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Thin-Walled Structures 40 (2002) 329–353www.elsevier.com/locate/tws

Buckling strength of the cylindrical shell andtank subjected to axially compressive loads

Seung-Eock Kim *, Chang-Sung Kim

Department of Civil and Environmental Engineering, Construction Tech. Research Institute, Sejong

University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea

Received 7 June 2001; received in revised form 23 October 2001; accepted 23 October 2001

Abstract

This paper aims to develop practical design equations and charts estimating the bucklingstrength of the cylindrical shell and tank subjected to axially compressive loads. Both geometri-

cally perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluatebuckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the para-metric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases.These results are different from those in the case of columns. The buckling strength of theperfect tank placed on an extremely soft foundation and a stiff foundation increases by up to1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strengthof the shell and tank decreases significantly as the amplitude of initial geometric imperfectionincreases. Convenient and sufficiently accurate design equations and charts used for estimatingbuckling strength are provided. © 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Cylindrical shell; Tank; Buckling strength; Initial geometric imperfection; Design equation

and chart

1. Introduction

Cylindrical shells and tanks with very thin walls are susceptible to buckling whenthey are subjected to axially compressive loads. Analytical and experimental research

* Corresponding author.

E-mail address: [email protected] (S.-E. Kim).

0263-8231/02/$ - see front matter © 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 6 6 - 0

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330 S.-E. Kim, C.-S. Kim / Thin-Walled Structures 40 (2002) 329–353

on the buckling of cylindrical shell structures has been performed mainly in the field

of mechanics and aeronautics. Goto and Shang [1] studied cylindrical shells with

thickness-to-diameter ratio of less than 350 and height-to-diameter ratio greater than

3. Their study was limited to relatively thick perfect shells that failed by plasticbuckling. Mandra and Mazzolani [2] researched cylindrical shells with diameter-to-

thickness ratio less than 400 subjected to axial compression. They found that the

thin cylindrical shells were very sensitive to initial geometric imperfections. Kim

and Kardomateas [3] analyzed an orthotropic shell. Chryssanthopulos et al. [4] stud-

ied stiffened cylinders. Soldatos [5] researched shells with non-circular cross-section.Donell and Wan [6], Miller [7] and Singer [8] performed experimental studies for

relatively thick cylindrical shells with diameter-to-thickness ratio of less than 400.

Their test results showed the cylindrical shells buckle locally where initial geometric

imperfections are large.

Melerski [9] performed linear elastic analysis of cylindrical tanks. Peek [10] ana-

lyzed unanchored liquid storage tanks under lateral loads. Lau and Zeng [11]

presented a simplified mathematical model for modeling the flexible bottom plate in

an unanchored cylindrical tank. Malhotra [12] carried out uplifting analysis of cylin-

drical tanks. Peek and El-Bkaily [13] studied postbuckling behavior of unanchored

steel tanks under lateral loads. Nam and Lee [14] studied the unsymmetrically loaded

cylindrical tank on an elastic foundation.

The aim of this paper is to investigate the buckling strength of the cylindrical

shell and tank with diameter-to-thickness ratio of greater than 800, which are, in

general, used for large-scale storage of liquid. Generally, buckling analysis may bedivided into bifurcation and load–deflection analysis. Bifurcation analysis is used for

perfect systems, while load–deflection analysis is used for imperfect systems. This

paper presents the buckling strength of the geometrically perfect cylindrical shell

and tank with a wide range of height-to-diameter and diameter-to-thickness ratios.

The geometrically imperfect cylindrical shell and tank are also studied. Practical

design equations and charts estimating buckling strength of the cylindrical shells andtanks are proposed.

2. Geometrically perfect cylindrical shell

In this section, the buckling strength of the geometrically perfect cylindrical shell

is presented. The analytical solution of the shell is widely known and brie fly intro-

duced herein. Numerical analysis is also performed. An appropriate element typeand the necessary number of elements to be used in numerical analysis are rec-

ommended by comparing analytical and numerical results.

