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Engineering, Technology & Applied Science Research Vol. 11, No. 1, 2021, 6708-6713 6708 
 

www.etasr.com Benzeguir et al.: Reliability of Buried Pipes in Heterogeneous Soil Subjected to Seismic Loads 

 

Reliability of Buried Pipes in Heterogeneous Soil 

Subjected to Seismic Loads 
 

Hichem Benzeguir 

Department of Civil Engineering 
University Djillali Liabes of Sidi Bel Abbes 

Sidi Bel Abbes, Algeria 

h_benzeguir@yahoo.fr 

Djamel Nedjar 

Department of Civil Engineering 

University of Sciences and Technology of Oran 
Oran, Algeria 

djamel_nedjar@yahoo.fr 

Sidi Mohamed Elachachi 

Civil and Environmental Engineering Department 
University of Bordeaux 

Talence, France 

sm_elachachi@yahoo.fr 

Mohamed Bensafi 

Department of Civil Engineering 

University of Sciences and Technology of Oran 
Oran, Algeria 

moha_bensafi@yahoo.fr  
 

 

Abstract- Dysfunctions and failures of buried pipe networks, like 
sewer networks, are studied in this paper from the point of view 

of structural reliability and heterogeneity of geotechnical 

conditions in the longitudinal direction. Combined soil spatial 

variability and Peak Ground Acceleration (PGA) induce stresses 

and displacements. A model has been developed within the frame 

of geostatistics and a mechanical description of the soil–structure 
interaction of a set of buried pipes with connections resting on the 

soil by a two-parameter model (Pasternak model). Structural 

reliability analysis is performed considering two limit states: 

Serviceability Limit State (SLS), related to large "counter slope" 

in a given pipe, and Ultimate Limit State (ULS), corresponding to 
bending moment. 

Keywords-reliability index; soil-structure interaction; spatial 

variability; SLS; ULS; seismic action 

I. INTRODUCTION  

Pipes that carry various substances need to be designed in a 
way capable of reducing the damage caused by ground 
displacements induced by earthquakes. The geoenvironmental 
effect has attracted concern on the performance of the buried 
pipes because of the associated hazards [1]. Most buried 
pipelines in seismic areas have sustained substantial damage in 
the past due to earthquake events [2]. Dysfunctions and failures 
of buried pipe networks, like sewer networks, are mainly 
caused by the heterogeneity of geotechnical conditions in the 
longitudinal direction and of the applied (seismic) action. 
Combined soil defects (differential settlements along the pipe, 
landslides, voids surrounding the pipe, etc.) and Peak Ground 
Acceleration (PGA) induce stresses (which lead to an Ultimate 
Limit State-ULS) and displacements (which constitute a 
violation of the Serviceability Limit State-SLS). It is worth 
noting that the influence of the variability of the soil is not 
reflected in current European standards [3]. Authors in [4] 
presented a comprehensive literature review on the seismic 
behavior of buried pipelines and underground structures 

summarizing the recent research. Structural response to ground 
motion during earthquake cannot be accurately predicted 
because of the complexity of the structural properties and 
ground motion parameters [5]. In this paper, the response of 
buried pipe is investigated considering the seismic excitation 
by selecting real earthquake data. 

II. SOIL PIPE SYSTEM MODELING 

A. Pasternak Model 

Among the models that describe the behavior of a beam 
resting on a soil and their interaction, the Pasternak model [6] 
attracts the most interest. In the soil-pipe interaction, the soil 
opposes on the components of a sewer network (pipes) a 
distributed force R(x) (in N/m) given by (1): 

���� = ����.�	
�    (1) 
where p(x) is the stress under the pipe (Pa), and Dext the 
external diameter of the pipe (m). According to the Pasternak 
model the stress is expressed as: 

���� = �
.���� − �� �
�
�
�    (2) 

where kw is the coefficient of subgrade reaction (or Winkler 
coefficient in N/m

3
 or Pa/m), ks is the shear coefficient (N/m) 

and w(x) the vertical pipe displacement (and thus the settlement 
of the soil).  

