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Seismic fragility analysis of a buried pipeline structure considering uncertainty of soil parameters
International Journal of Pressure Vessels and Piping 175 (2019) 103932
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International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Seismic fragility analysis of a buried pipeline structure considering uncertainty of soil parameters
T
Sungsik Yoona, Do Hyung Leeb, Hyung-Jo Junga,* a
Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea b Department of Civil, Environmental and Railroad Engineering, Paichai University, 155-40 Baejae-ro, Seo-gu, Daejeon, 35345, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Buried pipeline structure API 5L X65 Beam on nonlinear Winkler foundation model Uncertain soil parameters Nonlinear time history analysis Damage state Seismic fragility curve
In this study, the seismic fragility analysis for a buried gas pipeline of API 5L X65 was performed considering the uncertainty of soil parameters. For this purpose, nonlinear time history analyses were carried out for the pipeline considering soil-pipeline interaction represented by beam on nonlinear Winkler foundation model. A total of 12 input ground motions were employed and four different analytical cases were considered to evaluate the effect of the uncertainty of soil parameters. The four cases proved that uncertainty of soil parameters needs to be taken into account for the assessment of the pipeline response. Using results of the nonlinear time history analyses, seismic fragility analyses were conducted in accordance with three damage states. Analytical predictions indicated that seismic fragility curves with the uncertainty of soil parameters exhibit higher failure probability than those without the uncertainty. It is thus concluded that the present study is promising for the exploration of the seismic fragility analysis considering the uncertainty of soil parameters.
1. Introduction Recently many earthquakes have caused severe damage to lifeline structures such as gas, electricity, and water supply. Especially, underground facilities (e.g., gas and water pipeline) have been reported to exhibit more severe vulnerability against earthquakes. The Loma Prieta earthquake with a magnitude of 6.9 in San Francisco (1989) caused interruption of gas supply to 15,000 demand facilities. Moreover, the destruction of main gas pipelines led to a significant damage to critical buildings due to fire following the earthquake [1]. The Kobe earthquake (1995) of 6.9 magnitude resulted in electric sparks and gas fires at about 500 sites, and the direct and indirect total damage was estimated at approximately 200 billion dollars [2]. In addition, there was an interruption of gas supply to 100,000 people and the damage to main gas companies was estimated as 25 million dollars due to the ChiChi earthquake (1999) with a magnitude of 7.6 [3]. In order to evaluate the failure probability of buried pipelines, numerous researches have been conducted in terms of using various seismic intensity measures. Probability of failure was represented as one of the followings, i.e., Peak Ground Acceleration (PGA) [4], Modified Mercalli intensity (MMI) [5], and Peak Ground Velocity (PGV) [6] considering empirical fragility relation. In addition, Peak Ground Deformation (PGD), Arias Intensity (AI), Spectral Acceleration (SA),
*
Spectral Intensity (SI), maximum ground strain, and PGV 2/PGA were also utilized as damage indicators to represent the seismic fragility function [7–9]. In line of the above, many researches have also been carried out to evaluate the failure probability of buried pipelines in terms of using numerical analysis. Newmark and Rosenblueth [10] showed the deformation of pipeline using free-field ground deformation and curvature. Wang and Yeh [11] proposed a simplified seismic analysis method for fault movements. In their research, the parametric response analysis was performed with regard to various fault movements, crossing angle of earthquake, various burial depths and pipeline diameters. Moreover, Lee et al. [12] performed seismic behavior of buried gas pipeline according to different burial depth, buried type, and boundary conditions. More recently, Xie et al. [13] performed numerical analysis on the buried high-density polyethylene pipeline subjected to normal faulting. For considering soil-structure interaction, both 3-D and 2-D models were utilized and the results of numerical simulation were compared with centrifuge test. Uncertainty of soil variables can affect the response of buried structures due to geotechnical diversity such as inherent variability, and insufficient data. Since the geotechnical diversity is generally originated from many disparate sources, the parameters characterizing a soil can be estimated through a probabilistic approach. Vanmarcke [14] proposed a method for modeling the natural variability of soil
Corresponding author. E-mail address: [email protected] (H.-J. Jung).
https://doi.org/10.1016/j.ijpvp.2019.103932 Received 16 April 2019; Received in revised form 24 June 2019; Accepted 12 July 2019 Available online 13 July 2019 0308-0161/ © 2019 Published by Elsevier Ltd.
International Journal of Pressure Vessels and Piping 175 (2019) 103932
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corresponding force-displacement relationship of the soil spring in each direction. In Fig. 1, solid line is a general load-displacement relation of the surrounding soil domain in each direction, while dashed line is a simplified bilinear curve to represent the relationship. Soil mass and stiffness due to a burial depth are taken into account in the calculation of the force-displacement relationship of the springs. Detailed description of the relationship is discussed below. The maximum spring force per unit length of the pipeline in the axial direction is defined by
properties considering the statistical characteristics of soil profiles. Phoon and Kulhawy [15] suggested the uncertainty of soil parameter using in-situ measurement and transformation model according to soil and test types. Besides, various studies have also been conducted on the soil-structure interaction considering input parameters with uncertainty. Jin et al. [16] conducted seismically excited response variability of multi-story structure considering soil-structure interaction and uncertainty of soil parameters. Chaudhuri and Gupta [17] investigated seismic response of secondary systems considering uncertainty on soil parameters. Moreover, Gallage et al. [18] evaluated seismic response of buried segmented pipeline considering effect of uncertainty of the soil parameters. Imanzadeh et al. [19] estimated coefficient of the soil reaction modulus of buried pipeline taking into account the effect of uncertainty in soil and structure parameters. However, there were few studies on the seismic response of buried pipeline using nonlinear time history analysis including both material inelasticity and geometry nonlinearity. Most of the previous studies have mainly focused on the seismic fragility relationships between seismic intensity and corresponding structural performance by past earthquakes. Moreover, regarding soil uncertainty, most of the studies were limited to above-ground structures. This implies that salient soil features such as internal friction angle, unit weight and cohesion were not taken into account in the studies. In view of the above, the seismic fragility analysis of a buried pipeline has been conducted in terms of using nonlinear time history analysis and considering uncertainty of soil parameters. For this purpose, buried pipe and surrounding soil are modeled by beam elements and equivalent springs, respectively, and soil-pipeline interaction is represented by beam on nonlinear-Winkler foundation (referred to as BNWF hereafter) model. To analyze the sensitivity of the structural response to soil uncertainty, the first order-second moment (FOSM) analysis is performed and the relative variance is calculated according to the seismic intensity. Lastly, but by no means at least, seismic fragility curves are derived and compared according to the soil parameters.
Tu = πDαc + πDHγ‾
1 + KO tan δ 2
α = 0.608 − 0.123c −
0.274 0.695 + 3 c2 + 1 c +1
(1)
(2)
where α is an adhesion factor, D is outer diameter of pipeline, c is the soil cohesion representative of the soil backfill, H is distance to centerline of pipe from ground surface, γ‾ is effective unit weight of soil, K O is coefficient of pressure at rest, δ is interface friction angle between pipe and soil (=fφ) , φ is internal friction angle, and f is coating factor relating soil-pipeline interface. Calculated displacement corresponding to the maximum soil force is 3 mm, 5 mm, 8 mm, and 10 mm for dense sand, loose sand, stiff clay, and soft clay, respectively, as suggested by the American Lifeline Alliances [20]. The maximum transverse soil spring force per unit length of the pipe is given by
Pu = Nch cD + Nqh γHD ‾ Nch = a + bx +
(3)
c d + ≤ 9 (0 for c = 0) (x + 1)2 (x + 1)3
Nqh = a + bx + cx 2 + dx 3 + ex 4 (0 for φ = 0)
(4) (5)
where Nch and Nqh are horizontal bearing capacity factor for clay and sand, respectively and a, b, c, d, and e are empirical constants that were determined from published empirical results in Table 1 (Nqh can be interpolated to obtain intermediate values of internal friction angle between 20° and 45°), and x is ratio between depth of buried distance (H) and outer diameter (D). The displacement corresponding to the maximum soil force is determined by
2. BNWF model for soil-pipeline interaction In the present study, the BNWF model was employed to simulate the soil-pipeline interaction in the following nonlinear seismic response analysis of a buried pipeline. In the BNWF model, the interaction between soil and pipeline can be modeled as a discrete nonlinear spring (e.g., symmetric or asymmetric elastoplastic curve). American Lifeline Alliances [20] proposed the relationship between force and displacement of the equivalent soil spring based on the results from the field experiments and laboratory tests. The force-displacement relationship of the spring in each direction is assumed to be linearly elastic-perfectly plastic, hence bilinear. Fig. 1 illustrates the BNWF model and
D Δp = 0.04 ⎛H + ⎞ ≤ 0.1D − 0.15D 2⎠ ⎝
(6)
The maximum vertical bearing spring force per unit length of the pipe is determined by
Qd = Nc cD + Nq γHD + Nγ γ ‾
D2 2
Fig. 1. Representation of soil-pipeline interaction using the BNWF model [25,26]. 2
(7)
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Table 1 Empirically estimated horizontal bearing capacity factor. Factor
φ (degree)
x
a
b
c
d
Nch Nqh
0° 20°
H/D H/D
6.752 2.399
0.065 0.439
−11.063 −0.03
7.119
–
1.059 (10−3)
−1.754 (10−5)
Nqh
25°
H/D
3.332
0.839
−0.09
5.606 (10−3)
−1.319 (10−4)
Nqh
30°
H/D
4.565
1.234
−0.089
4.275 (10−3)
−9.159 (10−5)
Nqh
35°
H/D
6.816
2.019
−0.146
7.651 (10−3)
−1.683 (10−4)
Nqh
40°
H/D
10.959
1.783
0.045
−5.425 (10−3)
−1.153 (10−4)
Nqh
45°
H/D
17.658
3.309
0.048
−6.443 (10−3)
−1.299 (10−4)
the ground surface are neglected. Second, the inner pressure due to the fluid flow inside the pipeline is not taken into account. Third, the ground water level is set below the pipeline structure. Therefore, the uplift effect of the pipeline due to buoyancy is not considered, and hence the effective unit weight (γ‾ ) and unit weight (γ ) of the soil are identical. Fourth, the soil condition is assumed as a dense silty soil of Southern California. This assumption is somewhat intuitive. Since soil classification in most of Korean seismic design codes is derived from UBC [22], soil conditions as in the case of UBC are employed. It is qualitatively acknowledged that soil condition can be varied along the pipeline length. However, the current study is to investigate nonlinear response and corresponding seismic fragility analyses of the pipeline, which causes quantitative difficulty even for a single soil condition. Thus, the present study is focused on the dense silty of single condition. Subsequently, constant soil profile along the pipeline length is assumed in the numerical modeling. Finally, uncertainty is considered for soil only. This is due to the fact that the pipeline has relatively small material uncertainty since it is mainly a mass production. However, in the case of soil, there are many of uncertainty factors, such as measurement errors, inherent variability and etc. Subsequently, uncertainty of soil parameters are taken into account in the present analyses.