2.1. Analytical solution

If a cylindrical shell simply supported at the ends is uniformly compressed in theaxial direction as shown in Fig. 1, the general solution for very small displacements

can be given in the following form [15,16]:

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331S.-E. Kim, C.-S. Kim / Thin-Walled Structures 40 (2002) 329 – 353

Fig. 1. Cylindrical shell subjected to axial load.

u A sin nq cosmp x

H , (1)

v B sin nq cosmp

H (2)

and

w C sin nq cosmp

H , (3)

where A, B and C are constants; H is the height of the cylindrical shell; and n and m

are the buckling number of circumferential and longitudinal half-waves, respectively.When the simply supported conditions of w 0 and d2w / d x2 0 are used at the

ends, the critical stress is obtained as

s cr N x

t

R

S

E

(1n2), (4)

where

R (1n2) l4 a (n2 l2)4(2 n)(3n) l4n2 2 l4(1n2) l2n4(7

n) l2n2(3 n) n42n6,

S l2(n2 l2)2 2

1n l2 1n

2 n2[1 a (n2 l2)2]

2n2 l2

1n

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332 S.-E. Kim, C.-S. Kim / Thin-Walled Structures 40 (2002) 329 – 353

2a

1n l2 1n

2 n2[n2 (1n) l2,

a t 2

12 R2,

l mRp

H ,

N x is the axial force, E is Young’s modulus, n is Poisson’s ratio, t is the thickness

of the shell, and R is the radius of the shell. Theoretically, the critical stress of Eq.

(4) has an infinite number of solutions as the values of m and n vary. The minimum

critical stress among these is determined as the buckling stress. One dif ficulty in

using Eq. (4) is that the m and n values leading to the buckling stress are unknownuntil a large number of critical stresses are calculated and compared. As a result,

the equation essentially requires a lot of calculations of critical stresses depending

on the values of m and n in order to get the lowest critical stress, i.e. buckling stress.

Assuming that many buckling waves (m) form along the length of the cylinder,

the value of l2 becomes large. Then, Eq. (4) can be simplified in the following form:

s cr N x

t

1n2

E a (n2 l2)2

l2

(1n2) l2

(n2 l2)2. (5)

When the value of n in Eq. (5) is equal to zero, axisymmetric buckling occurs, andEq. (5) is simplified as

s cr N x

t Dm2p 2

tH 2

E

R2 D

H 2

m2p 2, (6)

where D Et 3 / [12(1n2)] is the flexural rigidity. Since s cr is a continuous function

of mp / H , the minimum value of Eq. (6) can be written in the following form:

s cr Et

R 3(1n2)

. (7)

Analytical solutions given by Eqs. (4)–(7) will be used for the benchmark values

in selecting an appropriate numerical model in Section 2.2.

2.2. Numerical analysis

Herein, an appropriate numerical model including the element type and necessary

number of elements in numerical analysis is presented. A parametric study is perfor-med over a wide range for the cylindrical shell. An accurate design equation estimat-

ing the buckling strength is proposed.

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2.2.1. Analysis method

Buckling analysis, in general, may be divided into bifurcation and load–deflection

analysis as shown in Fig. 2. Bifurcation buckling occurs when a maximum axial

compressive stress becomes equal to the buckling stress. The bifurcation analysiswas carried out using ABAQUS, a commercial finite element analysis program

[17,18]. The equation of bifurcation analysis is expressed in the following form:

([K ] l[S ]){y } 0, (8)

where [K ] is the stiffness matrix of the system, [S ] is the stress stiffness matrix, lis an eigenvalue determining buckling load (or load factor), and {y } is an inherent

vector determining buckling mode. If the load applied to the structure is Q N , the

critical buckling load is lQ N . This bifurcation analysis can capture both the buckling

stress and the failure mode of structures.

