Analytical Fragility Surfaces and Global Sensitivity Analysis of Buried Operating Steel Pipeline Under Seismic Loading

Abstract

The structural integrity of buried pipelines is threatened by the effects of Permanent Ground Deformation (PGD), resulting from seismic-induced landslides and lateral spreading due to liquefaction, requiring accurate analysis of the system performance. Analytical fragility functions allow us to estimate the likelihood of seismic damage along the pipeline, supporting design engineers and network operators in prioritizing resource allocation for mitigative or remedial measures in spatially distributed lifeline systems. To efficiently and accurately evaluate the seismic fragility of a buried operating steel pipeline under longitudinal PGD, this study develops a new analytical model, accounting for the asymmetric pipeline behavior in tension and compression under varying operational loads. This validated model is further implemented within a fragility function calculation framework based on the Monte Carlo Simulation (MCS), allowing us to efficiently assess the probability of the pipeline exceeding the performance limit states, conditioned to the PGD demand. The evaluated fragility surfaces showed that the probability of the pipeline exceeding the performance criteria increases for larger soil displacements and lengths, as well as cover depths, because of the greater mobilized soil reaction counteracting the pipeline deformation. The performed Global Sensitivity Analysis (GSA) highlighted the influence of the PGD and soil–pipeline interaction parameters, as well as the effect of the service loads on structural performance, requiring proper consideration in pipeline system modeling and design. Overall, the proposed analytical fragility function calculation framework provides a useful methodology for effectively assessing the performance of operating pipelines under longitudinal PGD, quantifying the effect of the uncertain parameters impacting system response.

1. Introduction

Buried continuous pipelines are vulnerable to the effects of Permanent Ground Deformation (PGD) resulting from seismic-induced landslides, faulting, and lateral spreading due to liquefaction. The pipeline response depends on its orientation with respect to the direction of the ground movement, which is generally a combination of transverse and longitudinal ground movement, occurring perpendicular and parallel to the pipeline axis, respectively. Transverse PGD induces predominant shear and bending stresses at the margins of the PGD zone, similar to the fault-crossing problem investigated by many researchers during the last 50 years [1,2,3,4,5]. Conversely, the longitudinal PGD results in axial compression and tension in the pipeline that may ultimately lead to local buckling and tensile rupture, respectively. According to previous investigations [6,7,8,9], buried continuous pipelines are significantly more vulnerable to PGD in the longitudinal direction, requiring a detailed analysis of the system performance under this hazard.

The response of buried continuous pipelines subjected to longitudinal PGD depends on the pipe deformation capacity under operating conditions, as well as the soil–structure interaction, amount of ground movement δ and its spatial extent L_b, and pattern of ground deformation (Figure 1).

The rigid block pattern, defined by a downslope movement δ within the soil block length L_b, induces localized relative soil–pipeline displacement at the margins, resulting in the largest pipe strains when compared to other distributive geometries considered by O’Rourke and Nordberg [7]. Consequently, this pattern has been extensively employed within simple analytical or more complex numerical methods to assess pipeline performance under longitudinal PGD [7,10,11,12,13,14,15,16,17,18,19].

These analytical structural models are more computationally efficient, compared to a complex numerical analysis approach, which requires further expertise of the engineer to analyze the models for use in routine engineering applications [20].

The existing analytical models for assessing the deformation of buried continuous pipelines subjected to longitudinal PGD consider the pipe material either as linear elastic [7,10] or inelastic, following a Ramberg–Osgood stress–strain relationship that is equal in tension and compression [8,11,19]. Herein, two conditions were established for which the pipeline deformation demand was assessed, depending on whether the soil block length L_b is short (case I) or long enough (case II) to fully mobilize the soil reaction along the pipeline under the imposed ground displacement (Figure 2). The pipeline design force was computed as the minimum between the ultimate soil reaction transferable to the pipe over a length of L_b/2 (case I) and the force computed assuming that the pipeline is fully compliant with the soil (case II), as recommended by the 2005 ALA guideline [12]. This conventional model predicts that half the total applied soil load is resisted in tension and half in compression, because of the assumed symmetric configuration of the soil–pipeline system, including the material constitutive relationships in tension and compression. Consequently, for a given PGD demand (d, L_b), the estimated peak pipeline strain magnitude in tension is equal to that in compression, as shown in Figure 2. This is not representative of pipelines made of materials with different stress–strain relationships for tension and compression under uniaxial or biaxial stress state conditions, resulting in asymmetric pipeline loading and strain demand in the tensile and compressive ground deformation zone.

Therefore, to accurately assess pipeline performance under longitudinal PGD, it is necessary to adopt analytical methods considering the asymmetric nonlinear material behavior of the operating pipeline for tension and compression as well as the elastoplastic axial soil–pipeline interaction [5,21].

Analytical models can be efficiently implemented within a seismic fragility analysis framework to evaluate the structural performance considering the uncertainties of the system parameters. Pipeline fragility assessments require proper consideration of the variability of the physical characteristics of the soil–pipe system, including the material strength properties, the soil cover depth, and the pipe operational loads such as temperature differences and internal pressure [22,23,24,25].

These uncertainties are considered within a probabilistic framework like the Monte Carlo Simulation (MCS), employing a large number of soil–pipeline system samples, generated by random sampling of the input system parameters, based on their probability distribution.

