Buckling 2

1 Copyright 2009 by ASME

Proceedings of PVP2009 2009 ASME Pressure Vessels and Piping Division Conference

July 26-30, 2009, Prague, Czech Republic

PVP2009-77227

BUCKLING CHARACTERISTICS OF FLOATING ROOF PONTOONS IN ABOVEGROUND STORAGE TANKS

SUBJECTED TO BOTH COMPRESSIVE AND BENDING LOAD

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 structural problems as evidenced by sinking failure in large diameter tanks at a refinery in Tomakomai, Japan. The pontoons of the floating roofs might be buckled due to circumferential bending moment 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 authors had reported the buckling strength of the pontoons subjected to circumferential bending load first and that of the pontoons subjected to both circumferential and radial bending load next in the previous papers. This paper presents the buckling strength of the pontoons subjected to both circumferential bending load and circumferential compressive load. The axisymmetric shell finite element method is used in the analysis. Linear elastic bifurcation buckling analysis is carried out and the buckling characteristics of the pontoon both with and without ring stiffeners are investigated.

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 type" and "double deck type". The single deck floating roofs, considered herein, consists of a thin circular plate called "deck" attached at the edge to a buoyant ring of box shaped cross section called "pontoon". Seven single deck floating roof had sunk in the 2003 Tokachi-Oki earthquake at a

refinery in Tomakomai, Japan. Theses floating roofs deformed to leak oil on them due to the liquid sloshing, and they lost buoyancy to sink. The 1st natural period of liquid sloshing in oil storage tanks usually has the range between 5 seconds and 15 seconds. When the predominant period of earthquake is within this range, liquid sloshing occurs and single deck floating roofs which are floating on the liquid surface deform. The seven single deck floating roofs had sunk slowly spending several days in the 2003 Tokachi-Oki earthquake. It is presumed that the small failures in the welded joints or in the stress concentrated parts of the pontoon were caused due to the sloshing, and next these failures expanded gradually in the sinking process spending several days. In the sloshing, the pontoon of the roof is subjected to the circumferential bending moment 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 view point of the single-deck floating roof sinking. The pontoon also deforms ovally and compressively at the sloshing, and is subjected to radial bending load and circumferential compressive load simultaneously together with circumferential bending load. The authors had investigated the elastic buckling characteristics of the pontoons with and without ring stiffeners subjected to circumferential bending load using axisymmetric shell finite element analysis [1,2]. Next, the influences of radius and thicknesses of the pontoon on buckling due to radial bending load were investigated, and the buckling due to both radial and circumferential bending was analyzed [3]. In this

Proceedings of the ASME 2009 Pressure Vessels and Piping Division Conference PVP2009

July 26-30, 2009, Prague, Czech Republic

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paper as the next stage of this research, the influences of radius and thicknesses of the pontoon on buckling due to circumferential compressive load are investigated, and the buckling due to both circumferential bending load and circumferential compressive load is analyzed. Linear elastic bifurcation buckling analyses are carried out using the axisymmetric shell finite element method .

NOMENCLATURE A Cross sectional area of pontoon [Bdm] Nonlinear strain-displacement transformation matrix {dm} Nodal displacement vector E Youngs modulus F Circumferential compressive force Fcr- Circumferential buckling compressive force Fcr-0 Circumferential buckling compressive force without ring stiffeners Ir Moment of inertia of pontoon cross section [KLm] Small displacement stiffness matrix [Km(N0)] Initial stress stiffness matrix [Km(N1)] Initial stress stiffness matrix [km(N0)] Elemental initial stress stiffness matrix [km(N1)] Elemental initial stress stiffness matrix Lcr Limit length of ring stiffener LD Lower edge difference Li Inner rim height Lo Outer rim height LP Pontoon width LR Ring stiffener length Mcr- Circumferential buckling bending moment of pontoon Mcr-0 Positive circumferential buckling bending moment of pontoon without ring stiffeners Mcr-00 Negative circumferential buckling bending moment of pontoon without ring stiffeners Mr Radial bending moment M Circumferential bending moment m Circumferential wave number Ni(i=0,1) Initial membrane force for m=0 NSi(i=0,1) Initial meridional membrane force for m=0 Ni(i=0,1) Initial circumferential membrane force for m=0 ri Inner rim radius (r, , z) Cylindrical coordinates s Meridional coordinate of shell element 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 u Tangential displacement of shell element urm Radial displacement of nodal point for m uzm z-direction displacement of nodal point for m um Circumferential displacement of nodal point for m v Circumferential displacement of shell element