Pasternak’s idealization considers the soil as being a system 
of identical but mutually independent, closely spaced, discrete, 
linearly elastic springs related by an incompressible "shear 
layer" which is defined as a layer of linear-elastic material of 
unit thickness that resists vertical shear forces only [6]. Thus it 
is a refinement of the well known Winkler model which suffers 
from not describing shear influence. The use of a two 
parameter model to characterize the soil's response under 
loading can appear a too simplified concept. However, such a 

Corresponding author: Hichem Benzeguir



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simplification seems coherent if one takes into account the 
variability and uncertainties related to the characterization of 
the soil. The spatial correlations which will be introduced 
further will ensure, in fact, a coherence of displacements, like it 
exists in a continuous medium. One should note that the two 
coefficients kw and ks are not soil-specific parameters. They are 
also affected by the rigidity of the pipe. In fact, these stiffness 
parameters depend on several factors, such as the length and/or 
the width (or diameter) of the pipe, the laying depth, the type of 
material used, and the type of the pipe bed. The value of these 
coefficients can only be approached by semi-empirical 
methods. For a same set of values, a parametric study, 
conducted in [7], illustrated the fact that kw varies from 1 to 3. 
To the best of our knowledge the literature has not given ways 
on how to identify ks in practice from physical and geometrical 
data [8]. In order to integer the damping of the system, the 
Pasternak model is adapted and expanded to a Kelvin-Voigt 
model by adding dashpots. The stiffness matrix of the soil–pipe 
system has been derived in [9] by using the energy method. We 
will consider two relative soil-pipe stiffness ratios, rp and rks 
which are defined by (3) and (4) and can be considered as 
governing parameters: 

�� = �� 	 �
���
��

5
    (3) 

��� = ����� !"�     (4) 
 

where Ep, I, Dext, and L and are respectively the pipe's Young 
modulus, the pipe's moment of inertia, the exterior diameter, 
and the length of the pipe. 

B. Joint-connection Stifness 

Sealing between pipes is ensured by joints made of cement 
mortar or more frequently of elastomer. The rigidity of these 
joints is as variable as the technologies and geometries 
employed: it can be very weak (flexible joints) or very high 
(welded joints). It is difficult to identify realistic numerical 
values of the joint stiffness even if some laboratory 
experiments have focused on this question [10]. In this work, 
we assume a continuity of vertical displacements at joint 
connection and contrary to a model of continuous beam, a 
model was developed which enables introducing 
discontinuities of rotation between the ends of the pipes. The 
joints between two adjacent pipes are assimilated to rotation 
springs with stiffness Rj relating to the proportionality between 
the bending moment M applied to the joint and the variation of 

the angle of rotation ∆ϕ: 

# = �$.∆&    (5) 
To take into account this joint stiffness, the pipe-joint 

stiffness ratio rjoint is introduced which is defined by: 

�$'()� = *+�,-.    (6) 
To summarize, the soil-pipe interaction system, one can say 

that it is governed by a geometric parameter L (pipe's length) 
and three relative (dimensionless) stiffness ratio parameters: rks 

(compression to shear), rp (soil to pipe), and rjoint (connection to 
pipe). 

III. MODELING THE SPATIAL VARIABILITY OF SOIL AND ITS 
EFFECTS:THE CORRELATION LENGTH 

In many common geotechnical problems, the variability of 
soil is only one among many sources of uncertainty (others are 
for example reduced sampling measurement or model errors). 
It can be accounted for by taking conservative values of the soil 
parameters, even if geotechnicians need long practice to be able 
to justify the choice of these values [11]. In fact, the variability 
of soils cannot be reduced to case-by-case variability – as a 
result of its natural or man-made fabric (deposit processes, 
compaction processes). The soil properties can be considered 
as spatially structured. Thus, tools like autocorrelation 
functions or semi-variograms appear to be appropriate for 
modeling. One must then identify the standard deviation or 
coefficient of variation of the studied property as well as the 
correlation length lc (i.e. the distance above which the local 
properties at two points can be assumed to be independent). 
The first consequence of the spatial correlation is that the 
representative value of any soil property depends on the 
volume concerned by the problem to be solved. This question 
has been analyzed in detail during the drafting of Eurocode-7 
but the code writers have limited themselves to general 
considerations, without prescribing any formal method: the 
representative value is only said to be a characteristic value 
defined as a cautious estimate of the parameter governing the 
studied limit state [12]. Nevertheless, accounting for spatial 
correlation has direct consequences on the safety of designs. A 
simple illustration of the effects of spatial variability is that of 
the rotation (tilting) of a foundation of length L resting on a 
heterogeneous elastic soil and supporting a uniform loading. It 
was shown [13] that the magnitude of the rotation depends on 
the correlation length in the horizontal direction: it tends 
towards zero when lc is very small (which corresponds to very 
quickly varying properties, thus homogeneous at L scale) or 
very large (which corresponds to very slowly varying 
properties, thus also homogeneous at L scale) and it is 
maximum for an intermediate range. This illustrates a 
consequence of the spatially correlated variation of soil 
properties: tilting occurs only when the soil is not 
homogeneous and its magnitude depends both on the scatter in 
the soil properties (linearly) and on the correlation length, with 
a "worst case" for a particular range of lc values. Since the role 
of the longitudinal variability of the filling appears essential, 
we chose to model it by using the theory of the local average of 
a random field developed in [14]. The random field of the 
coefficients kw or ks is defined by three properties: its average 

value 
w
k (resp. 

s
k ), its variance 

2

w
k  (resp. 

