Pvp2010-25092-Lower Bound Buckling Load of a Floating Roof Pontoon

1 Copyright 2010 by ASME

Proceedings of the ASME 2010 Pressure Vessels and Piping Division / K-PVP Conference PVP2010

July 18-22, 2010, Bellevue, Washington, USA

PVP2010-25092

LOWER BOUND BUCKLING LOAD OF A FLOATING ROOF PONTOON

Shoichi Yoshida Yokohama National University

Yokohama, Japan

ABSTRACT The 2003 Tokachi-Oki earthquake caused severe damage to oil storage tanks due to liquid sloshing. Seven single-deck floating roofs had experienced sinking failures in large diameter tanks at a refinery in Tomakomai, Japan. The pontoons of the floating roofs might be buckled due to bending load during the sloshing. The content in the tank was spilled on the floating roof from small failures which were caused in the welding joints of pontoon bottom plate by the buckling. Then the floating roof began to lose buoyancy and sank into the content slowly. The elastic buckling of the pontoon is important from the viewpoint of the single-deck floating roof sinking. The authors had reported the buckling strength of the pontoons subjected to bending and compressive loads in the published literatures. The axisymmetric shell finite element method for linear elastic bifurcation buckling was used in the analysis. The buckling characteristics of the pontoon both with and without ring stiffeners were investigated. The initial geometrical imperfection may diminish the buckling load. This paper presents the lower bound buckling load according to the reduced stiffness method proposed by Croll and Yamada. The lower bound buckling loads of the pontoon subjected to circumferential bending load are evaluated from the axisymmetric finite element analysis which includes the reduced stiffness method. INTRODUCTION The floating roofs are used in large aboveground oil storage tanks to prevent evaporation of the content. They are welded steel structure and are classified into two basic types, single-deck and double-deck structures. The single-deck floating roofs, which are considered herein, consists of a thin circular plate called a deck attached to a buoyant ring of box-shaped cross section called a pontoon. Seven single-deck floating roof had sunk in the 2003 Tokachi-Oki earthquake at a refinery in Tomakomai, Japan. The floating roofs deformed to leak oil due to the liquid sloshing as shown in Fig.1, and they lost buoyancy to sink

slowly spending several days. It is presumed that small failures in welded joints or in stress concentrated parts of the pontoon were caused due to the sloshing, and next these failures expanded gradually in the sinking process.

Floating roofFloating roof

Fig.1 Sloshing of floating roof tank

In the sloshing, the pontoon of floating roof is subjected to the circumferential bending load, which is concave downward on the maximum wave height side and upward on the minimum wave height side. The buckling may occur on either roof plate or bottom plate of the pontoon, on which circumferential compressive stress acts. The elastic buckling of the pontoon is important from the viewpoint of the single-deck floating roof sinking. The author had investigated the elastic buckling characteristics of the pontoons with and without ring stiffeners subjected to several combined bending load and compressive load due to the sloshing using the axisymmetric shell finite element analysis [1~4]. They were linear elastic bifurcation buckling analyses for non-axisymmetric buckling modes under axisymmetric loading. Though the sloshing load is non-axisymmetric, it was assumed to be axisymmetric in the analyses for simplification. In addition, the previous analyses were for the pontoons without initial geometrical imperfection. In thin shell structures, it is well known that the theoretical buckling load is different from the experimental buckling load. The reason of the difference is the geometrical imperfection of the shell. The design formula for buckling load is usually

Proceedings of the ASME 2010 Pressure Vessels & Piping Division / K-PVP Conference PVP2010

July 18-22, 2010, Bellevue, Washington, USA

PVP2010-250


derived from experimental lower bound loads. On the other hand, the reduced stiffness method proposed by Croll and Yamada [5~7] gives the lower bound buckling load theoretically. In this method, the lower bound is assumed to be obtained to eliminate the linear membrane strain energy from the quadratic components of the variation of potential energy in equilibrium condition. The shape of geometrical imperfection is not required to define in this method. This paper presents the lower bound buckling loads of the floating roof pontoon subjected to circumferential bending load. The axisymmetric shell finite element method which is the same as the previous studies [1~4] is used. The results of the analyses are evaluated to the lower bound using the reduced stiffness method.