2.2.2. Selection of analysis model

The dimensions of the cylindrical shell used were diameter of 20 m, height of 40

m and thickness of 0.025 m. The resulting diameter-to-thickness and height-to-diam-

eter ratios are 800 and 2, respectively. Material properties were E

2 × 1011 N / m2, n 0.3 and s y 3.2 × 108 N / m2. When a full model was used,

ABAQUS did not predict the buckling mode and buckling strength accurately. Thus

a half-model in height was used in this study (Fig. 3). The simply supported and

the symmetric boundary conditions were used for the bottom and top, respectively.

In the following, an appropriate analytical model was selected by comparing theanalytical with the numerical results.

2.2.2.1. Element type Three-dimensional shell elements offered in ABAQUS can

be divided into four-node (S4R, S4R5) and eight-node (S8R, S8R5) shell elements

in view of the number of nodes per element, and five- (S4R5, S8R5) and six-degrees-

Fig. 2. Buckling analyses.

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Fig. 3. Modeling of cylindrical shell: (a) full model; (b) half-model.

of-freedom (dof) (S4R, S8R) shell elements in view of the number of degrees of

freedom per element. The 5-dof shell element has three displacement and two

rotational components. The 6-dof shell element has three displacement and three

rotational components. The critical stresses obtained by numerical analyses using

these element types (Fig. 4) were compared with the theoretical stresses calculated

by Eq. (4). When element type S8R was used, the error in the buckling stress wasless than 1.4%. When elements S4R and S4R5 were used, the buckling stresses were

overestimated by up to 15% even though a suf ficient number of elements was used.

Thus element type S8R was selected in numerical analysis.

2.2.2.2. Number of element Buckling strength was obtained using different num-

bers of elements in both circumferential and axial directions. Analysis results were

compared with the theoretical ones given by Eq. (4) (Fig. 5). When the number of

elements in the circumferential direction increased from 12 to 32, the maximum error

reduced from 6.5% to 0.8%. It was preferable to keep the number of elements to

more than 20 in order to obtain accurate buckling strength.

Buckling stress was less sensitive to the number of elements in the axial direction

than to the number in the circumferential direction. When the number of elements

in the axial direction increased from 10 to 40, the maximum error reduced from

0.8% to 0.4%. When 32 elements in the circumferential direction and 40 elementsin the axial direction were used, the buckling strength estimated was very accurate,

within an error of 0.4%.

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Fig. 4. Critical stresses associated with mode shape corresponding to element type.

Fig. 5. Buckling stresses associated with different numbers of elements.

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2.2.2.3. Buckling mode The first, second and third mode shapes obtained by

numerical analysis are shown in Figs. 6–8, respectively. The first mode occurs in

m 1 and n 5, the second mode occurs in m 3 and n 9, and the third mode

occurs in m 5 and n 11. An axially symmetric buckling mode, one of higher-order modes, is shown in Fig. 9. This buckling mode occurs in m 10 in the axial

direction and n 0 in the circumferential direction. The buckling modes show good

agreement with the theoretical ones.

2.2.3. Parametric study

Buckling stresses were evaluated for cylindrical shells with diameter-to-thickness

ratio ranging from 800 to 2000 and height-to-diameter ratio from 0.5 to 3. Material

properties used were the same as before. The buckling stresses corresponding to the

various diameter-to-thickness and height-to-diameter ratios are presented as the ratio

of buckling stress to elastic modulus in Table 1. The axisymmetric buckling mode

(i.e. n 0, m ) occurred in the italicized area in Table 1. Their buckling stresses

were almost the same as the theoretical ones calculated by Eq. (7). The m and n

type buckling mode occurred in the non-italicized area in Table 1 and their buckling

stresses are very close to theoretical ones obtained by Eq. (4). Figs. 10 and 11 show

the buckling stresses with respect to the various diameter-to-thickness and height-

to-diameter ratios. The buckling stress decreases significantly as the diameter-to-

Fig. 6. First mode shape of cylindrical shell (m 1, n 5).