The seismic demands are compared to the structural capacity values associated with a certain performance limit state, allowing to estimate the associated probability of exceedance as a function of the level of seismic intensity measure, IM [26,27]. The latter is defined by a group of representative seismic ground motion parameters, e.g., the peak ground displacement δ, and the length of the PGD zone L_b. Conversely, the structural limit states are defined on the basis of an adequate Engineering Demand Parameter (EDP), e.g., the peak longitudinal strain, describing the system performance [28,29].

The analytical fragility relationships represent a useful design tool, allowing us to identify precise locations of pipeline damage, which is fundamental in evaluating pipeline seismic performance and prioritizing resource allocation for mitigative or remedial measures in spatially distributed lifeline systems.

To efficiently evaluate the seismic fragility of buried operating steel pipelines under longitudinal PGD, this study developed a new analytical model considering the asymmetric nonlinear response of the operating pipeline for tension and compression. The proposed model was validated through detailed finite element analysis, demonstrating its capacity to accurately evaluate pipeline system response under varying operational and seismic loads. The analytical formulation was further implemented within a fragility function calculation framework based on a Monte Carlo Simulation (MCS), assessing the probability of the pipeline exceeding the performance limit states, conditioned to the ground displacement hazard. Furthermore, to quantify the influence of the uncertain input parameters on the predicted pipeline performance, this study conducted Global Sensitivity Analysis (GSA), based on the Sobol’s variance decomposition method, using MCS to calculate the Sobol’s sensitivity indices.

First, this paper presents the methodology adopted to evaluate the seismic fragility of buried pipelines subjected to longitudinal PGD, using an accurate analytical model considering the asymmetric pipeline material behavior for tension and compression under varying service loads.

Then, the obtained fragility analysis results are carefully discussed and compared to data reported in other research publications, highlighting the main factors influencing the seismic performance of buried operating pipelines, considering the system uncertainties. Finally, the conclusion section highlights the contributions of the present paper to the state-of-the-art practice and research of pipeline seismic design, suggesting further important perspectives related to the addressed issues.

2. Methodology

This section presents the adopted methodology to assess the seismic fragility of buried operating pipelines subjected to longitudinal PGD. First, the developed analytical model is described, allowing us to evaluate the performance of buried operating pipelines under longitudinal PGD, as a function of the system nonlinearities and operational loads. Second, it defines the pipeline performance limit states evaluated through structural analysis. Then, the analytical model results are compared to the finite element analysis results, demonstrating the validity of the proposed analytical procedure. Finally, this section presents a probabilistic framework, based on Monte Carlo Simulation (MCS), to evaluate the fragility functions and quantify the effect of system uncertainties.

2.1. Analytical Model for Assessment of the Continuous Pipeline Response Under Longitudinal PGD

This section describes the developed analytical model for assessing the performance of buried continuous pipelines under longitudinal PGD, considering the nonlinear material constitutive relationships, force equilibrium, and displacement compatibility along the pipe–soil system. This algorithm allows implementation of the asymmetric nonlinear stress–strain relationship for the pipe material for tension and compression, as well as the elastoplastic axial soil–pipeline interaction, accurately assessing the response of the system subjected to this geohazard.

The ground deformation is idealized as a rigid-block movement, defined by a downslope movement δ over a block length L_b, resulting in a tension crack of width δ at the upslope end and a compression ridge over a distance δ on the downslope end (Figure 3a). The soil displacement assumes a constant value δ within the PGD zone, while being zero on either side of it.

The moving soil block tends to pull the pipe along with it, resulting in localized relative soil–pipeline displacement at the margins of the PGD zone, and associated resistance forces on either side of the sliding block head and toe. This results in a maximum pipeline axial force at the head and maximum axial compression force at the toe, decreasing linearly thereupon due to the sliding soil friction (f_s). Beyond this zone, the relative soil–pipeline displacement is negligible, and the pipeline displacement matches that of the ground (case II). As the soil displacement increases reaching a critical value δ = δ_cr, the relative soil–pipeline displacement and associated friction reaction are mobilized over the entire soil block length, and the pipeline deformation remains constant thereafter (case I).

Evidently, the region of the soil–pipeline system beyond the soil block behaves like a pull-out test under tension (region I) and compression (region IV), with the end displacement (ΔL) applied at the pipe points underlying the tension crack and compression bulge, respectively (Figure 3a). The total pipeline elongation at each side of the soil block head i.e., in region I (U_p1) and region II (U_p2), is equal to the magnitude of the overall pipeline contraction at each side of the toe of the sliding block, i.e., in region III (U_p3) and region IV (U_p4).

The strain demand was calculated using the analytical model developed by [30], evaluating the structural response of a continuous buried pipeline, subjected to a pullout force on one end (x = 0). The resulting pipeline displacement was obtained by integrating the axial strains associated with the axial force distribution along the pipeline, considering the beam on elastic foundation theory for the static friction length, and the force equilibrium for the frictional sliding length (Figure 3b). This model employs the theory of plasticity for modeling the pipe material, based on the associated flow rule with the von Mises yield criterion and isotropic strain hardening. Compared to the conventional model [8], the proposed analytical solution accounts for the initial axial thermal strains and biaxial stress state in the pipe due to internal pressure, as shown in Figure 4.