w Normal displacement of shell element m Rotation angle of nodal point for m Poissons ratio m Eigenvalue and buckling load parameter for m min Minimum of m -F Initial circumferential membrane stress for m=0 -M Initial circumferential membrane stress for m=0 Subscript m Circumferential wave number 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 "deck" attaches to the box shaped ring called "pontoon", as shown in Fig.1. 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 10 mm. The deck thickness is about 4.5 mm. 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 radial plates called "bulkhead". Each compartment has liquid tight requirement. API Standard 650 Appendix C [4] gives the minimum requirement for the floating roof design. However, the sloshing motion has not been considered in the design of the floating roofs.

Seal

Pontoon

Deck

Shell Bottom

Bulkhead Outer rim

Inner rim

Detail of Pontoon

Pontoon bottom plate

Pontoon roof plate

Deck

Pontoon

Deck

Fig.1 Single-Deck Floating Roof

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.2. The pontoon is subjected to the circumferential bending moment 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.

Deck

Pontoon

Max. Sloshing directionBuckling

Buckling

Fig.2 Buckling Portion

Fig.3 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[5]. Because of small leakage, this floating roof had remained afloat. Fig.4 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. 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 of Fig.3 and Fig.4 were almost identical immediately after the earthquake. It is concluded that the collapse of the single-deck floating roof as shown in Fig.4 did not occur at the earthquake, but it occurred during the sinking process.

Fig.3 Pontoon Buckling Failure(5)

Fig.4 Floating Roof that Sank in Oil The pontoon also deforms compressively due to the sloshing, and is subjected to circumferential compression load simultaneously together with circumferential bending load. The buckling of the pontoon usually occurs due to circumferential bending load because the flexural rigidity of the pontoon is less in this direction. ANALYSIS Axisymmetric Shell Finite Element The axisymmetric shell finite element used in this analysis is a conical frustum element as shown in Fig.5. In this element, the tangential displacement u and the circumferential displacement v are assumed to be linear and the normal displacement w to be cubic with regard to s, where s is the elemental coordinate. The strain-displacement relation based on the Kirchhoff-Loves assumption is given by the Novozhilovs equation as follows[6];

( )

++

+

+

++

=

vrcossin

sv

rcosw

rsin

sw

r12

sw

rsinv

rcosw

r1

sw

sinrv

svu

r

sinucoswr1v

r1

su

22

2

22

2

2

2

2

s

s

s

s

(1)

where, i and i (i= s, , s) express the strain and the change of curvature. The elastic stress-strain relation based on the Hookes law is as follow;



( )

=

s

s

s

s

2

2

22

2

s

s

s

s

24t1.sym

012t

012t

12t

0002

10000100001

1Et

MMMNNN

(2)

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 displacement vector can be defined by the following formula;

{ }

=

mcosmcosdmsindmcosd

msinmcos

d

m

rm

m

zm

m (3)

where, dim(i=z, , r) and m are the i direction displacement and the rotation angle, and m is the circumferential wave number.

dz L

r

z

s w

u

j

i

dr =dw/ds

r

v

: Nodal point

Fig.5 Axisymmetric Shell Finite Element Linear Bifurcation Buckling Analysis The virtual work principle gives the equation of bifurcation buckling for the circumferential wave number