NOMENCLATURE [Bdm] Nonlinear strain-displacement transformation matrix dr-m Radial displacement for m dz-m z-direction displacement for m d-m Circumferential displacement for m {dm} Nodal displacement vector or buckling mode E Youngs modulus fRS RS factor fRSm RS factor for m Ir Moment of inertia of pontoon cross section [KLm] Small displacement stiffness matrix [KMm] Membrane component of small displacement stiffness matrix [Km] Initial stress stiffness matrix [km] Elemental initial stress stiffness matrix LD Lower edge difference Li Inner rim height Lo Outer rim height LP Pontoon width LR Ring stiffener length M Circumferential bending moment Mcr Circumferential buckling bending moment Mcr-i Mcr with i ring stiffeners M*cr Lower bound circumferential buckling bending moment M*cr-i M*cr with i ring stiffeners Mcrm Circumferential buckling bending moment for m Mcrm-i Mcrm with i ring stiffeners M*crm Lower bound circumferential buckling bending moment for m M*crm-i M*crm with i ring stiffeners Mi Bending moment of i-direction m Circumferential wave number Ni Membrane force of i-direction NS0 Initial meridional membrane force for m=0 N0 Initial circumferential membrane force for m=0 ri Inner rim radius (r, , z) Cylindrical coordinates s Elemental coordinate t Shell thickness

ti Inner rim plate thickness tL Pontoon bottom plate thickness to Outer rim plate thickness tR Thickness of ring stiffener tu Pontoon roof plate thickness U2b Linear bending strain energy U2m Linear membrane strain energy u Tangential displacement of shell element um Tangential displacement of shell element for m V2m Nonlinear membrane strain energy v Circumferential displacement of shell element vm Circumferential displacement of shell element for m w Normal displacement of shell element wm Normal displacement of shell element for m Rotation angle of nodal point m Rotation angle of nodal point for m Ni Membrane force of i-direction for virtual displacement Mi Bending moment of i-direction for virtual displacement i Strain of i-direction for virtual displacement i Change of curvature of i-direction for virtual displacement Variation of total potential energy i Strain of i-direction m Eigenvalue or buckling load parameter for m min Minimum of m * RS Buckling parameter Total potential energy Poissons ratio Initial circumferential membrane stress for m=0 Elemental angle [ ] Coordinate transformation matrix i Change of curvature of i-direction Subscript -i The number of ring stiffeners Subscript m Circumferential wave number Superscript (L) Liner term Superscript (N) Nonlinear term Superscript * Lower bound

SINGLE DECK FLOATING ROOF STRUCTURE The single-deck floating roof is made of mild steel, and its members are jointed together by the welding. The thin circular plate called a deck attaches to a buoyant ring of box-shaped cross section called a pontoon, as shown in Fig.2. The pontoon consists of the inner rim, the outer rim, the pontoon roof and the pontoon bottom. Both the plates of the pontoon roof and the pontoon bottom have the range between 4.5 mm and 6 mm in thickness, and both the inner rim and the outer rim are about 12 mm. The deck thickness is about 4.5 mm (3/16 inch) regardless of the tank diameter. The pontoon is usually a wide and shallow trapezoidal shape. The width of the pontoon which equals to the length between the inner rim and


the outer rim is several meters in large tank. It is divided into a number of compartments in the pontoon by the radially arranged plates called a bulkhead. Each compartment has leakproof. API Standard 650 Appendix C [8] gives the minimum requirement for the floating roof design. However, the sloshing motion has not been considered in the floating roof design. The sloshing in cylindrical oil storage tanks occurs due to relatively long period earthquake motion in which predominant period is 5 seconds to 15 seconds. The 1st natural period of the tank is usually within this range. In the sloshing, the direction of the maximum and the minimum wave height side of the tank is the earthquake excitation direction as shown in Fig.3. The pontoon is subjected to the circumferential bending load which is concave downward on the maximum wave height side and upward on the minimum wave height side during the sloshing. The buckling may occur on either the roof plate or the bottom plate of the pontoon, where the circumferential compression stress acts.