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Fig. 7. Second mode shape of cylindrical shell (m 3, n 9).

thickness ratio increases (Fig. 10), while the buckling stress decreases slightly as the

height-to-diameter ratio increases (Fig. 11). These trends are different from those in

the case of columns. This is because buckling of a cylindrical shell is governed by

not only the axial buckling mode (m) but also the circumferential buckling mode

(n). The number of half-waves in the circumferential direction (n) decreases as the

height-to-diameter ratio increases.

2.2.4. Design equation

A design equation estimating the buckling strength of geometrically perfect cylin-

drical shells was developed by using regression analysis on the results in Table 1.The following design equation predicts buckling strength accurately with approxi-

mately 1% errors:

s cr E 1.19 H

D0.0256 t

D. (9)

3. Geometrically perfect cylindrical tank

The cylindrical tank, i.e. the cylindrical shell with a bottom plate, was studied.

Since a theoretical solution could not be obtained due to its complexity, numerical

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Fig. 8. Third mode shape of cylindrical shell (m 5, n 11).

Fig. 9. Axisymmetric buckling mode shape.

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Table 1

Buckling stresses of perfect cylindrical shell (s cr × 103 / E )

D / t H / D

0.5 1 2 3

Numerical Eq. (9) Numerical Eq. (9) Numerical Eq. (9) Numerical Eq. (9)

analysis analysis analysis analysis

800 1.5013 1.5141 1.4920 1.4875 1.4421 1.4613 1.4330 1.4462

900 1.3586 1.3459 1.3240 1.3222 1.3195 1.3000 1.2960 1.2855

1000 1.2111 1.2113 1.1875 1.1900 1.1755 1.1691 1.1689 1.1570

1100 1.1021 1.1012 1.0706 1.0818 1.0541 1.0628 1.0390 1.0480

1200 1.0170 1.0094 0.9885 0.9916 0.9581 0.9742 0.9450 0.96421300 0.9365 0.9318 0.9230 0.9154 0.8814 0.8993 0.8650 0.8889

1400 0.8654 0.8652 0.8152 0.8500 0.8221 0.8351 0.8035 0.8264

1500 0.8075 0.8075 0.7955 0.7933 0.7730 0.7794 0.7525 0.7713

1600 0.7610 0.7571 0.7421 0.7438 0.7334 0.7307 0.7120 0.7231

1700 0.7200 0.7125 0.6965 0.7000 0.6950 0.6877 0.6827 0.6806

1800 0.6819 0.6729 0.6570 0.6611 0.6574 0.6495 0.6490 0.6427

1900 0.6355 0.6375 0.6250 0.6263 0.6250 0.6153 0.6248 0.6089

2000 0.6054 0.6056 0.6000 0.5950 0.5901 0.5845 0.5899 0.5785

Fig. 10. Buckling stresses for various thickness and shape ratios.

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Fig. 11. Buckling stress for various shape and thickness ratios.

analysis was used. The numerical model was the same as that in the case of the

geometrically perfect shell, except that the bottom plate was also being modeled. A

half-model consisting of element type S8R was used. The numbers of elements used

were 32 in the circumferential and 40 in the axial direction. The material properties

used were the same as those previously. The effect of the soft and the stiff foundation

shown in Figs. 12 and 13 was studied.

Fig. 12. Tank on soft foundation.

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Fig. 13. Tank on stiff foundation.

3.1. Soft foundation

The extremely soft foundation was assumed to get a lower-bound buckling strength

of the cylindrical tank. The ideal boundary condition was modeled by simple supports

at the nodal points on the circumferential edges of the tank bottom. Parametric studies

were performed for tanks with diameter-to-thickness ratio ranging from 800 to 2000

and height-to-diameter ratio ranging from 0.5 to 3. Identical thicknesses of the wall

(t w) and bottom plate (t b) were used. The buckling stresses are listed in Table 2.The buckling strength of the perfect cylindrical tank on the soft foundation

increased by up to 1.6% compared with the cylindrical shell. Thus, the same design

equation [Eq. (9)] can be used in estimating the buckling strength of the geometri-

cally perfect tank placed on a soft foundation.

