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The socio-economic and environmental impact, in case of severe damage on Natural Gas (NG) pipeline networks, highlights the importance of a rational assessment of the structural integrity of this infrastructure against seismic hazards. Up to date, this assessment is mainly performed by employing empirical fragility relations, while a limited number of analytical fragility curves have also been proposed recently. The critical review of available fragility relations for the assessment of buried pipelines under seismically-induced transient ground deformations, presented in the first part of this paper, highlighted the need for further investigation of the seismic vulnerability of NG pipeline networks, by employing analytical methodologies, capable of simulating effectively distinct damage modes of this infrastructure. In this part of the paper, alternative methods for the analytical evaluation of the seismic vulnerability of buried steel NG pipelines are presented. The discussion focuses on methods that may appropriately simulate buckling failures of buried steel NG pipelines since these constitute critical damage modes for the structural integrity of this infrastructure, when subjected to seismically-induced transient ground deformations. Salient parameters that control the seismic response and vulnerability of buried pressurized steel pipelines and therefore should be considered by the relevant analytical methods, such as the operational pressure of the pipeline, the geometric imperfections of the pipeline walls, the trench backfill properties, the site characteristics and the spatial variability of the seismic ground motion along the pipeline axis, are thoroughly discussed. Finally, a new approach for the assessment of buried steel NG pipelines against seismically-induced buckling failures is introduced. Through the discussion, recent advancements in the field are highlighted, whilst acknowledged gaps are identified, providing recommendations for future research.
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A Critical Review on the Vulnerability Assessment of Natural Gas 1
Pipelines Subjected to Seismic Wave Propagation. Part 2: Pipe 2
Analysis Aspects 3
4
Grigorios Tsinidis1, Luigi Di Sarno2, Anastasios Sextos3, and Peter Furtner4
5
6
1University of Sannio, Italy & Vienna Consulting Engineers ZT GmbH, Austria 7
2University of Sannio, Italy & University of Liverpool, United Kingdom 8
3University of Bristol, United Kingdom & Aristotle University of Thessaloniki, Greece 9
4Vienna Consulting Engineers ZT GmbH, Austria 10
11
Corresponding Author: Dr Grigorios Tsinidis, Vienna Consulting Engineers ZT GmbH, 12
Austria, Untere Viaduktgasse 2, 1030, Vienna, email: tsinidis.grigorios@gmail.com 13
14
Abstract: The socio-economic and environmental impact, in case of severe damage on Natural 15
Gas (NG) pipeline networks, highlights the importance of a rational assessment of the 16
structural integrity of this infrastructure against seismic hazards. Up to date, this assessment is 17
mainly performed by implementing empirical fragility relations, while a limited number of 18
analytical fragility curves have also been proposed recently. The critical review of available 19
fragility relations for the assessment of buried pipelines under seismically-induced transient 20
ground deformations, presented in the first part of this paper, highlighted the need for further 21
investigation of the seismic vulnerability of NG pipeline networks, by employing analytical 22
methodologies, capable of simulating effectively distinct damage modes of this infrastructure. 23
In this part of the paper, alternative methods for the analytical evaluation of the seismic 24
vulnerability of buried steel NG pipelines are presented. The discussion focuses on methods 25
that may appropriately simulate buckling failures of buried steel NG pipelines since these 26
constitute critical damage modes for the structural integrity of this infrastructure, when 27
subjected to seismically-induced transient ground deformations. Salient parameters that control 28
the seismic response and vulnerability of buried pressurized steel pipelines and therefore 29
should be considered by the relevant analytical methods, such as the operational pressure of the 30
pipeline, the geometric imperfections of the pipeline walls, the trench backfill properties, the 31
site characteristics and the spatial variability of the seismic ground motion along the pipeline 32
axis, are thoroughly discussed. Finally, a new approach for the assessment of buried steel NG 33
pipelines against seismically-induced buckling failures is introduced. Through the discussion, 34
recent advancements in the field are highlighted, whilst acknowledged gaps are identified, 35
providing recommendations for future research. 36
Keywords: Natural gas pipelines; fragility; soil-pipe interaction; transient ground 37
deformations; steel pipelines; buckling 38
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1. Introduction 1
Earthquake-induced damages on Natural Gas (NG) and fossil-fuel pipeline networks may lead 2
to significant downtimes, which in turn may result in high direct and indirect economic losses, 3
not only for the affected area and state, but also internationally. Moreover, severe damages 4
may trigger ignition or explosions with life-treating consequences and significant effects on the 5
environment. The above aspects highlight the importance of simple, yet efficient, seismic 6
analysis and vulnerability assessment methods to be used for the design of new NG networks, 7
as well as the evaluation of the vulnerability and resilience of existing networks. Up to date, the 8
seismic vulnerability assessment of this infrastructure is mainly performed by implementing 9
empirical fragility relations. A limited number of analytical fragility curves that compute 10
probabilities of failure in the ‘classical sense’ have also been proposed recently (Lee et al., 11
2016; Jahangiri and Shakid, 2018). A critical review of available fragility relations for buried 12
pipelines, subjected to seismically-induced transient ground deformations, was presented in the 13
first part of this paper (Tsinidis et al. 2019). Through the detailed discussion, a general lack of 14
analytical fragility relations for a rigorous seismic vulnerability assessment of buried steel NG 15
pipelines was highlighted. This was partly attributed to the absence of optimum seismic 16
Intensity Measures IMs for this infrastructure, which may correlate effectively with diverse 17
seismically-induced damage modes. The above knowledge shortfalls highlight the need for 18
efficient analytical methodologies, which may allow for a thorough investigation of the 19
vulnerability of steel buried NG pipelines against distinct seismically-induced damage modes. 20
In this context, a thorough overview of alternative analytical methodologies for the 21
vulnerability assessment of buried steel NG pipelines under seismically-induced transient 22
ground deformations, is presented in this part of the paper. The discussion focuses on the 23
buckling failures, which constitute critical damage modes for the structural integrity of steel 24
buried pipelines subjected to transient ground deformations. Salient parameters that control the 25
seismic response and vulnerability of buried steel NG pipelines and therefore should be 26
considered by the relevant analytical methods are thoroughly discussed. Finally, a new 27
approach for the vulnerability assessment of steel buried NG pipelines against seismically-28
induced buckling failures is introduced. Through the discussion, recent advancements in the 29
field are highlighted, whilst acknowledged gaps are identified. 30
31
2 Analytical methods to define the EDP-IM relationship 32
A critical step in the quantitative seismic vulnerability assessment of any element at risk, e.g. a 33
buried steel NG pipeline, is the development of structure-specific analytical fragility curves 34
that provide a functional relationship between the EDP and the selected seismic IM on the 35
basis of predictions of relevant numerical analyses (Jalayer et al., 2017; Bakalis & 36
Vamvatsikos, 2018). Various approaches have been proposed in the literature for this purpose, 37
including the incremental dynamic analysis (IDA), the multiple-stripe analysis and the cloud 38
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analysis. In the framework of an IDA, a series of dynamic analyses are conducted, by 1
employing an adequate nonlinear model of the investigated structural system and using a suite 2
of accelerograms, which are progressively scaled upwards in amplitude, so as to cover a wide 3
range of seismic IM levels. Through these analyses the evolution of EDP with increasing 4
seismic intensity level is reported, in the so-called IDA curves (Vamvatsikos and Cornell, 5
2002). A similar approach is followed in multiple-stripe analysis. However, in this type of 6
analysis the selected ground shaking motions do not necessarily scaled to reach various 7
intensity levels. Instead use of different sets of scaled or unscaled records at each IM level is 8
made, in an effort to reflect the site-specific seismic hazard at each IM level (Jalayer and 9
Cornell, 2009). In cloud analysis un-scaled or scaled ground seismic motions are used. 10
Typically, only a single record will correspond to each seismic IM level, while scaled records 11
may be used to capture higher order damage states (Bakalis and Vamvatsikos, 2018; Miano et 12
al., 2018). This approach will result for a cloud of points in the EPD-IM plot. A probabilistic 13
relation between the EDP and the IM is then established on the basis of a regression analysis 14
(Jalayer et al., 2015). 15
Regardless of the selected method, a critical step in the fragility assessment of a structure is the 16
development of an adequate numerical methodology that will account for the majority of 17
salient parameters that control the seismic response of the structure, being at the same time 18
computationally efficient. Both Lee et al. (2016) and Jahangiri and Shakib (2018) performed 19
IDA analyses to examine numerically the seismic vulnerability of buried steel NG pipelines 20
and propose analytical fragility relations. However both studies disregarded critical parameters 21
that affect the seismic response and vulnerability of buried steel NG pipelines. 22
23
3 Parameters affecting the response of buried steel pipelines under transient 24
ground deformations 25
3.1 Soil-pipe interaction effects 26
The response of embedded civil infrastructure subjected to seismic wave propagation is 27
manifested by the transient ground displacements induced by the surrounding ground, while 28
the inertia-related effects are of minor -if not negligible- importance (Hashash et al., 2001). 29
This observation, which is valid even for large embedded structures, such as tunnels, has been 30
reflected to various design codes and guidelines for embedded infrastructures (EN 1998-4, 31
2006) and may further imply that the inertial soil-structure interaction effects are minor and 32
insignificant. Based on this remark, various pseudo-static analysis methods have been 33
developed for embedded structures, such as tunnels and pipelines (ALA, 2001; FHWA 2009; 34
ISO 23469 2005). Actually, several recent studies have compared the predictions of simplified 35
pseudo-static analyses methods and approaches for tunnels against full dynamic analyses 36
results, as well as experimental data from dynamic centrifuge tests, revealing reasonably good 37
comparisons (e.g. Tsinidis et al., 2015; Lanzano et al., 2015; Tsinidis et al., 2016). Since the 38
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mass of buried pipelines is much lower than that of extended embedded structures, the effect of 1
inertial SSI effects is expected to be even less important for buried pipelines. 2
Depending on the type of the dominant seismic wave (S-waves, P-waves or surface waves) and 3
the alignment of the pipeline axis to the ray path, pipelines may be subjected to either pure 4
axial compression or tension, bending in the horizontal or vertical plane, or a combination of 5
the above. The differential movement between the pipe and the surrounding ground, the 6
distinct response of the spatially variable soil deposits that a pipeline may cross, as well as the 7
spatially variable ground seismic motion along the pipeline axis, are among the parameters that 8
may induce a non-uniform shear stress field along the soil-pipe interface, which in turn may 9
lead to non-uniform axial stresses on the pipeline. The latter may impose axial buckling or 10
tensile rupture failures on the pipeline. Flexural stresses may be caused on an embedded pipe 11
by a direct introduction of soil curvature on it. These stresses can cause flexural failures, such 12
as bending buckling or excessive ovalization of the pipe section. Generally, the axial strains 13
that are induced on straight embedded pipelines due to transient ground deformations are larger 14
than the bending ones and actually dominate the response (Hindy and Novak, 1979). Bending 15
strains are generally higher than axial strains near pipe bends (Karamanos, 2016). 16
17
3.2 Geologic / geotechnical conditions of the site and distance to the seismic source 18
Not surprisingly, the geologic, geomorphic and geotechnical conditions of the site of interest, 19
affect significantly the transient ground deformations that are being induced on a buried 20
pipeline and therefore its seismic response and vulnerability. Regardless of the selected method 21
of analysis, a reliable knowledge of the degree of soil stiffness variation along the pipeline 22
axis, as well as of the soil stiffness gradient with depth, or the potential inclination of the 23
bedrock and other topographic characteristics, such as basins, hill crests and toes and 24
embankments, are parameters that may affect considerably the transient ground deformations 25
along the pipeline axis. 26
If the soil site of interest is relatively uniform, strains on extended pipelines can arise only by 27
the apparent propagation velocity of the seismic waves, or by large relative soil-pipe motion, 28
when the seismic shaking amplitude is large. Even under these circumstances, the expected 29
strain levels on the pipe are commonly smaller than the yield strain of the steel grades that are 30
