2000-comparison of design methods for a tank-bottom annular plate and concrete ringwall
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2000-Comparison of Design Methods for a Tank-bottom Annular Plate and Concrete RingwallTRANSCRIPT
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Comparison of design methods for a tank-bottom annular plate andconcrete ringwall
T.Y. Wu, G.R. Liu
Department of Mechanical and Production Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore, Singapore 119260
Received 7 March 2000; revised 27 June 2000; accepted 3 August 2000
Abstract
This paper studies three design methods for a bottom annular plate of large storage tanks and the design of the concrete ring wall. It is
shown that the annular-plate width, thickness, and the width that will project beyond the outer surface of the shell can be calculated
analytically. The design of the annular bottom plate and the concrete ringwall is discussed and the need for an accurate representation
for the interaction between the bottom plate and the foundation is emphasised. q 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Storage tanks; Tank-bottom; Tank foundation; Concrete ringwall; Annular plate
1. Introduction
Risks of storage tank failure and environmental contam-
ination are just two of the problems facing oil storage. As a
result, more attention should be given to tank design and
operation standards in addition to stringent safety controls,
since the majority of oil and its products are stored in large
capacity tanks. Many large above ground storage tanks have
been built with vertical cylindrical shells and ¯at bottoms
that rest directly on simply prepared subgrades and are
supported on a ringwall beneath the tank shells. Large, verti-
cal, cylindrical, single-, and ¯at-bottomed, unanchored,
welded steel storage tanks under hydrostatic loads are
considered, as shown in Fig. 1. Design calculation methods
for the tank-bottom annular plate are discussed. This plate is
normally designed empirically, there being no recognized
code, standard or common practice for design.
The way in which the bottom plate and the lowest tier of
the tank shell interrelate is important in relation to the
design of the bottom annular plate and the concrete ring
wall. The tank design standards for various countries [1±
5] only stipulate three dimensions for the annular plate. The
®rst is a minimum nominal width of the annular plates, such
as 1.8 m in Chinese Standard SH3046-92, 1.8 m (72 in.) in
API 650, or 500 mm in British Standard BS2654. The
second is a minimum nominal thickness, which is deter-
mined by the nominal plate thickness of the ®rst shell
course, and also related to the hydrostatic test stress in the
®rst shell course in API 650. The third is a minimum width
that projects beyond the outside edge of the weld attaching
the bottom to the shell plate or beyond the outer surface of
the shell plates. It is 25 mm (1 in.) in API 650 and 50 mm in
BS2654 and SH3046-92. The one-foot method or variable-
design-point method can be used to calculate the shell thick-
ness. However, no code or standard in any of the countries
listed uses equations to calculate the stresses of the bottom
annular plate to determine the above-mentioned three
dimensions. This paper sets out a discussion and recommen-
dation for the design equations of the bottom annular plates.
The designer of the concrete ringwall must ®rst obtain the
data on superimposed loadings to be used for the ringwall
design. These data are not provided in the current standards.
The forces applied on the ringwall by the tank body are
obtained if the ¯exion mechanism of the annular plate is
clearly known.
2. Three design methods for the tank-bottom annularplate
The American Petroleum Institute published a stress
calculation method for the annular plate in 1968 [6].
Later, the Academy of Sciences of China (ASC) produced
an alternative approach in 1979 [7]. Recently, Wu [8]
proposed a more accurate method. They all considered a
vertical cylindrical tank with a ¯at bottom that rests directly
on simply prepared soils and is supported on a concrete
ringwall.
The stress calculation of the tank-bottom annular plate
International Journal of Pressure Vessels and Piping 77 (2000) 511±517
0308-0161/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.
PII: S0308-0161(00)00055-7
www.elsevier.com/locate/ijpvp
E-mail address: [email protected] (T.Y. Wu).
