Engineering Structures 289 (2023) 116284
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Seismic responses of liquid storage tanks subjected to vertical excitation of
near-fault earthquakes
Jie-Ying Wu a, b, Qian-Qian Yu a, b, Qi Peng c, Xiang-Lin Gu a, b, *
a
Key Laboratory of Performance Evolution and Control for Engineering Structures (Tongji University), Ministry of Education, 1239 Siping Road, Shanghai 200092,
China
b
Department of Structural Engineering, College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China
c
School of Civil Engineering, Changsha University of Science & Technology, Changsha 410114, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Vertical seismic component
Near-fault earthquakes
LNG inner tank
Hydrodynamic pressure
Sloshing height
This study aims to evaluate the effect of vertical seismic component on dynamic responses of liquid storage tanks,
with a special focus on near-fault earthquakes. A shaking table test was performed on a 1:25 scaled steel tank,
which was loaded by both far- and near-fault ground motions with horizontal unidirectional, vertical unidi­
rectional, and combined horizontal and vertical bi-directional excitations. Seismic responses, including sloshing
height and hydrodynamic pressure were recorded and compared. Subsequently, a refined finite element (FE)
model of the scaled tank model was established by utilizing the structured Arbitrary-Lagrange-Eulerian (S-ALE)
solver and Fluid-Structure Interaction (FSI) algorithm, which was verified by comparison with the test results.
Afterward, the modeling methodology was further applied to a prototype large-scale liquefied natural gas (LNG)
inner tank for seismic analysis. In addition, seismic responses of the LNG inner tank were compared with the
recommendations given in seismic design codes for cylindrical tanks. The experimental and numerical findings
showed that both the convective and impulsive motions of liquid was amplified by vertical excitations, especially
under near-fault earthquakes. The peak sloshing heights under near- and far-fault earthquakes were increased by
14–142% and 8–56% due to the existence of vertical action, respectively. The LNG inner tank was more
vulnerable to inelastic buckling due to the presence of vertical seismic component. Moreover, the effect of
vertical excitations of near-fault earthquakes on hydrodynamic pressure and tank stresses was underestimated by
the current seismic design codes.
1. Introduction
As a part of lifeline facilities, damages to liquefied natural gas (LNG)
storage tanks under earthquakes may lead to significant economic losses
and environmental pollution. Especially under near-fault earthquakes,
violent liquid sloshing induced by long-period velocity pulse would
cause liquid spillage, increasing risks of fire and explosion. The seismic
design of liquid storage tanks is generally acknowledged that responses
caused by vertical ground motions are significantly less than that of
horizontal ones (such as standards of America [1], European [2], New
Zealand [3] and Japan [4]). However, according to the near-fault
earthquakes, it is found that the peak acceleration of vertical seismic
component may be as high as, or even exceed, that of horizontal com­
ponents, e.g., Northridge earthquake (1994) [5], Chi-Chi earthquake
(1999) [6] and Turkey earthquake (1999) [7]. The underestimation of
vertical seismic components is particularly perilous in the near-fault
events [7–8], such as the Kocaeli earthquake occurred in Turkey that
46 cylindrical tanks were damaged and six tanks burned due to the se­
vere liquid sloshing. The spilled liquid caused the fire spread to adjacent
tanks and burned for three days [9]. Therefore, the influence of vertical
action of near-fault earthquakes on dynamic behavior of liquid storage
tanks needs more attention.
Extensive studies have been conducted on seismic responses of liquid
storage tanks subjected to horizontal ground motions [10–11]. An early
attempt to predict the response of a liquid storage tank was performed
by Housner [12]. The seismic loads caused by liquid inside was divided
into convective and impulsive components, representing sloshing of
liquid in the upper part of the tank and high frequency vibration of the
fluid–structure system, respectively. Due to the thin shell of storage
tanks, the tank wall exhibited considerable deformation under earth­
quakes, which affected the impulsive motion of liquid [13–15]. Based on
* Corresponding author.
E-mail address: gxl@tongji.edu.cn (X.-L. Gu).
https://doi.org/10.1016/j.engstruct.2023.116284
Received 12 December 2022; Received in revised form 16 March 2023; Accepted 3 May 2023
Available online 12 May 2023
0141-0296/© 2023 Elsevier Ltd. All rights reserved.
J.-Y. Wu et al.
Engineering Structures 289 (2023) 116284
Nomenclature
G
pf
convective design response spectrum acceleration
coefficient
Ai
impulsive design response spectrum acceleration
coefficient
Af
acceleration coefficient for sloshing wave height
calculation
vertical earthquake acceleration coefficient
Av
Ag(t)
ground acceleration time-history in the free-field
response acceleration (relative to its base) of a simple
Af(t)
oscillator
Acn(t)
acceleration time-history of the response of a single degree
of freedom oscillator
Av(t)
vertical ground acceleration time-history
dmax
maximum sloshing height obtained from Eurocode 8
D
tank diameter
EEdx EEdy EEdz action effect due to X-, Y- and Z-directional seismic
components, respectively.
h
height of the monitoring point to the tank base
H
height of liquid level,
I1
modified Bessel functions of order one
′
I1
derivative of modified Bessel functions of order one
J1
Bessel function of the first order
Nh
tank hydrostatic membrane force
Ni
impulsive hoop membrane force in tank shell
convective hoop membrane force in tank shell
Nc
pi
spatial pressure distribution of rigid impulsive component
Ac
pc
pvr
PGA
PGD
PGV
R
r, θ, z
SF
( )
Sad Tj
( )
Sar Tj
SDS
Se(Tc1)
Tg
Tp
Y
ρ
δs
σT
the spring-mass analogy proposed by Housner [12], an additional de­
gree of freedom was added to take into account the relative wall
deformation with respect to the ground [16]. In recent years, valuable
insight into seismic responses of storage tanks has been gained,
involving the effect of parameters such as floating roof [17–20], liquid
level [21–23], foundation stiffness [24–27] and excitation frequency
[28]. Amiri and Sabbagh-Yazdi [17] showed that the presence of tank
roof increased tank natural frequencies. Park et al. [29] performed
shaking table tests on a cylindrical storage tank and found that the
response of the tank was governed by beam-type and oval-type vibra­
tions. The vibration tests were carried out on a 1/10 reduced scaled
model, and the result revealed that the oval-type vibration affected the
distribution shape and magnitude of hydrodynamic pressure [30].
Ormeño et al. [24] concluded that the flexible base was beneficial to
axial compressive stresses of the tank but might have an adverse impact
on displacements and accelerations. A shaking table test was performed
on a 1:14 scaled model of LNG tank by Zhang et al. [31], it was shown
that liquid motion increased the stress of the inner steel tank, but
mitigated vibrations of the outer concrete tank.