used in NG pipeline applications. In the presence of strong soil heterogeneities or special 31
topographic characteristics along the pipeline axis, increased ground deformations and 32
subsequent straining of the pipeline are expected, due to effects of the above site characteristics 33
on the wavefield. Generally, these site conditions may alter significantly the frequency content 34
of the ground seismic motion, leading to an amplification of the ground displacement 35
grandients. 36
The level of knowledge of the soil site characteristics determines the selection of the adequate 37
analysis or assessment method for a buried pipeline. Evidently, for uniform soil sites the use of 38
complex and sophisticated numerical models and tools is not required. For complex sites, soil 39
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response analyses, preferably in 2D or 3D, in combination with the use of soil-pipe interaction 1
models that can capture the axial response of the pipe and the relevant potential failures (e.g. 2
shell buckling) are recommended. Field surveys may contribute in gathering information 3
related to the site characteristics. Obviously, the above procedure is applicable in the design of 4
new pipeline networks, as well as for the pre-seismic assessment of existing networks, since it 5
is associated with a high computational effort and requires significant time to be performed. 6
As highlighted in the first part of this paper, the proximity of the site of interest to the seismic 7
source determines the dominating seismic waves on the site, as well as their frequency content. 8
A pipeline located near the seismic source is expected to be stricken by vertically propagated 9
body waves (e.g. S-waves) of high amplitude and frequency content. On the other hand, a 10
buried pipeline in a site away from the earthquake source is expected to be stricken by a 11
combination of nearly-vertically propagated body waves and surface waves (e.g. R-waves). 12
The predominant wavelength of the surface waves is expected to control the axial straining of 13
the pipeline (O’ Rourke M.J. and Hmadi, 1988). Summarizing, the distance of the site of 14
interest to the seismic source is expected to affect the characteristics of the seismic ground 15
motion and therefore should be considered in the relevant analysis or assessment framework. 16
The identification of the expected dominant seismic waves for the seismic vulnerability 17
assessment of an extended NG pipeline network could be informed by the results of a 18
preliminary seismic hazard analysis, the latter carried out for carefully selected seismic 19
scenarios. 20
21
3.3 Internal pressure 22
Steel pipelines used in transmission NG networks are commonly pressurized to high levels, 23
reaching pressures as high as 8-9 MPa. For this uniform pressure level, steel pipelines develop 24
large initial circumferential tensile stresses, which interact with axial straining caused by 25
potential seismically-induced transient ground deformations, leading to a rather complicated 26
axial response. The restriction of buried pipelines by the surrounding trench soil alters the 27
combined effects of the above loading conditions, further complicating the axial response. The 28
combined effects of internal pressurization and axial compression of free and restrained 29
cylindrical steel shells has been investigated before, both numerical and experimentally 30
(Timoshenko and Gere, 1961; Yun and Kyriakidis, 1990; Paquette and Kyriakides, 2006; 31
Kyriakides and Corona, 2007). To further elaborate on the axial response of steel NG pipelines, 32
a series of compression static analyses were performed by the authors on above ground and 33
embedded segments of API 5L X65 steel pipelines. Various parameters that affect the axial 34
response of steel pipelines, namely the diameter over thickness (D/t) ratio of the pipe segment, 35
the internal pressure level, the existence of imperfections on the walls of the pipe segment 36
(further discussed in the following section) and the characteristics of the trench soil 37
surrounding the buried pipe segments, were considered. The analyses were performed by 38
employing the finite element code ABAQUS (ABAQUS, 2012). Figure 1 illustrates 39
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representative numerical models of pipe segments developed in the context of this short study. 1
The pipeline segments were simulated with inelastic shell elements, while the trench soil, 2
surrounding the embedded segments, was simulated by means of linear elastic solid elements. 3
The pipe-soil interface was simulated by means of an advanced hard contact interaction model, 4
available in ABAQUS (2012). The shear response of the soil-pipe interface was obeying a 5
classical Coulomb friction model, described by a friction coefficient μ = 0.3. The segments 6
were subjected to a pure axial stress that was introduced in terms of axial deformation on the 7
one end-side of the segment, with the other end-side being constrained. More details about the 8
simulation may be found in Tsinidis et al. (2018). 9
During axial compressive loading the examined segments are subjected to plastic buckling, 10
which results in a drop of their axial stiffness and a significant increase of axial deformations. 11
Contrary to elastic shell buckling, where the collapse is sudden, in plastic buckling the failure 12
is separated from the first instability (Kyriakides and Corona, 2007). This response was also 13
verified herein. 14
Figure 2 compares average axial load-deformation paths computed for above ground and 15
equivalent embedded pipe segments. The results, which refer to segments of steel pipelines 16
with diameters 406.4 mm (16 in) and 1219.2 mm (48 in) and radius over thickness ratios R/t = 17
19.7 and 25.6, respectively, are presented in normalized forms. In particular, the axial loading 18
is normalized by the yield axial load of the section: 2
oy
PRt
(Paquette and Kyriakides, 19
2006), where the yield stress σy = 448 MPa. The normalized axial load is plotted against the 20
average axial shortening xl
, where
x
is the axial deformation of the segment and l is the 21
initial length of the segment. The trench soil in case of embedded pipes is characterized by a 22
density ρ = 1.8 t/m
3, a shear modulus G = 23.4 MPa, and a Poisson's ratio, v = 0.33. The 23
consideration of internal pressure results in a general lowering of the axial load-displacement 24
path. In parallel, the internal pressure leads the limit loading to occur at progressively higher 25
axial shortening levels xl
. This response, which has been verified experimentally by 26
Paquette and Kyriakides (2006), is attributed to the plastic interaction of the two loading 27
conditions, i.e. internal pressure and axial compression, acting on the segment. Generally, the 28
confinement, which is offered by the trench soil, ‘stabilizes’ the axial response of embedded 29
segments, leading to an increase of the axial load-deformation response compared to the one of 30
the equivalent above ground segments, i.e. higher critical loadings and ‘critical’ shortening 31
levels are identified for embedded segments compared to the ones of the equivalent above 32
ground segments. Finally, the critical stresses and the ‘critical’ axial shortening levels are both 33
decreasing with increasing R/t ratio. The main observation is that the internal pressure level can 34
affect considerably the seismic response and vulnerability of NG steel pipelines and therefore, 35
this parameter should be considered by the selected analysis method. 36
37
38
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3.4 Geometric imperfections of steel pipelines 1
The axial compression response of thin-walled steel cylindrical shells is known to be highly 2
affected by geometric imperfections. Actually, these imperfections may reduce significantly 3
the actual buckling critical loading of a steel cylindrical shell, compared to the one predicted by 4
theoretical elastic solutions (NASA, 1968). Operations related to manufacturing process, girth 5
welding, transportation and laying may lead to deviations of the pipeline walls from perfect 6
geometry. These imperfections, which are referred as geometric imperfections, are commonly 7
considered in numerical analyses by linearly superposing eigenmode shapes, the latter obtained 8
by an eigenvalue buckling analysis of the examined pipe shell. An alternative approach is to 9
simulate the imperfection by means of a geometric stress-free perturbation in the initial 10
geometry and mesh of the examined shell (e.g. Yun and Kyriakidis, 1990; Psyrras at al., 2019). 11
Variations of pressure on the pipeline walls caused by the surrounding trench soil along the 12
axis of the pipeline may also results in a kind of ‘load’ imperfections. 13
Yun and Kyriakidis (1990) reported that the presence of even low-magnitude axisymmetric 14
geometric imperfections may reduce significantly the bifurcation and limit stress and strain of a 15
steel pipeline, compared to those predicted for a ‘perfect’ geometry. This observation was 16
verified by the results of the short numerical parametric study, discussed in the previous 17
section. In particular, the study considered both ‘perfect’ segments and segments with 18
geometry perturbation. For the latter cases, a stress-free, biased axisymmetric imperfection was 19
assumed at a short zone of 1.0 m, which was set at the middle of the segment. The imperfection 20
was defined following Paquette and Kyriakides (2006) and Psyrras et al. (2019), having a 21
maximum amplitude of 10 % of the pipe wall thickness (i.e. w/t = 0.1). The imperfection level 22
was based on relevant specifications from NG pipeline manufactures. ArcelorMittal for 23
example specifies a manufacturing tolerance for the walls of API-5L X65 pipelines in the range 24
of + 15% to -12.5% (ArcelorMittal 2018). As seen in Figure 2, the critical loadings of 25
imperfect segments, as well as the axial shortening levels, where these loadings are observed 26
(i.e. ‘critical’ axial shortening levels), are both lower compared to those predicted for the 27
equivalent ‘perfect’/ segments. The differences on the computed paths of perfect and 28
equivalent imperfect segments are generally higher for non-pressurized segments, while they 29
decrease significantly with the increase of internal pressure. Actually, for pressurized pipe
30
segments with the maximum allowable operational pressure (i.e. p = 0.72 × py) the differences 31
on the loading-deformation paths of equivalent perfect and imperfect segments are negligible 32
for 0.5 0.8
xl
(depending on the R/t ratio of the examined segment). Additionally, the 33
differences between the axial response of perfect and imperfect buried segments are smaller 34
compared to those revealed for above ground segments, particularly for higher levels of 35
internal pressure. 36
The amplitude of initial geometric imperfection may also be a critical parameter for the axial 37
response of steel pipelines. Figure 3a compares average axial force-deformation paths 38
computed for above ground segments of a steel pipeline with diameter D = 406.4 mm and 39
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radius over thickness ratios R/t = 19.7, considering various levels of geometric imperfection 1
amplitudes. The comparisons are plotted for both non-pressurized and pressurized segments. 2
The axial response of the segments reduces with increasing imperfection amplitude (i.e. lower 3
responses are reported for w/t = 0.2). This is generally more evident in case of non-pressurized 4
segments (i.e. for p = 0). The pressurization of the segments reduces the detrimental effect of 5
initial geometric imperfections. Similar conclusions are drawn for the equivalent buried 6
segments (Figure 3b). The confinement that is offered by the trench soil reduces the effect of 7
imperfection on the axial response of the examined segments, particularly for the cases of 8
pressurized segments. Actually, for the pressurized segment with the maximum allowable 9
operational pressure level (p = 0.72 × pmax), the axial force-deformation paths predicted by the 10
analyses of equivalent perfect and imperfect segments are quite similar, if not identical, even in 11
the post-buckling regime. 12
Evidently, the geometric imperfections of the walls of a steel pipe are affecting significantly its 13
response under axial compressive straining and therefore they should be considered by the 14
selected design of vulnerability assessment method. The only way to simulate accurately these 15
imperfections is by means of shell elements with a quite refined mesh. The available analytical 16
studies on the seismic vulnerability assessment of NG pipelines (Lee et al., 2016; Jahangiri and 17
Shakib, 2018) have modelled the examined pipelines by means of inelastic beam elements and 18
fibers, disregarding the critical effect of geometric imperfections. 19
20
3.5 Trench soil 21
The properties of the trench soil, surrounding a buried pipeline, are another critical parameter 22
that might affect the seismic response and vulnerability of a steel pipeline. After the installation 23
of the pipeline, the excavated trench is backfilled, with the backfill material being compacted. 24
The compaction might increase the lateral earth pressure coefficient to values higher than 25
unity, which subsequently may lead to an increasing frictional resistance of the soil-pipe 26
interface. This increasing shear resistance of the soil-pipe interface may lead to higher axial 27
stresses and strains on the pipeline under seismically-induced transient ground deformations. In 28
this context, a critical issue for the rigorous assessment of buried steel pipelines is the 29
simulation of the shear response of the soil-pipe interface. The shear response of the soil-pipe 30
interface is commonly simulated through the classical Coulomb model, which correlates the 31
shear stresses developed along the soil-pipe interface with the soil normal pressures acting on 32
the pipe, through the friction coefficient μ. Generally, the friction coefficient of the soil-pipe 33
interface varies along the axis of a long pipeline while it may fluctuate during ground seismic 34
shaking. However, for steel pipelines without external coating it is bounded to the following 35
limits, μmin= 0.3 and μmax= 0.8. These limits are actually derived from the linear relation 36
between the friction coefficient μ and friction angle φ of the trench soil, i.e. 37