Seal

Pontoon

Deck

Shell Bottom

Bulkhead Outer rim

Inner rim

Detail of Pontoon

Pontoon bottom plate

Pontoon roof plate

Deck

Pontoon

Deck

Fig.2 Single-deck floating roof

Deck

Pontoon

Max. Sloshing directionBuckling

Buckling

Fig.3 Buckling location

Fig.4 shows the pontoon buckling in a 130,000 m3 crude oil tank at the 1999 Chi-Chi earthquake in Taiwan. Crude oil was inundated into the pontoon from ruptured pontoon bottom plate, and spilled into the deck from ruptured inner rim-to-deck

joint [9]. Because of small leakage, this floating roof had remained afloat. Fig.5 shows the failure of the single-deck floating roof in a 100,000 m3 crude oil tank at the 2003 Tokachi-Oki earthquake in Japan [3]. The floating roof had damaged due to the sloshing, and it sank into the content. This picture was taken after draining oil and cleaning the tank several months after the earthquake. The large deformation and failure of the floating roof is presumed to be caused during the sinking process. This roof remained afloat without collapse for several days after the earthquake according to aerial photos. The author presumes that the damage condition of both the single-deck floating roofs in Fig.4 and Fig.5 were almost identical immediately after the earthquake. It is concluded that the collapse of the single-deck floating roof as shown in Fig.5 did not occur at the earthquake, but it occurred during the sinking process.

Fig.4 Pontoon buckling failure [9]

Fig.5 Floating roof that sank in oil [3]

ANALYSIS Axisymmetric Shell Finite Element The axisymmetric shell finite element used in this analysis is a conical frustum element as shown in Fig.6. In this element, the tangential displacement u and the circumferential displacement v are assumed to be linear function and the normal displacement w to be cubic function with respect to s, where s is the elemental coordinate.

Pontoon

Deck

Deck

Pontoon

Spilled oil Spilled oil


The strain-displacement relation based on the Kirchhoff-Loves assumption is given by the Novozhilovs equation [10], and the linear term is expressed by Eq.(1) and the nonlinear term is expressed by Eq(2) [11]:

( )( )( )( )( )( )

( )

++

+

+

++

=

vrcossin

sv

rcosw

rsin

sw

r12

sw

rsinv

rcosw

r1

sw

sinrv

svu

r

sinucoswr1v

r1

su

22

2

22

2

2

2

2

Ls

Ls

Ls

Ls

L

Ls

(1)

=

cosvwsw

r1

cosvwr21

sw

21

2

2

2

)N(s

)N(

)N(s

(2)

where, i and i (i = s, , s) express the strain and the change of curvature. The superscript (L) and (N) denote a linear term and a nonlinear term, respectively. The elastic stress-strain relation based on the Hookes law can be written as:

( )

=

s

s

s

s

2

2

22

2

s

s

s

s

24t1.sym

012t

012t

12t

0002

10000100001

1Et

MMMNNN

(3)

where, E, and t are Youngs modulus, Poissons ratio and thickness. Ni and Mi (i = s, , s) are the membrane forces and the bending moments. The displacements in elemental coordinate are expressed using the Fourier expansion as follows:

=

=

m

m

m

m

0m wvu

mcos0000mcos0000msin0000mcos

wvu

(4)

dzL

r

z

s w

u

j

i

dr

=dw/ds

r

v

: Nodal point

Fig.6 Axisymmetric shell finite element

where m is a term of Fourier series, i.e., a circumferential wave number. The relation of the displacements between the global coordinate and the elemental coordinate is defined as follows:

=

m

mr

m

mz

m

m

m

m

ddd

10000cos0sin00100sin0cos

wvu

(5)

where, di-m(i = z, , r) and m are the i-direction displacement in global coordinate and the rotation angle. Eq.(5) can be simply written as follows: { } [ ]{ }mm du = (6) where [ ] is the coordinate transformation matrix. Linear Bifurcation Buckling Analysis The virtual work principle gives the equation of bifurcation buckling for the circumferential wave number

2m under axisymmetric load as follows: [ ] [ ]( ){ } 0dKK mmmLm =+ (7) where [KLm] , [Km], {dm} and m are the small displacement stiffness matrix, the initial stress stiffness matrix, the displacement vector and the buckling load parameter, respectively. The elemental initial stress stiffness matrix is expressed by the following forms:

[ ] [ ] [ ] edm0

0sT

Vdmm dVBN0

0NBk

e

=

(8)

where Ns0 is the initial meridian membrane forces and N0 is the initial circumferential membrane forces of the shell element. These are axisymmetric forces while the sloshing load is non-


axisymmetric. It is assumed that the pontoon buckling due to the sloshing are approximately obtained from the axisymmetric load. [Bdm] is the nonlinear strain-displacement transformation matrix. Eq.(7) is the equation for a eigenvalue problem, and is solved for the circumferential wave number 2m . The buckling load is derived from the minimum eigenvalue m. The eigenvector {dm} corresponding to the eigenvalue m becomes the buckling mode. The computer code based on the theory mentioned above was developed by the author. It was verified to investigate the application to several problems [12,13].

INITIAL STRESS The hatching area of Fig.7 is the pontoon cross section and is discretized into 1500 to 1600 axisymmetric shell finite elements. Point G is the centroid of the cross section. The deck plate is not modeled in the analysis. The bulkhead plates can not be taken into consideration in the analytical model because of the axisymmetric analysis. Fig.8 shows the ring stiffener which is LR in length and tR in thickness, and it attaches to both the pontoon roof plate and the pontoon bottom plate at regular intervals. The pontoon is subjected to circumferential bending moment M as illustrated in Fig.9

tu

tL

ti to

Li Lo

Lp

LD

rG

ri

Fig.7 Cross section of pontoon

u

Fig.8 Ring stiffener

M

M

Fig.9 Circumferential bending load

In the elastic bifurcation buckling analysis, the initial circumferential stresses are applied to the pontoon cross section as shown in Fig.10. The initial circumferential stress which is equivalent to the circumferential bending moment M is given by: z

IM

r

= (9) where, Ir is the moment of inertia of the pontoon cross section through the centroid G, z is the distance from r axis through the centriod. The initial meridional membrane forces in the Eq.(8) is defined by the following equation in this analysis. 0N 0S = (10) The initial circumferential membrane forces is defined by: tN 0 = (11)

The circumferential buckling bending moment Mcr- is calculated by the following formulas:

MM mincr = (12) In this equation, min is the minimum value of the minimum eigenvalues m in each circumferential wave number m. The circumferential buckling bending moment Mcrm for m is defined by: MM mcrm = (13) Mcr is the minimum value of Mcrm.

Fig.10 Initial stress due to circumferential bending moment


REDUCED STIFFNESS METHOD The virtual displacements u, v, w are non axisymmetric, and are given to axisymmetric shell structure which is equilibrium condition under axisymmetric load. In this condition, the strains and the changes of curvature of the shell are written as follows:

+++++

=

)L(s

)L(

)L(s

)N(s

)L(s

)N()L(0

)N(s

)L(s0s

s

s

s

s

(14)

where i(L), i(L) (i= s, , s) are the linear strain and the change of curvature, i(N) (i= s, , s) is the nonlinear strain due to the virtual displacements. The membrane forces and the bending moments due to the virtual displacements can be also written in the following form.