Nc = [cot(φ + 0.001)] ⎧exp [π tan(φ + 0.001)]tan2 ⎛45 + ⎨ ⎝ ⎩
φ + 0.001 ⎞ − 1⎫ ⎬ 2 ⎠ ⎭ (8)
φ Nq = exp (π tan φ)tan2 ⎛45 + ⎞ 2⎠ ⎝
(9)
Nγ = exp (0.18φ − 2.5)
(10)
where Nc , Nq , and Nγ are the bearing capacity factors and γ is total unit weight of soil. Corresponding displacement at Qd is 0.1D for granular soils and 0.2D for cohesive soils. The maximum vertical uplift spring force per unit length of pipe is given by
Qu = Ncv cD + Nqv γHD ‾
(11)
H Ncv = 2 ⎛ ⎞ ≤ 10 (0 for c = 0) ⎝D⎠
(12)
φH ⎞ ≤ Nq (0 for φ = 0) Nqv = ⎛ 44 ⎝ D⎠
(13)
e
where Ncv and Nqv are the vertical uplift factor for clay and sand, respectively and Nq is bearing capacity factor described in Eq. (9). Corresponding displacement at Qu is calculated by 0.01H–0.02H for dense to loose sands (< 0.1D) and 0.1H–0.2H for stiff to soft clays (< 0.2D). A more detailed description of the equations are provided in the American Lifeline Alliance, ASCE [20].
3.2. Uncertainty of soil parameters The three soil parameters that control the soil stiffness in the BNWF model were selected for the uncertainty of soil conditions. The internal friction angle, cohesion, and unit weight were considered as random variables with Gaussian distribution, while remaining parameters such as coefficient of pressure and coating factor were set as fixed parameter conditions. Table 3 shows the mean value and coefficient of variation (COV) of soil input parameters obtained by previous research results [23]. The COV value of soil parameters is assumed as 10% in the present study. Previous experimental work [24] indicated that the COV values showed a deviation in the three parameters. Considering the COV results of various soil types (sand, clay, silt), 10% of COV seemed to be reasonable, particularly in the absence of sufficient experimental data with regard to dense silt sand. The assumption that the three soil parameters are independent may have a significant influence on the results of random sampling. Various studies have been conducted to measure correlation between the parameters. However, no such relation has been estimated. Therefore, based on limited work [23], correlation matrix between the parameters for dense silty soil is suggested and summarized in Table 4. The suggestion given in Table 4 is supported by sound engineering judgment and relatively good agreement obtained between sensitivity and nonlinear time history analysis. There is certainly room for improvements
3. Numerical model of a buried pipeline 3.1. Numerical modeling assumption In order to carry out the nonlinear time history analysis of a buried gas pipeline, API 5L X65 which is widely utilized in South Korea was selected [12]. The material properties of the pipeline are shown in Table 2. The cross section of the pipeline is a circular steel hollow section with an outer diameter of 762 mm and an inner diameter of 727 mm. The pipeline is assumed to be a rough steel and thus friction coefficient of 0.8 is used. A straight pipeline with a length of 1.2 km and a distance of 1.881 m from the ground to the center line of pipe crosssection was employed in the subsequent analyses. As for the boundary condition, both ends of the pipeline were assumed as fixed end constraints. This can be attributed to the fact that the behavior of the pipeline does not depend on the constraint condition [21]. For numerical modeling of the buried pipeline, the following five conditions were assumed. First, the live load and dead load applied on Table 2 Material properties of API 5L X65 gas pipeline. Mass density (kg/m3 )
Elastic modulus (GPa)
Poisson's ratio
Yield strength (MPa)
Outer diameter (mm)
Thickness (mm)
Coefficient of friction
7850
210.7
0.3
445
762
17.5
0.8
3
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4. Input ground motions and eigenvalue analysis
Table 3 Mean and coefficient of variation for soil parameters. Parameter
Mean
COV (%)
Distribution
Internal friction angle Cohesion Unit weight
38° 5 kPa
10 10 10
Gaussian Gaussian Gaussian
18 kN/m3
4.1. Input ground motions Due to random nature of earthquakes, input ground motions have a lot of uncertainties regarding magnitude, duration, frequency contents, peak ground acceleration, spectral intensity, soil condition and so on. Subsequently, it is onerous to predict the seismic structural response according to a specific hazard level deterministically, although it is not completely impossible. To get over this, special care should be needed in selecting the input ground motions covering as many uncertainties as possible. For this purpose, input ground motion suites including various characteristics of seismic waves were selected in the present study [27]. Table 6 shows the selected 12 input ground motions. As observed in Table 6, various levels of peak ground acceleration (PGA) were taken into account. Since South Korea is classified as medium to low seismicity area, extreme event cases are intentionally excluded in the selection of the input ground motions. In the subsequent analyses, these PGAs were scaled from 0.1g up to 1.5g with an interval of 0.2g. Rationale behind this selection can be supported by Fig. 2. Fig. 2 shows acceleration response spectra of the 12 motions in axial, transverse, and vertical directions. As plotted in Fig. 2, frequency contents of the input motions are relatively well-distributed and the maximum spectral accelerations at the maximum amplifications exhibit various levels. It is thus postulated that current selection seems to be reasonable enough to consider uncertainty of input ground motion.
Table 4 Correlation matrix between the soil parameters. Correlation matrix
Cohesion
Internal friction angle
Unit weight
Cohesion Internal friction angle Unit weight
1 −0.7 0
−0.7 1 0.3
0 0.3 1
as more correlation results become available.
3.3. Numerical model of buried pipeline To perform earthquake response analysis considering soil-pipeline interaction, commercial software ABAQUS was utilized [25,26]. ABAQUS provides two-dimensional or three-dimensional pipe-soil interaction element to represent the nonlinear behavior of buried gas pipeline and surrounding soil. The pipe-soil interaction model is simply expressed as a constitutive model that defines the stiffness of the soil region without discretizing the soil domain into finite elements. Fig. 1 shows a graphical representation of the pipe-soil interaction model. The pipe-soil interaction element is defined using far-field node and pipeline node, and the behavior of pipe-soil interaction element is determined by constitutive equation proposed by ASCE [26]. In this model, input earthquake is propagated to the buried pipeline structure through an equivalent spring that reflects the characteristics of the soil profile. In this study, the 3-D pipe-soil interaction element (PSI34) has been utilized to conduct the soil-structure interaction analysis. The PSI34 element represents nonlinear constitutive behavior by modeling the interaction of buried pipeline structure and surrounding soil region [26]. For numerical simulation, the 1.2 km buried pipe and soil domain consist of 1000 numbers of pipe element (PIPE31) and PSI34 elements, and the damping ratio of the system is assumed as 5% to critical damping using the Rayleigh damping coefficient. Detailed description of the pipe-soil element is described in ABAQUS manual [25]. For considering the uncertainty of soil parameters, the random samples were generated by using Latin Hypercube Sampling (LHS) techniques. From obtained soil parameters, the maximum spring force and corresponding displacement were calculated using Eqs. (1)–(13). For a soil region, the stiffness (slope of the load-displacement curve) for tension and compression has the same slope because such region has an infinite domain in both axial and transverse directions. However, since the soil domain corresponding to the burial depth is not symmetric, the stiffness in vertical direction has two different values. Table 5 shows the maximum spring force and corresponding displacement for Southern LA regions, when each soil parameter has mean value.
4.2. Eigenvalue analysis An eigenvalue analysis was conducted for the straight pipeline embedded in dense silty sand using Lanczos algorithm [28]. The first three modes of vibrations are evaluated in vertical and transverse directions. The first three natural frequencies in the vertical direction were 14.507, 14.575 and 14.689 Hz, respectively, and those in the transverse direction were 15.669, 15.732 and 15.836 Hz, respectively. Meanwhile, lower modes of vibration were not observed in the axial direction. The eigenvalue analysis results indicate that natural frequencies between the modes are so close in both directions. Fig. 3 shows the representative first to third mode shapes in each direction. Moreover, the mass participation factors become more than 90% when the first to third mode are included in the analysis. It is therefore expected that higher natural frequency effects can be of significance, which stresses the importance of the nonlinear analysis conducted in the current study. 5. Nonlinear time history analysis 5.1. Results of seismic response analysis In this section, the variation of structural responses due to soil parameter uncertainties was investigated. For this purpose, an efficient Monte Carlo Simulation (MCS) employing LHS [29] was performed and nonlinear time history analysis was conducted by scaling PGA of the input ground motions. Choice of the LHS lies in the fact that in comparison with other random sampling methods, the LHS provides statistically significant results with fewer samples. Numerical analyses were carried out for the following four cases: 1) variation of all parameters considering correlation matrix; 2) variation of internal friction angle only (cohesion and unit weight are mean values); 3) variation of cohesion only (internal friction angle and unit weight are mean values); 4) variation of unit weight only (internal friction angle and cohesion are mean values). For nonlinear time history analysis, the maximum strain response of the pipeline was investigated for the above four cases. In general, if the boundary conditions of the buried pipeline were fixed support, the maximum axial strain usually occurred at the ends. Therefore, the
Table 5 Maximum spring force and corresponding displacement. Spring direction
Maximum force (N/ mm)
Corresponding displacement (mm)
Axial Transverse vertical (uplift) vertical (downward)
45.16 377.89 73.81 1897.09
3 90.48 18.81 76.2
4
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Table 6 Selected input ground motions. Earthquake (Year)
Station
PGA (g)
Magnitude
Mechanism
EQ1 EQ2 EQ3
Gulf of Aqaba (1995) Borrego (1968) Cape Mendocino (1992)
0.109 0.130 0.178
7.2 6.63 7.01
Strike slip Strike slip Reverse
EQ4 EQ5 EQ6
Chi-Chi (1999) Chi-Chi (1999) Coalinga-01 (1983)
0.263 0.198 0.194
7.62 7.62 6.36
Reverse Oblique Reverse Oblique Reverse
EQ7 EQ8 EQ9 EQ10 EQ11 EQ12
Imperial Valley (1979) Kocaeli (1999) Landers (1992) Loma Prieta (1989) San Fernando (1971) Tabas (1978)
Eliat El Centro array #9 Eureka-Myrtle & West CHY014 ILA067 Park field Fault Zone 1 Delta Fatih Indio-Coachella Canal Hayward-BART Whittier Narrows Dam Ferdows
0.351 0.187 0.109 0.159 0.107 0.108
6.53 7.51 7.28 6.93 6.61 7.35
Strike slip Strike slip Strike slip Reverse Oblique Reverse Reverse
Fig. 2. Acceleration response spectra of the selected input ground motions.
maximum axial strain occurring around the fixed ends was adopted in this study. Fig. 4 shows a representative case of the axial strain distribution at a critical time step subjected to San Fernando earthquake of 1.5g of PGA. The maximum strain was observed around the fixed end, and the strain distribution was gradually decreased with the distance away from the fixed end. This indicates that weak region of the buried pipeline can be a joint connecting each pipeline. Fig. 5 shows the mean and standard deviation of the normalized strain responses according to the number of samples subjected to PGAs of 0.1g, 0.5g, 0.9g, and 1.5g. The mean and standard deviation of the response were normalized to the response for number of sampling of 60.