0.5 0.9 tan
, which was proposed by O’Rourke and Hmadi (1988) and is commonly 38
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adopted in practice. For typical sand backfills the soil friction angle can range between 29o and 1
41-44o, yielding to above limits for the friction coefficients. It is worth noticing that friction 2
coefficient of the soil-pipe interface may be affected by the level of scour of the external face 3
of the steel pipeline since the scour reduces the frictional resistance along the soil-pipe 4
interface. Additionally, the existence of external pipe coating might lead to different friction 5
coefficients for the soil-pipe interface. ALA (2001) suggests the interface angle δ to be 6
estimated as follows: δ = f φ, where φ the internal friction of the soil and f a coating parameter 7
depending on the coating material (e.g. 0.6, 0.9 and 1.0 for Polyethylene, Coal Tar and 8
Concrete, respectively). The friction coefficient μ may then be computed as μ = tanδ. 9
The dimensions and stiffness of the trench soil are other parameters that might also affect the 10
seismic response of a buried pipeline. The commonly used assumption of an infinitely large 11
trench may lead to an underestimation of the actual response of a pipeline subjected to lateral 12
ground seismic motion, as highlighted by Kouretzis et al. (2013) and Chaloulos et al. (2015, 13
2017). 14
Figure 4 elaborates on the effect of trench soil stiffness on axial compression response of 15
embedded steel pipelines. In particular, the axial force-deformation paths computed for a steel 16
pipeline with diameter D = 1066.8 mm (40 in) and ratio over thickness ratio R/t = 27.9, 17
embedded in trench soils with different stiffnesses, are compared. The comparisons are 18
provided for both non-pressurized and pressurized segments, with either considering or 19
neglecting the initial geometric imperfections of the segments walls. Solid lines correspond to 20
the results referring to the trench soil with reduced stiffness, while dashed lines stand for the 21
analyses, where higher soil stiffness was considered for the trench. The increase of the stiffness 22
of the trench soil, results in an increased stabilization of the pipe segments, which in turn, 23
results in an increase of their axial response. This observation is more evident for the non-24
pressurized pipe segments. 25
The trench soil compliance is commonly simulated in a simplified way via discrete soil springs 26
and dashpots. Despite the computational efficiency of such a simulation approach, potential 27
geometric nonlinearities along the soil-pipe interface response can not be thoroughly accounted 28
for, particularly when the effect of geometric imperfections on the pipe walls are considered as 29
a perturbation of the pipeline geometry and mesh. The proper selection of the stiffness and 30
damping coefficients for soil springs and dashpots elements, respectively, is another critical 31
issue that remains still not full addressed (see Section 4.2). The alternative simulation of the 32
trench soil by means of 3D solid elements in a coupled simulation of an ‘infinitely’ long soil-33
pipe configuration is computationally inefficient, particularly in the framework of a large 34
number of numerical analyses that are required for the seismic vulnerability assessment of 35
buried pipelines. Some other alternatives are discussed in the ensuing (Section 4.3). 36
37
38
39
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3.6 Spatial variation of the ground seismic motion along the pipeline axis 1
Ground seismic motion is varying in space and time in terms of wave amplitude, phase, 2
frequency characteristics and duration. Given the complicated nature of the mechanisms that 3
cause this phenomenon and the numerous parameters and uncertainties involved, the so-called 4
spatial variability of seismic ground motion can only be investigated stochastically. The factors 5
that contribute to the spatial variability of seismic ground motion can be grouped into three 6
main effects, i.e. the wave passage effect, the ray-path effect and the local site effect (Figure 5). 7
The wave passage effect is associated with the finite velocity, with which the seismic waves 8
are traveling. This phenomenon results in different arrival times of the seismic waves to 9
various locations of the site. The ray-path effect is the result of the continuous reduction of 10
coherency of seismic waves due to successive reflections and refractions that take place along 11
their propagation through heterogeneous soil sites or due to the superposition of wave 12
emanating from different points of the seismic source. The latter effect is also known as 13
extended source effect. Finally, the local soil conditions may affect significantly characteristics 14
of the ground seismic motion, such as the amplitude, frequency content and duration. 15
Evidently, the spatial variation of the ground seismic motion along the axis of an extended 16
structure, such as a NG pipeline, is expected to affect significantly its response. Indeed, the 17
spatial variation of the ground seismic motion across the length of extended structures, such as 18
bridges, was found to affect considerably their response (e.g. Sextos et al., 2003; Sextos and 19
Kappos, 2009; Zerva, 2009). A common approach to account for these phenomena is to 20
implement deterministic time history analyses, introducing spatially variable ground motions, 21
while random vibration analysis is another alternative. The effect of incoherent ground seismic 22
motion on the seismic response of pipelines has been investigated by Zerva et al. (1985) and 23
Zerva (1994). Using random vibration analysis on analytical models of continuous and 24
segmented pipelines, the researchers reported a higher level of stresses on the pipelines under 25
partially correlated motions, compared to those predicted for perfectly coherent motions. 26
Similar observations may be found in more recent studies (e.g. Lee et al., 2009). 27
Soil inhomogeneities, special subsurface geomorphic conditions or irregular topography (e.g. 28
hills, canyons, valleys etc) may cause local site effects, which may amplify significantly the 29
ground seismic motion (e.g. Scandella and Paolucci, 2010; Gelagoti et al., 2010; Riga et al., 30
2018), resulting in high ground deformations on the pipelines. The critical effect of 31
inhomogeneous soil sites on the pipeline seismic response has been reported by numerical, 32
analytical and experimental studies (e.g. Hindy and Novak, 1979; Nishio et al. 1983; Liang 33
1995). Recently, Psyrras and Sextos (2018) proposed some idealized cases of soil sites, which 34
might be crucial for the seismic response of a crossing pipeline, i.e. a site consisting of two 35
horizontally adjacent soil layers with highly distinct properties and a site with a soft alluvial 36
valley of trapezoid shape, laying within a stiff soil or soft rock. The researchers highlighted 37
that for these site conditions, an amplification of the transient ground deformations is expected 38
near the soil boundaries (i.e. at the vertical interface of the two soil layers or at the valley 39
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edges) under vertically propagated seismic S-waves, which may lead to significant axial or 1
bending strains on a buried steel pipeline crossing these sites, depending on the position of the 2
pipeline axis compared with the polarization of the seismic waves. In a more recent study, 3
Psyrras et al. (2019) examined the potential of buckling of buried steel NG pipelines, crossing 4
the above idealized soil sites, under seismically-induced transient ground deformations, 5
highlighting that under particular circumstances, an appreciable axial stress concentration may 6
be observed near the soil discontinuities, which may even lead to buckling failures. 7
8
3.7 Elbows and other restrictions 9
Elbows are pipe bends, which under ground seismic shaking reveal a more complex behaviour 10
compared to the straight parts of the pipeline and may generally affect the response of buried 11
pipelines. Various numerical and experimental studies have focused on the response of these 12
elements under seismic and static loading conditions, the latter associated to seismically-13
induced permanent ground deformations, such as faulting (e.g. Shinozoka and Koike, 1979; 14
Saberi et al., 2013; Hamada et al., 2000; Yoshizaki et al., 2003; Karamitros et al., 2016; 15
Karamanos, 2016, among others). 16
Connections of the pipelines with stiffer structures, e.g. metering or pressure reduction stations, 17
may impose restrictions on the pipelines during ground seismic shaking, leading to significant 18
increases of axial or bending stresses locally. The effects of these restrictions on the seismic 19
vulnerability of steel NG pipelines have not been thoroughly studied. 20
21
3.8 Summary 22
A series of parameters that may affect the response and therefore the vulnerability of buried 23
steel NG pipelines were reported in this section. The consideration of the above parameters in 24
relevant vulnerability studies depends on the capabilities of the numerical method that is 25
employed each time. 26
27
4 Analysis methods for the seismic vulnerability assessment of buried steel 28
NG pipelines 29
30
Various methodologies may be found in the literature for the seismic analysis and vulnerability 31
assessment of buried pipelines, which can generally be classified as methods that neglect the 32
soil-pipe interaction (SPI) phenomena and methods that account for the SPI phenomena in 33
either a simplified or more detailed manner. A brief overview of the available methods is 34
presented in ensuing, focusing on the applicability and efficiency of each method in the 35
framework of a seismic vulnerability assessment study. A more detailed review of available 36
methods may be found elsewhere (e.g. Datta et al. 2001; Psyrras and Sextos, 2018). 37
38
-12-
4.1 Analysis methods neglecting the soil-pipe interaction phenomenafree-field 1
methods 2
The main assumption of these analysis methods is that the embedded structure, the buried 3
pipeline herein, is forced to conform perfectly to the movement of the surrounding ground; in 4
other words, the pipe strains are identical to the ones of the surrounding ground. These 5
approaches commonly adopted in the practice for buried structures, whose stiffness is much 6
smaller than the one of the surrounding ground (Newmark, 1967; Wang, 1993). Since the soil-7
structure interaction effects are totally ignored, these methods are commonly referred as ‘free-8
field’ analysis methods (Wang, 1993; Hashash et al., 2001). In his pioneer work, Newmark 9
(1967) developed analytical relations for the evaluation of the axial strain and the curvature of 10
a straight buried pipe, neglecting the SPI effects. The relations (Equations 1 and 2), are based 11
on the assumption that a straight pipe is embedded in a homogeneous, isotropic, infinite and 12
elastic soil medium, which is subjected to constant plane seismic wave, the latter propagated in 13
parallel with the pipeline axis with a velocity c. 14
1uu
x
ct