+++++

=

)L(s

)L(

)L(s

)N(s

)L(s

)N()L(0

)N(s

)L(s0s

s

s

s

s

MMM

NNNNNNNN

MMMNNN

(15)

where Ni(L), Mi(L) (i= s, , s) are the linear membrane force and the bending moment, Ni(N) (i= s, , s) is the nonlinear membrane force due to the virtual displacements. The variation of total potential energy can be written as: L+++= 321 (16) where ( )L,3,2,1ii = is the i-th order component of the variation of total potential energy with respect to the virtual displacement. The linear component 1 = 0 is the equilibrium and the quadratic component 2 = 0 is the buckling. 2 is decomposed as follows: M2b2M22 VUU ++= (17) where U2b is the linear bending strain energy, U2m is the linear membrane strain energy and V2m is the nonlinear membrane strain energy. These can be written in the following equations.

( ) ( ) ( ) ( ) ( ) ( )( )dANNN21U

A

Ls

Ls

LLLs

LsM2 ++= (18)

( ) ( ) ( ) ( ) ( ) ( )( )dAMMM21U

A

Ls

Ls

LLLs

Lsb2 ++= (19)

( ) ( ) ( ) ( )( )dANNNN21V

A0

NN00s

NS

Ns0sM2 +++= (20)

The buckling condition is obtained as follows: 0VUU M2b2M2 =++ (21) where is the buckling load parameter, and is given as:

M2

b2M2

VUU += (22)

The linear membrane strain energy U2m seems to reduce due to the initial geometrical imperfection. In the reduced stiffness method [5~7], U2m is eliminated from the quadratic component of the variation of total potential energy 2 as follows: 0VU M2

*b2 =+ (23)

where * is the RS buckling load parameter, and is given as:

M2

b2*

VU= (24)

In this paper, RS factor fRS is defined as:

b2M2

b2*

RS UUUf +==

(25) In the finite element analysis, RS factor for each circumferential wave number fRSm is derived form the following equation:

{ } [ ] [ ]( ){ }{ } [ ]{ }mLmTmmMmLm

Tm

RSm dKddKKdf = (26)

where {dm} is the buckling mode vector and [KMm] is the membrane component of small displacement stiffness matrix. The lower bound buckling load for each circumferentil wave number Mcrm* is expressed as: crmRSmcrm

* MfM = (27) The lower bound buckling load Mcr* becomes the minimum value of Mcrm*.

ANALYTICAL MODEL The pontoon is made of mild steel, and Young's modulus E and Poisson's ratio are 200 GPa and 0.3, respectively. In this analysis, the circumferential bending moment M where the pontoon roof plate is subjected to compressive stress is defined as positive as shown in Fig.9. Table 1 shows the basic analytical conditions of the pontoon. The cross sectional area


is A = 6.43104 mm2, and the moment of inertia is Ir = 6.54109 mm4 in this pontoon without ring stiffener. This is almost equivalent to the pontoon which has 70 m in diameter. The ring stiffeners are attached to both the pontoon roof plate and the pontoon bottom plate at regular intervals.

Table 1 Analytical condition Radius r i 30 mHeight L i 450 mmPlate thickness t i 12 mmHeight L o 900 mmPlate thickness t o 12 mmWidth L p 4 mRoof plate thickness t u 6 mmBottom plate thickness t L 6 mm

150 mmLength L R 120 mmThickness t R 10 mm

200 GPa0.3

Ring Stiffener

Pontoon

Lower edge difference L D

Young's modulus EPoison's ratio

Inner rim

Outer rim

LOWER BOUND BUCKLING LOAD Table 2 shows the buckling load without initial geometrical imperfection Mcr-i which is derived from Eq.(12), the minimum value of the RS factor fRSm-i and the lower bound buckling load ratio M*cr-i/Mcr-i. M*cr-i is the lower bound buckling load obtained by the reduced stiffness method and the notation i denotes the number of ring stiffeners. The buckling load without initial geometrical imperfection Mcrm-i for each circumferential wave number m is shown in Fig.11. The minimum of Mcrm-i becomes the buckling load Mcr-i which is shown in Table 2. The RS factor fRSm-i for m is shown in Fig.12. The minimum value of fRSm-i without ring stiffener (i = 0) is 0.733 at m = 6, that with one ring stiffener ( i = 1 ) is 0.478 at m = 12 and that with two ring stiffeners ( i = 2) is 0.256 at m = 16 as shown in Table 2. The RS factor fRSm-i is less than 1.0 and it decreases largely in the circumferential wave number m less than 30. Also fRSm-i decreases with increasing the number of ring stiffeners.