As observed in general, the response variation gradually converged as the number of samples increased. In addition, while the variation of the normalized mean value according to the number of samples was not significant, that of the standard deviation value was clearly observed, particularly up to 40 of number of sampling. This implies that a proper number of sampling can be of utmost importance in balancing analysis cost and accuracy. Fig. 5 also shows that the normalized mean and standard deviation values tend to converge around 50 samplings. Therefore, 60 samplings were employed to represent the seismic fragility curve of the pipeline in the present study. Fig. 6 shows the distribution of the maximum strain response 5
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Fig. 3. Representative mode shapes and mass participation factor.
Fig. 4. General trend of strain distribution of buried pipeline with fixed ends.
according to different levels of PGA intensities for both fixed soil parameters with mean values (as listed in Table 3) and consideration of the uncertainty of soil parameters. As the intensity of the PGA increases, the buried pipeline shows a large variation in the maximum strain response due to nonlinearity. It is worthy of noting that whereas 96 analyses were carried out for the model without consideration of soil uncertainty, 5760 analyses were conducted for each soil variable in case of considering soil uncertainty. Figs. 7 and 8 show the mean and standard deviation of the maximum strain response of the pipeline according to PGA intensities. The mean value of the maximum strain was increased when soil uncertainty was considered. However, the difference due to the uncertainty of each soil parameter was not significant. Meanwhile, the standard deviation of the maximum strain exhibited a discrepancy between the responses, resulting in relatively low response for the case of cohesion. In
particular, nonlinearity was observed at 0.5g, where the standard deviation of maximum strain was abruptly increased. Based on observation of Figs. 7 and 8, unit weight seemed to be the most salient parameter affecting the strain response of the pipeline. Further investigation has been carried out in order to verify the influence of each soil parameter quantitatively, and is discussed in the following section. 5.2. Sensitivity analysis The FOSM analysis was carried out to investigate the significance of soil parameters affecting nonlinear seismic response of the buried pipeline [30]. For this purpose, each soil parameter is assumed to be statistically independent each other and sensitivity analysis was performed for each soil parameter within a range of the mean ± one standard deviation. In addition, to simplify the relationship between the 6
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Fig. 5. Convergence of normalized strains for mean (left) and standard deviation (right).
7
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Fig. 6. Maximum strain response of the pipeline with and without considering uncertainty of soil parameters. N
structural response and the soil parameters, it is assumed that the variables and responses have a linear relationship. Let's assume that F is the structural response function of X having N random variables, and Y is the structural response to the random variable. Thus,
Y = F (X1 , X2 , ⋯, XN ).
Y ≅ F (μ X1 , μ X2 , ⋯, μ XN ) +
∑ (Xi − μ Xi ) i=1
∂F (X1 , X2 , ⋯, XN ) . ∂Xi
(15)
The mean of the response function μY represents the structural response when N random variables have a mean value, and the standard deviation of response function σY2 stands for variation of the response due to the variance of N random variables. Thus, the mean and standard deviation of the response function Y are calculated by following expressions.
(14)
In order to calculate the equation above numerically, Taylor expansion is applied to the function F and higher order terms are ignored. Thus, the above equation can be simplified as
μY ≅ F (μ X1 , μ X2 , ⋯, μ XN ) 8
(16)
International Journal of Pressure Vessels and Piping 175 (2019) 103932
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Fig. 7. Average strain response of pipeline. N
σY2 ≅
2
importance of the soil parameters was calculated using Eqs. (14)–(18). As shown in Fig. 10, unit weight has a significant effect on the buried pipeline, while cohesion has a little impact on the pipeline behavior. In the intermediate PGA range, the relative variance of internal friction angle and cohesion increases slightly, and the variance of unit weight decreases abruptly. Nonetheless, unit weight can be considered as a dominant factor affecting the pipeline response. In the case of unit weight, the soil mass is determined, which has a great influence (relative variance: 0.7) on the determination of the inertia force when the input ground motion occurs. On the other hand, the internal friction angle has an important influence (relative variance: 0.25) on the determination of soil stiffness. Therefore, the dynamic behavior of the buried pipeline is confined and can affect the deformation of the structure depending on the soil stiffness. Since cohesion (relative variance: 0.05) has little effect on soil mass and stiffness, it does not significantly affect the behavior of sandy soil with uncertainty. This is particularly true since the unit weight has the greatest effect on the mass and equivalent stiffness of spring in comparison with other parameters.
∑ σX2i ⎛ ∂F (X1, X2, ⋯, XN ) ⎞ ⎜
∂Xi
⎝
i=1
+
⎟
N
N
i
j
⎠
∑ ∑ ρ Xi Xj ∂F (X1, ⋯, XN ) ∂F (X1, ⋯, XN ) ∂Xi
∂Xj
(17)
Since the soil parameters are assumed to be independent ⎛⎜ρ Xi Xj =0), ⎝ the partial derivative of X for the response function F can be represented using the central difference method. Thus, the variance of the response function can be calculated numerically as given by equation (18). 2
⎛ ∂F (X1 , X2 , ⋯, XN ) ⎞ ∂Xi ⎝ ⎠ ⎜
⎟
2
F (μ1, ⋯, Xi + Δ Xi , μN ) − F (μ1, ⋯, Xi − Δ Xi , μN ) ⎞ = ⎜⎛ ⎟ 2Δ Xi ⎠ ⎝
(18)
Fig. 9 shows the normalized maximum strain response of the pipeline according to variation of soil parameters at four levels of PGAs. In Fig. 9, the maximum strains were normalized to the minimum value of soil parameters. For small PGA of 0.1g, the pipeline response remains essentially in elastic range, regardless of soil parameters. However, the response excurses inelastic range as the PGA increases. In particular, significant increase is observed in the response at 0.9g. This can be attributed to the frequency characteristics of the applied input ground motions. Particular emphasis is placed on the response at 1.5g. Normalized strain response at 1.5g is in general less than that of 0.5g and 0.9g. This can be due to the fact that the pipeline strains experience inelastic response mainly in intermediate PGA range as depicted in Fig. 8. Based on the structural response shown in Fig. 9, the relative
6. Seismic fragility curve 6.1. Definition of damage state The seismic fragility curve represents the conditional failure probability that an event exceeds the damage state for a certain level of earthquake. Damage state can be determined by a statistical method for predicting seismic intensity and physical response of structures induced by previous earthquakes. The need for quantitative evaluation of the damage state for a buried pipeline has been continually increased. However, very few studies have been available regarding the quantitative definition of damage state. In addition, almost all of the studies
Fig. 8. Standard deviation of strain response of pipeline. 9
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Fig. 9. Normalized strain with regard to soil parameters for individual input ground motions.
order to derive seismic fragility curve for the buried pipeline. They suggested the damage state through the maximum strain of the pipeline utilizing the seismic risk analysis for a buried pipeline network. Damage state is classified as major, moderate, and minor corresponding to
do not have standard criteria for strain-based pipeline damage states representing intact, leakage and breakage. Amongst the very few studies defining the damage state of the buried pipeline, the damage state proposed by Shinozuka et al. [31] is employed in the present study in 10
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complete loss (break), partial loss (leakage), and no loss (including minor leakage). Table 7 shows the damage state of the buried pipeline used in this study. In Table 7, εp represents the maximum strain of the buried pipeline under earthquake loading, and εy stands for the yield strain of the buried pipeline. 6.2. Seismic fragility analysis The results of nonlinear time history analyses have been utilized to derive the seismic fragility curve for the buried pipeline according to the damage states. In order to represent the seismic fragility curve, the method proposed by Shinozuka et al. was adopted [32,33]. In the methodology, the seismic fragility curve of the structure was assumed to be a log-normal distribution relationship between peak ground acceleration and failure probability. Empirical evaluation of seismic fragility curves in terms of structural damage experienced by previous earthquakes has proved the validity of the logarithmic normal distribution. The maximum likelihood method was employed to calculate two variables (median and log standard deviation) of the log-normal distribution, and the likelihood function is defined by
Fig. 10. Relative variance of each soil parameter according to PGA intensity.
N
Table 7 Damage state of the buried pipeline proposed by Shinozuka et al. [31].