(1) 15
22
22 2
1vv
x
ct



(2) 16
The above relations may be used to compute the axial strain and curvature that are imposed on 17
the pipeline, as a result of the propagation of any type of seismic wave (S-, P- or R-wave) 18
under any incidence angle. Based on Newmark’s approach, St John and Zahrah (1987) 19
presented a series of solutions for the computation of maximum ground axial strains and 20
curvatures caused by various types of seismic waves, which by assumption can directly be used 21
as predictions for the pipe strains. Newmark’s approach is recommended by Eurocode 8 (CEN, 22
2006) for the seismic design and assessment of buried pipelines, as long as the ground is 23
considered stable, elastic and homogeneous. Despite the simplicity of this framework, the 24
‘free-field’ methods may be considered inadequate for the design and assessment of ‘stiff’ 25
pipelines, i.e. pipelines with low radius over thickness (R/t) ratios, like the steel pipelines 26
commonly used in NG applications, as they may render quite conservative results in terms of 27
pipe strains. Additionally, critical damage modes of steel pipelines can not be simulated with 28
these methods, while salient parameters affecting the seismic response and vulnerability of 29
buried steel pipelines, namely the internal pressure, potential geometric imperfections of the 30
pipe walls and the trench soil and soil-pipe interface characteristics, may not be considered. In 31
this context, the implementation of ‘free-field’ analysis methods in the seismic vulnerability 32
assessment of buried pipelines is considered rather inadequate. 33
34
4.2 Analysis methods that consider the soil-pipe interaction phenomena 35
When the pipeline stiffness is comparable to that of the surrounding ground, SPI phenomena 36
might take place, leading to differences between the pipe and the surrounding ground 37
-13-
movements and hence affecting the pipe response. Some early studies reported a rather 1
favourable effect of SPI on the seismic response of pipelines that were embedded in uniform 2
soil sites (e.g. Hindy and Novak, 1979). Actually, St John and Zahrah (1987) proposed 3
reduction factors on analytical relations that were initially developed for the computation of 4
internal forces of long underground structures (e.g. pipelines, tunnels) neglecting the SPI 5
effects, in order establish relations that account for these effects. However, for buried pipelines 6
crossing heterogeneous soil sites, the consideration of the SPI effects was found to lead to 7
higher responses (Hindy and Novak, 1979). 8
9
4.2.1 Beam or shell element pipe models on soil springs 10
The simplest and most commonly used approach to account for the SPI effects is the beam-on-11
nonlinear-Winkler-foundation (BNWF) model (Figure 6a). In this analysis framework, the pipe 12
is simulated by means of beam elements, whilst discrete nonlinear translational springs and 13
dashpots of appropriate stiffness and damping are used to account for the soil compliance. The 14
latter are actually applied in the three orthogonal directions. In a one-dimensional 15
representation of the SPI interaction problem, the equations of motion of a pipeline, subjected 16
independently to ground displacement time histories
g
utand
g
vt, in the transverse 17
horizontal and the axial direction, respectively, are given as: 18
24
24
0
rel
hhrel
u
uu
mc EI ku
ttt


 (3) 19
22
22
0
rel
aarel
v
vv
mc EAku
ttt


 (4) 20
where u and v are the time-dependent pipe displacement components in the transverse 21
horizontal and axial direction, m is the distributed mass of the pipe, EI and EA are the flexural 22
and axial stiffnesses of the pipe cross-section, kh, ka are the equivalent soil springs constants 23
per unit length of the pipeline in the horizontal transverse and axial directions, ch and ca are the 24
equivalent soil dashpots constants per unit length of the pipeline in the horizontal transverse 25
and axial directions and rel g
uuu , rel g
vvv
. The response of the pipe, in terms of strains 26
may be obtained by solving the above equations in the time domain. BNWF models have been 27
widely used by practitioners for design purposes, as well as by researchers in the framework of 28
studies related to the seismic analysis and design of buried pipelines. For instance, Mavridis 29
and Pitilakis (1996) used a BNWF model to highlight the significant effects of soil-pipe 30
interaction on the axial response of buried pipelines. Nourzadeh and Takada (2013) used a 31
dynamic BNWF model to investigate the response of buried steel NG distribution pipelines 32
under seismically-induced transient ground deformations. Papadopoulos et al. (2015) employed 33
a BNWF model to perform 3D dynamic analyses of a buried steel NG pipeline and investigate 34
the effect of the spatial variability of ground seismic motion on its response. In the latter study, 35
the seismic ground deformation time histories that were imposed on the 3D BNWF model were 36
-14-
actually computed along the pipeline length via a separate 2D visco-elastic soil response 1
analysis of the examined soil site. 2
Since dynamic SSI effects can be ignored in case of buried pipelines, the above equations 3
degenerate to the following quasi-type forms: 4

