P m-1 s 2m s 
Table 1 
where symbols have been used the following 
p: 
R: 
radius of plastified zone 
liner radius 
LINER 
, o1 - o0 
€ =---
R 
• allim • t p =--
R 
173 
<p: 
e: 
po: 
radial pressure p at whích plastification 
starts 
parameters reflecting soil properties 
according to Mohr-Coulomb criterium 
interna! pressure 
interna! friction angle 
soil cohesion 
initial soil stress (assumed isotropic) 
radial displacement corresponding to 
p¡ 
v s Es: Poisson coefficient and Y oung Modulus 
ofsoil 
E1: liner hoop deformation 
01: liner radial displacement 
&: initial soil displacement before liner is 
introduced 
t: liner thickness 
E,: liner Young modulus 
O'JJim: liner limit stress 
Generally it is more realistic for reinforced concrete liners to 
use a combination of the material laws, as those presented in 
figure 4 and to limit not the stresses in the liner but the 
concrete deformation because the ultimate state will be 
concrete crushing (for instance Eu = 0.003) 
Fivur~ 4 
In figure 5 we show the diagram for a particular situation in 
which the liner has been modeled taking into account the 
nonlin11~r rnnc.rete behaviour law. (Ref 10) 
5 Reliability level 11 methods 
As it is well known (ref 6) in addition to limit states it is 
necessary to choose the basic variables which statistif 
properties are defmed through their mean values and 
covariances. ·In the examples we ha ve been treating, 6 
variables have been chosen related to the soil (Es, Ys, Ys· e, <p 
and H) and other 6 for the liner ( o0 , f~k t R, ros, roe) where the 
meaning of each symbol has been defmed in previous 
paragraphs, except ffis and m: that are the ratios of steel 
reinforcement u sed as bars or solid pro files respective! y. 
U sing the Hasofer and Lind definition (ref 7) the 
reliability index for every limit state is related to the mínimum 
distance of the origin to the failure surface when the variables 
are standardized and make independent after the adequate 
transformations. 
BEHAVlOR FOR MEAN VALUES 
P, 5.88 
--SOIL 
--UNER 
• • • o • • • • • • ·: • • • • • • • • • • • • ~ • - • • • • o • • • • • 
Z.88 
1.88 ............. : ........... . 
.88~~~~~~~~~~~-r~~~~~~~ 
.88 1.Z8 Z.18 3.68 
DISPLACEMENT (m) 
Figure 5 
1.88 &.88 
t -z 
On the other hand, the coordinates of the point at the 
surface (design point) (~a1 ~a2 •••••• ~aN) are used to obtain 
the sensitivities related to the different variables as well as to 
define weighting factors affecting to values of the stochastic 
variables. 
To obtain the design point we have used the typical 
Racwitz approach but also a surface response method (ref 11) 
based on the point estimation (Rosenblueth, (8)). 
In this method for functions of n variables it is 
necessary to obtain estimations at 2° points defined as 
combinations of the mean plus or minus the standard 
derivation of every variable. Using the results of those 
estimates a surface can be adjusted (the simplest case being an 
hyperplane) and ~ and a¡ computed to obtain the desired 
results. 
In this case it is very easy to show that if the plane 
representing the failure plane is expressed as 
where Zj are the typified variables (which values are ±1) then 
!wi 
a o 
_1 __ 
N 
!w¡z~ 
a¡ = N 
where N is the number of point estimations. 
Then the reliability index, the director cosines ak and 
the design point zk • are given by 
~ = a o ~ 
<X.k = 
a k [faf 
174 
Detailed values for mean values and covariance matrices ot 
basic variables can be found in ref 9, related to a road-tunnel 
in different soil conditions. As an example of the results figure 
6 shows. 
YOUNG MODULUS 
¡,00~--------------:------¡ 
-- SOIL FAILURE 
-- STRUCTURE FAILURE 
0:: \.50 
o 
1-
~ 
u. 
(.') 
z 1.00 
¡::: 
I (.') 
~ 
0.50 .......................... . 
o. oo l-,.-.-....,......,~---......... -.--..¡-.----ro..,.....,.-;-......-T""'T-.-;--.---r"'-r-...-j 
o.eo 1.011 :.no c,_nn 
COEF. OF VARIATION 1 1" 
SOIL DENSITY 
• 
101
u.oo y------.---~--:----~--~~----:11 
-- SOlLFAILURE 
-- STRUCTURE FAILURE 
.... > .....•... ·=· .. .. 1 ••••• 
10.75 
11.00 \.00 L.OO ],RQ t.no 
COEF. OF VARIATION 
Figure 6 
In the probabilistic finite element method one of tne 
procedures to o btain the sensitivity is the development in a 
Taylor series round the point under consideration by solving 
sequencially (reference) the systems 
where K is the stiffness matrix of the system, f the load vector 
and u the displacement vector, x is the vector of basic 
stochastic variables and superindex o means that all values are 
particularized for the basic variable set corresponding to the 
design point. 
In this way it is possible to connect the reliability 
modulus with a general representation of the problem. 