L=
∏ [F (IMi)]xi [1 − F (IMi)](1−xi) i=1
Damage state
Structural response (Maximum strain)
Minor
εp ≤ 0.7εy
Moderate
0.7εy ≤ εp ≤ εy εp ≥ εy
Major
F (IMi ) = Φ ⎜⎛ ⎝
ln(IMi )/ ck ⎞ ⎟ ξk ⎠
(19)
(20)
where F (IMi ) is the seismic fragility function following standard normal distribution when the pipeline is subjected to the seismic intensity of
Fig. 11. Comparisons of seismic fragility curves with and without considering uncertainty of soil parameters. 11
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Table 8 Median and log-standard deviation for seismic fragility curves. Variable
Moderate Major
Fixed soil
All parameters
Internal friction angle
Cohesion
Unit weight
ck
ξk
ck
ξk
ck
ξk
ck
ξk
ck
ξk
0.642 0.917
0.657 0.654
0.513 0.813
0.61 0.591
0.527 0.832
0.595 0.599
0.535 0.832
0.592 0.593
0.525 0.822
0.601 0.592
Fig. 12. Comparisons of seismic fragility curve (a) moderate damage state (b) major damage state.
standard deviation of seismic fragility curve are lower than those for fixed conditions of soil parameters. Lower mean values in the seismic fragility curves indicate higher failure probability, hence more vulnerable to earthquakes. This stresses the importance of the soil uncertainty in the fragility curves. In addition, lower log-standard deviation points out that the failure probability increases rapidly for a certain hazard-level of earthquakes, hence again a significance of including the soil uncertainty [34]. As discussed, the failure probability increases rather abruptly with the consideration of the soil uncertainty. This result implies that deformation of the buried pipeline may arrive at the moderate damage state even for a relatively small PGA, leading to an alteration in the fragility curves hence higher probability of failure. Moreover, the fragility curves corresponding to major damage state show that failure probability with consideration of the soil uncertainty increases rather rapidly than that without taking into account the soil uncertainty, particularly at 0.7g of PGA. This also confirms a tendency obtained for the moderate damage state. Meanwhile, a little difference is achieved in
IMi , and ck and ξk represent the median and log-standard deviation of the lognormal distribution for the k-th damage state, respectively. To obtain the ck and ξk that maximize the likelihood function, the partial differential of each parameter can be expressed as
∂ln(L) ∂ln(L) = =0 ∂ck ∂ξk
(21)
where k represents the number of defined damage states, and it was classified as two states (major and moderate) in the current study because there was no minimum strain in the minor state. Fig. 11 compares the seismic fragility curves according to uncertainty of soil parameters compared with fixed soil parameters (without uncertainty) when 12 input seismic loads were applied. The median and log-standard deviation of each seismic fragility curve are summarized in Table 8. The solid line shows the seismic fragility curve for the fixed conditions of the soil parameters, while the dashed line exhibits the seismic fragility curve considering the soil uncertainty. When the soil uncertainty is taken into account, mean value and log12
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between the fragility curves for the three soil parameters as displayed in Fig. 12. This can be attributed to the damage states. The damage states adopted herein do not seem to reflect a physical behavior of the pipeline under consideration precisely. Nevertheless, it is seemed that the present results may give a useful information with regard to the seismic fragility analysis of the buried pipeline, particularly for the inclusion of the soil uncertainty. It is thus worthy of noting that a better result is expected to achieve when more realistic damage states are available.
[2] C. Scawthorn, P.I. Yanev, 17 January 1995, Hyogo-ken Nambu, Japanese earthquake, Eng. Struct. 17 (3) (1995) 146–157. [3] W.W. Chen, B.J. Shih, C.W. Wu, Y.C. Chen, Natural gas pipeline system damages in the Ji-Ji earthquake (The City of Nantou), Proceedings of the Sixth International Conference on Seismic Zonation, Palm Springs, Riviera Resort, CA, 2000. [4] T. Katayama, K. Kubo, N. Sato, Earthquake damage to water and gas distribution systems, Proceedings of the US National Conference on Earthquake Engineering, Oakland, CA, 1975. [5] R. Eguchi, Seismic vulnerability models for underground pipes, Proceedings of Earthquake Behavior and Safety of Oil and Gas Storage Facilities, Buried Pipelines and Equipment, American Society of Mechanical Engineers, New York, 1983, pp. 368–373. [6] M. O'Rourke, G. Ayala, Pipeline damage due to wave propagation, J. Geotech. Eng. 119 (9) (1993) 1490–1498. [7] T.D. O'Rourke, S. Toprak, Y. Sano, Factors affecting water supply damage caused by the Northridge Earthquake, Proceedings of the 6th U.S National Conference on Earthquake Engineering, Seattle, WA, 1998. [8] M. O'Rourke, E. Deyoe, Seismic damage to segmented buried pipe, Earthq. Spectra 20 (4) (2004) 1167–1183. [9] O. Pineda-Porras, M. Ordaz, A new seismic intensity parameter to estimate damage in buried pipelines due to seismic wave propagation, J. Earthq. Eng. 11 (5) (2007) 773–786. [10] N. Newmark, E. Rosenblueth, Fundamentals of Earthquake Engineering, PrenticeHall Inc., New Jersey, 1971. [11] L.R.L. Wang, Y.H. Yeh, A refined seismic analysis and design of buried pipeline for fault movement, Earthq. Eng. Struct. Dyn. 13 (1) (1985) 75–96. [12] D.H. Lee, B.H. Kim, H. Lee, J.S. Kong, Seismic behavior of a buried gas pipeline under earthquake excitations, Eng. Struct. 31 (5) (2009) 1011–1023. [13] X. Xie, M.D. Symans, M.J. O'Rourke, T.H. Abdoun, T.D. O'Rourke, M.C. Palmer, H.E. Stewart, Numerical modeling of buried HDPE pipelines subjected to normal faulting: a case study, Earthq. Spectra 29 (2) (2013) 609–632. [14] E.H. Vanmarcke, Probabilistic modeling of soil profiles, J. Geotech. Eng. Div. 103 (11) (1977) 1227–1246. [15] K.-K. Phoon, F.H. Kulhawy, Characterization of geotechnical variability, Can. Geotech. J. 36 (4) (1999) 612–624. [16] S. Jin, L. Lutes, S. Sarkani, Response variability for a structure with soil–structure interactions and uncertain soil properties, Probabilistic Eng. Mech. 15 (2) (2000) 175–183. [17] S.R. Chaudhuri, V. Gupta, Variability in seismic response of secondary systems due to uncertain soil properties, Eng. Struct. 24 (12) (2002) 1601–1613. [18] C. Gallage, R. Pathmanathan, K. Jayantha, Effect of soil parameter uncertainty on seismic response of buried segmented pipeline, 1st International Conference on Geotechnique, Construction Materials and Environment, Tsu City, Mie, Japan, 2011. [19] S. Imanzadeh, A. Denis, A. Marache, Effect of uncertainty in soil and structure parameters for buried pipes, Proceedings of 4th International Conference on Site Characterization, Porto de Galinhas, Pernambuco, Brazil, 2012. [20] ALA, (Ameriean Lifeline Alliance), Guidelines for the Design of Buried Steel Pipe, American Society of Civil Engineers, 2001. [21] T. Datta, E. Mashaly, Seismic response of buried submarine pipelines, J. Energy Resour. Technol. 110 (4) (1988) 208–218. [22] UBC, Uniform building code, Int. Conf. Build. Offic. (1997) Chapter 16, division IV. [23] P. Raychowdhury, Effect of soil parameter uncertainty on seismic demand of lowrise steel buildings on dense silty sand, Soil Dyn. Earthq. Eng. 29 (10) (2009) 1367–1378. [24] A.L. Jones, S.L. Kramer, P. Arduino, Estimation of Uncertainty in Geotechnical Properties for Performance-Based Earthquake Engineering, Pacific Earthquake Engineering Research Center, College of Engineering, University of California, 2002. [25] H. Hibbit, B. Karlsson, E. Sorensen, ABAQUS User Manual, Simulia, Providence, RI, 2012version 6.12. [26] J. Audibert, D. Nyman, T. O'Rourke, Differential Ground Movement Effects on Buried Pipelines, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, (1984), pp. 150–183. [27] D.H. Lee, B.H. Kim, S.-H. Jeong, J.-S. Jeon, T.-H. Lee, Seismic fragility analysis of a buried gas pipeline based on nonlinear time-history analysis, Int. J. Steel Struct. 16 (1) (2016) 231–242. [28] C. Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, United States Governm. Press Office Los, Angeles, CA, 1950. [29] M. Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29 (2) (1987) 143–151. [30] F.S. Wong, First-order, second-moment methods, Comput. Struct. 20 (4) (1985) 779–791. [31] M. Shinozuka, S. Takada, H. Ishikawa, Some aspects of seismic risk analysis of underground lifeline systems, J. Press. Vessel Technol. 101 (1) (1979) 31–43. [32] M. Shinozuka, M.Q. Feng, H.-K. Kim, S.-H. Kim, Nonlinear static procedure for fragility curve development, J. Eng. Mech. 126 (12) (2000) 1287–1295. [33] M. Shinozuka, M.Q. Feng, J. Lee, T. Naganuma, Statistical analysis of fragility curves, J. Eng. Mech. 126 (12) (2000) 1224–1231. [34] M. Shinozuka, M. Feng, H. Kim, T. Ueda, Statistical Analysis of Fragility Curves, Technical Report at Multidisciplinary Center for Earthquake Engineering Research, NY, USA, 2002.
7. Concluding remarks In the present study, seismic fragility analyses have been performed for a buried gas pipeline considering uncertainty of soil parameters. To achieve this objective, nonlinear time history analyses have been carried out using selected 12 earthquake ground motions. The BNWF model was employed to represent soil-pipeline interaction, and the LHS scheme was adopted to take into account the uncertainty of soil parameters. In addition, sensitivity analyses have been performed to evaluate the effect of soil parameters on the response of the pipeline. Fragility analyses for the pipeline have also been conducted using the nonlinear time history analysis results. Based on analytical predictions obtained in the study, following conclusions are discussed.
• Nonlinear time history analyses of the pipeline show that strain
• •
response considering the uncertainty of soil parameters is greater than that without the uncertainty. In addition, the behavior of buried pipeline taking into account the uncertainty of soil parameters indicates that unit weight is a dominant parameter amongst others. Sensitivity analyses reveal that relative importance is calculated in the strain response for unit weight parameter. This result is similar to that obtained in the nonlinear time history analyses, and stresses importance and necessity of the sensitivity analysis in association with the soil uncertainty. Seismic fragility curves considering the soil uncertainty show higher failure probability than those without the uncertainty. In addition, failure probability considering the soil uncertainty increases rapidly in comparison with that without the uncertainty. It is therefore noteworthy that the present fragility analyses with the soil uncertainty are expected to enable the assessment of the failure probability of the pipeline to be captured accurately, particularly considering a difficulty of access to the underground pipeline.
In all, as demonstrated in the above discussion, analytical observation is affected by the uncertainty of soil parameters. The predictions presented in the current study however are derived in terms of using three soil parameters only due to shortage of available geotechnical data. It is thus promising that the present study can further be improved when more information on the geotechnical properties of soil become available, hence correlation between soil parameters. Besides, more improvement is also anticipated when realistic damage state reflecting physical behavior of the pipeline is suggested. Acknowledgements This work was supported by the National Research Foundation Korea (NRF) Grant funded by the Korean government (MSIP) (No. 2017R1A5A1014883). This work was also financially supported by Korea Ministry of Land, Infrastructure and Transport (MOLIT) as Innovative Talent Education Program for Smart City. References [1] Moore Dames, The Loma Prieta earthquake: impact of lifeline systems, J. Disaster Recovery 3 (2) (1999) 8.