3.2. Stiff foundation

The extremely stiff foundation was assumed to get an upper-bound buckling

strength. The ideal boundary condition was modeled by simple supports at the nodal

Table 2

Buckling stresses of perfect cylindrical tank on soft foundation (s cr × 103 / E )

D / t H / D

0.5 1 2 3

800 1.5192 1.5155 1.4551 1.4351

1100 1.1022 1.0958 1.0702 1.0456

1400 0.8659 0.8648 0.8353 0.80461700 0.7203 0.7134 0.7025 0.6898

2000 0.6061 0.6031 0.6005 0.5905

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points on the circumferential edges as well as on the bottom plate. The ranges of the

diameter-to-thickness and height-to-diameter ratios studied were the same as before.

Identical thicknesses of the wall (t w) and bottom plate (t b) were used. The calculated

buckling stresses are listed in Table 3. The buckling strength of the perfect tank placed on the stiff foundation increased by a maximum of 5.6% compared with that

on the soft foundation.

A design equation for estimating buckling strength of the geometrically perfect

cylindrical tank was developed by using regression analysis on the results listed in

Table 3. The following design equation predicts the buckling strength with suf ficientaccuracy within approximately 1% error:

s cr E 1.28 H

D0.0256 t

D. (10)

4. Geometrically imperfect cylindrical shell

This section examines the buckling stress of the geometrically imperfect cylindri-

cal shell. The load–defection analysis was used.

4.1. Analysis method

The load–deflection analysis method accounts for initial geometric imperfections,

which could be determined by a linear superposition of buckling eigenmodes. The

initial geometric imperfection is expressed in the following form:

xi i 0

M

wii, (11)

where i is the ith mode shape and wi is the associated scale factor.

Non-linear static analysis, including both material and geometric non-linearity,

Table 3

Buckling stresses of perfect cylindrical tank on stiff foundation (s cr × 103 / E )

D / t H / D

0.5 1 2 3

800 1.5353 1.5192 1.5135 1.5130

1100 1.1133 1.1042 1.1010 1.0905

1400 0.8737 0.8676 0.8650 0.83401700 0.7243 0.7151 0.7132 0.7007

2000 0.6107 0.6078 0.6060 0.6056

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was used to obtain the ultimate load capacity. The Newton–Raphson solution tech-

nique and default convergence tolerances were used for non-linear analyses.

4.2. Analysis model

The numerical model is the same as that in the case of the geometrically perfect

shell. The half-model with element type S8R was used. Simply supported and sym-

metric boundary conditions were used for the bottom and the top, respectively. The

numbers of elements used were 32 in the circumferential and 40 in the axial direction.The material properties were the same as those used previously. The parametric

studies were performed for diameter-to-thickness ratios of 800, 1400 and 2000 and

for height-to-diameter ratio from 0.5 to 3. The first buckling mode shape obtained

by eigenvalue analysis was used as the initially imperfect shape of the shell. The

ratio of the magnitude of initial imperfection to wall thickness (d 0 / t ) varied from 0

to 3.

4.3. Analysis results

Cylindrical shell structures, in general, are very sensitive to the amplitude of the

initial geometric imperfection. The buckling stresses associated with the different

diameter-to-thickness and height–diameter ratios, and the magnitudes of initial

imperfection, are presented as the ratio of buckling stress to elastic modulus in Table

4 and Figs. 14–17. It was observed that the cylindrical shells with the axisymmetricbuckling mode were much more sensitive to initial imperfection than those with the

non-axisymmetric buckling mode. The buckling strength of a geometrically imperfect

cylindrical shell can be estimated by linear interpolation using the data in Table 4

and Figs. 14–17.