Although the dynamic behavior of liquid storage tanks under hori­
zontal earthquakes has been widely assessed, the response under vertical
earthquakes has not been fully understood. Vertical seismic energy is
transmitted to the liquid storage tank in the forms of horizontal hy­
drodynamic pressure, which mainly results in an amplified hoop stress
on the tank wall. Haroun and Tayel [32] proposed a prediction model to
analyze the responses of elastic, cylindrical tanks under vertical earth­
quakes. Results showed that the hoop stress caused by the vertical
earthquake was comparable to the hydrostatic hoop stress. Li et al. [33]
performed shaking table tests on a scaled cylindrical oil-storage tanks
and addressed that the magnification effect in the vertical direction was
substantial. Seismic responses of vertically excited liquid storage tanks
considering both the rigid and flexible foundations were further inves­
tigated [34–35]. The soil-structure interaction reduced the effect of
vertical excitations, and such a reduction was related to shear wave
specific gravity
spatial pressure distribution of flexible impulsive
component
spatial pressure distribution of convective component
spatial pressure distribution of vertical ground
acceleration
peak ground acceleration
peak ground displacement
peak ground velocity
tank radius
components of cylindrical coordinates with origin at the
center of the tank
scale factor of seismic ground motions
acceleration design spectrum
acceleration spectrum of the record ground motion
design, 5% damped, spectral response acceleration
parameter at short period (T = 0.2 s) based on ASCE 7
method
elastic response spectral acceleration at the 1st convective
mode of the fluid for damping a value appropriate for the
sloshing response
predominant period of the ground motion
period of the velocity pulse
distance from liquid surface to analysis point
liquid density
maximum sloshing height obtained from API 650
total combined hoop stress in the shell
velocity of the soil and height-to-radius ratio of the tank. Similarly,
Haroun and Abou-Izzeddine [36–37] conducted a comprehensive study
and found that the reduction magnitude of soil-structure interaction on
seismic responses was related to the tank geometries. The responses of
concrete rectangular storage tanks subjected to vertical ground motions
was analyzed by Kianoush and Chen [38]. It was concluded that the base
shear due to vertical acceleration reached 45% of that due to horizontal
accelerations. Ghaemmaghami and Kianoush [39] investigated the
seismic behavior of rectangular tanks considering both the impulsive
and convective liquid motions. The effect of vertical acceleration on
tank response was found to be less significant when horizontal and
vertical excitations were considered together. Kang et al. [40] evaluated
hydrodynamic pressures on a rigid cylindrical tank under both hori­
zontal and vertical excitations by employing computational fluid dy­
namics analysis. It was shown that the vertical motion significantly
increased the hydrodynamic pressure on the tank wall and changed the
pressure distribution.
The studies described above was mainly concentrated on far-fault
earthquakes. In recent years, potential damage resulted from nearfault earthquakes is attracting more attention. Near-fault earthquakes
are known for long-period pulse characteristics due to forward rupture
directivity and fling-step effects, which contain large energy content in
the low frequency range and produce significant residual displacements
[41–42]. Liquid storage tanks belong to the class of structures with long
fundamental period, thus the near-fault earthquakes could introduce
more devastating responses to tanks than far-fault earthquakes with
equal intensity [43]. Kalogerakou et al. [44] examined rigid liquidcontaining tanks under near- and far-fault earthquakes. The response
of the second convective mode under near-fault earthquakes was
approximately three times larger than that under far-fault conditions. In
addition, current provisions may lead to a significant underestimation
on peak sloshing height when the directivity pulse of near-fault earth­
quakes had substantial content near the frequency of the second
convective mode. By conducting a shaking table test, Zhou et al. [45]
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J.-Y. Wu et al.
Engineering Structures 289 (2023) 116284
analyzed sloshing responses of vertical storage tanks under action of
near-fault ground motions. Results showed that the sloshing height had
a certain linear relationship with the PGA, and was obviously larger
under near-fault earthquakes [46]. Although the source type and dis­
tance dependent near-fault factors to the design spectrum have been
introduced in recent seismic design codes, they pay little attention to
physical characteristics of near-fault earthquakes. In addition, an
experimental and numerical investigation on seismic responses of LNG
inner tanks under vertical action of near-fault earthquakes is less re­
ported. Therefore, it is necessary to conduct further studies on seismic
behaviors of LNG inner tanks subjected to vertical excitations to provide
assistance for structural seismic design.
In this study, a preliminary investigation on the effect of vertical
seismic excitations on the responses of liquid storage tanks was per­
formed through a shaking table test, and a 1:25 scaled steel tank was
excited by X-, Z-, XZ-directional ground motions. Seismic responses
including sloshing height and hydrodynamic pressure were detected and
compared. Subsequently, a refined finite element (FE) model of the
scaled steel tank was established by utilizing the commercial software
LS-DYNA [47]. The modelling methodology was validated by compari­
son with the test result and was further applied to the prototype largescale LNG inner tank. Series of near- and far-fault ground motions
were performed in the cases of Z-, XY-, XYZ-directional excitations to
examine the effect of vertical component on dynamic behavior of LNG
inner tanks. Finally, the responses of LNG inner tanks were compared
with the recommendations given in the current seismic design codes for
cylindrical liquid storage tanks.
table, a discontinuous retaining ring was attached at the top of the tank,
and a thin steel plate was set around the bearing platform. The me­
chanical properties of Q345 steel were obtained through tensile coupon
tests [49], with average elastic modulus, yield strength and tensile
strength of 2.12 × 105, 413.33 and 502.33 MPa, respectively. The
concrete of the bearing platform was measured with elastic modulus and
cubic compressive strength of 4.12 × 104 and 50.51 MPa, respectively.
The shaking table used could output seismic excitations in three di­
rections simultaneously, with a frequency range of 0.1–100 Hz and
maximum displacements of 100, 75 and 50 mm in the X, Y, and Z di­
rections, respectively. The layout of instruments and directions of input
excitations are displayed in Fig. 2. The pressure on the tank wall was
detected by a CY200 intelligent digital pressure sensor (500 Hz, 0.1% FS,
max. 50 kPa), and the hydrodynamic pressure was obtained by sub­
tracting the hydrostatic pressure from the recorded data. Five pressure
transducers were arranged along the height on the west side of the wall,
i.e., monitoring points P1 − P5. A non-intrusive dynamic displacement
measuring and analysis system (sampling frequency ≥ 4000 Hz, accu­
racy ≥ 0.1 mm), which consisted of a charge-coupled device lens,
markers and an acquisition system, was applied to measure the liquid
sloshing height. In this test, the markers were attached on a buoy at 300
mm from the tank wall along the X direction (Fig. 2a). The near-infrared
light emitted by the markers could be captured by the charge-coupled
device lens, thereby recording spatial coordinates of the markers
through the acquisition system.
Two ground motions of near-fault earthquakes, i.e., Chi-Chi 1529
(recorded from station TCU102) and Chi-Chi 1505 (recorded from sta­
tion TCU068) were selected, with velocity pulse periods of 9.63 and
12.29 s, respectively. For comparison, a far-fault ground motion was
extracted, i.e., El Centro waves (recorded from Imperial Valley earth­
quakes in 1940). In order to focus on the effect of velocity pulse on the
liquid convective motion [44], the ground motions were time scaled
according to the similarity factor of convective period (i.e., 1/5). The
horizontal component of ground motions was input along the east–west
direction (X direction), and the vertical direction was set as Z direction.
According to the Chinese standard GB 50011-2010 [51], the PGA ratio
of X and Z seismic component was set as 1:0.65. The test was carried out
according to scenarios listed in Table 1. Taking the scenarios 10–12 as
examples, the acceleration time history and corresponding spectrums
output by the shaking table are displayed in Fig. 3. The dominant fre­
quency (6–12 Hz and 16–21 Hz in horizontal and vertical components of
El Centro earthquake, respectively, and 0.1–4 Hz in both the horizontal
and vertical components of Chi-Chi earthquake) indicated that the
seismic signals reproduced by the shaking table was believed to be
reliable in the experimental study.