2m under axisymmetric loading of m=0 as follows; [ ] ( )[ ] ( )[ ]( ){ } 0dNKNKK m1mm0mLm =++ (4) 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

Vdm0m dVBN0

0NBNk

e

= (5)

( )[ ] [ ] [ ] edm1

1sT

Vdm1m dVBN0

0NBNk

e

= (6) where, Ns0, Ns1 are the initial meridian membrane forces and N0, N1 are the initial circumferential membrane forces of the shell, respectively. These are axisymmetric forces while the sloshing load is non-axisymmetric. It is assumed that the buckling characteristics of the pontoon due to the sloshing are approximately obtained from the axisymmetric load. The nonlinear strain-displacement transformation matrix [Bdm] is derived from the following equation.

( )

( ) [ ]

=

mcosmcosdmsindmcosd

B

rmsinvmcosw

r1

smcosw

m

rm

m

zm

dmmm

m

(7)

Eq.(4) 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. Ns0, N0 are the membrane forces which are investigated the influence on the buckling load, and Ns1, N1 are those which are calculated the buckling load. The computer code based on the theory mentioned above was developed by the author. It was verified to investigate the application to several problems[7,8]. ANALYTICAL CONDITIONS The hatching area of Fig.6 is the pontoon cross section and is discretized into 1500 to 1600 axisymmetric shell finite elements. 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.7 shows the ring stiffener which is usually attached to both the pontoon roof plate and the pontoon bottom plate at regular intervals.

tu

tL

tito

Li Lo

Lp

LD

rG

ri

Fig.6 Cross Section of Pontoon



u

Fig.7 Ring Stiffener

In the elastic bifurcation buckling analysis, the initial circumferential stresses are applied to the pontoon cross section as shown in Fig.8. The initial circumferential stress -M which is equivalent to the circumferential bending moment M is given by: z

IM

rM

= (8) The initial circumferential stress -F which is equivalent to the circumferential compressive force F is given by:

A

FF

= (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 and A is the cross sectional area. The initial meridional membrane forces in the Eq.(5) and Eq.(6) are defined by the following equation in this analysis. 0NN 1S0S == (10) When F is the force which is investigated the influence on the buckling load, and M is the moment which is calculated the buckling load, the initial circumferential membrane forces are defined by: tN,tN M1F0 == (11) When M is the moment which is investigated the influence on the buckling load, and F is the force which is calculated the buckling load, the initial circumferential membrane forces are defined by: tN,tN F1M0 == (12)

The circumferential buckling bending moment Mcr- and the circumferential buckling compressive force Fcr- are calculated by the following formulas: MM mincr = (13) FF mincr = (14) In this equation, min is the minimum value of the minimum eigenvalues m in each circumferential wave number m. 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. This is almost equivalent to the pontoon which has 70 m in diameter.

-M

(a)Circumferential Bending Moment

-F

(b) Circumferential Compressive Force

Fig.8 Initial Stress on Pontoon

M

M

Mr

Mr

F

F Fig.9 Loads Applied to Pontoon

Table 1 Basic Analytical Conditions 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 mm