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International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Seismic fragility analysis of a buried pipeline structure considering uncertainty of soil parameters
T
Sungsik Yoona, Do Hyung Leeb, Hyung-Jo Junga,* a
Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea b Department of Civil, Environmental and Railroad Engineering, Paichai University, 155-40 Baejae-ro, Seo-gu, Daejeon, 35345, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: Buried pipeline structure API 5L X65 Beam on nonlinear Winkler foundation model Uncertain soil parameters Nonlinear time history analysis Damage state Seismic fragility curve
In this study, the seismic fragility analysis for a buried gas pipeline of API 5L X65 was performed considering the uncertainty of soil parameters. For this purpose, nonlinear time history analyses were carried out for the pipeline considering soil-pipeline interaction represented by beam on nonlinear Winkler foundation model. A total of 12 input ground motions were employed and four different analytical cases were considered to evaluate the effect of the uncertainty of soil parameters. The four cases proved that uncertainty of soil parameters needs to be taken into account for the assessment of the pipeline response. Using results of the nonlinear time history analyses, seismic fragility analyses were conducted in accordance with three damage states. Analytical predictions indicated that seismic fragility curves with the uncertainty of soil parameters exhibit higher failure probability than those without the uncertainty. It is thus concluded that the present study is promising for the exploration of the seismic fragility analysis considering the uncertainty of soil parameters.
1. Introduction Recently many earthquakes have caused severe damage to lifeline structures such as gas, electricity, and water supply. Especially, underground facilities (e.g., gas and water pipeline) have been reported to exhibit more severe vulnerability against earthquakes. The Loma Prieta earthquake with a magnitude of 6.9 in San Francisco (1989) caused interruption of gas supply to 15,000 demand facilities. Moreover, the destruction of main gas pipelines led to a significant damage to critical buildings due to fire following the earthquake [1]. The Kobe earthquake (1995) of 6.9 magnitude resulted in electric sparks and gas fires at about 500 sites, and the direct and indirect total damage was estimated at approximately 200 billion dollars [2]. In addition, there was an interruption of gas supply to 100,000 people and the damage to main gas companies was estimated as 25 million dollars due to the ChiChi earthquake (1999) with a magnitude of 7.6 [3]. In order to evaluate the failure probability of buried pipelines, numerous researches have been conducted in terms of using various seismic intensity measures. Probability of failure was represented as one of the followings, i.e., Peak Ground Acceleration (PGA) [4], Modified Mercalli intensity (MMI) [5], and Peak Ground Velocity (PGV) [6] considering empirical fragility relation. In addition, Peak Ground Deformation (PGD), Arias Intensity (AI), Spectral Acceleration (SA),
*
Spectral Intensity (SI), maximum ground strain, and PGV 2/PGA were also utilized as damage indicators to represent the seismic fragility function [7–9]. In line of the above, many researches have also been carried out to evaluate the failure probability of buried pipelines in terms of using numerical analysis. Newmark and Rosenblueth [10] showed the deformation of pipeline using free-field ground deformation and curvature. Wang and Yeh [11] proposed a simplified seismic analysis method for fault movements. In their research, the parametric response analysis was performed with regard to various fault movements, crossing angle of earthquake, various burial depths and pipeline diameters. Moreover, Lee et al. [12] performed seismic behavior of buried gas pipeline according to different burial depth, buried type, and boundary conditions. More recently, Xie et al. [13] performed numerical analysis on the buried high-density polyethylene pipeline subjected to normal faulting. For considering soil-structure interaction, both 3-D and 2-D models were utilized and the results of numerical simulation were compared with centrifuge test. Uncertainty of soil variables can affect the response of buried structures due to geotechnical diversity such as inherent variability, and insufficient data. Since the geotechnical diversity is generally originated from many disparate sources, the parameters characterizing a soil can be estimated through a probabilistic approach. Vanmarcke [14] proposed a method for modeling the natural variability of soil
Corresponding author. E-mail address: [email protected] (H.-J. Jung).
https://doi.org/10.1016/j.ijpvp.2019.103932 Received 16 April 2019; Received in revised form 24 June 2019; Accepted 12 July 2019 Available online 13 July 2019 0308-0161/ © 2019 Published by Elsevier Ltd.
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corresponding force-displacement relationship of the soil spring in each direction. In Fig. 1, solid line is a general load-displacement relation of the surrounding soil domain in each direction, while dashed line is a simplified bilinear curve to represent the relationship. Soil mass and stiffness due to a burial depth are taken into account in the calculation of the force-displacement relationship of the springs. Detailed description of the relationship is discussed below. The maximum spring force per unit length of the pipeline in the axial direction is defined by
properties considering the statistical characteristics of soil profiles. Phoon and Kulhawy [15] suggested the uncertainty of soil parameter using in-situ measurement and transformation model according to soil and test types. Besides, various studies have also been conducted on the soil-structure interaction considering input parameters with uncertainty. Jin et al. [16] conducted seismically excited response variability of multi-story structure considering soil-structure interaction and uncertainty of soil parameters. Chaudhuri and Gupta [17] investigated seismic response of secondary systems considering uncertainty on soil parameters. Moreover, Gallage et al. [18] evaluated seismic response of buried segmented pipeline considering effect of uncertainty of the soil parameters. Imanzadeh et al. [19] estimated coefficient of the soil reaction modulus of buried pipeline taking into account the effect of uncertainty in soil and structure parameters. However, there were few studies on the seismic response of buried pipeline using nonlinear time history analysis including both material inelasticity and geometry nonlinearity. Most of the previous studies have mainly focused on the seismic fragility relationships between seismic intensity and corresponding structural performance by past earthquakes. Moreover, regarding soil uncertainty, most of the studies were limited to above-ground structures. This implies that salient soil features such as internal friction angle, unit weight and cohesion were not taken into account in the studies. In view of the above, the seismic fragility analysis of a buried pipeline has been conducted in terms of using nonlinear time history analysis and considering uncertainty of soil parameters. For this purpose, buried pipe and surrounding soil are modeled by beam elements and equivalent springs, respectively, and soil-pipeline interaction is represented by beam on nonlinear-Winkler foundation (referred to as BNWF hereafter) model. To analyze the sensitivity of the structural response to soil uncertainty, the first order-second moment (FOSM) analysis is performed and the relative variance is calculated according to the seismic intensity. Lastly, but by no means at least, seismic fragility curves are derived and compared according to the soil parameters.
Tu = πDαc + πDHγ‾
1 + KO tan δ 2
α = 0.608 − 0.123c −
0.274 0.695 + 3 c2 + 1 c +1
(1)
(2)
where α is an adhesion factor, D is outer diameter of pipeline, c is the soil cohesion representative of the soil backfill, H is distance to centerline of pipe from ground surface, γ‾ is effective unit weight of soil, K O is coefficient of pressure at rest, δ is interface friction angle between pipe and soil (=fφ) , φ is internal friction angle, and f is coating factor relating soil-pipeline interface. Calculated displacement corresponding to the maximum soil force is 3 mm, 5 mm, 8 mm, and 10 mm for dense sand, loose sand, stiff clay, and soft clay, respectively, as suggested by the American Lifeline Alliances [20]. The maximum transverse soil spring force per unit length of the pipe is given by
Pu = Nch cD + Nqh γHD ‾ Nch = a + bx +
(3)
c d + ≤ 9 (0 for c = 0) (x + 1)2 (x + 1)3
Nqh = a + bx + cx 2 + dx 3 + ex 4 (0 for φ = 0)
(4) (5)
where Nch and Nqh are horizontal bearing capacity factor for clay and sand, respectively and a, b, c, d, and e are empirical constants that were determined from published empirical results in Table 1 (Nqh can be interpolated to obtain intermediate values of internal friction angle between 20° and 45°), and x is ratio between depth of buried distance (H) and outer diameter (D). The displacement corresponding to the maximum soil force is determined by
2. BNWF model for soil-pipeline interaction In the present study, the BNWF model was employed to simulate the soil-pipeline interaction in the following nonlinear seismic response analysis of a buried pipeline. In the BNWF model, the interaction between soil and pipeline can be modeled as a discrete nonlinear spring (e.g., symmetric or asymmetric elastoplastic curve). American Lifeline Alliances [20] proposed the relationship between force and displacement of the equivalent soil spring based on the results from the field experiments and laboratory tests. The force-displacement relationship of the spring in each direction is assumed to be linearly elastic-perfectly plastic, hence bilinear. Fig. 1 illustrates the BNWF model and
D Δp = 0.04 ⎛H + ⎞ ≤ 0.1D − 0.15D 2⎠ ⎝
(6)
The maximum vertical bearing spring force per unit length of the pipe is determined by
Qd = Nc cD + Nq γHD + Nγ γ ‾
D2 2
Fig. 1. Representation of soil-pipeline interaction using the BNWF model [25,26]. 2
(7)
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Table 1 Empirically estimated horizontal bearing capacity factor. Factor
φ (degree)
x
a
b
c
d
Nch Nqh
0° 20°
H/D H/D
6.752 2.399
0.065 0.439
−11.063 −0.03
7.119
–
1.059 (10−3)
−1.754 (10−5)
Nqh
25°
H/D
3.332
0.839
−0.09
5.606 (10−3)
−1.319 (10−4)
Nqh
30°
H/D
4.565
1.234
−0.089
4.275 (10−3)
−9.159 (10−5)
Nqh
35°
H/D
6.816
2.019
−0.146
7.651 (10−3)
−1.683 (10−4)
Nqh
40°
H/D
10.959
1.783
0.045
−5.425 (10−3)
−1.153 (10−4)
Nqh
45°
H/D
17.658
3.309
0.048
−6.443 (10−3)
−1.299 (10−4)
the ground surface are neglected. Second, the inner pressure due to the fluid flow inside the pipeline is not taken into account. Third, the ground water level is set below the pipeline structure. Therefore, the uplift effect of the pipeline due to buoyancy is not considered, and hence the effective unit weight (γ‾ ) and unit weight (γ ) of the soil are identical. Fourth, the soil condition is assumed as a dense silty soil of Southern California. This assumption is somewhat intuitive. Since soil classification in most of Korean seismic design codes is derived from UBC [22], soil conditions as in the case of UBC are employed. It is qualitatively acknowledged that soil condition can be varied along the pipeline length. However, the current study is to investigate nonlinear response and corresponding seismic fragility analyses of the pipeline, which causes quantitative difficulty even for a single soil condition. Thus, the present study is focused on the dense silty of single condition. Subsequently, constant soil profile along the pipeline length is assumed in the numerical modeling. Finally, uncertainty is considered for soil only. This is due to the fact that the pipeline has relatively small material uncertainty since it is mainly a mass production. However, in the case of soil, there are many of uncertainty factors, such as measurement errors, inherent variability and etc. Subsequently, uncertainty of soil parameters are taken into account in the present analyses.