Herein, the pipe material is assumed to have a piecewise stress–strain curve, defined by the engineering strain–stress values (ε_i, σ_i), either in tension (ε > 0) or in compression (ε < 0), where E_i = (σ_i − σ_i−1)/(ε_i − ε_i−1) is the slope of the i-th segment constituting the pipe multi-linear stress–strain relationship (Figure 4). Assuming the plastic behavior of the pipe steel material within the Von Mises plasticity with isotropic hardening (Figure 4), the nominal stress–strain curve (ε_i, σ_i) is derived from the true strain–true stress response (ε_t,i, σ_t,i), as a function of the tensile coupon test data in terms of true strain and true stress (ε°_i, σ°_i), as well as the operating loads, as indicated in

Table 1. Closed-form analytical solution for the axial true stress–true strain response of the steel pipeline subjected to internal pressure P_i and temperature variations ΔT [30].

i	σt,i	εt,i
0	${\sigma }_0=\nu {\sigma }_{\theta }-E\alpha \Delta T$	ε0 = 0
1	${\sigma }_{t,1}={\sigma }_{\theta }/2\pm \sqrt{{\left({\sigma }_1^o\right)}^2-3{\sigma }_{\theta }^2/4}$	${\epsilon }_{t,1}={\epsilon }_0+{\displaystyle \frac{{\sigma }_{t,1}-{\sigma }_0}{E}}$
>1	${\sigma }_{t,i}={\sigma }_{\theta }/2\pm \sqrt{{\left({\sigma }_i^o\right)}^2-3{\sigma }_{\theta }^2/4}$	${\epsilon }_{t,i}={\epsilon }_{t,i-1}+{\displaystyle \frac{\left({\sigma }_{t,i}-{\sigma }_{t,i-1}\right)}{E_i^o}}-{\displaystyle \frac{\sqrt{3}{\sigma }_{\theta }}{2H_i^o}}\arctan \left({\displaystyle \frac{2\sqrt{3}{\sigma }_{\theta }\left({\sigma }_{t,i}-{\sigma }_{t,i-1}\right)}{3{\sigma }_{\theta }^2+\left(2{\sigma }_{t,i}-{\sigma }_{\theta }\right)\left(2{\sigma }_{t,i-1}-{\sigma }_{\theta }\right)}}\right)$

. Herein, σ_θ = P_iD/2t denotes the hoop stress in the pressurized pipeline, E the Young’s modulus of the pipe material, ν the Poisson’s ratio, E°_i = (σ°_i − σ°_i−1)/(ε°_i − ε°_i−1) the tangent modulus of the i-th segment constituting the pipe multi-linear true stress–true strain relationship, and H°_i = E⋅E°_i/(E − E°_i) the plastic modulus of the true stress–true plastic strain curve (flow curve).

The pipe–soil interaction is modeled using an elastic-perfectly plastic force–displacement relationship, defined by the maximum soil friction force per unit length of the pipeline f_s, and the relative soil–pipeline displacement at onset of friction sliding u₀, where k = f_s/u₀ is the rigidity of friction interaction.

The analytical formulation for evaluating the buried pipeline displacement ΔL subjected to a pull-out force at the head (F = F_t,max) and toe (F = F_c,max) of the sliding soil block is indicated in

Table 2. The analytical formulation for evaluating the pipeline response (ΔL, F) at the head and toe of the soil block, behaving like a pull-out test for tension (+) and compression (-), respectively [30].

i	F-Interval	Pipe Displacement ΔL	Pipe Axial Force F
1	$\left|F-A{\sigma }_0\right|\le \sqrt{AE{f}_s{u}_0}$	$\pm \sqrt{A{E}_1/k}\cdot \epsilon$	$A{\sigma }_0+A{E}_1\left(\epsilon -{\epsilon }_0\right)$
	$A{\sigma }_1^{-}\le F\le A{\sigma }_1^{+}$	$\pm {\displaystyle \frac{u_0}{2}}\pm {\displaystyle \frac{A{E}_1}{2f_S}}{\epsilon }^2$
>1	$F<A{\sigma }_1^{-}\vee F>A{\sigma }_1^{+}$	$\Delta L_{i-1}\pm {\displaystyle \frac{A{E}_i}{2f_S}}\left({\epsilon }^2-{\epsilon }_{i-1}^2\right)$	$A{\sigma }_{i-1}+A{E}_i\left(\epsilon -{\epsilon }_{i-1}\right)$

. Herein, the unanchored length (L_a) of the pipeline is assumed to be sufficiently long so that the pipeline response is unaffected by far-end boundary conditions.

The magnitude of the total pipeline displacement under tension (U_p,max = U_p1 + U_p2 ≅ 2ΔL) and compression (U_p,max = U_p3 +U_p4 ≅ 2ΔL) are equal to the ground displacement δ until the soil friction mobilizes over the entire soil block (case II), remaining constant thereafter, U_p,max = δ_cr − u₀ (case I). This allows us to directly assess the amount of ground displacement δ inducing a maximum pipeline strain ε either in tension or compression, at the soil block head and toe, respectively, using the following equation:

δ(ε) = 2ΔL(ε),   for δ < δ_cr

(1)

The evaluation of the critical soil displacement δ_cr as a function of the soil block length L_b is described in detail in Section 2.4. Most importantly, Equation (1) allows us to determine the amount of critical ground displacement (δ_cr,i) corresponding to the achievement of the level of pipe axial strain associated with a certain performance limit state, either for tension or compression.