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requires a proper description of the interaction between the
bottom and the foundation, as shown in Fig. 1. The bottom
of a storage tank is a large steel membrane constrained by
the base and the shell. As a product is added to the tank, the
weight of the product presses the bottom inner plates into
complete contact with the foundation. Therefore, the inner
bottom plates can be assumed as non-stressed because of
uniform support from the foundation. Only a minimum
nominal thickness, exclusive of any corrosion allowance,
is speci®ed for the inner bottom plates in many codes or
standards. Since the concrete ringwall cannot ¯ex down-
ward and the shell applies a large bending moment to the
bottom plate at the shell-bottom junction, the only possible
movement of the bottom annular plate is to ¯ex upward, i.e.
to lift it off the concrete ringwall. The uplift hypothesis for
the bottom annular plate is reasonable and adopted by all the
three calculation methods mentioned above. To simplify the
calculation, the ®rst shell plate tier is represented as a long
half cylinder under hydrostatic pressure, as shown in Fig. 2.
This simpli®cation is accurate from an engineering view-
point [9]. The compatibility conditions at the tank shell-
bottom junction are the continuities of the respective displa-
cement and rotation angle. The three methods are simply
presented below.
2.1. The API method [6]
As shown in Fig. 3, a small part of the annular plate lifts
clear of the foundation for some distance S. The bottom
plate ¯exural bending is of nonlinear nature with a changing
boundary condition, since a different hydrostatic load will
cause a changing uplift distance. The outer edge of the
annular plate is simply supported with a zero vertical displa-
cement. The compatibility conditions are used at the shell-
bottom attachment, where the shell horizontal displacement
is zero. The boundary conditions at the inner support point
are zero for both vertical displacement and rotation angle,
assuming the ringwall has suf®cient width to support the
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517512
Nomenclature
A1;A2;A3;A4 coef®cients
C distance from shell plate centre to outer edge of bottom annular plate
D shell diameter
Db Et31=�12�1 2 y 2��
Ds Et3s =�12�1 2 y 2��
E steel elasticity modulus, 206 GPa
G tank shell weight per length of circumference
H tank water-®lling height
Kb elastic coef®cient of the tank's ¯exible foundation
Ks Ets=R2
L the design width of the annular plate
M0 bending moment at the tank shell-bottom junction
M2 bending moment of annular plate at one speci®c point
P tank-bottom water pressure, P � HgQ0 shear force at the tank shell-bottom
q2 supporting force per square metre in ASC method
R shell radius
R1 supporting force of annular plate at bottom plate periphery
R2 shear force of annular plate at one speci®c point
S distance for which the annular plate lifts clear of the foundation in API and Wu methods
ts shell wall thickness of the lowest tier
t1 bottom annular plate thickness
t2 bottom inner-plate thickness
w horizontal displacement of the shell-bottom junction
bb
���������Kb=4Db
4p
bs
���������Ks=4Ds
4p
ub rotation angle of the bottom annular plate at the shell-bottom junction
us rotation angle of the shell plate at the shell-bottom junction
g speci®c weight of water, 9.81 kN/m3
ss minimum yield stress of the bottom annular plate
smax theoretical maximum bending stress of the bottom annular plate
s20 annular-plate bending stress at the point 20 mm from the shell's inner wall inward of the tank centre
y Poisson's ratio, 0.3
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annular plate. The calculation formulae are as follows:
M0 � �GC 1 0:25P�S2 2 C2���C2 2 S2��S2 2 3C2� �1�
R2 � �0:5P�S2 2 C2�1 GC 1 M0�S
�2�
us � g�Hbs 2 1�Ks
1M0
2bsDs
; �3�
ub � 1
Db
R2�S 2 C�22
2P�S 2 C�3
6
" #: �4�
Assuming a value of S, then M0;R2, us and ub are calculated
from the above equations, respectively. When a certain S
results in the same us and ub; this S corresponds to this
combination of conditions, along with M0;R2, us and ub:
Therefore, the stresses of the annular plate can be calculated
analytically. It should be noted that the relationship between
M0 and ub is nonlinear rather than linear as taken in Ref. [6].