Table 2 Lower bound buckling load ratio M cr-i

(kN-m)i =0 57.20 (39) 0.733 (6) 0.975 (38)i =1 171.37 (69) 0.478 (12) 0.995 (68)i =2 370.94 (101) 0.256 (16) 0.564 (14)

Number ofRing Stiffeners M

*cr-i /M cr-i

Minimumf RSm-i

(Circumferential wave number m )

The lower bound buckling load ratio M*crm-i/Mcr-i for each circumferential wave number m is shown in Fig.13. The minimum value of M*crm-i becomes the lower bound buckling load M*cr-i which is shown as the ratio in Table 2. The ratio M*crm-2/Mcr-2 with two ring stiffeners decreases largely at m = 14. The circumferential wave number m of the buckling load Mcr-i is identical to that of the lower bound buckling load M*cr-i both without and with one ring stiffener. On the other hand, in the pontoon with two ring stiffeners, m is 101 in Mcr-2 and m is 14 in M*cr-2 as shown in Table 2, and they are not identical.

Circumferential bending

10

100

1000

10000

0 20 40 60 80 100 120Circumferential wave number m

Mcr

m-i

(kN

-m)

No ring ( i=0 )One ring ( i=1 )Two rings ( i=2 )

Fig.11 Buckling load


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120

Circumferential wave number m

RS

fact

or f

RSm

-i


Fig.12 RS factor



0.1

1.0

10.0

100.0


M* c

rm-i

/Mcr

-i


Fig.13 Lower bound buckling load ratio

The linear membrane strain energy U2M derived from Eq.(18) and the linear bending strain energy U2b derived from Eq.(19) are shown in Fig.14~Fig.16. Fig.14 shows the energies of the pontoon without ring stiffeners, Fig.15 shows the energies with a ring stiffener and Fig.16 shows the energies with two ring stiffeners. Those strain energies are calculated assuming the maximum buckling mode to 1 mm and those are divided by the circular constant . U2b increases with increasing the circumferential wave number m. On the other hand, U2M increases first and then it decreases with increasing m. The grade of an increase and decrease of U2M becomes large with increasing the number of ring stiffeners.

Circumferential bending ( i =0 )

10

100

1000

10000

100000

1000000

0 20 40 60 80 100 120


Stra

in e

nerg

y (N

-mm

)

U2bU2M

Fig.14 Strain energy of the pontoon without ring stiffener

Circumferential bending ( i =1 )

100

1000

10000

100000

1000000


Stra

in e

nerg

y (N

-mm

)

U2bU2m

Fig.15 Strain energy of the pontoon with a ring stiffener

Cicumferential bending ( i =2 )

100

1000

10000

100000

1000000

0 20 40 60 80 100 120


Stra

in e

nerg

y (N

-mm

)

U2bU2m

Fig.16 Strain energy of the pontoon with two ring stiffeners

The buckling modes of the pontoon with two ring stiffeners are shown in Fig.17. These modes are illustrated on the section = 0, and they distribute cosm along the circumferential direction. The buckling mode of the circumferential wave number m = 12 deforms in whole area of the pontoon roof plate. The buckling mode of m = 26 shows the local buckling which deforms in the pontoon roof plate between the ring stiffeners. The linear membrane strain energy U2M becomes the maximum at m = 20 in Fig.16. The buckling mode of m = 20 is just transforming into the local buckling. Because the buckling mode transforms into local buckling at m = 20 with increasing m, the linear membrane strain energy U2M becomes the maximum and then it decreases with increasing m.