Nc = [cot(φ + 0.001)] ⎧exp [π tan(φ + 0.001)]tan2 ⎛45 + ⎨ ⎝ ⎩
φ + 0.001 ⎞ − 1⎫ ⎬ 2 ⎠ ⎭ (8)
φ Nq = exp (π tan φ)tan2 ⎛45 + ⎞ 2⎠ ⎝
(9)
Nγ = exp (0.18φ − 2.5)
(10)
where Nc , Nq , and Nγ are the bearing capacity factors and γ is total unit weight of soil. Corresponding displacement at Qd is 0.1D for granular soils and 0.2D for cohesive soils. The maximum vertical uplift spring force per unit length of pipe is given by
Qu = Ncv cD + Nqv γHD ‾
(11)
H Ncv = 2 ⎛ ⎞ ≤ 10 (0 for c = 0) ⎝D⎠
(12)
φH ⎞ ≤ Nq (0 for φ = 0) Nqv = ⎛ 44 ⎝ D⎠
(13)
e
where Ncv and Nqv are the vertical uplift factor for clay and sand, respectively and Nq is bearing capacity factor described in Eq. (9). Corresponding displacement at Qu is calculated by 0.01H–0.02H for dense to loose sands (< 0.1D) and 0.1H–0.2H for stiff to soft clays (< 0.2D). A more detailed description of the equations are provided in the American Lifeline Alliance, ASCE [20].
3.2. Uncertainty of soil parameters The three soil parameters that control the soil stiffness in the BNWF model were selected for the uncertainty of soil conditions. The internal friction angle, cohesion, and unit weight were considered as random variables with Gaussian distribution, while remaining parameters such as coefficient of pressure and coating factor were set as fixed parameter conditions. Table 3 shows the mean value and coefficient of variation (COV) of soil input parameters obtained by previous research results [23]. The COV value of soil parameters is assumed as 10% in the present study. Previous experimental work [24] indicated that the COV values showed a deviation in the three parameters. Considering the COV results of various soil types (sand, clay, silt), 10% of COV seemed to be reasonable, particularly in the absence of sufficient experimental data with regard to dense silt sand. The assumption that the three soil parameters are independent may have a significant influence on the results of random sampling. Various studies have been conducted to measure correlation between the parameters. However, no such relation has been estimated. Therefore, based on limited work [23], correlation matrix between the parameters for dense silty soil is suggested and summarized in Table 4. The suggestion given in Table 4 is supported by sound engineering judgment and relatively good agreement obtained between sensitivity and nonlinear time history analysis. There is certainly room for improvements
3. Numerical model of a buried pipeline 3.1. Numerical modeling assumption In order to carry out the nonlinear time history analysis of a buried gas pipeline, API 5L X65 which is widely utilized in South Korea was selected [12]. The material properties of the pipeline are shown in Table 2. The cross section of the pipeline is a circular steel hollow section with an outer diameter of 762 mm and an inner diameter of 727 mm. The pipeline is assumed to be a rough steel and thus friction coefficient of 0.8 is used. A straight pipeline with a length of 1.2 km and a distance of 1.881 m from the ground to the center line of pipe crosssection was employed in the subsequent analyses. As for the boundary condition, both ends of the pipeline were assumed as fixed end constraints. This can be attributed to the fact that the behavior of the pipeline does not depend on the constraint condition [21]. For numerical modeling of the buried pipeline, the following five conditions were assumed. First, the live load and dead load applied on Table 2 Material properties of API 5L X65 gas pipeline. Mass density (kg/m3 )
Elastic modulus (GPa)
Poisson's ratio
Yield strength (MPa)
Outer diameter (mm)
Thickness (mm)
Coefficient of friction
7850
210.7
0.3
445
762
17.5
0.8
3
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4. Input ground motions and eigenvalue analysis
Table 3 Mean and coefficient of variation for soil parameters. Parameter
Mean
COV (%)
Distribution
Internal friction angle Cohesion Unit weight
38° 5 kPa
10 10 10
Gaussian Gaussian Gaussian
18 kN/m3
4.1. Input ground motions Due to random nature of earthquakes, input ground motions have a lot of uncertainties regarding magnitude, duration, frequency contents, peak ground acceleration, spectral intensity, soil condition and so on. Subsequently, it is onerous to predict the seismic structural response according to a specific hazard level deterministically, although it is not completely impossible. To get over this, special care should be needed in selecting the input ground motions covering as many uncertainties as possible. For this purpose, input ground motion suites including various characteristics of seismic waves were selected in the present study [27]. Table 6 shows the selected 12 input ground motions. As observed in Table 6, various levels of peak ground acceleration (PGA) were taken into account. Since South Korea is classified as medium to low seismicity area, extreme event cases are intentionally excluded in the selection of the input ground motions. In the subsequent analyses, these PGAs were scaled from 0.1g up to 1.5g with an interval of 0.2g. Rationale behind this selection can be supported by Fig. 2. Fig. 2 shows acceleration response spectra of the 12 motions in axial, transverse, and vertical directions. As plotted in Fig. 2, frequency contents of the input motions are relatively well-distributed and the maximum spectral accelerations at the maximum amplifications exhibit various levels. It is thus postulated that current selection seems to be reasonable enough to consider uncertainty of input ground motion.
Table 4 Correlation matrix between the soil parameters. Correlation matrix
Cohesion
Internal friction angle
Unit weight
Cohesion Internal friction angle Unit weight
1 −0.7 0
−0.7 1 0.3
0 0.3 1
as more correlation results become available.
3.3. Numerical model of buried pipeline To perform earthquake response analysis considering soil-pipeline interaction, commercial software ABAQUS was utilized [25,26]. ABAQUS provides two-dimensional or three-dimensional pipe-soil interaction element to represent the nonlinear behavior of buried gas pipeline and surrounding soil. The pipe-soil interaction model is simply expressed as a constitutive model that defines the stiffness of the soil region without discretizing the soil domain into finite elements. Fig. 1 shows a graphical representation of the pipe-soil interaction model. The pipe-soil interaction element is defined using far-field node and pipeline node, and the behavior of pipe-soil interaction element is determined by constitutive equation proposed by ASCE [26]. In this model, input earthquake is propagated to the buried pipeline structure through an equivalent spring that reflects the characteristics of the soil profile. In this study, the 3-D pipe-soil interaction element (PSI34) has been utilized to conduct the soil-structure interaction analysis. The PSI34 element represents nonlinear constitutive behavior by modeling the interaction of buried pipeline structure and surrounding soil region [26]. For numerical simulation, the 1.2 km buried pipe and soil domain consist of 1000 numbers of pipe element (PIPE31) and PSI34 elements, and the damping ratio of the system is assumed as 5% to critical damping using the Rayleigh damping coefficient. Detailed description of the pipe-soil element is described in ABAQUS manual [25]. For considering the uncertainty of soil parameters, the random samples were generated by using Latin Hypercube Sampling (LHS) techniques. From obtained soil parameters, the maximum spring force and corresponding displacement were calculated using Eqs. (1)–(13). For a soil region, the stiffness (slope of the load-displacement curve) for tension and compression has the same slope because such region has an infinite domain in both axial and transverse directions. However, since the soil domain corresponding to the burial depth is not symmetric, the stiffness in vertical direction has two different values. Table 5 shows the maximum spring force and corresponding displacement for Southern LA regions, when each soil parameter has mean value.
4.2. Eigenvalue analysis An eigenvalue analysis was conducted for the straight pipeline embedded in dense silty sand using Lanczos algorithm [28]. The first three modes of vibrations are evaluated in vertical and transverse directions. The first three natural frequencies in the vertical direction were 14.507, 14.575 and 14.689 Hz, respectively, and those in the transverse direction were 15.669, 15.732 and 15.836 Hz, respectively. Meanwhile, lower modes of vibration were not observed in the axial direction. The eigenvalue analysis results indicate that natural frequencies between the modes are so close in both directions. Fig. 3 shows the representative first to third mode shapes in each direction. Moreover, the mass participation factors become more than 90% when the first to third mode are included in the analysis. It is therefore expected that higher natural frequency effects can be of significance, which stresses the importance of the nonlinear analysis conducted in the current study. 5. Nonlinear time history analysis 5.1. Results of seismic response analysis In this section, the variation of structural responses due to soil parameter uncertainties was investigated. For this purpose, an efficient Monte Carlo Simulation (MCS) employing LHS [29] was performed and nonlinear time history analysis was conducted by scaling PGA of the input ground motions. Choice of the LHS lies in the fact that in comparison with other random sampling methods, the LHS provides statistically significant results with fewer samples. Numerical analyses were carried out for the following four cases: 1) variation of all parameters considering correlation matrix; 2) variation of internal friction angle only (cohesion and unit weight are mean values); 3) variation of cohesion only (internal friction angle and unit weight are mean values); 4) variation of unit weight only (internal friction angle and cohesion are mean values). For nonlinear time history analysis, the maximum strain response of the pipeline was investigated for the above four cases. In general, if the boundary conditions of the buried pipeline were fixed support, the maximum axial strain usually occurred at the ends. Therefore, the
Table 5 Maximum spring force and corresponding displacement. Spring direction
Maximum force (N/ mm)
Corresponding displacement (mm)
Axial Transverse vertical (uplift) vertical (downward)
45.16 377.89 73.81 1897.09
3 90.48 18.81 76.2
4
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Table 6 Selected input ground motions. Earthquake (Year)
Station
PGA (g)
Magnitude
Mechanism
EQ1 EQ2 EQ3
Gulf of Aqaba (1995) Borrego (1968) Cape Mendocino (1992)
0.109 0.130 0.178
7.2 6.63 7.01
Strike slip Strike slip Reverse
EQ4 EQ5 EQ6
Chi-Chi (1999) Chi-Chi (1999) Coalinga-01 (1983)
0.263 0.198 0.194
7.62 7.62 6.36
Reverse Oblique Reverse Oblique Reverse
EQ7 EQ8 EQ9 EQ10 EQ11 EQ12
Imperial Valley (1979) Kocaeli (1999) Landers (1992) Loma Prieta (1989) San Fernando (1971) Tabas (1978)
Eliat El Centro array #9 Eureka-Myrtle & West CHY014 ILA067 Park field Fault Zone 1 Delta Fatih Indio-Coachella Canal Hayward-BART Whittier Narrows Dam Ferdows
0.351 0.187 0.109 0.159 0.107 0.108
6.53 7.51 7.28 6.93 6.61 7.35
Strike slip Strike slip Strike slip Reverse Oblique Reverse Reverse
Fig. 2. Acceleration response spectra of the selected input ground motions.
maximum axial strain occurring around the fixed ends was adopted in this study. Fig. 4 shows a representative case of the axial strain distribution at a critical time step subjected to San Fernando earthquake of 1.5g of PGA. The maximum strain was observed around the fixed end, and the strain distribution was gradually decreased with the distance away from the fixed end. This indicates that weak region of the buried pipeline can be a joint connecting each pipeline. Fig. 5 shows the mean and standard deviation of the normalized strain responses according to the number of samples subjected to PGAs of 0.1g, 0.5g, 0.9g, and 1.5g. The mean and standard deviation of the response were normalized to the response for number of sampling of 60.