2. Shaking table test
2.1. Description of the LNG inner tank
The prototype structure is a 160,000 m3 LNG storage tank, which
consists of a reinforced concrete outer tank and a steel inner tank. The
steel inner tank is the primary vessel for LNG, with a diameter and
height of 80.00 and 35.43 m, respectively. The tank wall is divided into
ten layers from bottom to top, with each thickness of 24.9, 22.4, 19.8,
17.3, 14.7, 12.2, 12.0, 12.0, 12.0, 12.0 mm, respectively. The bottom
plate is divided into an annular plate and a center plate, with the
thicknesses of 6 and 5 mm, respectively. The inner tank is made of
A553M Type I (9%Ni) steel [48], of which the elastic modulus and yield
strength are 204 GPa and 585 MPa, respectively. The cryogenically
cooled LNG with a density of 450 kg/m3 is stored in the inner tank, and
the normal working liquid level is 31.83 m.
2.2. Scaled tank model and test set-up
2.3. Experimental results and discussion
A shaking table test was performed on a 1:25 scaled steel tank , with
a diameter and height of 3.20 and 1.42 m, respectively. More details of
the similarity design have been discussed by Yu et al [49]. According to
structural systems of full containment LNG storage tanks [50], a rein­
forced concrete slab with sand infilled was manufactured as the bearing
platform, on which the tank model without bottom anchorages was
placed (Fig. 1). In order to prevent liquid from spilling to the shaking
2.3.1. Response under vertical ground motion
(1) Sloshing height.
Fig. 4 displays time history of the sloshing wave height under vertical
excitations with PGAs of 0.13 and 0.26 g. The liquid exhibited nonlinear
sloshing during the vertical excitations, with a convective period of
about 2 s. The sloshing height increased significantly with the seismic
intensity. The peak sloshing heights in S1, S2 and S3 were 6, 15, and 16
mm, respectively, whereas that reached 34, 39 and 40 mm in S4, S5 and
S6. In addition, the sloshing height under vertical component of Chi-Chi
waves was 1.2–2.5 times larger than that under El Centro waves, which
was due to the long-period velocity pulse contained in Chi-Chi waves.
(2) Hydrodynamic pressure.
Due to the larger pressure, time histories of hydrodynamic pressure
acting on the tank wall bottom (monitoring point P5) in S4-S6 are shown
in Fig. 5. The trend of hydrodynamic pressure was consistent with that of
vertical ground accelerations, and the peak hydrodynamic pressure and
peak acceleration occurred simultaneously (approximately at 2.8, 7.2
and 7.4 s in S4, S5 and S6, respectively). Since the hydrodynamic
Fig. 1. Test set-up of the scaled storage tank.
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Engineering Structures 289 (2023) 116284
Fig. 2. Schematic diagram of the test model and instrument layout.
under Chi-Chi waves. In the scenarios examined in this study, the
sloshing height under Chi-Chi waves was 4.9–19.7 times of that under El
Centro waves.
An abnormally high sloshing height was observed under the Zdirectional excitation of El Centro wave (0.4 g), which might be due to
the disturbance of excitations in the previous scenario. Under Z-direc­
tional excitations of Chi-Chi waves with PGAs of 0.2 and 0.4 g, the peak
sloshing height was 24–39% and 29–61% of that under X-directional
excitations, respectively. Comparing results under X- and XZ-directional
excitations, it was shown that the peak sloshing height was increased
due to the inclusion of vertical seismic components, especially for nearfault earthquakes. For example, the peak sloshing height increased from
137 to 156 mm due to the vertical component of Chi-Chi 1505 waves
(0.4 g), whereas that increased from 12 to 13 mm under El Centro waves.
It was because more obvious convective motion was excited due to the
long-period velocity pulse contained in Chi-Chi waves. In this test, the
peak sloshing height was increased by 24.8% and 8.4% on average
under Chi-Chi and El Centro waves due to the presence of vertical
seismic component, respectively.
(2) Hydrodynamic pressure.
In order to analyze the effect of vertical seismic component on hy­
drodynamic pressure, Fig. 8 shows distribution of peak hydrodynamic
pressure along the normalized tank height (ratio of monitoring point
height to total height, h/H) under seismic waves combined in different
directions. It was shown that, the hydrodynamic pressure under Zdirectional excitations was almost linearly distributed with respect to
the normalized height, whereas that showed a nonlinear distribution
subjected to X- and XZ-directional excitations. This is due to the fact that
the hydrodynamic pressure under vertical excitations was mainly caused
by the liquid impulsive motion, and was only related to liquid density,
vertical acceleration and height [32]. However, the liquid convective
motion was significant under X- and XZ- directional excitations, espe­
cially for near-fault earthquakes, consequently to a nonlinear pressure
distribution caused by convective and impulsive motions of liquid. The
ratios of peak hydrodynamic pressure under Z-directional excitation to
that under X-directional excitation are averaged in Table 2. With the
normalized height from 0.1 to 0.5, the hydrodynamic pressure gener­
ated under Z-directional excitations was 39–56% of that under the Xdirectional, which should be appropriately accounted for in the seismic
design of liquid storage tanks.
The hydrodynamic pressure acting on the tank wall under combined
X- and Z-directional excitations was further analyzed. In the case of El
Centro waves, the presence of vertical seismic component resulted in a
pronounced increase in peak hydrodynamic pressure at normalized
height from 0.1 to 0.4. The peak hydrodynamic pressure at normalized
Table 1
Testing scenarios.
Scenario
Earthquake
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
El Centro
Chi-Chi 1529
Chi-Chi 1505
El Centro
Chi-Chi 1529
Chi-Chi 1505
El Centro
Chi-Chi 1529
Chi-Chi 1505
El Centro
Chi-Chi 1529
Chi-Chi 1505
El Centro
Chi-Chi 1529
Chi-Chi 1505
El Centro
Chi-Chi 1529
Chi-Chi 1505
PGA set (g)
PGA recorded (g)
X
Z
X
Z
−
−
−
−
−
−
0.20
0.20
0.20
0.20
0.20
0.20
0.40
0.40
0.40
0.40
0.40
0.40
0.13
0.13
0.13
0.26
0.26
0.26
−
−
−
0.13
0.13
0.13
−
−
−
0.26
0.26
0.26
−
−
−
−
−
−
0.22
0.25
0.25
0.23
0.25
0.23
0.40
0.45
0.42
0.43
0.42
0.39
0.12
0.13
0.12
0.28
0.27
0.26
−
−
−
0.12
0.13
0.12
−
−
−
0.22
0.26
0.25
pressure at monitoring point P5 was mainly caused by the liquid
impulsive motion, and this motion varied in synchronism with the
vertical acceleration [32]. The peak hydrodynamic pressure in S4, S5, S6
were 2.20, 2.32 and 2.64 kPa, respectively, approximately reaching 1/3
of the hydrostatic pressure (7.95 kPa). This indicated that the vertical
liquid motion would exert considerable horizontal pressure on the tank
wall, which should be considered in the seismic design. Furthermore,
spectrums of the hydrodynamic pressure are plotted in Fig. 6. Under El
Centro wave, the liquid was dominated by impulsive motion, with fre­
quencies of 16–21 Hz. It was because the vertical component of El
Centro waves had a predominant frequency of 16 Hz (Fig. 3). In contrast,
both the convective (0–4 Hz) and impulsive motions (16–21 Hz) were
excited under Chi-Chi waves. It was shown that the vertical excitation
might not only cause liquid sloshing in upper part of the tank, but also
induce coupled vibrations of the tank and liquid with high frequency. In
addition, the convective motion was more significant due to the longer
period of velocity pulse of Chi-Chi 1505 wave.