2

s
k ) and its scale (or 

length) of correlation lc (resp. lcs). These scales are related to a 
function of correlation ρ(τ) in (7) where τ points out the 
distance between two points, and which describes the spatial 

structure of correlation of the properties: ρ(τ)  differs whether 
the properties vary more or less quickly while deviating from a 
given point. This correlation length (length from which the 
correlation between soil properties tend to disappear) depends 
on the characteristic (modulus, porosity, water content, etc.) 
and on the direction (horizontal or vertical). 



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/�0� = exp	�−2 |6|78 )    for  0 9 :;    (7)     
The soil is subdivided in several zones. The random field 

value in each zone is thus a random variable whose value is 
estimated by the average of the space field over the zone. The 
local average and the variance in zone i of length Di fulfil (8): 

�<�
��(�= = �
>>>>    (8a) 
Var<�
��(�= = �
BBBBCD��(�    (8b) 

Equation (8b) shows that the local variance Var [kw(Di)] 
depends on the length Di of zone i while following a variance 
reduction function γ(Di). 

D��(� = 2E78�FG
C E�F78 − 1 I exp	�−

�F
78�G    (9) 

γ(Di) is a measure of the variance reduction due to the 
averaging of the random process according to the length of the 
studied zone. In this work, kw and ks follow a lognormal 
distribution. In order to show the effect of both the mean values 
of the Winkler and shear coefficients and their correlation 
length, six cases were considered (Table I) and applied to an 
example of a sewer section (set defined between two manholes) 
and which will be presented in further detail. The section is 
made of 20 pipes of unit length L=3m. For each computation, 
the outputs which are linked to limit states are processed all 
along the section and their worst value is kept. Finally the 
Cumulative Density Function (CDF) of these parameters 
(stress, displacement, bending moment) is drawn and used for 
comparison. Figure 1 shows the CDF of the maximum bending 
stress in a concrete sewer's section. 

TABLE I.  VALUES OF kw, ks, lc AND lcs 

parameters kw (KN/m
3
) lc (m) ks (KN/m) lcs (m) 

Case A 10 3 1 30 

Case B 10 30 1 3 

Case C 10 3 1 3 

Case D 10 3 10 3 

Case E 10 30 10 3 

Case F 10 3 10 30 

 
Fig. 1.  CDF bending stress for the 6 different cases presented in Table I. 

Two observations can be made: 

• Not taking into account the shear effect is not conservative 
(case C compared to case F) being given that the mean 
stress varies more than 20%, 

• The correlation length of the Winkler coefficient seems to 
have more importance than the correlation length of the 
shear coefficient (case B compared to A and C cases or case 
E compared to D and F cases). 

IV. EQUATIONS OF MOTION 

The soil resistance to the pipe motion is generated by the 
relative motion u between the pipe and the soil. The pipe 
resistance comes from the absolute displacement U (Figure 2). 
The governing equation of the system is [15]:  

J#� I #�'7KLMNO I JP� I P�'7KLMQO I RS� I S
 I S�TUMV = 
<#�'7=UWXN V I <P�'7WXQ = I RS� I S�TUWV    (10) 

in which Mp and Msoil are respectively the mass matrix of the 
pipe and of the soil, Kp, Kw and Ks the stiffness matrices of the 
pipe and the soil, and Cp and Csoil the damping matrices of the 
pipe and of the soil respectively. We note that the damping 

matrices are constructed from the Rayleigh damping. , , U, U U& &&

are respectively the vectors of absolute displacement, velocity 

and acceleration, u, ,  ,  and u u u& && are the vectors of relative 

displacement, velocity and acceleration, and ,  ,  and 
s s s

u u u& && us 

are the vectors of ground displacement, velocity, and 
acceleration of the soil. Denoting the total stiffness as [K] = 
[Kp] + [Kw]+ [Ks], the total mass as [M] = [Mp]+[Msoil], the total 
damping as [C] = [Cp] + [Csoil], and the absolute displacement 
as {U} = {us} + {u}, the equations of motion in terms of the 
relative displacement can be rewritten as: 

<#=UWN V I <P=UWQV I YSZUWV =  
−J#�KUWXN V − JP�WXQ K − RS�TUWXV    (11) 

These equations of motion are solved for each time history 
of the ensemble of real earthquake records. The boundary 
conditions can be considered either as fixed ends or as free 
ends if we want to eliminate their effects. 