As observed in general, the response variation gradually converged as the number of samples increased. In addition, while the variation of the normalized mean value according to the number of samples was not significant, that of the standard deviation value was clearly observed, particularly up to 40 of number of sampling. This implies that a proper number of sampling can be of utmost importance in balancing analysis cost and accuracy. Fig. 5 also shows that the normalized mean and standard deviation values tend to converge around 50 samplings. Therefore, 60 samplings were employed to represent the seismic fragility curve of the pipeline in the present study. Fig. 6 shows the distribution of the maximum strain response 5
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Fig. 3. Representative mode shapes and mass participation factor.
Fig. 4. General trend of strain distribution of buried pipeline with fixed ends.
according to different levels of PGA intensities for both fixed soil parameters with mean values (as listed in Table 3) and consideration of the uncertainty of soil parameters. As the intensity of the PGA increases, the buried pipeline shows a large variation in the maximum strain response due to nonlinearity. It is worthy of noting that whereas 96 analyses were carried out for the model without consideration of soil uncertainty, 5760 analyses were conducted for each soil variable in case of considering soil uncertainty. Figs. 7 and 8 show the mean and standard deviation of the maximum strain response of the pipeline according to PGA intensities. The mean value of the maximum strain was increased when soil uncertainty was considered. However, the difference due to the uncertainty of each soil parameter was not significant. Meanwhile, the standard deviation of the maximum strain exhibited a discrepancy between the responses, resulting in relatively low response for the case of cohesion. In
particular, nonlinearity was observed at 0.5g, where the standard deviation of maximum strain was abruptly increased. Based on observation of Figs. 7 and 8, unit weight seemed to be the most salient parameter affecting the strain response of the pipeline. Further investigation has been carried out in order to verify the influence of each soil parameter quantitatively, and is discussed in the following section. 5.2. Sensitivity analysis The FOSM analysis was carried out to investigate the significance of soil parameters affecting nonlinear seismic response of the buried pipeline [30]. For this purpose, each soil parameter is assumed to be statistically independent each other and sensitivity analysis was performed for each soil parameter within a range of the mean ± one standard deviation. In addition, to simplify the relationship between the 6
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Fig. 5. Convergence of normalized strains for mean (left) and standard deviation (right).
7
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Fig. 6. Maximum strain response of the pipeline with and without considering uncertainty of soil parameters. N
structural response and the soil parameters, it is assumed that the variables and responses have a linear relationship. Let's assume that F is the structural response function of X having N random variables, and Y is the structural response to the random variable. Thus,
Y = F (X1 , X2 , ⋯, XN ).
Y ≅ F (μ X1 , μ X2 , ⋯, μ XN ) +
∑ (Xi − μ Xi ) i=1
∂F (X1 , X2 , ⋯, XN ) . ∂Xi
(15)
The mean of the response function μY represents the structural response when N random variables have a mean value, and the standard deviation of response function σY2 stands for variation of the response due to the variance of N random variables. Thus, the mean and standard deviation of the response function Y are calculated by following expressions.
(14)
In order to calculate the equation above numerically, Taylor expansion is applied to the function F and higher order terms are ignored. Thus, the above equation can be simplified as
μY ≅ F (μ X1 , μ X2 , ⋯, μ XN ) 8
(16)
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Fig. 7. Average strain response of pipeline. N
σY2 ≅
2
importance of the soil parameters was calculated using Eqs. (14)–(18). As shown in Fig. 10, unit weight has a significant effect on the buried pipeline, while cohesion has a little impact on the pipeline behavior. In the intermediate PGA range, the relative variance of internal friction angle and cohesion increases slightly, and the variance of unit weight decreases abruptly. Nonetheless, unit weight can be considered as a dominant factor affecting the pipeline response. In the case of unit weight, the soil mass is determined, which has a great influence (relative variance: 0.7) on the determination of the inertia force when the input ground motion occurs. On the other hand, the internal friction angle has an important influence (relative variance: 0.25) on the determination of soil stiffness. Therefore, the dynamic behavior of the buried pipeline is confined and can affect the deformation of the structure depending on the soil stiffness. Since cohesion (relative variance: 0.05) has little effect on soil mass and stiffness, it does not significantly affect the behavior of sandy soil with uncertainty. This is particularly true since the unit weight has the greatest effect on the mass and equivalent stiffness of spring in comparison with other parameters.
∑ σX2i ⎛ ∂F (X1, X2, ⋯, XN ) ⎞ ⎜
∂Xi
⎝
i=1
+
⎟
N
N
i
j
⎠
∑ ∑ ρ Xi Xj ∂F (X1, ⋯, XN ) ∂F (X1, ⋯, XN ) ∂Xi
∂Xj
(17)
Since the soil parameters are assumed to be independent ⎛⎜ρ Xi Xj =0), ⎝ the partial derivative of X for the response function F can be represented using the central difference method. Thus, the variance of the response function can be calculated numerically as given by equation (18). 2
⎛ ∂F (X1 , X2 , ⋯, XN ) ⎞ ∂Xi ⎝ ⎠ ⎜
⎟
2
F (μ1, ⋯, Xi + Δ Xi , μN ) − F (μ1, ⋯, Xi − Δ Xi , μN ) ⎞ = ⎜⎛ ⎟ 2Δ Xi ⎠ ⎝
(18)
Fig. 9 shows the normalized maximum strain response of the pipeline according to variation of soil parameters at four levels of PGAs. In Fig. 9, the maximum strains were normalized to the minimum value of soil parameters. For small PGA of 0.1g, the pipeline response remains essentially in elastic range, regardless of soil parameters. However, the response excurses inelastic range as the PGA increases. In particular, significant increase is observed in the response at 0.9g. This can be attributed to the frequency characteristics of the applied input ground motions. Particular emphasis is placed on the response at 1.5g. Normalized strain response at 1.5g is in general less than that of 0.5g and 0.9g. This can be due to the fact that the pipeline strains experience inelastic response mainly in intermediate PGA range as depicted in Fig. 8. Based on the structural response shown in Fig. 9, the relative
6. Seismic fragility curve 6.1. Definition of damage state The seismic fragility curve represents the conditional failure probability that an event exceeds the damage state for a certain level of earthquake. Damage state can be determined by a statistical method for predicting seismic intensity and physical response of structures induced by previous earthquakes. The need for quantitative evaluation of the damage state for a buried pipeline has been continually increased. However, very few studies have been available regarding the quantitative definition of damage state. In addition, almost all of the studies
Fig. 8. Standard deviation of strain response of pipeline. 9
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Fig. 9. Normalized strain with regard to soil parameters for individual input ground motions.
order to derive seismic fragility curve for the buried pipeline. They suggested the damage state through the maximum strain of the pipeline utilizing the seismic risk analysis for a buried pipeline network. Damage state is classified as major, moderate, and minor corresponding to
do not have standard criteria for strain-based pipeline damage states representing intact, leakage and breakage. Amongst the very few studies defining the damage state of the buried pipeline, the damage state proposed by Shinozuka et al. [31] is employed in the present study in 10
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complete loss (break), partial loss (leakage), and no loss (including minor leakage). Table 7 shows the damage state of the buried pipeline used in this study. In Table 7, εp represents the maximum strain of the buried pipeline under earthquake loading, and εy stands for the yield strain of the buried pipeline. 6.2. Seismic fragility analysis The results of nonlinear time history analyses have been utilized to derive the seismic fragility curve for the buried pipeline according to the damage states. In order to represent the seismic fragility curve, the method proposed by Shinozuka et al. was adopted [32,33]. In the methodology, the seismic fragility curve of the structure was assumed to be a log-normal distribution relationship between peak ground acceleration and failure probability. Empirical evaluation of seismic fragility curves in terms of structural damage experienced by previous earthquakes has proved the validity of the logarithmic normal distribution. The maximum likelihood method was employed to calculate two variables (median and log standard deviation) of the log-normal distribution, and the likelihood function is defined by
Fig. 10. Relative variance of each soil parameter according to PGA intensity.
N
Table 7 Damage state of the buried pipeline proposed by Shinozuka et al. [31].