4
4hg
u
EI k u u
t

(5) 5

2
2ag
v
EA k v v
t

(6) 6
Evidently, solving the above equations requires a rational selection of the soil springs 7
constants. Several soil spring models have been proposed the last decades on the basis of 8
experimental and numerical studies, to account for the soil compliance on buried pipelines in 9
both the axial and the horizontal transverse directions (O’ Rourke M.J. and Wang, 1978; 10
Selvadurai, 1985; El Hmadi and O’Rourke M.J., 1988). Other studies provided relations for the 11
evaluation of the ultimate soil resistance forces to lateral, vertical or oblique pipeline 12
movements (Audibert and Nyman, 1977; Nyman, 1984; O’Rourke M.J. and El Hmadi, 1988; 13
Hsu et al., 2001). A rather detailed summary of these studies may be found in Psyrras and 14
Sextos (2018). 15
The most commonly used relationships for the computation of soil springs for buried pipelines 16
are those summarized in ALA (2001) guidelines (Table 1). The discrete soil springs are 17
actually defined in four principal directions, i.e. axial, vertical uplifting, vertical bearing and 18
lateral, following Hansen and Christensen (1961) and Trautmann and O’Rourke (1983) and 19
adopt an elasto-plastic bilinear curve to account for the trench soil and soil-pipe interface non 20
linear response. It is worth noticing that these springs are defined for pipelines embedded in a 21
uniform soil deposits. 22
The simulation of the pipeline as an equivalent beam may lead to a rough estimation of the 23
axial and bending deformations and strains that are imposed on it by seismically-induced 24
transient ground deformations. The pipe material nonlinearity, which is a crucial parameter in 25
the seismic vulnerability assessment, may be considered by means of fibers (Lee et al., 2016; 26
Jahangiri and Shakib, 2018). However, with this simulation approach, it is not possible to 27
replicate accurately potential seismically-induced damage modes, described in the first part of 28
the paper, e.g. local buckling, ovalization etc. Additionally, it is possible to account for the 29
hoop stresses caused by the pressurization of the steel pipeline, which as seen in Section 3.3, 30
interact with the axial straining on the pipe during ground seismic shaking, affecting its axial 31
response. Moreover, critical aspects that may affect the axial compressive response of steel NG 32
pipelines, such as the geometric imperfections of the pipe walls, may not be considered. On the 33
contrary, a shell model of the pipeline allows for an adequate simulation of various damage 34
modes, as well as for the consideration of most of the salient parameters described in Section 3. 35
In this context, several studies have employed shell pipe models in the analysis of the seismic 36
response and vulnerability of pipelines. In the simplest case, the soil-pipe relative movements 37
-15-
may be neglected and therefore the seismic excitation may be directly be applied on the 1
pipeline; in this case the analysis is degenerated in a structural analysis of the shell pipeline. 2
Kouretzis et al. (2006) followed such as simulation approach, in an effort to validate closed-3
form solutions for the evaluation of strains caused by seismic S-waves on steel pipes. A more 4
accurate alternative, includes the consideration of the soil compliance by means of soil springs, 5
the latter being introduced along the perimeter of the shell model (e.g. Figure 6b). As an 6
example, Lee et al. (1984) used an elasto-plastic cylindrical shell model of a pipe to examine 7
its structural stability under seismic wave propagation, simulating the soil compliance by 8
means of soil springs. The springs were introduced only in the normal direction of the pipe, 9
neglecting the SPI effects in the critical axial direction. In an effort to examine thoroughly the 10
parameters under which a buried pipeline, subjected to compressive axial loading, can bifurcate 11
plastically in either beam- or shell model buckling, Yun and Kyriakidis (1990) used both beam 12
models of pipelines with large-deflection kinematics being accounted for, as well as inelastic 13
shell models, with geometric imperfections being considered. The SPI effects in the later case 14
were considered in a similar fashion with Lee et al. (1984), i.e. by means of normal soil 15
springs, disregarding the axial ones. 16
The extended length of the buried pipelines, in addition to the need for very refined shell 17
meshes in order to capture potential damage modes, such as local buckling failures, are 18
expected to increase considerably the computational cost of this type of analyses compared to 19
the ‘classical’ BNWF models. A potential solution towards reducing the relevant computation 20
cost is the use of hybrid models, similar to the one illustrated schematically in Figure 6c. In this 21
modelling approach, the critical parts of an examined pipeline, e.g. locations where a buried 22
steel pipeline is crossing identified geotechnical discontinuities that may result in an amplified 23
response under transient ground deformations, are modelled in a detailed fashion, by means of 24
shell elements, while the rest pipeline is simulated by means of beam elements. The latter parts 25
are connected rigidity to the detailed shell model parts. The soil compliance is simulated along 26
the whole length of the pipeline by means of soil springs. An example of this simulation 27
approach is provided by Saberi et al. (2013), who used a hybrid beam-shell pipe model to 28
evaluate the response of steel pipe bends. 29
Regardless of the simulation approach that is used for the pipeline, the determination of 30
impedance functions (e.g. springs and dashpots) for long underground structures, such as 31
buried pipelines, is a quite delicate problem (Pitilakis and Tsinidis, 2014). The few analytical 32
relations that may be found in the literature are defined on the basis of simplified elasto-plastic 33
idealizations of the actual nonlinear soil-pipe response, while they do not account for the soil 34
heterogeneities along the pipeline axis. Moreover, most of the available relations were derived 35
under the assumption of monotonic loading conditions, neglecting the actual cyclic nature of 36
seismic excitation and the hysteretic characteristics of the soil-pipe system. The effect of 37
coupling of the directional components of the relative soil-pipe motion, on the pipeline 38
response and vulnerability, is commonly disregarded by the available soil spring relations. 39
-16-
Along these lines, further investigation is needed towards the proposal of soil springs or macro-1
elements that could simulate adequately the soil compliance on buried pipelines. 2
An additional issue of the pipe shell on soil springs models is related with the introduction of 3
local forces on shell elements at the locations of the soil springs. These forces may alter the 4
distribution of stresses and strains on the pipeline wall and even lead to inaccurate buckling 5
responses. This problem might be more intense in case of coarse mesh of the pipeline. Along 6
these lines, particular emphasis should be placed on the appropriate meshing of the pipeline 7
and simulation of soil compliance by means of soil springs. 8
9
4.2.2 3D continuum models of the pipe-trench soil system 10
An alternative method to consider the soil compliance on buried pipelines in a numerical 11
analysis framework is by simulating part of the surrounding ground, as a continuum trench-like 12
model that encloses the buried pipeline. In this framework, the pipeline is simulated by means 13
of shell elements, whilst solid elements are used for the surrounding trench soil. This 14
simulation allows the use of contact elements or sophisticated contact models that may account 15
rigorously for the potential geometrical nonlinear phenomena along the soil-pipe interface 16
during loading, including sliding, or even separation of the pipe from the surrounding trench 17
soil. This analysis approach was used by some researchers to simulate the response of buried 18
pipelines subjected to permanent ground deformations due to faulting (e.g. Vazouras et al., 19
2010; Vazouras et al., 2012; Vazouras et al., 2015; Vazouras and Karamanos, 2017; Sarvanis et 20
al., 2017). 21
The use 3D continuum models of the pipe-trench soil system allows for critical parameters that 22
affect the axial response of buried steel NG pipelines, such as the potential geometric 23
imperfections of the pipeline walls, or the internal pressure of the pipeline, to be effectively 24
considered. The main shortfall of this approach is the higher computational cost, compared to 25
that of beam- or shell-on-soil-springs models. Indeed, implementing 3D continuum models in a 26
vulnerability assessment framework of buried NG pipelines under seismically-induced 27
transient ground deformations, is computationally inefficient, particularly in case, where a 28
large number of full dynamic time histories analyses are expected to be conducted within an 29
IDA, multi-stripe or cloud analysis. However, the approach may be applicable in studies, 30
where the effects of transient ground deformations on the response and vulnerability of the 31
buried NG pipelines are investigated in a pseudo-static manner. A representative example is 32
provided by Psyrras and Sextos (2018), who examined the potential of buckling failures of 33
buried steel NG pipelines that cross heterogeneous soil deposits, when subjected to 34
seismically-induced transient ground deformations. The methodology, which is further detailed 35
in Psyrras et al. (2018) and Psyrras et al. (2019), consists of two separate steps. The in-plane 36
ground deformations that are expected to be imposed on the pipeline during ground seismic 37
shaking are initially computed along its axis, via 2D elastic or visco-elastic response analyses 38
of pre-selected soil sites, neglecting the effects by the presence of the pipeline. The researchers 39
-17-
implemented in their analyses the idealized sites discussed above (i.e. sites consisted of two 1
horizontally adjacent soil layers of highly distinct characteristics and stiff soil/rock sites that 2
included a soft alluvial valley of trapezoidal shape in the middle). An ‘adequately’ long 3D 3
continuum pipe-trench soil model is used in a second step to evaluate the pipeline response 4
under critical soil displacement patterns, identified from the soil response analyses. The pipe 5
model is simulated by shell elements and is encased by a near-field trench-like soil continuum 6
model. According to the researchers, a trench soil model of reduced dimensions around the 7
pipe, starting from a few meters below the ditch line and reaching up to the ground level, 8
suffices for the proper simulation of the soil compliance since the typical burial depth of NG 9
pipelines is reduced and the SSI effects on buried pipelines are generally not important (see 10
also Section 3.1). In the particular study, the trench soil model was simulated by means of 11
elastic solid elements with equivalent soil properties being assign on them (i.e. soil degraded 12
stiffness), the latter being estimated by the separate 2D soil response analyses of the first step. 13
The displacement patterns are introduced statically on the soil volume and transmitted on the 14
pipeline through the soil-pipe interface, which is simulated by means of advanced interaction 15
models. Based on this analysis framework, Psyrras et al. (2019) reported that under particular 16
circumstances and ground motion characteristics buckling damages may be observed on steel 17
NG pipelines, at locations, where the properties of the surrounding ground change drastically. 18
Evidently, the pseudo-static simulation of the seismic loading, i.e. the transient ground 19
deformations herein, is computationally more efficient compared to a dynamic simulation in 20
the framework of full time histories analyses. However, an ‘adequately long’ 3D soil-pipe 21
continuum model is required in order to replicate the actual SPI phenomena, accounting for the 22
‘anchorage’ length of the pipeline on the surrounding trench soil and its effect on the 23
transmitted stresses on the pipeline through the soil-pipe interface. Additionally, there is a 24
requirement of fine meshes of the pipe and the trench soil, to adequately resolve the buckling 25
modes of the pipeline, which may potentially be caused by seismically-induced ground 26
deformations on the pipeline. The above aspects reduce the applicability of this analysis 27
approach in seismic vulnerability studies, where a large number of analyses are mandatory, in 28
order to establish rigorous analytical fragility curves. A potential solution is to use continuum 29
models with refined meshes at critical locations (e.g. near the assumed geotechnical 30
discontinuities) and much coarser mesh seeds elsewhere. Even with this meshing strategy, the 31
computational times might be quite significant. 32
33
4.2.3 3D hybrid continuum models of the pipe-trench soil system 34
At the time of writing, the authors of this paper were expending efforts to evaluate the 35
vulnerability of buried steel NG pipelines against local buckling failures, potentially induced 36
by seismic transient ground deformations near geotechnical discontinuities. In the framework 37
of this investigation, a 3D hybrid continuum model of the pipe-trench soil system was 38
developed, which may be generally used in relevant studies for the seismic vulnerability 39
-18-
assessment of buried steel pipelines. More specifically, the study focused on idealized soil-pipe 1
configurations, consisting of a buried steel NG pipeline crossing perpendicularly a vertical 2
geotechnical discontinuity with an abrupt change on the soil properties. The analysis 3
framework, which resembles, to some extent the one of Psyrras et al. (2019), consists of two 4
steps. A 3D trench-like continuum soil model, encasing a cylindrical shell model of the 5
pipeline, is initially developed in ABAQUS (2012), in order to compute the axial compressive 6
response of the pipeline under an increasing level of axial relative ground displacement, caused 7
by vertically propagated S-waves near a geotechnical discontinuity. The analysis focuses on the 8
axial ground displacements and disregards the vertical ones that might be observed near 9
geotechnical discontinuities since the former introduce much higher strains on the pipeline. 10
Figure 7 illustrates the typical numerical model layout. The pipeline is simulated by means of 11
inelastic shell elements, while solid elastic elements are used to model the trench soil. The 12
properties of the trench soil elements correspond to average equivalent properties of the soil, 13
estimated via the 1D soil response analyses in the separate analysis step, as described in the 14
ensuing. The distance between the side boundaries of the trench model and the pipe edges is set 15
equal to one pipe diameter, whereas the distance between the pipe invert and the bottom 16
boundary of the trench model is set equal to 1.0 m. Evidently, the distance between the pipe 17
crown and ground surface is defined on the basis of the adopted burial depth of the examined 18
pipeline. The mesh is refined at the middle section of the model, which corresponds to the 19
location, where the properties of the site are supposed to change abruptly, thus resulting in a 20
‘step’ on the axial ground deformations imposed on the pipeline during the seismic wave 21
propagation (i.e. δu in Figure 7). Typical static boundary conditions are applied at the bounding 22
soil surfaces. In particular, the bottom boundary of the soil model is fixed in the vertical 23
direction, while the side-boundaries of the soil model are fixed in the horizontal direction. The 24
ground surface is set free. An advanced contact model is implemented for simulation of the 25
soil-pipeline interface, with the sliding behaviour being controlled by the classical Coulomb 26
friction model. The length of the 3D pipe-soil trench model is set equal to 20 times the 27
diameter of the pipeline, while nonlinear springs, acting parallel to the pipeline axis, are 28
introduced at both end sides of the pipeline, in order to account for the effect of the infinite 29
pipeline length on the response of the examined pipeline-trench soil configuration. The force-30
displacement relation of the nonlinear springs is given as follows: 31
32
max
2
0
max max max max max max
for
+2 for
xx
s
xx
s
ssss
EA k
FD
EA m
km k k k k
 