2.3.2. Responses under horizontal and vertical ground motions
(1) Sloshing height.
The peak sloshing heights under X-, Z- and XZ-directional excitations
are displayed in Fig. 7. Due to the long-period velocity pulse and small
dominant frequency, the liquid exhibited more significant sloshing
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J.-Y. Wu et al.
Engineering Structures 289 (2023) 116284
Fig. 3. Acceleration time history and acceleration spectrum output by the shaking table.
Fig. 4. Time histories of sloshing height under vertical ground motions (PGAs of 0.13 and 0.26 g).
height of 0.1 was increased by 32% and 33% under excitations of 0.2
and 0.4 g, respectively, and that at normalized height of 0.4 was
increased by 14% and 20%, respectively. However, vertical seismic
component exhibited little effect on the pressure acting on the wall with
normalized height larger than 0.5. This is associated with the fact that
the liquid was dominated by impulsive motion under El Centro waves. In
the case of Chi-Chi 1529 waves with PGA of 0.2 g, the presence of
vertical seismic component resulted in 24–33% increases in the peak
hydrodynamic pressure at normalized heights from 0.1 to 0.65. Since
the sloshing magnitude was relatively small under vertical action, the
pressure at the monitoring point P1 was essentially unchanged. Simi­
larly, due to presence of vertical component of Chi-Chi 1505 waves with
PGAs of 0.2 and 0.4 g, the peak hydrodynamic pressure on the tank wall
with normalized heights from 0.1 to 0.65 was increased by 24–33% and
10–34%, respectively. In summary, the vertical component of far-fault
earthquakes mainly increased the hydrodynamic pressure exerted on
the tank wall with normalized heights of 0.1–0.4, whereas that on the
underwater tank wall was increased integrally under vertical actions of
near-fault earthquakes.
3. Numerical modeling and validation of the large-scale LNG
inner tank
3.1. Modeling methodology
The Arbitrary-Lagrangian-Eulerian (ALE) method, combines the ad­
vantages of both the Lagrangian and Eulerian formulations [52–53], is
feasible to simulate the nonlinear liquid sloshing, and explore
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J.-Y. Wu et al.
Engineering Structures 289 (2023) 116284
Fig. 5. Time histories of hydrodynamic pressure under vertical ground motions (PGA = 0.26 g).
and that of the prototype was 42 m in both the X and Y directions and 37
m in the Z direction. The control card *ALE_MULTI_MATERIAL_GROUP
was activated to define the ALE material groups for interface recon­
struction. Then, the keyword *ALE_STRUCTURED_MESH_VOLUME_­
FILLING was adopted to fill materials into the S-ALE mesh, i.e., air, and
water or LNG. Considering the cylindrical tank in this study, the volume
filling operation was performed in two steps. Firstly, the entire S-ALE
mesh was infilled with air by setting the variable GEOM to ALL, and the
variable AMMGTO corresponded to the *ALE_MULTI_MATER­
IAL_GROUP card ID of the air. In the second step, a cylindrical fluid
domain was filled to the S-ALE mesh by setting the GEOM to CYLINDER.
The spatial position and height of the cylinder was determined by two
endpoints, e.g., NID1 (0, 0, 0) and NID2 (0, 0, 1) for the test tank
(Fig. 9a), and the radius of the cylinder was defined by variables of
RAD1 and RAD2. With respect to the Lagrangian part, both the tank wall
and base plate were modelled by utilizing four-node shell elements with
full integration (Fig. 9b). The stiffeners were established using HughesLiu beam elements with cross-section integration, and were connected to
the tank wall by means of sharing nodes. The concrete bearing platform
was created by eight-node hexahedral solid elements with one-point
integration.
To simulate the interaction between the S-ALE part and Lagrangian
structures (Fig. 9c), the control card *ALE_STRUCTURED_FSI was acti­
vated with a penalty formulation coupling method, which could auto­
matically detect and cure fluid leakage. The variables of LSTRSID and
ALESID corresponded to the part ID of the Lagrangian structure and SALE mesh, respectively, and the MCOUP was the ID of the material to be
coupled. In order to define the coupling pressure as a function of the
penetration, the penalty factor was set to a curve which consisted of two
points, i.e., (0, 0) and (0.01, 10132.5) [47]. It is noticed that the normal
of Lagrangian segment should be pointed toward the coupled fluids. The
successive contact and separation between the tank bottom and concrete
bearing platform was realized by the algorithm *CON­
TACT_AUTOMATIC _NODE_TO_SURFACE with static and dynamic fric­
tion coefficients of 0.50 and 0.45, respectively [54]. In order to avoid
undesirable oscillation in contact, a viscous damping coefficient of 20%
was specified using the VDC option, and a segment-based contact was
activated by setting SOFT = 2.
For material models, the keywords *MAT_NULL (MAT_009) and
*EOS_LINEAR_POLYNOMIAL were adopted to describe the liquid and
air, which has been widely utilized in literature [56–57]. In order to
capture the nonlinear behavior of the tank, such as elephant’s foot and
diamond buckling [58], an elastic–plastic material, i.e., *MAT_PLAS­
TIC_KINEMATIC (MAT_003), was utilized for the tank wall and bottom
plate. The concrete for bearing platform was assigned with nonlinear
fracture mechanism through constitutive model *MAT_CSCM_CON­
CRETE (MAT_159). The parameters of material model and state equa­
tion used for the FE model are listed in Table 3.
The boundary condition at the bottom of the bearing platform was
realized by the keyword *BOUNDARY_SPC_SET, and the translational
degree of freedom in the seismic excitation directions were released. The
Fig. 6. Spectrum of hydrodynamic pressure at monitoring point P5 (PGA =
0.26 g).
Z
X
XZ
Fig. 7. Peak sloshing height under Z-, X- and XZ-directional excitations (PGAs
of 0.2 and 0.4 g).
interaction between the flexible tank and liquid inside by employing the
Fluid Structure Interaction (FSI) algorithm [54–55]. In this study, a new
solver, i.e., structured Arbitrary-Lagrangian-Eulerian (S-ALE) method
updated in LS-DYNA R9.0 [47] was implemented, which has identical
fundamental theory as the ALE method, but exhibits characteristics of
low memory consumption and high computational efficiency [50]. In
order to create the S-ALE part, several control points were firstly
determined
by
using
the
control
card
*ALE_STRUCU­
TRED_MESH_CONTROL_POINTS. Subsequently, spacing parameters
defined separately in the X, Y and Z directions were input into the
control card *ALE_STRUCTURED_MESH to generate a 3D mesh and
invoke the S-ALE solver. For the test tank model, the length of the S-ALE
part was 6 m in both the X and Y directions and 2 m in the Z direction,
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Engineering Structures 289 (2023) 116284
X
Z
XZ
X
Z
XZ
X
Z
XZ
X
Z
XZ
X
Z
XZ
X
Z
XZ
Fig. 8. Distribution of peak hydrodynamic pressure along height under Z-, X- and XZ-directional excitations (PGAs of 0.2 and 0.4 g).
results of sloshing patterns, and time histories of sloshing height and
hydrodynamic pressure obtained from test and simulation were
compared. Fig. 11 shows the sloshing patterns under El Centro waves
and Chi-Chi waves. Due to the short predominant period, the liquid
exhibited slight oscillation under El Centro waves. In contrast, the liquid
underwent significant sloshing under Chi-Chi waves because of the longperiod velocity pulse. Typically, the sloshing patterns of liquid level
obtained from the numerical simulation agreed well with the experi­
mental results. For quantitative comparison, time histories of sloshing
height in scenarios with pronounced liquid sloshing, i.e., S17 and S18,
were selected and compared (Fig. 12). The curves of experimental result
were discontinuous due to the light interference in record of nearinfrared light by using the charge-coupled device lens. The sloshing
height time histories obtained from the S-ALE approach were in good
agreement with the test results, with deviation of peak sloshing height of
19% and 3% in S17 and S18, respectively.