2.2. Deformation Capacity and Performance Limit States for the Steel Pipeline

In this study, the pipeline response was assessed in terms of the critical ground displacement corresponding to the achievement of maximum allowable longitudinal compression and tension strains, associated with normal operability and pressure integrity performance goal.

2.2.1. Normal Operability Limit (NOL) State

It is expected that the pipeline will maintain its functionality after a seismic event, and the induced longitudinal strains are limited to avoid excessive distortion of the pipe cross-section impairing the passage of internal pigs for cleaning and inspections for material leakage. According to the 2001 ALA Guideline [31], the longitudinal compression strain limit, associated with the onset of local buckling, is given by the following:

$${\epsilon }_{c1}=0.5{\displaystyle \frac{t}{D\prime }}+3000{\left({\displaystyle \frac{pD}{2Et}}\right)}^2-0.0025$$

(2)

with,

$$D\prime ={\displaystyle \frac{0.5D}{1-\frac{3}{D}\left(D-D_{\min }\right)}},$$

(3)

where p is the difference between the pipe internal and external pressure, E is the elastic modulus of the pipe material, t is the pipeline thickness, D is the pipeline diameter, and D_min is minimum pipe diameter because of ovalization.

To maintain normal operability, the 2001 ALA Guideline recommends limiting the magnitude of longitudinal tensile strain to ε_t1 = 2%.

2.2.2. The Pressure Integrity Limit (PIL) State

It accepts significant pipe ovalization and wrinkle formation without loss of containment, which may subsequently result in pipe wall folding and associated excessive tensile strains, leading to crack initiation and ultimate rupture. To guarantee the pressure integrity performance requirement, the longitudinal compressive strain limit was evaluated as ε_c2 = 1.76 t/D. The allowable longitudinal tensile strain limit was assumed to be ε_t2 = 4% [31,32].

Evidently, the pipeline performance is controlled by its compressive behavior, being the magnitude of the strain limit for normal operability (NOL) for compression lower than that for tension (ε_c1 < ε_t1), for typical ranges of the diameter-to-thickness ratio (D/t) in onshore applications.

2.3. Validation of the Analytical Model

The proposed analytical model was validated against numerical simulation evaluating the response of buried operating pipeline under longitudinal PGD, using the finite element software ABAQUS/Standard [33]. First, the system performance was analyzed numerically within the beam on Winkler foundation theory. Then, the numerical results were compared to those obtained using the state-of-the-art analytical procedure [8,11,13] and the proposed analytical model, demonstrating the capacity of the latter to accurately evaluate the pipeline response.

The calculation example considers a X42 steel grade pipeline with diameter of 0.508 m and thickness of 7.1 mm, buried in dense sand with a cover depth H_c = 1.5 m, measured from the soil surface to the pipe crown. The analyzed soil–pipeline parameters are summarized in

Table 3. Pipe–soil system parameters.

Parameter	Value	Units
Pipe diameter, D	0.508	m
Pipe wall thickness, t	7.1	mm
Pipe elastic modulus, E	210	GPa
Poisson’s ratio, ν	0.3
Pipe yield stress, σy	290	MPa
Pipe cover depth, Hc	1.5	m
Soil density, γ	18.0	kN/m3
Soil friction angle, ϕ	40	°
Pipe–soil interface friction angle, δi	28	°
Soil pressure at rest, K0	1
Soil friction reaction per unit pipe length, fs	26.8	kN/m
Relative soil–pipe displacement at friction sliding, u0	3	mm

.

The pipeline performance was evaluated considering the presence and absence of service loads, including internal pressure P_i, and temperature variations ΔT, demonstrating the accuracy of the proposed analytical method to assess the system response under operating conditions.

It is assumed that the unpressurized pipeline (P_i = 0 MPa) has no temperature variation with respect to its installation temperature (ΔT = 0 °C). Conversely, the pressurized pipeline is assumed to operate at an average temperature of ΔT = 50 °C, compared to the pipe installation temperature, and internal pressure P_i = 4.4 MPa, which is 75% of the maximum allowed pressure, P_max, according to [34]:

$$P_{\mathit{max}}=0.72\cdot \left(2{\sigma }_y{\displaystyle \frac{t}{D}}\right)$$

(4)

2.3.1. Finite Element Analysis of the Soil–Pipeline System

Within the numerical approach, the pipeline was modeled using the PIPE22H beam element type implemented in Abaqus/Standard [33], which is particularly suitable to model long, slender pipelines with a thin-walled circular cross-section, allowing the possibility of specifying external or internal pressures. Instead, the longitudinal soil–pipeline interaction was modeled with uniaxial spring elements SPRING2 connected at each node of the pipeline on one end, while being assigned the far-field ground motion at the other end through the boundary conditions. The adopted mesh size for the beam pipe elements is 0.10 m, based on the mesh sensitivity study performed herein, assuring efficiency and accuracy of the numerical solution.