 


 

 
 

 

(7) 33
34
where: 35
-19-
s
Dk
EA
(8) 1
max
D
mEA
(9) 2
x
is the soil-pipe relative axial movement caused by the relative axial ground deformation δu, 3
max
is the maximum shear resistance that develops along the soil-pipe interface, ks is the shear 4
stiffness of the soil-pipe interface and EA is the axial stiffness of the pipeline cross section. As 5
already discussed, for cohesionless backfills, i.e. common trench soil conditions, the maximum 6
shear resistance depends on the adopted Coulomb friction coefficient μ and varies along the 7
perimeter of the pipe. On this basis, mean values of max
and ks should be computed, based on 8
numerical simulations of simple axial pull-out tests of the examined pipe from the trench soil. 9
It is worth noting that the proposed simulation of the end-boundaries of the pipeline is inspired 10
from a numerical model that was developed by Vazouras et al. (2015), in order to account for 11
the effect of the infinite length of buried pipeline when subjected to seismically-induced strike-12
slip faulting. The theoretical background and the necessary modifications that are required in 13
order to expand such a simulation approach in case of seismically-induced in-plane transient 14
ground deformations are presented in Appendix A. The selection of the length of the 3D hybrid 15
model is made on the basis of a sensitivity analysis, by comparing the stresses and strains 16
computed at the middle critical section of the pipeline by the 3D hybrid model, with relevant 17
predictions of an equivalent quite extended, almost ‘infinite’, 3D continuum model of the soil-18
pipe configuration subjected to the same axial ground deformation pattern. With reference to 19
the loading pattern of the 3D hybrid model; the stress state, associated with the gravity and the 20
internal pressure of the pipeline, is initially established within a general static step. The 21
selected ground displacement pattern is then introduced monotonically on the soil-pipeline 22
configuration, through the soil volume and the free ends of the nonlinear springs, in stepwise 23
ramp-type fashion, as per Figure 7. This loading condition results in an axial compressive 24
response of the pipeline, which may be traced for an increasing level of relative axial 25
displacement δu through a simple static analysis step or even better via a modified Riks 26
solution algorithm. Through this analysis, a correlation between the relative axial ground 27
displacement δu and the maximum compressive axial strain of the critical middle section of the 28
pipeline (i.e. near the assumed geotechnical discontinuity) may be established. 29
In a second step, critical relative axial ground deformation patterns δue are determined at the 30
pipeline’s depth, for selected sites, consisting of two laterally adjacent soil deposits of 31
dissimilar properties, subjected to ground seismic motions at their base, in the form of 32
vertically propagated S-waves. A series of separate 1D nonlinear soil response analyses of the 33
adjacent soil deposits are carried out for this purpose, under a variety of selected ground 34
seismic motions, in the framework of an IDA, multiple-stripe or cloud analysis. The outcome 35
of the soil response analyses, in terms of critical relative axial ground deformation patterns δue 36
-20-
and various seismic IMs at the burial depth of the pipeline or at bedrock conditions, is finally 1
correlated with the predicted straining of the pipeline, using the δu - maximum compressive 2
axial strain correlations computed by the 3D SPI analyses in the first step. 3
Since the response of the pipeline is computed for an increasing level of relative axial ground 4
displacement δu, the outcome of one 3D SPI analysis can be used to examine the axial straining 5
of the pipe under a variety of selected ground axial relative displacements δue, caused by 6
diverse ground seismic motions. This of course is possible under the assumption and 7
implementation of mean equivalent soil properties for the trench-soil, corresponding to the 8
strain-range that is anticipated by the selected ground seismic motions. The mean equivalent 9
soil properties (i.e. degraded soil stiffness) are actually estimated for the examined range of 10
strains, which are caused by the selected ground motions, via the 1D soil response analyses. 11
Additionally, with this analysis approach, crucial parameters that affect the axial response of 12
pipelines under compression, such as the initial geometric imperfections of the pipeline walls 13
or the level of internal pressure of the pipeline, can effectively be considered. 14
Figure 8 illustrates some representative results from 3D SPI analyses that were carried out 15
following the first step of the above analysis framework. In particular, contour diagrams of the 16
axial stresses, developed at the critical zone of the pipeline (i.e. the middle zone of the pipeline 17
model, where the soil properties are assumed to change), are plotted for two distinct steps of 18
the analysis, i.e. before major concentration of stresses and buckling failure at the zone and at 19
the end of the analysis, after buckling failure occurrence. The diagrams are plotted on the 20
deformed shapes of the pipelines, so as to highlight the buckling failures that occur for higher 21
levels of imposed relative axial ground deformations. Additionally, the figure portrays the 22
evolution of maximum compressive strain of the critical pipeline zone with increasing relative 23
axial ground deformation δu. The presented results correspond to a steel API 5L X65 pipeline 24
with diameter D = 762 mm and wall thickness t = 14.3 mm (i.e. R/t = 26.6), which is embedded 25
1 m bellow the ground surface, in a trench soil with density, ρ = 1.65 t/m3, shear modulus G = 26
37 MPa and Poisson ratio, v = 0.3 (trench soil A). A friction coefficient μ = 0.45 is considered 27
at the soil-pipe interface, while the nonlinear springs at the end-sides of the pipe are defined, as 28
per Equation 7. The results are provided for various levels of internal pressure for the pipeline 29
(i.e. p = 0, 4 MPa and 8 MPa), while both a ‘perfect’ and an equivalent imperfect pipeline are 30
considered. For the simulation of the imperfect pipeline, a stress-free, biased axisymmetric 31
imperfection is considered, having maximum amplitude of 10 % of the pipe wall thickness. It 32
is recalled herein that the level of imperfection is based on relevant specifications from NG 33
pipeline manufactures, e.g. ArcelorMittal (2018). This imperfection is applied over a short 34
zone of 2.0 m, centred at the middle of the pipeline model. 35
In line with the previous observations, both the pressurization level of the pipeline and the 36
initial geometric imperfections affect the axial response of the examined pipelines. In 37
particular, with increasing relative axial ground deformation δu, the pipeline tends to bend 38
upwards, i.e. towards the free ground surface. This response results in an early concentration of 39
-21-
compressive axial stresses at the invert part of the pipeline. The existence of geometric 1
imperfections is found to affect significantly the distribution of the axial stresses on the 2
pipeline. Actually, these stresses tend to distribute more uniformly across the lower part of the 3
perfect pipeline. On the contrary concentrations of stresses are observed at the imperfection 4
‘bulges’ of the imperfect pipeline. The pressure level of the pipeline tends to affect the 5
buckling patterns of the pipelines that take place under increased relative axial ground 6
deformations, i.e. 14 20
ucm
 for the examined cases. Inward deformations of the pipe 7
walls (i.e. deformations towards the pipe cavity) are observed for the non-pressurized (i.e. p = 8
0 MPa) or the low pressurized (i.e. p = 4 MPa) pipelines, while a combination of inward and 9
outward deformations (i.e. deformations towards the trench soil) are observed on the highly 10
pressurized pipelines (i.e. p = 8 MPa). The effects of the above parameters are evident on the 11
evolution of maximum compressive axial strain of the critical zone of the pipeline with the 12
increasing relative axial ground deformation δu. Higher stains are reported on the pressurized 13
pipelines even at low δu, compared to those predicted on the non-pressures pipelines. This 14
observation is related to the combining effect of the internal pressure and the axial compressive 15
straining of the pipeline caused by the ground movement, on axial response of the pipeline (see 16
also Section 3.3). Additionally, the pipeline with the geometric imperfection tends to 17
concentrate higher strains throughout the analysis compared to the equivalent ‘perfect’ 18
pipeline, with the differences between the two cases being as high as 15 %. 19
Similar comparisons of the computed axial response of the examined pipelines are presented in 20
Figure 9, referring to the case where the pipelines are embedded in a stiffer trench soil (density, 21
ρ = 1.9 t/m3, shear modulus G = 63 MPa, trench soil B) with a higher soil-pipe friction 22
coefficient be also considered, i.e. μ = 0.78. The examined pipelines exhibit a higher axial 23
response, both in terms of axial stresses and axial strains, compared to the previous case (i.e. 24
for trench soil A), with the buckling phenomena taking place in lower relative axial ground 25
deformations, i.e. 8 10
ucm
 . This increasing axial response is attributed to the higher axial 26
deformations that are transferred from the trench to the pipeline through the rougher interface. 27
Additionally, the higher confinement that is being offered by the surrounding trench soil 28
partially reduces the upward bending of the pipeline (i.e. bending towards the ground surface) 29
during the kinematic loading of the system, which in turn leads to an increased localization of 30
axial straining at the critical zone of the pipeline. 31
Inevitably, there are some limitations with the analytical methodology described herein. The 32
effect of inertial SPI and of the evolution of stresses and deformations on the pipeline response, 33
as well as time-dependent phenomena, such as fatigue and steel strength and stiffness 34
degradation due to cyclic loading, are all neglected. 35
36
37
38
39
-22-
4.3 Summary and identified challenges 1
The selection of a suitable methodology to evaluate the vulnerability of buried steel NG 2
pipelines against seismically-induced transient ground deformations, depends largely on the 3
damage mechanism of interest, as well as the desired degree of accuracy, since the physical 4
problem in its entirety is extremely complex and uncertain (Psyrras and Sextos, 2018). 5
Theoretically, detailed full dynamic analysis, making use of 3D numerical models of the soil-6
pipe configuration and accounting for the material nonlinearities of the soil and the pipe, as 7
well as for the geometric nonlinearities that might take place along the soil-pipe interface, 8
constitute the most rigorous way to evaluate the seismic response and vulnerability of buried 9
steel pipelines. However, the very large dimensions of the problem, the complexity of 10
simulating material and geometrical nonlinearities and the geometric imperfections of the 11
pipelines, the uncertainties in the definition of the characteristics of heterogeneous soil sites 12
and the inherently random varying ground seismic motion, render a fully 3D time history 13
analysis of the coupled pipeline-trench soil system computationally prohibitive. Along these 14
lines, some simplifications should always be made. Table 2 summarizes the most important 15
advantages and disadvantages of the presented ‘simplified’ methods, along with their potential 16
applicability for design or vulnerability assessment purposes. 17
‘Free-field’ analysis methods may be considered inadequate for the seismic vulnerability 18