Time histories of hydrodynamic pressure on monitoring points P3-P5
in S16-S18 are displayed in Figs. 13-15, respectively. The numerical
result of hydrodynamic pressure coincided well with that recorded
experimentally. For instance, the deviations of peak hydrodynamic
pressure of monitoring points P3-P5 in S16 were 9%, 24% and 8%,
respectively, and that in S17 were 10%, 11% and 15%, respectively. In
general, the FE model based on S-ALE approach was feasible to predict
the seismic responses of liquid storage tanks with good accuracy,
including peak value and shape of time history.
Table 2
Average ratios of peak hydrodynamic pressure under Z-directional excitation to
that under X-directional excitation.
h/H
0.10
0.40
0.50
0.65
0.77
Average ratio
0.56
0.43
0.39
0.27
0.24
gravity was applied to the whole model in terms of the gravitational
acceleration by using keyword *LOAD_BODY_Z, and acceleration time
histories in different directions were input at the model bottom through
keyword *BOUNDARY_PRESCRIBED_MOTION_SET.
A mesh size convergence analysis was performed to determine an
appropriate element size. For the FE model of the scaled tank, element
sizes of 0.06, 0.07, 0.08, 0.10 m were selected for both the Lagrangian
and ALE part, whereas that of the full-scale LNG tank were set as 0.7,
0.8, 1.0, 1.2 m. Taking S18 for an example, the comparisons of pressure
time histories at the monitoring point P5 are displayed in Fig. 10.
Considering the model accuracy and computational efficiency, the
element size of 0.07 m and 0.80 m was adopted for FE models of the
scaled tank and full-scale LNG tank, respectively.
3.2. Comparison between experimental and numerical results
In order to verify the accuracy of the FE model, such as modeling
process, and parameters of material models and contact algorithms, the
Fig. 9. FE model configuration for the test tank.
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Table 3
Parameters of material models and state equation used for the FE model.
MAT_003
MAT_159
Parameter
Steel
Parameter
Concrete
Density (kg/m3)
Yong’s modulus (GPa)
Poisson’s ratio
Yield strength (MPa)
Tangent modulus (MPa)
7850
200
0.3
413
533
Density (kg/m3)
RECOV
Unconfined compression strength (MPa)
Maximum aggregate size (mm)
2500
11
30
20
MAT_009
*EOS_LINEAR_POLYNOMIAL
Parameter
3
Density (kg/m )
Pressure cut-off (Pa)
Viscosity coefficient
(N⋅s/m2)
Water (LNG)
Air
Parameter
Water (LNG)
Parameter
Air
1000 (480)
100 (100)
0.87 × 10− 3
(0.87 × 10− 3)
1.18
10
1.84 × 10− 5
C0 (Pa)
101,325
(101325)
2.25 × 109
(2.25 × 109)
C4
C5
E0 (Pa)
V0
0.4
0.4
2.5 × 105
1.0
C1 (Pa)
Note: the values placed in brackets present the property of LNG.
Fig. 10. Comparison of pressure time histories of FE models with various element sizes.
Fig. 11. Comparison of sloshing patterns obtained from test and numerical simulation.
Fig. 12. Sloshing height time histories from experimental and numerical results in S17 and S18.
4. Dynamic response assessment of large-scale LNG inner tanks
of eight records were selected from the Pacific Earthquake Engineering
Research (PEER) Center [59], the near- and far-fault records were ob­
tained from the same earthquake. Pertinent information of the selected
ground motions is summarized in Table 4. Considering the two distin­
guished natural periods of the LNG inner tank (i.e., convective period of
9.80 s and impulsive period of 0.49 s calculated according to Ref. [2]),
4.1. Selection of seismic ground motions
A design response spectrum was generated for soil class D according
to API 650 with an exceedance probability of 2% in 50 years [1]. A total
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Engineering Structures 289 (2023) 116284
Fig. 13. Hydrodynamic pressure time histories from experimental and numerical results in S16.
Fig. 14. Hydrodynamic pressure time histories from experimental and numerical results in S17.
Fig. 15. Hydrodynamic pressure time histories from experimental and numerical results in S18.
selected records were scaled to match the target acceleration spectrum
in the period range of 0–10 s, in line with Ozsarac et al. for instance [55].
The ground motions were scaled by adopting a minimization of mean
squared error (MMSE) method as shown in Eq. (1). The spectrums of the
modified seismic wases were displayed in Fig. 16. In order to analyze the
effect of vertical seismic component, three loading types were adopted
for seismic waves, i.e., Z-directional, XY-directional and XYZ-directional
excitations.
( )
( )
∑n
j=1 Sad Tj × Sar Tj
SF =
(1)
( )2
∑n
j=1 Sar Tj
( )
( )
where SF is the scale factor; Sad Tj and Sar Tj represent design
spectrum and spectrum of the record ground motion, respectively.
peak sloshing heights under Z-directional excitations of near-fault
earthquakes were relatively larger than that under far-fault earthquakes.
Subjected to XYZ-directional excitation of the near-fault records, the
peak sloshing height under near-fault records of Chi-Chi earthquake and
Kocaeli earthquake were obviously larger, with values of 3.25 and 2.93
m, respectively. It was because the period of velocity pulse contained in
Chi-Chi earthquake (5.00 s) and Kocaeli earthquake (5.99 s) was closer
to that of liquid convective motion, leading to more pronounced slosh­
ing of the liquid. Note that there was an obviously large sloshing height
under far-fault records of Imperial earthquake, i.e., 3.82 and 3.53 m
under XY- and XYZ-directional excitations. This is due to the relatively
larger predominant period (4–5 s) of the ground motions. Generally, the
inclusion of vertical seismic component resulted in an increase in peak
sloshing height of approximately 14–142% under excitations of nearfault records, whereas that was about 8–56% under far-fault records.
It follows that the vertical seismic component would amplify the
convective motion of the liquid, especially for near-fault earthquakes.
4.2. Sloshing height
Fig. 17 displays the peak sloshing height under Z-, XY- and XYZdirectional excitations. It was shown that considerable sloshing heights
occurred under Z-directional excitations of near-fault earthquakes, e.g.,
a peak sloshing height of 0.7 and 0.6 m under Z-directional excitations of
Chi-Chi earthquake and Kocaeli earthquake, respectively. Generally, the
4.3. Hydrodynamic pressure
The peak hydrodynamic pressure acting on the four sides of the tank
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Table 4
Detailed information of the selected ground motions.