U2b U2M

U2b U2M

U2b U2M


Fig.17 Buckling mode of the pontoon with two ring stiffeners

CONCLUSIONS The initial geometrical imperfection may diminish the buckling load. The lower bound buckling loads of the floating roof pontoon of an oil storage tank subjected to circumferential bending load are evaluated from the axisymmetric finite element analysis and the reduced stiffness method. The shape of imperfection is not required to define in the reduced stiffness method. As a result, the lower bound buckling loads of the pontoon both without ring stiffeners and with one ring stiffener are almost the same as the buckling loads without geometrical imperfection. However, the lower bound buckling load of the pontoon with two ring stiffeners is 56% of the buckling loads without geometrical imperfection. In this paper, the finite element method is the linear elastic bifurcation buckling analysis for non-axisymmetric buckling mode under axisymmetric loading. Though the sloshing load is non-axisymmetric and theoretically has the circumferential wave number m=1 due to the horizontal seismic excitation, it is assumed to be axisymmetric, i.e., m=0, in this analysis. The verification of this assumption will be a future study.

REFERENCES [1]Yohida, S. and Kitamura, K., 2006, "Buckling of Single-

Deck Floating Roofs in Aboveground Oil Storage Tanks due to Circumferential Bending Load", PVP2006-ICPVT-

11-93696, Proceedings of 2006 ASME PVP Conference, Vancouver.

[2]Yoshida, S. and Kitamura, K., 2007, "Buckling of Ring Stiffened Pontoons of Floating Roofs in Aboveground Oil Storage Tanks", PVP2007-26252, Proceedings of 2007 ASME PVP Conference, San Antonio.

[3]Yoshida, S., 2008, "Buckling Characteristics of Floating Roof Pontoons in Aboveground Storage Tanks Subjected to Bending Load in Two Directions", PVP2008-61085, Proceedings of 2008 ASME PVP Conference, Chicago.

[4]Yoshida, S., 2009, "Buckling Characteristics of Floating Roof Pontoons in Aboveground Storage Tanks Subjected to Both Compressive and Bending Load", PVP2009-77227, Proceedings of 2009 ASME PVP Conference, Prague.

[5]Yamada, S., and Croll, J.G.A, 1989, "Buckling Behavior of Pressure Loaded Cylindrical Panels", Journal of Engineering Mechanics, 115(2), pp.327-344.

[6]Yamada, S., and Croll, J.G.A, 1993, "Buckling and Post-Buckling Characteristics of Pressure-Loaded Cylinders", Journal of Applied Mechanics, 60, pp.290-299.

[7]Yamada, S., and Croll, J.G.A, 1999, "Contribution to Understanding the Behavior of Axially Compressed Cylinders", Journal of Applied Mechanics, 66, pp.299-309.

[8]American Petroleum Institute, 2007, API Standard 650, Welded Steel Tanks for Oil Storage, 11th edition.

[9]Yoshida, S., Zama, S., Yamada, M., Ishida, K. and Tahara, T., 2001, "Report on Damage and Failure of Oil Storage Tanks due to the 1999 Chi-Chi Earthquake in Taiwan", Proceedings of 2001 ASME PVP Conference, Atlanta, PVP-Vol.428-2, pp.11-19.

[10]Zienkiewicz, O.C. and Taylor, R.L., 2002, The Finite Element Method for Solid and Structural Mechanics, 6th Edition, Vol.2, Elsevier, pp.510-513.

[11]Stricklin, J.A., et.al, 1968, "Analysis of Shells of Revolution by the Matrix Displacement Method", AIAA Journal, 6(12), pp.2306-2312.

[12]Yoshida, S. and Miyoshi, T., 1992, "Bifurcation Buckling of the Top End Closure of Oil Storage Tanks under Internal Pressure", Proceedings of ASME PVP Conference, New Orleans, 230, pp.111-115.

[13]Yoshida, S. and Tomiya, M., 1998, "Elastic-Plastic Buckling Analysis of the Uplifted Shell-to-Bottom Joint of Internally Pressurized Oil Storage Tanks using Axisymmetric Shell Finite Element Method", Proceedings of ASME PVP Conference, San Diego, 370, pp.121-128.

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Pvp2010-25092-Lower Bound Buckling Load of a Floating Roof Pontoon