5. Geometrically imperfect cylindrical tank

This section examines the buckling stress of the geometrically imperfect cylindri-

cal tank. The load–deflection analysis was used. The initial geometric imperfectionscould be determined by a linear superposition of buckling eigenmodes. The numeri-

cal model was the same as in the case of the geometrically perfect tank, except for

the geometric imperfections being modeled. Parametric studies were performed for

tanks with diameter-to-thickness ratios of 800, 1400 and 2000 and height-to-diameter

ratio ranging from 0.5 to 3. The first buckling mode shape obtained by eigenvalue

analysis was used as the initially imperfect shape of the tank. The ratios of the

magnitude of initial imperfection to wall thickness (d 0 / t ) varied from 0 to 3.

5.1. Soft foundation

The buckling stresses of the geometrically imperfect cylindrical tank placed on

the soft foundation are presented with respect to the imperfection ratio (d 0 / t ) in Table

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Table 4

Buckling stresses of geometrically imperfect shell (s cr × 103 / E )

H / D d 0 / t D / t

800 1400 2000

0.5 0.0 1.5013 0.8654 0.6054

0.1 0.9223 0.6029 0.4584

0.3 0.6145 0.4673 0.3489

1.0 0.3446 0.2461 0.1858

2.0 0.2803 0.2127 0.1508

3.0 0.2564 0.2015 0.1377

1.0 0.0 1.4912 0.8152 0.6040

0.1 1.9000 0.5405 0.3923

0.3 0.8270 0.4349 0.3061

1.0 0.5266 0.2436 0.1658

2.0 0.4127 0.1985 0.1387

3.0 0.3343 0.1785 0.1235

2.0 0.0 1.4421 0.8221 0.5901

1.0 1.0395 0.6233 0.3384

2.0 0.8530 0.5315 0.2824

3.0 0.7252 0.4584 0.2535

3.0 0.0 1.4330 0.8035 0.5899

1.0 1.1108 0.5386 0.2905

2.0 0.9631 0.4490 0.2140

3.0 0.8458 0.4012 0.1913

5 and Figs. 18–21. It was observed that the tanks with the axisymmetric buckling

mode were more sensitive to initial imperfection than those with the non-axisym-

metric buckling mode. The buckling strength of a geometrically imperfect tank placed on a soft foundation can be estimated by linear interpolation using the data

in Table 5 and Figs. 18–21.

5.2. Stiff foundation

The buckling stresses of the imperfect tank placed on the stiff foundation are

presented with respect to the imperfection ratio (d 0 / t ) in Table 6 and Figs. 22–25.

The buckling strength of a geometrically imperfect cylindrical tank placed on a stiff

foundation can be estimated by linear interpolation using the data in Table 6 and

Figs. 22–25.

6. Conclusions

This paper studied the buckling strength of the cylindrical shell and tank subjected

to axially compressive loads. The conclusions are as follows.

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Fig. 14. Buckling stresses of imperfect shell ( H / D 0.5).


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Table 5

Buckling stresses of geometrically imperfect tank on soft foundation (s cr × 103 / E )