NGA
Earthquake
Station
Tp (s)
Direction
PGA
(g)
PGV
(cm/s)
PGD
(cm)
PGV/PGA
Tg (s)
SF
1510
Near-fault records
Chi-Chi
(in 1999)
TCU075
5.00
E
N
V
003
273
DWN
022
292
UP
000
270
UP
0.33
0.26
0.23
0.29
0.19
0.47
0.57
0.99
0.76
0.26
0.14
0.19
109.56
36.08
50.92
34.94
41.68
10.56
76.13
67.35
27.83
44.63
32.64
14.12
96.6
31.58
25.02
9.36
11.61
3.46
41.90
24.33
7.59
41.16
29.77
5.77
0.34
0.14
0.23
0.12
0.22
0.02
0.14
0.09
0.04
0.18
0.24
0.08
3.03
0.49
3.28
1.24
0.56
0.07
0.35
0.34
0.14
3.41
6.83
0.44
1.22
E
N
V
225
315
DWN
2035
2125
UP
000
090
UP
0.15
0.19
0.08
0.13
0.08
0.06
0.62
0.45
0.33
0.19
0.16
0.13
20.81
20.72
7.59
15.60
13.68
4.17
28.78
31.39
9.73
19.17
15.35
8.75
10.16
11.41
6.83
13.26
6.85
3.16
6.26
3.92
1.47
22.72
14.10
7.22
0.14
0.11
0.10
0.12
0.17
0.07
0.05
0.07
0.03
0.10
0.10
0.07
2.31
1.72
1.56
4.55
4.55
5.12
0.21
0.37
0.24
0.56
0.54
0.14
2.55
159
Imperial Valley-06
(in 1979)
Agrarias
2.34
983
Northridge
(in 1994)
Jensen
3.54
1161
Kocaeli
(in 1999)
Gebze
5.99
CHY046
—
1208
Far-fault records
Chi-Chi
(in 1999)
163
Imperial Valley-06
(in 1979)
Calipatria
—
952
Northridge
(in 1994)
Beverly Hills
—
1160
Kocaeli
(in 1999)
Fatih
—
XY
0.60
2.33
4.97
1.03
2.16
wall (east, south, west, and north) under Chi-Chi earthquakes, Imperial
earthquake, Northridge earthquake and Kocaeli earthquake are extrac­
ted and plotted in Figs. 18-19, Figs. 20-21, Figs. 22-23 and Figs. 24-25,
respectively. Results under Z-directional excitations show that the hy­
drodynamic pressure generated by vertical action was generally linearly
distributed along height, with large values at normalized heights of
0.1–0.3 and 0.9 under partial scenarios. Such as in Fig. 22, the peak
hydrodynamic pressure exhibited large values of 0.36 × 105 and 0.20 ×
105 Pa at normalized heights of 0.1 and 0.9 under Z-directional excita­
tion, respectively, achieving 56% and 90% of that under XYZ-directional
excitation. This is resulted from that the hydrodynamic pressure exerted
on the tank wall could be divided into three parts, i.e., a long-period
component contributed by liquid convective motion (at normalized
height of approximately 0.1–0.3), a short-period component contributed
by the tank wall vibrations (at normalized height of about 0.9), and a
component contributed by liquid impulsive motion (distributed linearly
along height) [32]. In some cases, the inclusion of vertical seismic
component caused a significant increase in hydrodynamic pressure at
one side of the tank wall, but may also leaded to a decrease pressure on
the other side. For example, under Imperial earthquake (NGA = 159, in
Fig. 20), the inclusion of vertical seismic component resulted in 10–30%,
17–25% and 11–36% increases in the peak hydrodynamic pressure at
normalized heights from 0.2 to 0.6 for the south, west and north tank
walls, respectively, but decreases of 13–22% on the east side. With
respect to Northridge earthquake (NGA = 983, in Fig. 22), the addition
of vertical seismic component leaded to increases of 12–42% and
19–44% at normalized heights from 0 to 0.6 for the west and north tank
walls, but decreases of 5–16% and 13–40% for the east and south tank
walls. It was stem from that the rotation motion of liquid was influenced
by the action of vertical seismic component, and the liquid particles
moved following a curved path due to the Coriolis effects and centrifugal
forces [40]. As a result of the Coriolis effect, the addition of vertical
seismic component might change the location in which the maximum
hydrodynamic pressure appeared at the tank wall.
Comparing the near-fault and far-fault records, it can be seen that the
maximum hydrodynamic pressure under XYZ-directional excitations in
the near-fault events were larger than that under far-fault events. The
maximum hydrodynamic pressure under near-fault records of Chi-Chi
Fig. 16. Acceleration response spectrum of the selected ground motions.
Z
2.18
XYZ
Fig. 17. Peak sloshing height under Z-, XY- and XYZ-directional excitations.
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XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 18. Distribution of peak hydrodynamic pressure under near-fault records of Chi-Chi earthquake (NGA = 1510).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 19. Distribution of peak hydrodynamic pressure under far-fault records of Chi-Chi earthquake (NGA = 1208).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 20. Distribution of peak hydrodynamic pressure under near-fault records of Imperial earthquake (NGA = 159).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 21. Distribution of peak hydrodynamic pressure under far-fault records of Imperial earthquake (NGA = 163).
11
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XY
Z
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XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 22. Distribution of peak hydrodynamic pressure under near-fault records of Northridge earthquake (NGA = 983).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 23. Distribution of peak hydrodynamic pressure under far-fault records of Northridge earthquake (NGA = 952).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 24. Distribution of peak hydrodynamic pressure under near-fault records of Kocaeli earthquake (NGA = 1161).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 25. Distribution of peak hydrodynamic pressure under far-fault records of Kocaeli earthquake (NGA = 1160).
12
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XY
Z
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earthquake, Imperial earthquake, Northridge earthquake and Kocaeli
earthquake were 2.59 × 105, 1.16 × 105, 0.63 × 105 and 3.03 × 105 Pa,
respectively, whereas that under far-fault records were 0.82 × 105, 0.87
× 105, 0.49 × 105 and 0.41 × 105 Pa, respectively. Note that the
maximum hydrodynamic pressure under near-fault records of Chi-Chi
earthquake and Kocaeli earthquake were pronouncedly larger, reach­
ing approximately 2–7 times of that under other records. It was because
ground motions with higher PGV/PGA ratio exhibited wider
acceleration-sensitive region, resulting in more vibration modes of LNG
inner tanks to fall within the acceleration-sensitive region of the
response spectrum [60–61].
The average positive differences between the peak hydrodynamic
pressure under XY- and XYZ-directional excitations are shown in Fig. 26.
Due to the vertical action of near- and far-fault records, the peak hy­
drodynamic pressure was increased by 15–89%, and 28–72%, respec­
tively, which should be adequately accounted for in the seismic design of
LNG inner tanks.
whereas the elephant’s foot buckling occurred under XYZ-directional
excitation resulted from the larger hydrodynamic pressure. In addition
to cause a precipitation of buckling failure on the tank wall, the vertical
earthquake also promoted the development of plastic deformation. As
shown in Fig. 35c, the tank wall exhibited a more fully developed plastic
deformation under XYZ-directional excitations.