Inner rim

Outer rim

Pontoon

Lower edge difference L D



BUCKLING DUE TO COMPRESSIVE LOAD The buckling analyses for the pontoons without ring stiffeners are carried out. Only the circumferential compressive force F is applied to the pontoon to investigate the buckling characteristics due to F. The buckling characteristics due to the circumferential bending moment M[1,2] and the radial bending moment Mr[3] had been reported in the previous papers. The initial membrane forces in Eq.(4) are given by Eq.(12) with -M=0. Inner Rim Radius The elastic bifurcation buckling analyses are carried out for various sized pontoons which have the same cross section except for the inner rim radius ri in order to investigate the influence of ri on the circumferential buckling compressive force Fcr-. The pontoon data is written in Table 1 except for the inner rim radius. Fig.10 shows both the relation between the circumferential buckling compressive force ratio Fcr-/Fcr-0 and the inner rim radius ri and the relation between the circumferential wave number m and ri. Fig.10 also shows the relation between the circumferential buckling moment ratio Mcr-/Mcr-00 and ri[1]. Fcr-0 is the circumferential buckling compressive force of ri=30 m without ring stiffeners and is 189.2 kN. Mcr-00 is the negative circumferential buckling bending moment of ri=30 m without ring stiffeners, and is 55.70 kN-m. The negative bending moment acts on the pontoon in opposite direction of Fig.9, and is plotted at the absolute values in Fig.10 and Fig.11, 12 after mentioned. The circumferential buckling compressive force Fcr- slightly decreases similar to the circumferential buckling bending moment Mcr- if the inner rim radius ri increases. The circumferential wave number m of Fcr- is identical with that of Mcr-, and it increases with increasing ri. It is concluded that the inner rim radius ri hardly influences on the circumferential buckling compressive force Fcr-.

1)

0.98

0.99

1.00

1.01

1.02

20 25 30 35 40 45 50

Inner Rim Radius r i (m)

Fcr-/F

cr-

0 or

Mcr

-/M

cr-

00

15

30

45

60

75

Wav

e N

umbe

r m

Buckling Force RatioBuckling Moment RatioWave Number

Fig.10 Relation between Inner Rim Radius and Buckling Ratio

Rim Plate Thickness The elastic bifurcation buckling analyses are carried out for various sized pontoons which have the same cross section except for the rim thicknesses ti=to in order to investigate the influence of ti=to on the circumferential buckling compressive force Fcr-. The pontoon data is written in Table 1 except for the rim plate thicknesses. Fig.11 shows both the relation between the circumferential buckling compressive force ratio Fcr-/Fcr-0 and the rim thicknesses ti=to and the relation between the circumferential wave number m and ti=to with the range of 6 mm~14 mm. Fig.11 also shows the relation between the circumferential buckling moment ratio Mcr-/Mcr-00 and ti=to with the range of 6 mm~12 mm[1]. The result shows that Fcr- increases with increasing ti=to, and Fcr- of ti=to=12 mm is 25% greater than that of ti=to=6 mm. Mcr- of ti=to=12 mm is 13% greater than that of ti=to=6 mm. The circumferential wave number m of Fcr- is identical with that of Mcr-. It is concluded that the rim plate thicknesses ti=to fairly influences on the circumferential buckling compressive force Fcr-.

1)

0.7

0.8

0.9

1.0

1.1

6 8 10 12 14

Rim Plate Thickness t i & t o (mm)

Fcr-

/Fcr

-0

or

Mcr

-/M

-cr-0

0

32

34

36

38

40

Wav

e N

umbe

r m


Fig.11 Relation between Rim Plate Thickness and Buckling

Ratio Pontoon Roof and Bottom Plate Thickness The elastic bifurcation buckling analyses are carried out for various sized pontoons which have the same cross section except for the pontoon roof and bottom plate thicknesses tu=tL in order to investigate the influence of tu=tL on the circumferential buckling compressive force Fcr-. The pontoon data is written in Table 1 except for the pontoon roof and bottom plate thicknesses. Fig.12 shows both the relation between the circumferential buckling compressive force ratio Fcr-/Fcr-0 and the pontoon roof and bottom palate thicknesses tu=tL and the relation between the circumferential wave number m and tu=tL. Fig.12



also shows the relation between the circumferential buckling moment ratio Mcr-/Mcr-00 and tu=tL[1]. The result shows that Fcr- rapidly increases with increasing tu=tL. Fcr- of tu=tL=12 mm is 13.3 times greater than that of tu=tL=4.5 mm. Mcr- of tu=tL=12 mm is 15.7 times greater than that of tu=tL=4.5 mm. The circumferential wave number m of Fcr- is almost identical with that of Mcr-, and it decreases with increasing tu=tL. It is concluded that the pontoon roof and bottom plate thicknesses tu=tL greatly influences on the circumferential buckling compressive force Fcr-. This is similar to the circumferential buckling bending moment Mcr-.