 
Fig. 2.  Pipe and ground motion. 

V. PERFORMANCE FUNCTIONS AND LIMIT STATES 

Two performance functions are defined. One relates to the 
counterslope and corresponds to the SLS, by relating the 
sewer's hydraulic performance to the local counterslope. The 
presence of too high counterslopes harms the flow of the 
effluents and facilitates the clogging of the pipes by the 
sedimentation of the suspended particles. The corresponding 
limit state function is: 

[\X = P]*−P]X    (12) 



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where CSR is the maximum acceptable counterslope and CSS is 
the maximum simulated counterslope. The other performance 
function relates to the stresses and defines the equivalent of an 
ULS, by relating the cracking state (for a concrete pipe) to the 
bending stresses: 

[^ = _*−_X    (13) 
where _* is the yield tension stress of the concrete and _X is the 
maximum bending stress. We assume that CSS and _X  also 
follow lognormal distributions, whose means and standard 
deviations are obtained from coupling Monte-Carlo simulations 
and finite element method, and that CSR as _*  also follow 
lognormal distributions of means and standard deviations of 
4%± 0.8%, and 2± 0.3MPa respectively.  

It should be noted that there is not any regulation limiting 
the counterslope, that's why we have chosen a 4% arbitrary 
limit based on the design considerations: trespassing such a 
value leads to damages or dismantling of the joints and to 
defects of sealing. The ± 0.8% can be seen as a model 
uncertainty on this SLS criterion. The target values of the 
reliability index to be reached are 3.8 for the ULS and 1.5 for 
the SLS with a service life of 50 years. In the following, we 

will consider two reliability indexes βULS  and βSLS.  

VI. CASE STUDY 

A. Caracteristics of the Sewer’s Section 

Lets consider a reinforced concrete material for the pipe  
(Ep = 25000MPa) with an exterior diameter Dext = 1m, a 
thickness of 0.08m, made of a set of 20 pipes with L = 3m unit 
length. It rests on a soil whose Winkler coefficient is 10kN/m

3
 

and shear coefficient is 10kN/m. This type of configuration 
corresponds to a pipe-soil ratio rp = 0.678. It must be noted that 
the pipe geometry intervenes in the expression of inertia I and 
in that of the Winkler coefficients. The pipes are subjected to a 
deterministic uniform loading of 50kN/m and to the earthquake 
of El-Asnam (Algeria, 1980). The ratio between the length of 

correlation lc and the pipe length L is defined as λ. In the 
following sections are analyzed: 

• the effect of the length of correlation on the response of the 
system, 

• the effect of the relative soil-pipe stiffness, for an 

unfavorable length of correlation (λ # 1), 

• and the effect of the relative joint-pipe stiffness. 

The values of the obtained reliability β indexes (ULS or 
SLS) must be considered rather comparatively that in an 
absolute way because of:   

• the arbitrary character of the values retained for CSR as _*, 
• the type of the probabilistic distribution, 

• model uncertainties on soil characteristics (kw, ks). 

B. Effect of Correlation Length 

The analysis is carried out for lengths of correlation ranging 

from 0.03m to 1500m, (10
-3 
< λ < 5×10

2
). Two types of joints 

are considered: rigid (rjoint=10
+5
) and flexible (rjoint=10

-5
). 

Figures 3 and 4 show the ULS and SLS reliability indexes. It 
can be seen that there is a critical length of correlation and thus 

a critical value of λ which causes the most unfavorable effects 
in the structure. The evolution of reliability is not monotonous. 
The existence of such a limit value is one of the invariants of 
the problems of (heterogeneous) soil-structure interaction [13]. 

Reliability indexes grow when λ tends to zero or infinity. In 
these two situations, the soil tends to be homogeneous at the 
scale of the analysis, either because its variations are very fast 
are filtered by the pipe, or because its variations are very slow, 
inducing almost uniform values between close elements. We 
had seen that the rigid joints, and for a high coefficient of 

variation of kw, do not satisfy the ULS for λ between 0.01 and 
7, while the flexible joints provide much better results. On the 
other hand, if we consider the SLS, the flexible joints provide 
results a little worse than the rigid joints. One thus notes at this 
level a conflict in the choice of the "good pipe length" to 
answer the ULS and the SLS. 

 

 
Fig. 3.  Effect of the correlation length on the ultimate state reliability 

index for two values of the variation of coefficient kw and two of joints. 