Similar to the conclusion for hydrodynamic pressure, the tank stress
under mostly near-fault records were larger than that under far-fault
records. The maximum tank stresses under near-fault records of Impe­
rial earthquake, Northridge earthquake and Kocaeli earthquake were
494, 341 and 350 MPa, respectively, whereas that under far-fault re­
cords were 340, 323 and 279 MPa, respectively. The average positive
differences of tank stresses between the XY-directional and XYZ-direc­
tional excitations were extracted and plotted in Fig. 36. The vertical
near-fault records of Chi-Chi earthquake provided the most significant
enhancement on tank stresses, with average increases of 12–43%. In
general, due to the vertical action of near- and far-fault records, the tank
stresses were increased by 8–43% and 7–20%, respectively.
4.4. Tank stress
4.5. Performance comparison with code recommendations
The distribution of peak Von Mises stresses along height on the four
sides of the tank wall (east, south, west, and north) under Chi-Chi
earthquakes, Imperial earthquake, Northridge earthquake and Kocaeli
earthquake are extracted and plotted in Figs. 27-28, Figs. 29-30,
Figs. 31-32 and Figs. 33-34, respectively. The stress distribution along
height under vertical excitations was identical to that under horizontal
excitations, i.e., the maximum stress generally occurred at the tank wall
with normalized heights from 0.1 to 0.3. In partial loading scenarios, the
tank wall stresses generated by Z-directional excitation were compara­
ble to that under XYZ-directional excitation, e.g., under Imperial
earthquake (Fig. 30c), Northridge earthquake (Fig. 32a), and Kocaeli
earthquake (Fig. 34b). This indicates that the vertical earthquake had a
significant effect on the tank stress by transmitting the seismic energy to
tank wall in the forms of hydrodynamic pressure.
The tank wall stresses were obviously increased due to the presence
of vertical seismic component. For instance, in Fig. 27c, the stresses on
the west tank wall with normalized heights from 0.1 to 0.7 was
increased by 19–36% due to the existence of vertical action. In addition,
the LNG inner tank was more vulnerable to inelastic buckling when the
vertical seismic component considered, such as observed in Chi-Chi
earthquake (both the near- and far-fault records, Fig. 35a-b), Imperial
earthquake (near-fault records, Fig. 35c) and Northridge earthquake
(near-fault records, Fig. 35d). Under Chi-Chi earthquake and Northridge
earthquake, the tank remained elastic under XY-directional excitation,
In this section, a comparison was made between the seismic re­
sponses obtained from finite element method (FEM) and seismic design
codes of steel cylindrical liquid storage tanks for low temperature ser­
vice, i.e., EN 14620 [62], EN 2654 [63] and EN 7777 [64]. The reference
of tank seismic design in EN 14620 is made to Eurocode 8 [65], which
separately evaluates the pressure generated by vertical acceleration and
combines the action effects of seismic components with the rule of EEdx
+ 0.3 EEdy + 0.3 EEdz or 0.3 EEdx + 0.3 EEdy + EEdz, where EEdx (EEdy), EEdz
represent action effect due to horizontal and vertical seismic compo­
nent, respectively. The recommendations in EN 2654 are based on the
requirements in Appendix E of API 650 [1], which applies a vertical
earthquake acceleration coefficient Av to consider the effect of vertical
seismic component on the tank wall stress. Since the design methods of
EN 7777 and EN 2654 are relatively simple and conservative, the ap­
proaches presented in API 650 and Eurocode 8 were adopted.
The seismic action on the tank wall is classified as convective, rigid
impulsive and flexible impulsive components. The spatial pressure dis­
tribution of rigid impulsive pi, flexible impulsive pf, convective pc as well
as that due to vertical ground acceleration pvr are given by the expres­
sions [2]:
pi (ξ, ϛ, θ, t) = Ci (ξ, ϛ)ρHcosθAg (t)
∞
∑
(2)
dn cos(vn ϛ)Afn (t)
(3)
ψ n cosh(λn γζ)J1 (λn ξ)cosθAcn (t)
(4)
pf (ϛ, θ, t) = ρH ψ cosθ
n=0
pc (ξ, ζ, θ, t) = ρ
∞
∑
n=1
(5)
pvr (ϛ, t) = ρH(1 − ϛ)Av (t)
in which:
ξ = r/R; ζ = z/H; vn =
2n + 1
π; γ = H/R
n
(6)
where R and H are tank radius and height of liquid level, respec­
tively. ρ is the liquid density. r, θ, z are components of cylindrical co­
′
ordinates with origin at the center of the tank. I1 and I1 denote the
modified Bessel functions of order one and its derivative, respectively. J1
is Bessel function of the first order. Ag(t) is the ground acceleration timehistory in the free-field. Af(t) is the response acceleration (relative to its
base) of a simple oscillator having the period and damping ratio of mode
n. Acn(t) donates the acceleration time-history of the response of a single
degree of freedom oscillator. Av(t) is the vertical ground acceleration.
More details are described in A.2-A.3 in Appendix A of Eurocode 8.
Fig. 26. Average positive differences between the peak hydrodynamic pressure
under XY-directional and XYZ-directional excitations.
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Engineering Structures 289 (2023) 116284
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 27. Distribution of peak stress along height under near-fault records of Chi-Chi earthquake (NGA = 1510).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 28. Distribution of peak stress along height under far-fault records of Chi-Chi earthquake (NGA = 1208).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 29. Distribution of peak stress along height under near-fault records of Imperial earthquake (NGA = 159).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 30. Distribution of peak stress along height under far-fault records of Imperial earthquake (NGA = 163).
14
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XY
Z
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XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 31. Distribution of peak stress along height under near-fault records of Northridge earthquake (NGA = 983).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 32. Distribution of peak stress along height under far-fault records of Northridge earthquake (NGA = 952).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 33. Distribution of peak stress along height under near-fault records of Kocaeli earthquake (NGA = 1161).
XYZ
XY
Z
XYZ
XY
Z
XYZ
XY
Z
Fig. 34. Distribution of peak stress along height under far-fault records of Kocaeli earthquake (NGA = 1160).
15
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XY
Z
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XYZ
XY
Z
XYZ
XY
Z
XY
Z
XY
Z
XYZ
XYZ
Fig. 35. Stress distribution contours of the tank when the plastic deformation fully developed under X-, XY- and XYZ-directional excitations.
The total hoop stress is expressed as the sum of hydrostatic Nh,
impulsive Ni and convective hoop stresses Nc [1]. Considering the LNG
tank geometry, the hydrodynamic and hydrostatic hoop stresses are
determined by the following formulas:
[
(
( )2 ]
)
Y
Y
D
Ni = 8.48Ai GDH
tanh 0.866
− 0.5
(7)
H
H
H
[
]
Y)
1.85Ac GD2 cosh 3.68(H−
D
[
]
Nc =
cosh 3.68H
D
(8)
Nh = 0.24(H − 0.3)DG
(9)
as 0.9 [66]. Ac and Ai are convective and impulsive design response
spectrum acceleration parameters, respectively, of which details are
presented in Appendix E, E.4.6 of API 650. In order to provide consis­
tency results, the design spectra and corresponding parameters used in
Sec. 5.1 was utilized.