L=
∏ [F (IMi)]xi [1 − F (IMi)](1−xi) i=1
Damage state
Structural response (Maximum strain)
Minor
εp ≤ 0.7εy
Moderate
0.7εy ≤ εp ≤ εy εp ≥ εy
Major
F (IMi ) = Φ ⎜⎛ ⎝
ln(IMi )/ ck ⎞ ⎟ ξk ⎠
(19)
(20)
where F (IMi ) is the seismic fragility function following standard normal distribution when the pipeline is subjected to the seismic intensity of
Fig. 11. Comparisons of seismic fragility curves with and without considering uncertainty of soil parameters. 11
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Table 8 Median and log-standard deviation for seismic fragility curves. Variable
Moderate Major
Fixed soil
All parameters
Internal friction angle
Cohesion
Unit weight
ck
ξk
ck
ξk
ck
ξk
ck
ξk
ck
ξk
0.642 0.917
0.657 0.654
0.513 0.813
0.61 0.591
0.527 0.832
0.595 0.599
0.535 0.832
0.592 0.593
0.525 0.822
0.601 0.592
Fig. 12. Comparisons of seismic fragility curve (a) moderate damage state (b) major damage state.
standard deviation of seismic fragility curve are lower than those for fixed conditions of soil parameters. Lower mean values in the seismic fragility curves indicate higher failure probability, hence more vulnerable to earthquakes. This stresses the importance of the soil uncertainty in the fragility curves. In addition, lower log-standard deviation points out that the failure probability increases rapidly for a certain hazard-level of earthquakes, hence again a significance of including the soil uncertainty [34]. As discussed, the failure probability increases rather abruptly with the consideration of the soil uncertainty. This result implies that deformation of the buried pipeline may arrive at the moderate damage state even for a relatively small PGA, leading to an alteration in the fragility curves hence higher probability of failure. Moreover, the fragility curves corresponding to major damage state show that failure probability with consideration of the soil uncertainty increases rather rapidly than that without taking into account the soil uncertainty, particularly at 0.7g of PGA. This also confirms a tendency obtained for the moderate damage state. Meanwhile, a little difference is achieved in
IMi , and ck and ξk represent the median and log-standard deviation of the lognormal distribution for the k-th damage state, respectively. To obtain the ck and ξk that maximize the likelihood function, the partial differential of each parameter can be expressed as
∂ln(L) ∂ln(L) = =0 ∂ck ∂ξk
(21)
where k represents the number of defined damage states, and it was classified as two states (major and moderate) in the current study because there was no minimum strain in the minor state. Fig. 11 compares the seismic fragility curves according to uncertainty of soil parameters compared with fixed soil parameters (without uncertainty) when 12 input seismic loads were applied. The median and log-standard deviation of each seismic fragility curve are summarized in Table 8. The solid line shows the seismic fragility curve for the fixed conditions of the soil parameters, while the dashed line exhibits the seismic fragility curve considering the soil uncertainty. When the soil uncertainty is taken into account, mean value and log12
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between the fragility curves for the three soil parameters as displayed in Fig. 12. This can be attributed to the damage states. The damage states adopted herein do not seem to reflect a physical behavior of the pipeline under consideration precisely. Nevertheless, it is seemed that the present results may give a useful information with regard to the seismic fragility analysis of the buried pipeline, particularly for the inclusion of the soil uncertainty. It is thus worthy of noting that a better result is expected to achieve when more realistic damage states are available.
[2] C. Scawthorn, P.I. Yanev, 17 January 1995, Hyogo-ken Nambu, Japanese earthquake, Eng. Struct. 17 (3) (1995) 146–157. [3] W.W. Chen, B.J. Shih, C.W. Wu, Y.C. Chen, Natural gas pipeline system damages in the Ji-Ji earthquake (The City of Nantou), Proceedings of the Sixth International Conference on Seismic Zonation, Palm Springs, Riviera Resort, CA, 2000. [4] T. Katayama, K. Kubo, N. Sato, Earthquake damage to water and gas distribution systems, Proceedings of the US National Conference on Earthquake Engineering, Oakland, CA, 1975. [5] R. Eguchi, Seismic vulnerability models for underground pipes, Proceedings of Earthquake Behavior and Safety of Oil and Gas Storage Facilities, Buried Pipelines and Equipment, American Society of Mechanical Engineers, New York, 1983, pp. 368–373. [6] M. O'Rourke, G. Ayala, Pipeline damage due to wave propagation, J. Geotech. Eng. 119 (9) (1993) 1490–1498. [7] T.D. O'Rourke, S. Toprak, Y. Sano, Factors affecting water supply damage caused by the Northridge Earthquake, Proceedings of the 6th U.S National Conference on Earthquake Engineering, Seattle, WA, 1998. [8] M. O'Rourke, E. Deyoe, Seismic damage to segmented buried pipe, Earthq. Spectra 20 (4) (2004) 1167–1183. [9] O. Pineda-Porras, M. Ordaz, A new seismic intensity parameter to estimate damage in buried pipelines due to seismic wave propagation, J. Earthq. Eng. 11 (5) (2007) 773–786. [10] N. Newmark, E. Rosenblueth, Fundamentals of Earthquake Engineering, PrenticeHall Inc., New Jersey, 1971. [11] L.R.L. Wang, Y.H. Yeh, A refined seismic analysis and design of buried pipeline for fault movement, Earthq. Eng. Struct. Dyn. 13 (1) (1985) 75–96. [12] D.H. Lee, B.H. Kim, H. Lee, J.S. Kong, Seismic behavior of a buried gas pipeline under earthquake excitations, Eng. Struct. 31 (5) (2009) 1011–1023. [13] X. Xie, M.D. Symans, M.J. O'Rourke, T.H. Abdoun, T.D. O'Rourke, M.C. Palmer, H.E. Stewart, Numerical modeling of buried HDPE pipelines subjected to normal faulting: a case study, Earthq. Spectra 29 (2) (2013) 609–632. [14] E.H. Vanmarcke, Probabilistic modeling of soil profiles, J. Geotech. Eng. Div. 103 (11) (1977) 1227–1246. [15] K.-K. Phoon, F.H. Kulhawy, Characterization of geotechnical variability, Can. Geotech. J. 36 (4) (1999) 612–624. [16] S. Jin, L. Lutes, S. Sarkani, Response variability for a structure with soil–structure interactions and uncertain soil properties, Probabilistic Eng. Mech. 15 (2) (2000) 175–183. [17] S.R. Chaudhuri, V. Gupta, Variability in seismic response of secondary systems due to uncertain soil properties, Eng. Struct. 24 (12) (2002) 1601–1613. [18] C. Gallage, R. Pathmanathan, K. Jayantha, Effect of soil parameter uncertainty on seismic response of buried segmented pipeline, 1st International Conference on Geotechnique, Construction Materials and Environment, Tsu City, Mie, Japan, 2011. [19] S. Imanzadeh, A. Denis, A. Marache, Effect of uncertainty in soil and structure parameters for buried pipes, Proceedings of 4th International Conference on Site Characterization, Porto de Galinhas, Pernambuco, Brazil, 2012. [20] ALA, (Ameriean Lifeline Alliance), Guidelines for the Design of Buried Steel Pipe, American Society of Civil Engineers, 2001. [21] T. Datta, E. Mashaly, Seismic response of buried submarine pipelines, J. Energy Resour. Technol. 110 (4) (1988) 208–218. [22] UBC, Uniform building code, Int. Conf. Build. Offic. (1997) Chapter 16, division IV. [23] P. Raychowdhury, Effect of soil parameter uncertainty on seismic demand of lowrise steel buildings on dense silty sand, Soil Dyn. Earthq. Eng. 29 (10) (2009) 1367–1378. [24] A.L. Jones, S.L. Kramer, P. Arduino, Estimation of Uncertainty in Geotechnical Properties for Performance-Based Earthquake Engineering, Pacific Earthquake Engineering Research Center, College of Engineering, University of California, 2002. [25] H. Hibbit, B. Karlsson, E. Sorensen, ABAQUS User Manual, Simulia, Providence, RI, 2012version 6.12. [26] J. Audibert, D. Nyman, T. O'Rourke, Differential Ground Movement Effects on Buried Pipelines, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, (1984), pp. 150–183. [27] D.H. Lee, B.H. Kim, S.-H. Jeong, J.-S. Jeon, T.-H. Lee, Seismic fragility analysis of a buried gas pipeline based on nonlinear time-history analysis, Int. J. Steel Struct. 16 (1) (2016) 231–242. [28] C. Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, United States Governm. Press Office Los, Angeles, CA, 1950. [29] M. Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29 (2) (1987) 143–151. [30] F.S. Wong, First-order, second-moment methods, Comput. Struct. 20 (4) (1985) 779–791. [31] M. Shinozuka, S. Takada, H. Ishikawa, Some aspects of seismic risk analysis of underground lifeline systems, J. Press. Vessel Technol. 101 (1) (1979) 31–43. [32] M. Shinozuka, M.Q. Feng, H.-K. Kim, S.-H. Kim, Nonlinear static procedure for fragility curve development, J. Eng. Mech. 126 (12) (2000) 1287–1295. [33] M. Shinozuka, M.Q. Feng, J. Lee, T. Naganuma, Statistical analysis of fragility curves, J. Eng. Mech. 126 (12) (2000) 1224–1231. [34] M. Shinozuka, M. Feng, H. Kim, T. Ueda, Statistical Analysis of Fragility Curves, Technical Report at Multidisciplinary Center for Earthquake Engineering Research, NY, USA, 2002.
7. Concluding remarks In the present study, seismic fragility analyses have been performed for a buried gas pipeline considering uncertainty of soil parameters. To achieve this objective, nonlinear time history analyses have been carried out using selected 12 earthquake ground motions. The BNWF model was employed to represent soil-pipeline interaction, and the LHS scheme was adopted to take into account the uncertainty of soil parameters. In addition, sensitivity analyses have been performed to evaluate the effect of soil parameters on the response of the pipeline. Fragility analyses for the pipeline have also been conducted using the nonlinear time history analysis results. Based on analytical predictions obtained in the study, following conclusions are discussed.
• Nonlinear time history analyses of the pipeline show that strain
• •
response considering the uncertainty of soil parameters is greater than that without the uncertainty. In addition, the behavior of buried pipeline taking into account the uncertainty of soil parameters indicates that unit weight is a dominant parameter amongst others. Sensitivity analyses reveal that relative importance is calculated in the strain response for unit weight parameter. This result is similar to that obtained in the nonlinear time history analyses, and stresses importance and necessity of the sensitivity analysis in association with the soil uncertainty. Seismic fragility curves considering the soil uncertainty show higher failure probability than those without the uncertainty. In addition, failure probability considering the soil uncertainty increases rapidly in comparison with that without the uncertainty. It is therefore noteworthy that the present fragility analyses with the soil uncertainty are expected to enable the assessment of the failure probability of the pipeline to be captured accurately, particularly considering a difficulty of access to the underground pipeline.
In all, as demonstrated in the above discussion, analytical observation is affected by the uncertainty of soil parameters. The predictions presented in the current study however are derived in terms of using three soil parameters only due to shortage of available geotechnical data. It is thus promising that the present study can further be improved when more information on the geotechnical properties of soil become available, hence correlation between soil parameters. Besides, more improvement is also anticipated when realistic damage state reflecting physical behavior of the pipeline is suggested. Acknowledgements This work was supported by the National Research Foundation Korea (NRF) Grant funded by the Korean government (MSIP) (No. 2017R1A5A1014883). This work was also financially supported by Korea Ministry of Land, Infrastructure and Transport (MOLIT) as Innovative Talent Education Program for Smart City. References [1] Moore Dames, The Loma Prieta earthquake: impact of lifeline systems, J. Disaster Recovery 3 (2) (1999) 8.
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