The X42 steel grade pipe material model is defined within the von Mises plasticity theory with isotropic hardening, with Young modulus E = 210 GPa, and yield stress σ_y = 290 MPa. The elasto-plastic force displacement relationship of the soil springs is defined by the sliding soil friction force per unit length of the pipeline, f_s = 26.8 kN/m, and the relative soil–pipe displacement at the onset of friction sliding, u₀ = 3 mm, calculated according to the ALA guidelines [31], assuming compacted dense sand with friction angle ϕ = 40° (Table 3).

The length of the pipeline–soil system is equal to 1000 m, so the system response to the imposed ground displacement is representative of an infinitely long pipeline, unaffected by the far-end boundary conditions. The assumed soil block length is L_b = 300 m (case II), located at the center of the soil–pipeline system so that its midpoint lies on the pipeline bisector.

The numerical analysis was conducted in two consecutive steps. First, the internal pressure and temperature variations were applied in the pipeline, while the pipeline ends and the free ends of the soil springs were restrained in the longitudinal direction. Second, the soil block movement δ = 4 m was applied statically with a maximum step increment equal to Δδ = 1 mm at the free nodes of the soil springs within the soil block length, while outside of the moving block the soil nodes remain fixed. At each step increment, the nonlinear equilibrium equations were solved iteratively by the Newton–Raphson method, allowing us to assess the system performance at any level of applied ground displacement, until material failure.

2.3.2. Comparison Between Numerical and Analytical Solutions

Figure 5 shows a comparison between the numerical and analytical model results assessing the performance of the pressurized and unpressurized pipelines in terms of the maximum axial strains for tension and compression, as a function of the ground displacement δ. Overall, the peak pipe strain magnitudes in the tensile and compressive PGD zone increase monotonically with the ground movement, progressively reaching the NOL and PIL performance limit states at critical values of ground displacement δ_cr,i, based on the operating conditions (

Table 4. The critical ground displacement δ_cr,i corresponding to the achievement of the normal operability (NOL) and pressure integrity (PIL) limit states, for tension and compression, according to the numerical and analytical methods.

	Analysis Method	Normal Operability Limit		Pressure Integrity Limit
		Tension	Compression	Tension	Compression
		δt1 (m)	δc1 (m)	δt2 (m)	δc2 (m)
Pressurized pipeline	numerical	0.5706	0.0996	0.7835	0.4192
	analytical	0.5710	0.1001	0.7854	0.4226
	% difference	0.06%	0.51%	0.24%	0.81%
	conventional	0.3440	0.2327	0.5179	0.3988
	% difference	65.90%	−57.18%	51.30%	5.09%
Unpressurized pipeline	numerical	0.3656	0.2200	0.5420	0.5738
	analytical	0.3655	0.2199	0.5415	0.5773
	% difference	−0.02%	−0.06%	−0.08%	0.61%
	conventional	0.3511	0.2069	0.5249	0.3917
	% difference	4.12%	6.35%	3.24%	46.48%

).

The results obtained using the conventional model reported in [8,11,13] agree well with the numerical analysis only for the case of the unpressurized pipeline under tension, while diverging significantly for the pressurized pipeline, with a percent difference exceeding 5% (Table 4). Conversely, the comparison between the proposed analytical model and the numerical simulation results shows excellent agreement for both cases of pipeline operating conditions. Specifically, the percent difference between the pipeline performance results for these two methods does not exceed 0.9% (Table 4), demonstrating the greater accuracy of this analytical procedure to assess system response, compared to the conventional method.

Finally, the proposed analytical model, evaluating the response of the buried operating pipeline subjected to longitudinal PGD, can be implemented in most programming languages (e.g., Python, Matlab) for further parametric analyses.

2.4. Critical Soil Displacement (δ_cr) and Length (L_cr) for the Performance Limit States of the Operating Pipeline

As the soil displacement δ increases (case II), the maximum magnitude of the pipeline axial force and associated strain at both margins of the PGD zone increase monotonically (Equation (1)), until the soil reaction mobilizes fully over the entire soil block length L_b (case I). Herein, the total soil load (f_sL_b) over the PGD zone is resisted by the developed pipe axial force that is at a maximum at the head (F_t,max) and toe (F_c,max) of the sliding soil block. Therefore, the critical soil length L_cr, associated with full mobilization of the soil reaction (case I) for a given value of soil displacement δ, is directly proportional to the overall pipe load at the edges of the PGD zone:

L_cr = [F_t,max(δ/2) − F_c,max(δ/2)]/f_s

(5)

Consequently, the pipeline deformation demand will reach a certain limit state only if the soil displacement δ and block length L_b are equal or greater than the critical soil displacement δ_cr,i and length L_cr,i corresponding to that performance criterion, respectively.

Figure 6 shows the variation of the critical soil block length L_cr as a function of the ground displacement δ, indicating the critical values associated with the achievement of the performance limit states in the pressurized (P_i/P_max = 0.75, ΔT = 50 °C) and unpressurized pipelines (P_i/P_max = 0, ΔT = 0 °C). Furthermore, the distributions of the axial strain, stress, force, displacement, and soil friction reaction along the pressurized and unpressurized pipelines for increasing values of applied ground movement δ are presented in Appendix A, illustrating the different system responses for case I and case II.