The effect of vertical acceleration on the tank stress is considered in
terms of the vertical earthquake acceleration coefficient Av, and the total
hoop tensile stress on the tank wall σT is combined as:
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Nh ± N 2i + N 2c + (Av Nh /2.5)2
(10)
σT =
t
where the vertical earthquake acceleration coefficient Av shall be
taken as 0.47SDS, and SDS is the design, 5% damped, spectral response
acceleration parameter at short period (T = 0.2 s) based on ASCE 7
method. The allowable tensile stress limitation for the inner tank ma­
terial shall be determined in accordance with BS 7777 [64], i.e., 0.85
where D and H are the tank diameter and design liquid level,
respectively. Y is the distance from liquid surface to analysis point, and
the tank wall with normalized height of 0.1 was used for the analysis
point in this study. G is the specific gravity of the liquid, which is taken
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Engineering Structures 289 (2023) 116284
MPa, respectively. The hoop tank stresses with and without vertical
acceleration considered were comparable to each other, because the API
650 only takes into account the effect of vertical action on hydrostatic
pressure. In addition, the maximum hoop stresses calculated were far
less than that obtained from FEM. The reason is that the API 650 per­
forms a force reduction factor in the tank stress design to avoid the
overstrength on the material, ductility and damping [67], which was
taken as 3.5 and 2 for impulsive and convective motions of self-anchored
tanks, respectively. In the cases of the force reduction factors were not
considered, the hoop stresses predicted were below the code recom­
mendations (456 and 455 MPa for horizontal and combined excitations,
respectively), except for the event subjected to combined excitations of
near-fault records (482 MPa). This demonstrated that the effect of ver­
tical action of near-fault earthquakes on hydrodynamic pressure and
tank stresses was underestimated in Eurocode 8 and API 650.
Fig. 36. Average positive differences between the peak tank stress under XYdirectional and XYZ-directional excitations.
5. Conclusions
In this paper, the influence of vertical seismic component on dy­
namic responses of LNG storage tanks were experimentally and
numerically examined. A series of shaking table tests were conducted on
a 1:25 scaled steel tank under seismic excitations with different direc­
tional combinations. In terms of prototype large-scale LNG inner tanks,
the S-ALE and FSI algorithms were employed to realistically capture the
nonlinear liquid sloshing and hydrodynamic pressure on the tank wall.
Ground motions including near- and far-fault earthquakes were selected,
and responses such as sloshing height, hydrodynamic pressure and tank
stress were extracted and compared. The main conclusions are drawn as
follows:
times the yield stress for operating basis earthquake (OBE) seismic loads,
and 1.0 times of yield strength for safe shutdown earthquake (SSE)
seismic loads, i.e., 497.25 and 585 MPa for OBE and SSE seismic design,
respectively.
With respect to maximum sloshing height, both the API 650 and
Eurocode 8 present expressions based on assumption of small amplitude
wave motion:
δs = 0.5DAf (API 650)
(11)
dmax = 0.84RSe (Tc1 )/g(Eurocode 8)
(12)
where Af is the acceleration coefficient for sloshing waves height,
calculated according to E.7.2 in Appendix E of API 650. Se(Tc1) is the
elastic response spectral acceleration at the 1st convective mode of the
fluid for damping a vale appropriate for the sloshing response.
Table 5 summarized seismic responses of the LNG inner tank ob­
tained from FEM and codes under horizontal and combined horizontal
and vertical excitations. Generally, the peak sloshing height predicted
by the FEM were less than that recommended by codes, since the effect
of high-order sloshing modes was relatively small. However, under ChiChi earthquake (near-fault records) and Imperial earthquake (far-fault
records), the peak sloshing height reached 3.25 and 3.82 m, which
exceeded 74% and 104% of the API 650 recommendation, and 26% and
49% of the Eurocode 8 recommendation. It was stem from that the
period of velocity pulse contained in Chi-Chi earthquake (5.00 s) and the
predominant period of Imperial earthquake (4.97 s) were close to the
first convective period of the storage tank. This indicates that current
codes of tank seismic design may underestimate the sloshing height in
the cases of near-fault earthquakes containing long-period velocity pulse
and far-fault earthquakes with long predominant periods. In terms of
hydrodynamic pressure, the recommendations without vertical action
may lead to an underestimated result, especially for near-fault earth­
quakes. As for hoop stresses, the maximum values under horizontal and
combined excitations obtained from API 650 were 159.5 and 161.33
(1) Both the convective and impulsive motions were induced under
vertical actions. In the shaking table tests, the liquid was domi­
nated by impulsive motion with frequency of 18–21 Hz under
vertical excitation of El Centro waves, whereas that was domi­
nated by convective motion with frequency of 0–4 Hz under ChiChi waves.
(2) The vertical seismic components would amplify the convective
motion of the liquid, especially for near-fault earthquakes. In the
scenarios examined in this study, the peak sloshing height under
near- and far-fault records were increased by 14–142% and
8–56% due to the presence of vertical action, respectively.
(3) Resulting from the coupling excitations of horizontal and vertical
ground motions, the location in which the maximum hydrody­
namic pressure occurred would change in the cases of vertical
action considered. In general, the peak hydrodynamic pressure
was increased by 15–89% and 28–72% for the 160,000 m3 LNG
storage tank studied due to the vertical excitation of near- and
far-fault records, respectively.
(4) In addition to cause a precipitation of inelastic buckling failure on
the tank wall, the vertical earthquake also promoted the devel­
opment of plastic deformation. For the loading conditions
examined in this work, the increases on the tank stresses due to
Table 5
Seismic responses of the LNG inner tank obtained from FEM and codes.
Parameter
Peak sloshing height (m)
Maximum hydrodynamic pressure (×105 Pa)
Maximum hoop stress (MPa)
a
b
FEM
API 650
Horizontal
Combined
Horizontal
0.16–3.53a
(0.37–2.85)
0.39–0.66a
(0.48–2.11)
251 − 351a
(315–442)
0.25–3.82a
(0.47–3.25)
0.41–0.87a
(0.63–3.03)
288 − 383a
(327–482)
1.87b
Eurocode 8
Combined
161.33
The upper and lower (placed in brackets) values represent response range under far- and near-fault earthquakes, respectively.
The maximum sloshing height was calculated by using design spectral acceleration.
17
Combined
2.57b
—
159.5
Horizontal
0.43–0.71a
(0.31–0.89)
—
0.53–0.80a
(0.42–1.04)
J.-Y. Wu et al.
Engineering Structures 289 (2023) 116284
vertical action of near- and far-fault records was 8–43% and
7–20%, respectively.
(5) Both the API 650 and Eurocode 8 may give nonconservative
result in maximum sloshing height of liquid storage tanks under
near-fault earthquakes with long-period velocity pulse or farfault earthquakes with long predominant periods. Moreover,
the effect of vertical action of near-fault earthquakes on hydro­
dynamic pressure and tank stresses was underestimated in cur­
rent seismic design codes.
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CRediT authorship contribution statement
Jie-Ying Wu: Methodology, Writing – original draft, Writing – re­
view & editing. Qian-Qian Yu: Methodology, Writing – original draft,
Writing – review & editing. Qi Peng: Data curation, Validation, Formal
analysis. Xiang-Lin Gu: Methodology, Conceptualization, Writing –
review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgements
This research work was supported by the National Key R&D Program
of China (2022YFC3803000) and National Key R&D Program of China
(2017YFC1500700).
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