2.2. The ASC method [7]
The width of the concrete ringwall for many large tanks is
relatively narrow, compared with the distance S obtained
using the API method. Therefore, the inner support point
in Fig. 3 cannot satisfy its boundary conditions Ð zero
vertical displacement and zero rotation angle. The inner
point falls on the elastic foundation adjacent to the ringwall
in most cases. As shown in Fig. 4, the ASC method consid-
ers the effect of the inner ¯exible foundation. It assumes the
same distance, which falls on the ¯exible foundation and
bears a triangular reaction force, as the uplift distance.
However, it does not consider the section that projects
beyond the outer surface of the shell plates. The annular
plate is simply supported at the point just beneath the
shell and thus has a zero vertical displacement therein.
Another boundary or compatibility condition at the shell-
bottom attachment is that the ®rst shell plate and the annular
plate have the same rotation angle therein. The rotation
angle at the most inner support point is zero, and the vertical
displacement is q2=Kb. The calculation formulae are as
follows:
M0 �11
240
ts
t1
� �3
PS3 11
4b4s
2b2s P
2bs 1 �1 2 y�ts=�Rt2� 2 g
!1
2bs 1 �1 2 y�ts=�Rt2� 21
bs
217S
40
tst1
� �3;
�5�
M0 � 1
�1920Db=KbS4�2 49
47
62
960Db
KbS4
� �PS2
; �6�
q2 � 12
5P 1
2M0
S2
� �: �7�
The solution procedures are the same as above. A certain
value of S results in an identical M0 in Eqs. (5) and (6).
2.3. The Wu method [8]
In view of the special use of the concrete ringwall, Wu [8]
proposed a new method to account for the effect of the
elastic foundation. As shown in Fig. 5, Wu's method also
considers an uplift part with a length of S, as in the API
method. However, the inside part of the annular plate
toward the tank centre is considered as a semi-in®nite
plate ¯exibly supported by the foundation. The reaction
force of the subgrade is linear with the annular plate vertical
displacement, while the elastic coef®cient of the subgrade
adjacent to the ringwall is Kb. The shell horizontal displace-
ment is zero at the shell-bottom junction. The calculation
formulae have been obtained as follows:
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517 513
x
(H-x)γ
Q0
M0
y
Fig. 2. Shell forces.
Tank shell
Annular plate Inner plate
Groundlevel
Centre of theringwall and shell
Coarsegravel orcrushedstone Compacted
refill
Compactedclean sand
Fig. 1. The tank shell-bottom junction and foundation with a concrete
ringwall.
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M0 � 0:25P�2A1A4 2 A2A3�1 GC�A2C2 2 A1�A1 2 3A2C2
; �8�
ub � 1
Db
"R2
2�S 2 C�2 2
P
6�S 2 C�3
2M2�S 2 C�21
2b2b
�R2 1 2bbM2�#; �9�
M2 � 0:5PA4 2 GC 2 M0
A2
; �10�
R2 � 0:5P�S2 2 C2�1 GC 1 M2 1 M0
S: �11�
where
A1 � 3S2 1 bbS3 13S
bb
;
A2 � 1 1 bbS;
A3 � C4 2 S4 22S3
bb
16S
b3b
;
A4 � C2 2 S2 2S
bb
:
The shell rotation angle is calculated using Eq. (3). The
solution procedure is similar to the above. A certain value
of S results in the same us and ub. The bending moment of
the annular plate can be obtained according to Fig. 5.
M � R1x �x , C�; �12a�
M � R2�S 2 x�2 0:5P�S 2 x�2 2 M2 �C , x , S�;�12b�
M � 2e2bbx{M2�sin�bbx�1 cos�bbx��1 R2 sin�bbx�=bb}
�x . S�: (12c)
3. Discussion
3.1. Theoretical comparison of the three methods
With the exception of tanks founded on solid rock, hard-
core, or similar sub-bases, some settlement is bound to take
place. Large, and perhaps even moderate, irregularities in
settlement may lead to serious distortion of the bottom plate
and other important elements of the tank. Due to the rela-
tively small width of the concrete ringwall for many large
tanks, the inner support point in Fig. 3 falls on the inner
elastic foundation adjacent to the ringwall. Therefore, the
inner point cannot satisfy its boundary conditions Ð zero
vertical displacement and zero rotation angle. The API
method is not suitable for thin ringwall cases. However,
many crushed stone or gravel ringwalls have suf®cient
width to support the annular plate and meet the required
boundary conditions.