Clearly, the pipeline response depends on the operating loads, with the performance limit states in compression being reached for lower levels of critical soil displacement and length in the pressurized pipeline, compared to the unpressurized one.

Interestingly, the pressurized and unpressurized pipelines satisfy all the performance criteria for soil block lengths L_b that are shorter than 145 m and 254 m, respectively, for any value of the applied ground displacement, δ. Conversely, the pipeline will fail to satisfy any limit state for values of the soil block length L_b and ground displacement δ exceeding 280 m and 0.79 m, respectively (Figure 6).

Figure 7 and Figure 8 show the maximum pipe strain magnitude as a function of the ground deformation demand (δ, L_b), for the pressurized and unpressurized pipelines, respectively, highlighting the critical PGD values (δ_cr,i, L_cr,i) corresponding to the achievement of the performance limit states.

Clearly, the curve of the critical soil block length L_cr as a function of the ground displacement δ (Figure 6) separates the PGD demand characterizing case I of short soil block (δ > δ_cr ∨ L_b < L_cr) from that of case II of long soil block (δ < δ_cr ∨ L_b > L_cr). Hence, the strain isolines are defined by two half-lines running parallel to the δ and L_b axis, intersecting on the critical curve, as shown in Figure 7 and Figure 8.

2.5. Uncertainty Analysis

The deterministic analysis procedure adopted in the previous section assumes precise knowledge of the system parameters. However, all problem variables are characterized by a certain degree of uncertainty, including the material strength and the seismic loading. To quantify the effect of these uncertainties on the pipeline performance, this study develops a robust fragility analysis framework based on the Monte Carlo Simulation (MCS). The latter considers a large number of soil–pipeline system samples, generated by random sampling of the input system parameters, based on their probability distribution. The seismic demands are compared to the structural capacity values associated with each performance limit state, allowing us to estimate the associated probability of exceedance conditioned to the level of ground motion intensity measure, IM, represented by the vector (δ, L_b).

Table 5. Probabilistic characteristics of the input parameters.

	Parameter	Units	Distribution	Mean or Range	COV
Pipe	Internal pressure, Pi	KPa	Uniform	[0, 5846.4]
	Temperature variation, ΔT	°C	Uniform	[0, 50]
	Yield strength, σy	MPa	Normal	290	0.035
Soil	Cover depth, Hc	m	-	[1, 1.5, 2]
	Soil friction reaction per unit pipe length, fs	KN/m	Normal	[19.2, 26.8, 34.4]	0.3
	Relative soil–pipe displacement at friction sliding, u0	mm	Normal	3	0.3

summarizes the input variables and their probability distribution. To investigate the effect of the soil cover depth H_c, three different values between 1.0 m and 2.0 m were considered, representative of typical onshore pipeline installations. The soil strength parameters, i.e., the longitudinal soil reaction per unit length of pipeline f_s, and relative soil–pipe displacement at friction sliding, u₀ are assumed to follow a normal distribution, with the mean values derived according to [31], and a coefficient of variation COV = 30% [26,35,36], as indicated in Table 5. The uncertainty of the X42 steel grade pipe material was modeled considering that the yield strength follows a normal distribution, with a mean value σ_y = 290 MPa and a small COV = 3.5% (Table 5), representing the low variability of the steel properties.

The limit state functions for normal operability (NOL) and pressure integrity (PIL) for tension and compression are expressed in terms of the system demand (δ, L_b) and capacity (δ_cr,i, L_cr,i) corresponding to each performance criteria:

g_t1 = max(δ_t1 − δ, L_b,t1 − L_b)

(6)

g_c1 = max(δ_c1 − δ, L_b,c1 − L_b)

(7)

g_t2 = max(δ_t2 − δ, L_b,t2 − L_b)

(8)

g_c2 = max(δ_c2 − δ, L_b,c2 − L_b)

(9)

The probability of exceeding the normal operability (NOL) and the pressure integrity limit (PIL) state, conditioned to the PGD intensity measure level (δ, L_b), is given by the joint union of the associated damage states for tension and compression:

P[NOL|(δ, L_b)] = P[max(g_t1, g_c1) ≤ 0|(δ, L_b)]

(10)

P[PIL|(δ, L_b)] = P[max(g_t2, g_c2) ≤ 0|(δ, L_b)]

(11)

These probabilities can be effectively calculated using the Monte Carlo Simulation (MCS) method, as summarized in

Table 6. Summary of fragility function calculation framework, based on MCS.

