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: engp8773@nus.edu.sg (T.Y. Wu).

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

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.

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.

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.

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.

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

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[10] Chen D-F, Li X-T. Stress gauging and analysis of 10 £ 104 m3 ¯oat-

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[11] Wu TY. Stress analysis of storage tanks with their bottom annular

plates and center plates lap-welded. Oil Gas Storage Transp

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T.Y. Wu, G.R. Liu / International Journal of Pressure Vessels and Piping 77 (2000) 511±517 517