1)

0

2

4

6

8

4 6 8 10 12

Pontoon Plate Thickness t u & t L (mm)

Fcr-/F

cr-

0 or

Mcr

-/M

cr-

00

32

34

36

38

40

Wav

e N

umbe

r m


Fig.12 Relation between Pontoon Plate Thickness and Buckling

ratio Pontoon Width The elastic bifurcation buckling analyses are carried out for various sized pontoons which have the same cross section except for the pontoon width Lp in order to investigate the influence of Lp on the circumferential buckling compressive force Fcr-. The pontoon data is written in Table 1 except for the pontoon width. Fig.13 shows both the relation between the circumferential buckling compressive force ratio Fcr-/Fcr-0 and the pontoon width Lp and the relation between the circumferential wave number m and Lp. Fig.13 also shows the relation between the circumferential buckling moment ratio Mcr-/Mcr-0 and Lp. Mcr-0 is the positive circumferential buckling bending moment of Lp=4 m without ring stiffeners, and is 57.20 kN-m. The result shows that Fcr- decreases with increasing Lp, and Fcr- of Lp=5 m is 46.3% less than that of Lp=3 m. Mcr- decreases with increasing Lp similar to Fcr-. The circumferential wave number m of Fcr- is almost identical with that of Mcr-, and it decreases with increasing Lp. It is concluded that the pontoon width Lp fairly influences on the circumferential buckling compressive force Fcr-.

Buckling Mode The buckling mode of the pontoon of Table1 without ring stiffeners subjected to circumferential compressive force F is shown in Fig.14. This is the deformation at =0, and it distributes cosm along the circumferential direction. Both the pontoon roof and the pontoon bottom plate largely deform in the buckling mode. The same modes as Fig.14 are obtained in all cases analyzed in this chapter. Circumferential Wave Number The radial members of the pontoon can not be considered into the analytical model in this axisymmetric FEA. The bulkhead is especially important for the evaluation of pontoon strength. Three-dimensional analysis is required to obtain precise solutions. It is presumed that the buckling load obtained in this analysis does not differ from the actual buckling load, because the circumferential wave number m is larger than the number of bulkheads.

0.6

0.8

1.0

1.2

1.4

1.6

3.0 3.5 4.0 4.5 5.0

Pontoon Width L p (m)

F cr-/

F cr-0

or

Mcr

-/M

cr-

0

30

35

40

45

50

55

Wav

e N

umbe

r m


Fig.13 Relation between Pontoon Width and Buckling Ratio

ri=30 m, ti=to=12 mm, tu=tL=6 mm, Fcr-/Fcr-01.0

Fig.14 Buckling Mode of Pontoon without Ring Stiffeners due

to Compressive Force QUANTITY AND SIZE OF RING STIFFENER The relation between the circumferential buckling compressive forces Fcr- and both quantity and size of the ring stiffeners is investigated. Several ring stiffeners are attached to both the pontoon roof and bottom plate at regular intervals as shown in Fig.6. The thickness of the ring stiffener tR is 10



mm and its length LR varies as shown in Fig.7. The pontoon data is written in Table 1. Fig.15 shows the relation between the buckling compressive force ratio Fcr-/Fcr-0 and the ring stiffener length LR. Fcr- increases with increasing LR in the beginning, and then it becomes almost constant at a certain length. This length is defined as "limit length Lcr" in this paper. Lcr increases with increasing the number of the ring stiffeners, and Lcr is 57 mm in one ring stiffener and 90 mm in two ring stiffeners. When LR equals to Lcr, the circumferential buckling compressive force ratio Fcr-/Fcr-0 is 3.261 and 7.211 with one and two ring stiffeners, respectively. According to this result, the ring stiffener is effective on the buckling strength of the pontoon. Fig.16 shows the buckling modes at several lengths of a ring stiffener. Fig.17 shows the buckling modes in two ring stiffeners. These are the deformations at =0, and they distribute cosm along the circumferential direction. The buckling modes are transformed from the whole deformation of the pontoon roof and the bottom plate into the local deformation between the ring stiffeners or the rim plate and the ring stiffener at the limit length Lcr, if the ring stiffener length LR increases. Lcr depends on the thickness of ring stiffener tR and load types. In the circumferential bending moment M, Lcr is 60 mm in one ring stiffener and 80 mm in two ring stiffeners. When LR equals to Lcr, the circumferential buckling moment ratio Mcr-/Mcr-0 is 2.869 and 6.256 with one and two ring stiffeners, respectively [2].