Definition of the uncertain input parameters and their probability distributions. Generation of a sample set of the random variables in the system, considering their probability distributions. Evaluation of the pipeline displacement capacities for each strain limit corresponding to each performance criterion δcr,i, using Equation (1) and the functional relationships in Table 2: δcr,i = 2ΔL(εcr,i). Evaluation of the critical soil lengths Lcr,i, using Equation (5) and the analytical formulation in Table 2 for calculating the pipeline force at the head (Ft,max) and toe (Fc,max) of the soil block: Lcr,i = [Ft,max(δcr,i/2) − Fc,max(δcr,i/2)]/fs Evaluation of the limit state functions, as the difference between the calculated system capacity (δcr,i, Lcr,i) and demand (δ, Lb), using Equations (6)–(9). Evaluation of the indicator functions, I1i(δ, Lb) = max(gt1, gc1) ≤ 0, and I2i(δ, Lb) = max(gt2, gc2) ≤ 0, which are equal to the unity under unsatisfactory performance for the normal operability and pressure integrity limit state, respectively, and are zero otherwise. Repetition of steps (1) to (6) N times, to obtain N sample values of I1i(δ, Lb), I2i(δ, Lb), counting the unsatisfactory performance for the normal operability (NOL) and the pressure integrity limit (PIL) state, respectively. Evaluation of the probability of exceedance of the normal operability (NOL) and the pressure integrity limit (PIL) state, for a given PGD demand (δ, Lb), as the ratio between the total sum of I1i(δ, Lb) and I2i(δ, Lb) to the sample size N:
	$P\left[NOL|\left(\delta ,L_b\right)\right]={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{I}_{1i}\left(\delta ,L_b\right)}$	(12)
	$P\left[PIL|\left(\delta ,L_b\right)\right]={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{I}_{2i}\left(\delta ,L_b\right)}$	(13)

.

The described algorithm can be easily implemented within most programming languages, like Python [37], to evaluate the fragility surfaces representing the conditional probability of the system reaching a performance limit state, as a function of the seismic demand.

3. Fragility Surfaces

This section presents the fragility analysis results of the buried operating steel pipelines subjected to longitudinal PGD, obtained using the methodology described in Section 2.5.

The evaluated fragility surfaces for the buried pipeline at a cover depth H_c = 1.5 m are shown in Figure 9a and Figure 9b for the NOL and the PIL performance limit states, respectively.

Clearly, the probability of the pipeline exceeding the performance criteria increases for larger soil displacements δ and lengths L_b, being greater for the NOL, compared to the PIL limit state. The iso-probability lines are defined by two half-lines parallel to the δ and L_b axis (Figure 9). This is consistent with the pipeline deformation response observed for the deterministic analysis (Figure 7 and Figure 8), as schematically illustrated in Figure 10. Specifically, the median values of soil displacement (δ, L_b), corresponding to a 50% probability of reaching the NOL and PIL criteria are (0.13 m, 193.4 m) and (0.48 m, 267.4 m), respectively, as indicated in

Table 7. Comparison of pipeline performance based on deterministic and fragility analysis results.

		Deterministic Analysis			Fragility Analysis (Median Value)
	Hc (m)	1	1.5	2	1	1.5	2
NOL	δcr,1 (m)	0.31	0.22	0.17	0.18	0.13	0.10
	Lcr,1 (m)	356.5	254.9	198.3	270.2	193.4	151.7
PIL	δcr,2 (m)	0.76	0.54	0.42	0.67	0.48	0.38
	Lcr,2 (m)	388.9	278.0	216.3	372.5	267.4	208.4

. These critical values are less than the ones evaluated deterministically for the unpressurized pipeline (Figure 6), particularly for the NOL performance limit state (Table 7). This may result in under-designed pipeline systems when using the deterministic approach, highlighting the importance of accurate uncertainty analysis for a reliable infrastructure design.

To investigate the effect of the cover depth H_c on pipeline performance, the fragility functions were evaluated considering a minimum cover depth value of H_c = 1 m and a maximum of H_c = 2 m (Figure 11). This results in a proportional variation in the soil friction reaction per unit pipe length f_s [31] that directly controls the pipeline deformation demand (Table 2).

Clearly, the probability of the pipeline exceeding the performance limit states for a given amount of PGD (δ, L_b) increases with greater burial depths and associated soil friction reaction. Specifically, the median values of ground displacement intensity (δ, L_b), corresponding to a 50% probability of reaching the NOL criteria are (0.18 m, 270.2 m) and (0.10 m, 151.7 m) for the shallower (H_c = 1 m) and deeper (H_c = 2 m) soil cover depths, respectively (Table 7). Likewise, a critical PGD demand (δ, L_b) of (0.67 m, 372.5 m) and (0.38 m, 208.4 m) is needed to achieve a 50% probability of reaching the PIL criteria for the former (H_c = 1 m) and latter (H_c = 2 m) pipe burial condition, respectively.

This is consistent with current pipeline design guidelines prescriptions, recommending the use of shallow burial depths, light-weight backfill, and pipe coating with a low friction coefficient to minimize the intensity of soil–pipeline interaction, optimizing the system performance [14,32].

The obtained fragility surfaces permit us to assess the probability of the pipeline exceeding the performance criteria, conditioned to the PGD demand (d, L_b), considering the effect of system uncertainties, including the varying operational loads.

4. Global Sensitivity Analysis

This section presents the results of the global sensitivity analysis (GSA) conducted to quantify the uncertainty (variance) of the model prediction y = g(x₁, x₂, …, x_m), attributed to each input random variable x_i. This method has the advantage of considering the sensitivity over the entire input space, including the nonlinear interaction effects between the system variables, allowing us to identify those parameters that have the greatest influence on the model output.

The adopted GSA procedure is based on the Sobol’s variance decomposition method, in which the total variance of model output V = Var(y) is decomposed into component variances resulting from individual parameters V_i and their interactions V_ijk [38].