The ASC calculation is simpli®ed in terms of the effects
of the ¯exible foundation and does not consider the annular
plate's protruding length. In the ASC formulae, the bottom
plate's horizontal displacement is considered as [7]
w � �1 2 y�RQ0
Et2: �13�
This equation means that the bottom plate is regarded as a
circular plate with the bottom inner-plate thickness and with
a load of Q0 acting at its periphery. This equation does not
consider the friction force applied by the elastic foundation.
Due to the hydrostatic pressure, the bottom inner plate may
not have a horizontal displacement due to the friction force.
Indeed, the formulae for the API and Wu methods employ
zero horizontal displacement therein. However, there is
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517514
R2
P=Hγ
M0
G
R1
CS
Tank wall centre
Fig. 3. A force distribution of the API model for the tank-bottom plate.
P=Hγ
M0
R1
S/2
Tank wall centre
S/2
q2
Fig. 4. A force distribution of the ASC model for the tank-bottom plate.
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little in¯uence on the ®nal results whether this displacement
is taken as zero or given by Eq. (13), as veri®ed in Refs.
[8,9].
If the ringwall or subgrade has suf®cient width to support
the annular plate in such a way that the annular plate can be
completely supported by a rigid ¯at surface, the elastic
coef®cient Kb can be taken as large, for example 100
times the normal value. Theoretically, the API method can
be obtained using the Wu method when Kb tends to in®nity.
When a ¯at-bottom tank is designed without anchoring
the shell to a counterbalancing weight, the bottom is
designed to carry all the weight and pressure forces distrib-
uted on the bottom and to transfer the uplift forces from the
sidewall through the bottom plates. This case, as required in
section 3.11.3 of API standard 620, is also in the discussion
scope, if the shell is cylindrical.
3.2. Theoretical calculations and experimental data
analysis
To calculate the stresses of the tank-bottom annular plate
using either the ASC method or the Wu method, the elastic
coef®cient of the ¯exible foundation adjacent to the ringwall
is required. The value Kb � 39:24MN=m3 is long established
in engineering practice in China and is further discussed
below. In China, three tanks of representative volumes
20 000, 50 000, and 100 000 m3 were strain-gauged during
the water-®lling test, when they were ®rst constructed. This
paper examines these ®eld results to discuss the three
proposed methods.
Fig. 6 shows the annular plate bending stress distribution
of the 100 000-m3 tank and is representative of the bending
stresses using the Wu method. The other two methods have
a similar distribution. If the API method is used, there is no
bending stress at the distance S and beyond. The ASC
method does not consider the annular-plate part projecting
beyond the shell surface. From Fig. 6, it is clearly seen that
the annular-plate stresses reach a maximum value at the
inner point of the shell-bottom junction and then become
low and die out rapidly. The bending stresses at a distance of
20 mm to the inner surface of the shell wall were measured.
They are 520 MPa for the 50 000 m3 tank [7] and 430 MPa
for the 20 000 m3 tank. The 100 000 m3 tank value is
390.7 MPa, taken as an average of 416, 403.5, 389 MPa
measured at one position and 387.5, 375.5, and 370.1 MPa
measured at another position [10]. Using the raw data in
Table 1, we obtained the theoretical results shown in
Table 2, where the errors are calculated using the annular-
plates' theoretical and experimental bending stresses at a
distance of 20 mm to the inner surface of the shell. The
maximum annular plate bending stresses are between
yield and twice yield stress for the 50 000 and 20 000 m3
tanks.
It is clearly shown from Table 2 and Fig. 6 that, for the
100 000 m3 tank, the Wu method predicts more accurately
the maximum annular plate stresses than the other two
methods. This is probably because the whole settlement of
the subgrade and the ringwall is small, only a few centi-
metres. Therefore, the ¯exible foundation has a good
support to the bottom plate, and the empirical value Kb
applies to the calculations. The empirical Kb is not measured
adjacent to the ringwall, but is an average value
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517 515
M2
R2
M2
R2
P=Hγ P=Hγ
M0
G
R1
CS
Tank wall centre
Ù ∞x
x
Fig. 5. A force distribution of the Wu model for the tank-bottom plate.