assessment of buried steel NG pipelines since they can not be used for the simulation of 19
distinct damage modes, while they may not account for salient parameters controlling the 20
seismic response of buried steel pipelines. Simplified BNWF models have been proposed in 21
the literature to account for the effects of SPI on the seismic response and vulnerability of 22
buried pipelines. The simulation of the pipeline as an equivalent beam may result in a rough 23
estimation of the axial and bending deformations and strains. However, this analysis approach 24
does not allow for an accurate simulation of diverse damage modes of buried steel NG 25
pipelines, while again it is not possible to account for all the important parameters that might 26
affect the axial response of this infrastructure. The alternative use of shell-pipe models on soil 27
springs may be helpful towards a better simulation of critical damage modes of buried steel 28
pipelines. However, the accurate evaluation of the soil springs is a rather delicate problem that 29
remains still open. Moreover, the adequate meshing of the pipeline with shell elements is a 30
very important issue that should always be carefully accounted for. 31
The simulation of the soil compliance by means of a continuum model, enclosing the buried 32
pipeline, constitutes an alternative analysis approach. This type of SPI models, in combination 33
with a pseudo-static simulation of the seismically-induced transient ground deformations, may 34
provide a rather rigorous framework for the vulnerability assessment of buried steel NG 35
pipelines. This approach requires a series of separate soil response analysis to determine 36
critical soil displacement patterns that are used as input for the SPI analysis models. Obviously, 37
this analysis framework is associated with a higher computational cost compared to that of 38
beam- or shell-on-soil-spring models. An alternative to the later simulation approach, which 39
-23-
makes use of nonlinear springs to reduce the required length of the detailed 3D SPI models, 1
while accounting for the effect of pipeline infinite length, i.e. the 3D hybrid models, was 2
presented above. The method may account for critical parameters affecting the seismic 3
response and vulnerability of steel buried NG pipelines, while reducing significantly the 4
associated computational cost. 5
Regardless of the selected methodology, the ‘accurate’ simulation of the site conditions, as 6
well as the consideration of salient parameters affecting the pipeline axial response, are 7
expected to lead to more reliable observations regarding the seismic response of buried NG 8
pipelines, contributing towards the selection of the optimum seismic IM for each damage mode 9
and the development of reliable analytical fragility functions to be used in quantitative risk 10
assessment of NG networks. 11
The validation of the above methodologies against experimental results from rigorous test 12
campaigns or data from real cases in the field may provide evidence on the efficiency of each 13
analysis approach. A series of shaking table tests were recently carried out on a pipeline model, 14
embedded in an inhomogeneous soil site, at the shaking table of the University of Bristol, in 15
the framework of the Exchange-RISK project (http://www.exchange-risk.eu/). One of the 16
novelties of this testing campaign was the recording of the strains across the pipeline by means 17
of distributed fiber optic sensing. The results of this study are expected to shed light on seismic 18
response of buried pipelines that are embedded in similar soil sites, while they will constitute a 19
valuable dataset for validating the relevant analysis approaches. 20
21
5. Conclusions 22
Alternative analytical methods for the evaluation of the vulnerability of buried steel NG 23
pipelines under seismically-induced transient ground deformations were thoroughly presented 24
in this part of the paper. Particular emphasis was placed on the analytical tools for the 25
vulnerability assessment against seismically-induced buckling failure modes since these 26
constitute critical damage modes for the structural integrity of buried steel pipelines. Salient 27
parameters that control the seismic response and vulnerability of buried steel pipelines and 28
therefore should be considered in the relevant analytical methods were discussed, while a new 29
analysis approach for the assessment of buried steel NG pipelines was also introduced. The 30
main conclusions of this study may be summarized as follows: 31
The seismic response and vulnerability of buried steel NG pipelines is dominated by the 32
kinematic loading induced by the seismic movement of the surrounding ground. This 33
mechanism is actually causing the axial or bending straining on the pipeline, which in turn 34
may lead to distinct damage failures. The geologic and geotechnical conditions of the site, 35
the distance of the site from the seismic source, the internal pressure of the pipeline, the 36
potential geometric imperfections of the walls of the pipeline, the characteristics of the 37
trench soil, any potential restrictions of the pipeline (e.g. elbows), as well as the spatial 38
variation of the ground seismic motion along the pipeline axis, are among the parameters 39
-24-
that may affect the axial response of a buried steel NG pipeline caused by seismically-1
induced transient ground deformations. Along these lines, the above parameters should be 2
considered to the highest level possible by the numerical approaches that are implemented 3
for analytical evaluation the seismic vulnerability of this critical civil infrastructure. 4
The complexity of the seismic response and vulnerability of buried steel NG pipelines, in 5
addition to the high level of uncertainty on the definition of the above parameters, renders 6
the use of a 3D full dynamic time history analysis computationally prohibitive, particularly 7
in the framework of a seismic vulnerability analysis. Hence, the use of simplified models in 8
vulnerability assessment studies of NG pipeline networks is inevitable. BNWF models, 9
shell pipe models on soil springs or hybrid shell-beam pipe models on soil springs may be 10
used in the framework of dynamic time history analysis or pseudo-static analyses for the 11
vulnerability assessment of buried steel NG pipelines. These approaches offer 12
computational efficiency. The main shortcoming of the above analytical models is related 13
with the simulation of the soil compliance on the pipeline via ‘simplified’ soil springs. A 14
The use of 3D pipe-trench soil continuum models, combined with a pseudo-static 15
simulation of the seismically-induced transient ground deformations, computed separately 16
by means of a soil response analysis, may provide a more rigorous framework for the 17
vulnerability assessment of buried steel NG pipelines. The latter framework is associated 18
with a higher computational cost compared to that of a beam- or shell-on-soil-springs 19
models. However, this shortcoming may be partially resolved through the use of hybrid 20
boundaries at the end-sides of the pipe, so as to reduce the required length of the 3D 21
derailed model and the relevant computational cost, whist accounting for the effect of the 22
pipeline infinite length in an efficient fashion. Regardless of the selected analysis approach, 23
the validation of numerical tools with experimental data from rigorous testing campaigns or 24
recordings from real case studies, may contribute towards more reliable analysis 25
frameworks for the quantitative vulnerability assessment of buried steel NG pipelines. 26
27
Appendix A. Nonlinear springs for the simulation of an infinitely long 28
pipeline subjected to axial loading due to seismically-induced in-plane 29
relative axial ground deformations near geotechnical discontinuities 30
31
This appendix summarizes the theoretical background for the definition of the nonlinear 32
springs that are introduced at the end-boundaries of the detailed 3D SPI hybrid model, in order 33
to account for the effect of the infinite length of a buried pipeline, subjected to seismically-34
induced in-plane relative axial ground deformations near a geotechnical discontinuity, on its 35
axial response. The proposed simulation is inspired by a numerical model that was developed 36
by Vazouras et al. (2015) to account for the infinite length of buried pipeline when subjected to 37
seismically-induced strike-slip faulting. Given the differences between the two loading 38
-25-
mechanisms, some modifications are required in order to expand such a simulation approach 1
herein. 2
Figure A1 illustrates schematically the problem in hand. Only the right-hand side end boundary 3
of the hybrid SPI model is presented and discussed herein. A similar approach should be 4
followed for the left-hand side end boundary of the model. An infinitely long buried steel NG 5
pipeline of diameter D is considered to be embedded in a uniform elastic trench soil of density 6
ρ, Young’s modulus, Es, and Poisson’s Ratio, vs. The pipeline is made of steel, having a 7
Young’s Modulus, E, and Poisson’s Ratio, v. The trench soil volume, surrounding the pipe, is 8
subjected to a seismically-induced in-plane transient ground deformation, which is introduced 9
on the trench soil in a pseudo-static manner and corresponds to the maximum ground 10
deformation computed at the site of interested during ground seismic shaking from a separated 11
soil response analysis. This deformation is kept constant with the depth coordinate over the 12
trench soil domain, on the grounds that the depth of the truncated domain is small compared to 13
the maximum predominant wavelength of the impinging waves examined, hence the in-plane 14
motion of the soil particles does not vary significantly in depth (Psyrras et al., 2019). The 15
deformation yields in a uniform movement of the trench-soil parallel to the pipeline axis, i.e. us 16
in Figure A1. The soil movement mobilizes shear stresses along the soil-pipe interface, which 17
cause compressive or tensile axial deformations on the pipeline, i.e. up in Figure A1. 18
The shear stresses that are critical for the development of the axial deformations on the pipeline 19
are largely depending on the tangential behaviour of the soil-pipe interface, which is assumed 20
to follow the elasto-plastic law, as per Figure A2. The latter is defined on the basis of the shear 21
stiffness ks and maximum shear resistance τmax of the soil-pipe interface. For cohesionless soils, 22
which constitute a common backfill for trenches of pipelines, the above parameter may be 23
evaluated via a simple pull-out analysis of the pipeline from the trench soil (Figure A2), as per 24
Vazouras et al. (2015). 25
When the soil movement us is higher than a critical value, sliding will occur on part of the soil-26
pipe interface, i.e. along the sliding segment of the pipeline of length Ls, in Figure A1, while 27
the rest length of the infinite pipe will remain fully bonded to the surrounding ground, i.e. the 28
fixed segment of the pipeline, of length Le in Figure A1. 29
For very low relative pipe-trench soil displacements, the pipeline may be considered full 30
bonded with the surrounding trench soil and the shear stresses developed along the perimeter 31
of the pipeline that are lower the maximum shear resistance τmax and may be computed as 32
follows: 33
s
ku
(A1) 34
where u is the relative displacement on the pipe-soil interface caused by the soil movement us 35
and the pipe movement up (in this case u = us, since no sliding occurs) Considering the stress-36
strain relationship of the pipe material and the axial force equilibrium, the equilibrium equation 37
for the pipe segment yields into: 38
-26-
2
2
20
du u
dx
(A2) 1
where: 2
2
s
Dk
EA
(A3) 3
Accounting for the particular boundary conditions of the problem herein, the solution of 4
Equation A2 is given by the following equation: 5