Specifically, the first-order Sobol index S_i of an input variable x_i represents the fraction of the output variance attributed to x_i (S_i = V_i/V), while the higher order Sobol indices S_ijk indicate the impact of input parameter interactions on the output results (S_ijk = V_ijk/V). Finally, the total-order index S_Ti quantifies the overall effects of one input parameter x_i including its interactions with all other variables on the model output, and is defined as the sum of all sensitivity indices involving x_i (S_Ti = S_i + Σ_i≠jS_ij + Σ_i≠j≠kS_ijk + …).

To accurately estimate the first-order and total Sobol indices this study performs a double-loop Monte Carlo integration procedure [39].

The estimated influence of the input random variables on the achievement of the pipeline performance limit states, based on Sobol’s sensitivity indexes, is shown in Figure 12.

The soil block length L_b is the most influential parameter, with a first-order sensitivity index S_i of 51.2% and 38.1% for the NOL and PIL limit states, respectively, and a total sensitivity index S_Ti greater than 72% for both performance limit states. This is evident, since the length of the PGD zone L_b controls the pipeline deformation demand, where each limit state is achieved provided that the soil block length value exceeds the corresponding critical threshold (L_b ≥ L_cr,i).

The second most significant variable is the soil friction reaction per unit pipe length f_s, with a first-order sensitivity index S_i of 7.8% and 8.7% for the NOL and PIL limit states, respectively, and a total sensitivity index S_Ti greater than 32% for both damage levels. The third most influential parameter for the PIL limit state is the soil displacement δ, with an associated first-order sensitivity index S_i of 12.8% and a total sensitivity index S_Ti of 33.2%.

Interestingly, the pipeline operating temperature is the third most influential parameter for the NOL limit state, with a first-order sensitivity index S_i and a total sensitivity index S_Ti of 1.6% and 15.2%, respectively. This is consistent with the sensitivity analysis results reported in [25], quantifying the effect of uncertainties of the pipeline system subjected to longitudinal PGD, including the significant impact of operational loads.

While the first-order sensitivity indices for the internal pressure P_i, are negligible (S_i < 1%), the total sensitivity index S_Ti reaches 10.7% and 5.5% for the NOL and PIL limit states, respectively, highlighting the considerable interaction effects of P_i with the other input variables.

The least influential input random variables are the pipe yield strength σ_y and relative soil–pipe displacement at friction sliding u₀, with an associated total sensitivity index S_Ti less than 4.5% and 0.3%, respectively. This can be attributed to the low variability assumed for the steel yield strength, and the negligible effect of the elastic stiffness in the soil–pipeline interaction for a large PGD, respectively. This is consistent with the sensitivity analysis results reported in [25] and the simplified assumptions adopted in existing analytical models of buried pipelines under longitudinal PGD, which conservatively neglect the relative soil–pipeline displacement at friction sliding u₀ [11].

The use of both sensitivity indices allows a comprehensive understanding of the influence of the input variables on the performance limit state function, highlighting the importance of interaction effects between the soil–pipeline system parameters.

5. Conclusions

This study develops a new analytical model that accurately and efficiently evaluates the performance of buried operating pipelines under longitudinal PGD, considering the asymmetric pipeline response- to tension and compression under varying operational loads.

A further comparison of the proposed analytical model to the detailed finite element analysis results showed excellent agreement, demonstrating its capacity to accurately assess the pipeline response as a function of the system parameters, including the operational pressure and temperature variations. The analytical model was efficiently implemented within a robust fragility function calculation framework based on MCS, allowing us to assess the probability of exceedance of the pipeline performance limit states conditioned to the PGD demand (δ, L_b), considering the system uncertainties.

The evaluated fragility surfaces showed that the probability of the pipeline reaching the performance criteria increases for larger soil displacement δ and lengths L_b, as well as cover depths H_c, because of the greater mobilized soil reaction counteracting pipeline deformation. This requires implementation of proper engineering design solutions that minimize the risk of pipeline damage, for example, by adopting shallow soil cover depths, light weight backfill, and low-friction pipe coating.

The performed GSA allowed us to quantify the uncertainty of the pipeline performance assessment attributed to each random system parameter, considering the sensitivity over the entire input space, including the nonlinear interaction effects between variables. The PGD length (L_b) was the most influential parameter with respect to the exceedance of the NOL and PIL performance limit states, followed by the soil friction per unit pipe length (f_s) and ground displacement (δ), based on the first-order and total-order Sobol indices. This is consistent with the expected system performance, since the intensity of the PGD and soil–pipeline interaction directly control the pipeline deformation demand.

The significant total sensitivity indices of the temperature variation (ΔT) and internal pressure (P_i) for the NOL limit state demonstrated the importance of the effects of their interaction with the other input variables. Both the deterministic and uncertainty analysis results highlighted the impact of operational loads on pipeline performance, which need to be accurately considered in pipeline system modeling and design. The comparison between the deterministic and fragility analysis results showed that neglecting the variability of the system parameters, including operational loads, may result in under-designed pipelines, highlighting the importance of uncertainty analysis for a reliable infrastructure design.

Overall, the proposed analytical fragility function calculation framework provides a useful methodology for effectively assessing the performance of operating pipelines under longitudinal PGD, quantifying the effects of the critical parameters impacting system response.