Fig. 6. Bending stresses of the bottom annular plate for 100 000 m3 tank.
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accumulated over a considerable area and in various
subgrade Kb measurements. Annular plate stresses for the
20 000 and 50 000 m3 tanks are discussed in the section
below.
3.3. Excessive differential settlement
It is necessary to have the elastic coef®cient of the inner
¯exible foundation to apply the ASC and Wu methods.
Average load-bearing capacity or elastic coef®cient over a
considerable or whole area is usually used in the engineer-
ing design, but it may be deceptive. Its value near the ring-
wall may still be misleading. The reason is that these values
are obtained through directly testing the foundation. In this
way, we have ®rst assumed that the ringwall and its inner
foundation have the same or uniform settlement. However,
this is unlikely to be true, especially in large settlement
cases.
The annular-plate stresses obtained using the Wu method
are larger than those using the API method, since a weak
support is expected from the ¯exible foundation. As shown
in Table 2, the experimental annular plate stresses for the
20 000 and 50 000 m3 tanks are much larger than those
predicted by the Wu and API methods. This is most
probably caused by the excessive differential settlement
local to the inner wall of the ringwall, because the ®eld
settlement measurements show that the inner elastic founda-
tion has much more settlement than the ringwall and the
total settlement of the whole foundation is more than one
metre. In this case, the inner elastic foundation adjacent to
the ringwall would not support the annular plate as required.
The elastic coef®cient Kb would have been considerably
reduced to cause larger stresses in the annular plate. There-
fore, it is a gross mistake or misconception that the concrete
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517516
Table 1
Data for three test storage tanks
Nominal volume (km3) D (m) H (m) C (mm) L (m) ts (mm) t1 (mm) t2 (mm) G (kN/m) ss (Mpa)
20 40 15 60 0.8 24 9 6 20 240
50 60 18.04 60 0.8 32 12 9 26.3 350
100 80 20.22 116.25 2.4 32.5 21 12 29.7 500
Table 2
Theoretical calculation results for three test storage tanks
Tank (km3) Method C (mm) S (mm) M0 (kN) R1 (kN/m) R1=G M2 (kN) R2 (kN/m) smax (MPa) s20 (MPa) Error (%)
20 Wu 60 195.1 28.318 58.76 2.94 0.899 218.88 355.0 268.7 237.5
Wu 75.58 197.8 28.987 57.85 2.89 1.088 219.86 341.2 257.7 /
API 60 404.1 27.410 56.93 2.85 / 13.71 295.9 213.9 250.2
ASC / 605.6 26.541 48.61 2.43 / / 484.5 380.4 11.5
50 Wu 60 293.3 216.57 91.49 3.48 0.675 224.01 461.7 368.6 229.1
Wu 113.8 320.9 220.41 88.81 3.38 1.141 225.95 429.3 340.3 /
API 60 554.1 215.33 90.02 3.42 / 23.52 413.9 323.1 237.9
ASC / 778.9 213.59 75.95 2.89 / / 566.2 463.9 10.8
100 Wu 116.25 521.1 248.91 147.4 4.96 0.407 237.35 432.4 376.1 23.73
Wu 188.22 522.8 254.01 138.9 4.68 2.422 241.86 378.8 327.1 /
API 116.25 928.6 247.04 147.1 4.95 / 43.71 407.3 351.2 210.1
ASC / 1234. 239.93 136.7 4.60 / / 549.5 483.0 23.8
P=Hγ
G
R1
Bottomannularplate
Concreteringwall
Lateralpressure
Bottom pressure
Fig. 7. Typical soil pressure distributions on the concrete ringwall.
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ringwall should be constructed as strong as possible,
because this causes excessive differential settlement,
makes the annular-plate seemingly hanging, and results in
even much larger annular-plate stresses.