For instance we have used the previous approach with 
a Terzaghi rnethod in which , the liner is represented by 
monodimensional rnembers and the passive pressures by 
elastic springs (figure 7) while sorne active pressures are 
directly applied using sorne empirical rules (figure 8). This 
system is well suited for shallow tunnels or arcas near portals 
Figure 7 
NODAL DISPLACEMENTS 
NODE DISPL. X DISPL. Y ROTAT.Z 
1 o -0.03 o 
2 -0.0002059 -0.0298239 0.0002515 
3 -0.0003296 -0.0293388 0.0004534 
4 -0.0003213 -0.0286435 0.0005919 
5 -0.0001574 -0.0278221 0.0007247 
6 0.0002347 -0.0268497 0.0010117 
7 0.0011195 -0.0255626 0.0015959 
8 0.0028333 -0.0239318 0.001977 
9 0.0024892 -0.0235306 0.001977 
10 0.004561 -0.0219528 0.0014093 
11 0.0059476 -0.0212753 0.0005173 
12 0.0060363 -0.02111 -0.0005802 
13 0.0049981 -0.0209927 -0.0013695 
14 0.0031523 -0.0207702 -0.0017773 
15 0.0019959 -0.0174825 -0.0022707 
16 0.0008091 -0.0129022 -0.002992 
17 -0.0002168 -0.0074596 -0.0033018 
18 -0.0003004 -0.0027106 -0.0028H 
19 -0.0002074 -0.0006508 -0.0017927 
20 -0.0001054 0.0003652 -0.0007395 
21 o 0.0006382 o 
:..-- h _____...: 
• • 1 
; : 
1 J 
• 1 
. . 
• 1 
1 • 
Figure 8 
The next tables show the results obtained in an example using 
the previous defmition. In it, was calculated the reliability 
index of the tunnel using as a failure surface the points where 
the vertical displacement of the liner crown was 3 cm. The 
stochastics variables used were the stiffness of the springs. In 
those tables, the point over the failure surface, the sensitivities 
coefficients of the springs and the reliability index are shown 
as well as the weighting factors of the stochastics variables. 
FAILURE SURF ACE WEIGHTING FACTORS 
Sensitivity Coeff. Variable phi 
Alfa1 0.027 1 0.977 
Alfa2 0.010 2 0.992 
Alfa3 0.765 3 0.360 
Alfa4 0.030 4 0.975 
AlfaS 0.010 5 0.991 
Alfa6 0.010 6 0.992 
Alfa7 0.040 7 0.96(] 
AlfaS -0.035 8 1.029 
Alfa9 o.6os 9 0.49( 
Alfa10 0.111 10 0.907 
Alfa11 0.129 11 0.892 
Alfa12 0.095 12 0.921 
Alfa13 o.oi2 13 0.982 
Alfa14 -0.013 14 1.011 
Alfa15 -0.001 15 1.001 
Alfa16 0.00( 16 1.000 
Beta= 3.3453 
Pf(%)= 0.0411 
Table 11 
175 
It is also possible to model a general shape by a 2-D fmite 
element method by using the Panet approach as described by 
Leca and Clough (ref. 14), i.e.: representing the excavation by 
a softening ground of the central part of the mesh and the 
application of an initial stress state as indicated in figure 9. 
Figure 9 
6 Conclusions 
In the paper we have tried to show how different classical 
methods of structural reliability can be applied to the 
assesment of the safety of tunnel liners. Although more work 
is necessary it is hoped that this line of thought will improve 
the current measures of safety. In addition it is clear that the 
procedure can be used as an identification tool for back 
analysis fixing as limit state the variables to be adjusted. 
References 
[ 1] M. Panet: Calcul des tunnels par le méthode 
convergence-confinement presses de Ponts et chaussees. 
1995 
[2] Z.T. Bieniaski: Rock Mechanics Desing in Mining and 
Tunnelling. Balkema 1983 
176 
[3] Rabcewicz: The New Austrian Tunnelling Method. 
Water Power, Nov, Dec 1964 and January 1965 
[ 4] Rabcewicz: Principies of Dimensioning the supporting 
system for the N A TM. Water Power March 1973 
L5] Fritz: An analytical solution for axisymmetric tunnel 
problems in Elasto-Viscoplastic media. lnt. Jour. Num 
and analyt. Meth in Geomechanics. Vol 8 1980 
[6] Madsen, S. Krenk, N.C. Lind: Methods of structural 
safety. Prentice 1986 
[7] Hasofer, N.C. Lind: Exact and invariant second moment 
code format. Jour. Eng. Mech. Div. ASCE Vol 100 
1974 
[8] Rosenblueth: Two-point estimates in probability App. 
Math. Mode11. Vol. 5 1981 
[9] Gómez Lera; E. Alarcón: Introduction to structural 
reliability theory and sorne of its applications (in 
spanish). Repport 408. Instituto de Ciencia de la 
Construcción Eduardo Torroja. 1992 
[ 10] E. Laso, M.S. Gómez Lera, E. Alarcón: A level II 
reliability approach to tunnel support design. App. 
Math. Modell. Vol. 19, 1995 
[11] E.P. Box, N.R. Draper: Empirical Model-Building and 
response surfaces. App. Math. Modell. J. Wiley 1987 
[12] Kleiber, T.D. Hien: The stochastic fmite element 
method. Wiley 1992 
[ 13] Liu, T. Belytschko: Computational Mechanics of 
Probabilistic and reliability analysis. Elmpress Intern. 
1989 
[14] Leca, W. Clough: Preliminary design for NATM Tunnel 
Support in Soil. ASCE Jour. Geotech. Eng. Vol. 118 
1992 
[15] E. Laso: Application of level 11 structural reliability 
methods to tunnel liners. (In Spanish). Thechnical 
University of Madrid (UPM), 1991 


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Probabilistic Analysis of Tunnel Liners

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