x
ux ue
(A4) 6
The corresponding axial force along the pipe may then be evaluated as follows: 7

x
du
F x EA EA EAue
dx

  (A5) 8
For the limit case, at which sliding initiates at x = 0, i.e. at the end boundary of the hybrid 9
model, the relative displacement
0
x
sp
uuu
 becomes equal to the elastic slip 10
displacement maxes
uk
; hence, Equation A5 yields: 11
max
0
s
FEA
k
(A6) 12
When the relative soil-pipe displacement is higher than the elastic slip displacement, then 13
sliding occurs along a part of the pipeline, i.e. sliding segment of length Ls in Figure A1, while 14
the rest of soil-pipeline interface is responding elastically, exhibiting no sliding. It can be 15
shown that the equilibrium equation for the sliding segment is given by the following 16
expression: 17
22
max
22
00
du du
EA D m
dx dx

 (A7) 18
where 19
max
D
mEA
(A8) 20
Considering the distribution of the pipeline axial strain along its axis due to the assumed 21
kinematic loading, as well as the boundary conditions of the problem in hand, it can be shown 22
that the length of the pipe, along which sliding occurs, is given by the following expression: 23
2
max max max
12
sx
sss
Lm
mk k k



 


 

 

(A9) 24
25
Finally, the force at the end-boundary of the 3D SPI hybrid model, caused by the relative 26
displacement between the soil and the pipe is: 27
max
0maxs
s
FDLEA
k
 
 (A10) 28
-27-
Summarizing, the force-displacement relation of the nonlinear spring that replicates the effect 1
of an infinite long pipe is given by the following expressions: 2
max
2
0
max max max max max max
for
+ 2 for
xx
s
xx
s
ssss
EA k
FD
EA m
km k k k k
 
 


 

 
 

 

(A11) 3
4
The above relations and assumptions are valid for both tensile and compression axial loading 5
of the pipeline, caused by the ground axial movement. From a simulation perspective, to 6
account for the relative deformation δx at the side boundaries of the pipeline of the hybrid 7
model, one should introduce the ground deformation us pattern on both the volume of the 8
trench soil, as well as on the free end of the nonlinear springs (as per Figures 7 and A1). This is 9
the only actual modification of this simulation approach, compared to Vazouras et al. (2015). 10
The length of the detailed 3D SPI model should be selected on the basis of a short parametric 11
analysis, preceded the vulnerability analysis study. Generally, the length of the model should 12
be long enough, to avoid any bias of the results caused by potential boundary effects, and at the 13
same time, short enough, to avoid a misrepresentation of the actual kinematic loading induced 14
by the surrounding trench soil. 15
The above simulation approach is validated for a representative pipe-trench soil configuration 16
in the following, by comparing the results of the 3D hybrid SPI model, in terms of pipeline 17
Mises stresses and axial strains at the critical middle section of the pipeline (i.e. where soil 18
properties and hence ground deformation are changed), with the predictions of an equivalent 19
‘infinitely’ long 3D detailed model of the examined system. The comparisons refer to an API-20
X65 steel NG pipeline with outer diameter D = 914 mm (36 in), wall thickness t =12.7 mm (i.e. 21
radius over thickness ratio, R/t=36), yield strength σy = 448.5 MPa, Young’s modulus E = 210 22
GPa and Poisson’s ratio, v = 0.3. The pipeline is pressurized to a pressure level of 8 MPa and is 23
assumed to be embedded at burial depth of 1 m below the ground surface in a trench soil, 24
characterized by a density ρ = 1.65 t/m3, a Young’s modulus E = 42.9 MPa, and a Poisson’s 25
ratio, v = 0.3. The length of the 3D hybrid SPI model is set equal to 20 times the diameter of 26
the pipeline, with the end boundaries being simulated by means of nonlinear springs, as per 27
Equations A11. The length of the long ‘infinite’ model was set equal to 1000 D, to ensure that 28
its predictions were approaching those of a numerical model with infinite length. Figure A3 29
compares contour diagrams of the Mises stresses and axial strains distributions computed at the 30
critical central section of the pipeline as s result of a relative ground axial deformation δu = 20 31
cm, introduced at the middle of the sections. The comparisons refer to two friction coefficients 32
for the soil-pipe interface, i.e. μ = 0.3 and μ = 0.78. Clearly, for both soil-pipe interface 33
conditions, the 3D hybrid model results in very similar –if not identical – results with the long 34
-28-
3D SPI models, while a higher axial response of the pipeline is naturally reported for the 1
rougher soil-pipe interface. 2
3
Acknowledgements 4
This work was supported by the Horizon 2020 Programme of the European Commission under 5
the MSCA-RISE-2015-691213-EXCHANGE-Risk grant (Experimental and Computational 6
Hybrid Assessment of NG Pipelines Exposed to Seismic Hazard, www.exchange-risk.eu). This 7
support is gratefully acknowledged. 8
9
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List of Figures 1
(
a
)
(
b
)
Fixed boundary Fixed boundary
1 m
5DAxial deformation
Imperfection
3D
1 m
2D
2
Figure 1. Representative numerical models of (a) above ground and (b) embedded pipe segments 3
developed in ABAQUS to examine the effects of salient parameters on the axial response of NG 4
pipelines (adapted after Tsinidis et al., 2018). 5
6
0 0.5 1 1.5 2 2.5 3
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 406.4 mm (R/t= 19.7)
p=0
p=0.18×p
y
p=0.35×p
y
p=0.72×p
y
0 0.5 1 1.5 2 2.5
3
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 1219.2 mm (R/t=25.6)
p=0
p=0.23×p
y
p=0.46×p
y
p=0.72×p
y
0 0.5 1 1.5 2 2.5 3
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 406.4 mm (R/t= 19.7)
p=0
p=0.18×p
y
p=0.35×p
y
p=0.72×p
y
0 0.5 1 1.5 2 2.5 3 3
.5
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 1219.2 mm (R/t=25.6)
p=0
p=0.23×p
y
p=0.46×p
y
p=0.72×p
y
(a) (b)
(c) (d)
7
Figure 2. Average axial load- deformation paths computed for various levels of internal pressure for (a) 8
an above ground pipe segment with R/t = 19.7, (b) a buried pipe segment with R/t = 19.7, (c) an above 9
ground pipe segment with R/t = 25.6, (d) a buried pipe segment with R/t = 25.6 (py: pressure 10
corresponding to yield stress of the pipe, dashed lines: perfect segments, solid lines: segments with 11
initial geometric imperfection, w/t =0.1). 12
-33-
0 0.5 1 1.5 2 2.5 3
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 406.4 mm (R/t= 19.7)
p=0
p=0.72×p
y
0 0.5 1 1.5 2 2.5
3
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 406.4 mm (R/t= 19.7)
p=0
p=0.72×p
y
w/t=0
w/t=0
.1
w/t=0.2
w/t=0
w/t=0.1
w/t=0.2
(a) (b)
1
Figure 3. Average axial load-deformations paths of (a) above ground and (b) embedded pipe segments 2
of a steel pipeline with diameter D = 406.4 mm and radius over thickness ratios R/t = 19.7, computed 3
for various amplitudes of initial geometric imperfections of the pipe walls. 4
5
2
4
6
8
w/t=0, G
s
=23.4 Mpa
w/t=0.1,G
s
=23.4 Mp
a
w/t=0,G
s
=46,8 Mpa
w/t=0.1,G
s
=46,8 Mp
a
0 0.5 1 1.5 2 2.5 3 3.5
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 1066.8 mm (R/t = 27.9), p=0.72xp
ma
x
0 0.5 1 1.5 2 2.5 3 3
.5
x
/l (%)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
P/P
o
D = 1066.8 mm (R/t = 27.9), p=0
6
Figure 4. Effect of the stiffness of the trench soil on the average axial load-deformation paths computed 7
for non-pressurized or pressurized segments of a pipeline with diameter D = 1066.8 mm and radius over 8
thickness ratio R/t = 27.9) by either considering or neglecting the initial geometric imperfections. 9
10
1 2 3
wave front
(a)
1 2 3 : Pipeline
1 2 3
(b)
heterogeneity
(d)
1 2 3 1 2 3
(c)
fault
epicenter
A B
11