 
Fig. 4.  Effect of the correlation length on the service limit state reliability 

index for two values of the variation of coefficient kw and two of joints. 

C. Effect of the Joint-pipe Stiffness Ratio 

The effect of joints on the behavior of the system was 
analyzed by varying the joint-pipe ratio rjoint from 10

-5
 to 10

+5
. 

Figures 5 and 6 show the evolution of the reliability indexes for 
different coefficients of variation of Winkler coefficient. The 
strong influence of the joints' stiffness should be underlined 
since the rigid joints are penalizing with the ULS, by the fact 
that they necessarily induce more significant stresses. On the 
other hand, the flexible joints are penalizing with the SLS. An 
intermediate joint stiffness must be chosen in order to carry out 



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the best compromise, since reliability is completely assured for 
both ULS and SLS. Finally, one can also note that for a given 
value of rjoint, the ULS index reliability and the SLS index 
reliability decrease with the increase of kw coefficients of 
variation.   

 

 
Fig. 5.  Effect of joint-pipe stiffness ratio on ultimate limit state reliability 

index for four values of the variation of coefficient kw. 

 
Fig. 6.  Effect of joint-pipe stiffness ratio on the service limit state 

reliability index for four values of the variation of coefficient kw. 

D. Effect of the Soil-pipe Stiffness Ratio 

By maintaining the same set of pipes and soil parameters 

(λ = 1, rks = 1), we vary the soil-pipe stiffness ratio rp in the 
range between 0.4 and 1.7 which corresponds to a Winkler 
coefficient in the range from 100kPa to 100MPa for four values 
of the coefficients of variation (CVkw = 0.1, 0.2, 0.4, and 0.8). 
While taking a length of correlation lc close to the length of the 

pipe (λ # 1), we refer to the situation in the vicinity of the most 
unfavorable geotechnical conditions. Figures 7 and 8 present 
the values of the reliability indexes. Whatever the type of joint, 
reliability is lower for high soil-pipe stiffness ratio, situation 
corresponding to sections of very stiff pipes or low soil 
stiffness. In the presence of rigid joints (Figure 7) and high 
coefficient of variation of the soil's parameter, the Eurocode 
βULS is not respected (lower than 3.8). We can see that 
increasing the coefficient of variation induces reduction of the 
reliability of the pipe section. So, it is better to have flexible 
rather than rigid joints, it is better to privilege a lower rp to 
ensure the "good" reliability of the network.  

 

 
Fig. 7.  Effect of soil-pipe stiffness ratio on the service ultimate state 

reliability index for four values of the variation of coefficient kw. 

In terms of counterslopes (Figure 8), it is better to have 
rigid than flexible joints. To ensure the reliability of the sewer's 
section using rigid joints imposes that the ground is rather 
homogeneous. It can be also noted that it is more profitable to 
be in the presence of a relatively homogeneous soil whose 
mechanical characteristics are poor that in the presence of a 
better ground but whose heterogeneity is more pronounced. 

 

 
Fig. 8.  Effect of soil-pipe stiffness ratio on the service limit state 

reliability index for four values of the variation of coefficient kw. 

VII. CONCLUSION 

A model which includes a description of the soil's spatial 
variability, within the frame of geostatistics has been developed 
in this paper. It is based on a mechanical description of the 
soil–structure interaction between a set of segmented buried 
pipes and the soil support represented by the Pasternak model. 
The need to take into account the soil spatial variability is 
clearly recognized. Several conclusions are drawn: 

Soil heterogeneity induces effects (differential settlements, 
bending moments, stresses, and possible cracking) that cannot 
be predicted if homogeneity is assumed. These effects can 
significantly affect the SLS or ULS reliabilities. In addition to 
the effects of the soil stiffness, the magnitude of the induced 
stresses depends mainly on four factors or governing 
parameters:  



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• The magnitude of the soil variability (i.e. its coefficient of 
variation). 

• The soil-structure length ratio, which combines the soil 
fluctuation scale and a structural characteristic length 
(buried pipe length). A worst value, corresponding to the 
value leading (from a statistical point of view) to the 
(statistically) largest effects in the structure, can be found. 

• The soil-structure stiffness ratio. 

• The structure-connection (pipe-joint) stiffness ratio (relative 
flexibility). 

The principal benefit of such an approach is that it provides 
some new approaches for better considering phenomena such 
as the geometrical irregularities in the longitudinal profile 
during the control of soil compaction of sewer trench filling. 
This kind of approach can also give the experts new tools for 
better calibration of safety in soil-structure interaction 
problems, when the soil variability is an influential parameter. 

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