H / D d 0 / t D / t

800 1400 2000

0.5 0.0 1.5126 0.8664 0.6059

0.1 0.9938 0.6384 0.4058

0.3 0.6461 0.4560 0.3438

0.5 0.4933 0.3673 0.2880

1.0 0.3816 0.2800 0.2219

1.5 0.3517 0.2584 0.2001

2.0 0.3333 0.2498 0.1836

2.5 0.3176 0.7408 0.1757

3.0 0.3098 0.2362 0.1644

1.0 0.0 1.4912 0.8578 0.6040

0.1 0.9282 0.5686 0.4029

0.3 0.5747 0.4021 0.3014

0.5 0.4576 0.3170 0.2427

1.0 0.3645 0.2626 0.2001

1.5 0.3421 0.2299 0.1816

2.0 0.3261 0.2186 0.1677

2.5 0.3125 0.2058 0.1549

3.0 0.3018 0.1997 0.1499

2.0 0.0 1.4568 0.8369 0.5920

0.5 1.2880 0.7328 0.50641.0 1.1200 0.6480 0.4477

1.5 1.0030 0.5701 0.3849

2.0 0.8848 0.5024 0.3331

2.5 0.7728 0.4381 0.2917

3.0 0.6995 0.3888 0.2757

3.0 0.0 1.4330 0.8035 0.5899

0.5 1.2224 0.7060 0.4610

1.0 1.1002 0.6506 0.3818

1.5 1.0237 0.5811 0.3272

2.0 0.9502 0.5526 0.2792

2.5 0.9024 0.5347 0.2467

3.0 0.8518 0.5150 0.285

1. When elements S4R and S4R5 are used, the buckling stresses are overestimated

even though a suf ficient number of elements is used. When element type S8R is

used, the buckling stresses are very close to those calculated from the analytical

equation within an error of 1.4%.

2. When a full model is used, ABAQUS does not predict buckling mode and buck-

ling strength accurately. When a half-model in height with symmetric boundary

condition is used, ABAQUS can predict accurate buckling mode and strength.3. The axisymmetric buckling mode (n 0, m ) occurs when the height-to-

diameter ratio is less than 0.5 and the diameter-to-thickness ratio is greater than

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Fig. 18. Buckling stresses of imperfect tank on soft foundation ( H / D 0.5).

Fig. 19. Buckling stresses of imperfect tank on soft foundation ( H / D 1.0).

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Table 6

Buckling stresses of geometrically imperfect tank on stiff foundation (s cr × 103 / E )

H / D d 0 / t D / t

800 1400 2000

0.5 0.0 1.5353 0.8737 0.6107

0.1 1.0701 0.7028 0.5248

0.3 0.7928 0.5626 0.3853

1.0 0.5034 0.3824 0.2884

2.0 0.4413 0.3224 0.2426

3.0 0.3952 0.2875 0.2045

1.0 0.0 1.5192 0.8676 0.6078

0.1 1.0191 0.5494 0.4127

0.3 0.7225 0.4655 0.3159

1.0 0.4501 0.3042 0.2208

2.0 0.4106 0.2732 0.1992

3.0 0.3873 0.2536 0.1782

2.0 0.0 1.5135 0.8650 0.6060

0.1 0.9208 0.6011 0.4433

0.3 0.6359 0.4602 0.3312

1.0 0.4698 0.2836 0.1944

2.0 0.3893 0.2604 0.1783

3.0 0.3495 0.2265 0.1576

3.0 0.0 1.5130 0.8340 0.6056

1.0 0.6522 0.6528 0.20182.0 0.5207 0.5743 0.1564

3.0 0.4902 0.5084 0.1233

900. The buckling stress of these shells can be calculated by Eq. (7). The other

shells can be calculated by Eq. (4).4. The buckling stresses decrease significantly as the diameter-to-thickness ratio

increases, while they decrease slightly as the height-to-diameter ratio increases.

These trends are different from those of columns.

5. The buckling strength of the perfect tank placed on an extremely soft foundationand a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared

with that of the perfect shell.

6. Buckling strength decreases significantly as the amplitude of initial geometric

imperfection increases. The cylindrical shells and tanks with the axisymmetric

buckling mode are more sensitive to initial imperfection than those with the non-

axisymmetric buckling mode.

7. Convenient and suf ficiently accurate design equations to use in estimating buck-

ling strength of the perfect shell and tank are developed.

8. The buckling strength of the geometrically imperfect shell and tank can be esti-

mated by linear interpolation using data in Tables 4–6 and Figs. 14–25.9. The lower- and upper-bound buckling strength of the tank, corresponding to

extremely soft and stiff foundations, is also provided.

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Fig. 22. Buckling stresses of imperfect tank on stiff foundation ( H / D 0.5).


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