3.4. Concrete ringwall design
It is a long-standing practice that the ringwall centre coin-
cides with the tank shell centre. However, the point where
the annular plate applies loadings to the ringwall is the most
peripheral point of the annular plates, and it is not the tank
shell centre, as shown in Fig. 7. The soil pressure distribu-
tions on the concrete ringwall were measured [11]. It is
typical of Fig. 7 to re¯ect this pressure distribution. It is
clearly shown that the ringwall bottom pressure at the
outside is larger than that at the inside. This implies that
the annular plate applies load to the ringwall from the most
peripheral point of the annular plates. This implicitly proves
that the annular plate's protruding length is at work and it is
improper not to consider this width in the ASC method. Due
to a very narrow width of the ringwall, this force eccentri-
city will produce considerable torsion in the ringwall.
Therefore, it is not justi®ed that the ringwall centre
conforms to the tank shell centre.
The present design load of the concrete ring wall is taken
as the total weight of the tank shell and top, and it is G �29:724 kN=m in the 100 000 m3 tank. The actual force
applied to the concrete ring wall is R1 � 149:8 kN=m,
which is about ®ve times the present design load G. It is
apparent that the ringwall loading and its application point
should be considered in their real circumstances.
4. Conclusion
The design of the annular-plate width, thickness, and the
width that will project beyond the outer surface of the tank
shell is discussed, along with the design of the concrete
ringwall. A few remarks are listed as a conclusion to this
work:
1. The excessive differential settlement local to the inner
wall of the ringwall should be avoided in order to reduce
the stresses in the bottom annular plates. If the excessive
differential settlements in the 20 000 and 50 000 m3
tanks had not been that large, the annular plate bending
stresses would have been smaller as predicted by the Wu
method.
2. The annular-plate's bending stresses have a very local
characteristic. For example, they are very small beyond
the width 500 mm for the 100 000 m3 tank.
3. For the Wu and API methods, the wider the width that
will project beyond the outer surface of the shell is, the
smaller is the maximum stress in the annular plates.
4. The point where the annular plate applies loadings to the
concrete ringwall is the peripheral point of the annular
plate, and it is not the tank shell centre. Therefore, it is
not justi®ed that the concrete ringwall centre conforms to
the tank shell centre.
5. The ringwall loading and its application point should be
considered according to the API or Wu method.
References
[1] API Standard 650: Welded steel tanks for oil storage. The tenth
edition, November 1998, American Petroleum Institute, Washington,
DC.
[2] API Standard 653: Tank inspection, repair, alteration, and reconstruc-
tion. The second edition, December 1995, American Petroleum Insti-
tute, Washington, DC.
[3] API Standard 620: Design and construction of large, welded, low-
pressure storage tanks. The ninth edition, February 1996, American
Petroleum Institute, Washington, DC.
[4] BS 2654: Speci®cation for manufacture of vertical steel welded non-
refrigerated storage tanks with butt-welded shells for the petroleum
industry. The third edition, 1989.
[5] China Standard SH3046-92: Petro-chemical design speci®cation for
vertical cylindrical steel welded storage tanks. Petro-Chemical Indus-
try Press, Beijing 1992 (in Chinese).
[6] Denham JB, Russell J, Wills CMR. How to design a 600 000-Bbl.
tank. Hydrocarbon Processing 1968;47(5):137±42.
[7] Li G-C. Stress analysis of the stepped shell wall and bottom plate
of large cylindrical storage tanks. Mechanics and Practice
1979;1(4):38±41.
[8] Wu TY. More accurate method devised for tank-bottom annular plate
design. Oil Gas J 1996;94(21):81±3.
[9] Wu TY. A new stress analysis method for petroleum storage tanks
and its calculation veri®cation. Petro-Chemical Equipment
1997;26(5):15±19 (in Chinese).
[10] Chen D-F, Li X-T. Stress gauging and analysis of 10 £ 104 m3 ¯oat-
ing-roof tanks. Oil Gas Storage Transp 1988;7(6):29±36 (in Chinese).
[11] Wu TY. Stress analysis of storage tanks with their bottom annular
plates and center plates lap-welded. Oil Gas Storage Transp
1997;16(8):13±18 (in Chinese).
T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517 517