tR=10 mm, M

0

2

4

6

8

10

0 50 100 150 200

Length of Ring Stiffener L R mm

Buc

klin

g Fo

rce

Rat

io F

cr-

/Fcr

-0

One RingTwo Rings

Fig.15 Relation between Circumferential Buckling Moment

and Numbers of Ring Stiffener

(a) LR=55 mm, Fcr-/Fcr-03.228

(b) LR=57 mm, Fcr-/Fcr-03.261

(c) LR=120 mm, Fcr-/Fcr-03.437

Fig.16 Buckling Mode of Pontoon with a Ring Stiffener due to Compressive Force

(a) LR=85 mm, Fcr-/Fcr-06.917

(b) LR=90 mm, Fcr-/Fcr-07.211

(c) LR=120 mm, Fcr-/Fcr-07.535

Fig.17 Buckling Mode of Pontoon with Two Ring Stiffeners due to Compressive Force

BUCKLING DUE TO BOTH COMPRESSIVE AND BENDING LOAD Buckling Load The buckling of the pontoon with and without ring stiffeners due to the circumferential bending moment M is analyzed, while the circumferential compressive force F is also acting on the pontoon. In this analysis, the pontoon cross section data is written in Table 1. The ring stiffener thickness tR is 10 mm, and its length LR is 120 mm which is longer than the limit length Lcr. The initial membrane forces in Eq.(4) are given by Eq.(11).



Fig.18 shows the relation between the circumferential buckling moment ratio Mcr-/Mcr-0 and the circumferential compressive force ratio F/Fcr-0. The positive Mcr- is the buckling moment where the pontoon roof plate is compressive, and the negative Mcr- is the buckling moment where the pontoon roof plate is tensile. According to Fig.18, the relation between of Mcr-/Mcr-0 and F/Fcr-0 is almost linear. Buckling Mode The buckling modes of the pontoon with and without ring stiffeners are shown in Fig.19~Fig.21. These are the deformations at =0, and they distribute cosm along the circumferential direction. In these figures, they show the modes of (a)F/Fcr-i=0.5 and positive Mcr- and (b)F/Fcr-i=0.5 and negative Mcr-. Fcr-i(i=0,1,2) is the circumferential buckling compressive force with i ring stiffeners, and Fcr-1=3.437Fcr-0, Fcr-2=7.535Fcr-0. Fcr-i(i=1,2) of the ring stiffener length LR=120 mm is slightly greater than that of LR= Lcr. Since the ring stiffener length LR=120 mm is larger than the limit length Lcr, the buckling modes of the pontoon with ring stiffeners show the local buckling which deforms in the pontoon roof and bottom plate between the ring stiffeners or the ring stiffener and the rim plate.

-8

-6

-4

-2

0

2

4

6

8

0 2 4 6 8F /F cr-0

Mcr

-/M

cr-

0

Two RingsOne RingNo Ring

Fig.18 Relation between Circumferential Buckling Moment and Circumferential Compressive Force using Dimensionless

Form by Buckling Load without Ring Stiffeners

(a) F/Fcr-00.500, Mcr-/ Mcr-00.507

(b) F/Fcr-00.500, Mcr-/ Mcr-0-0.490

Fig.19 Buckling Mode of Pontoon without Ring Stiffeners due to both Bending Moment and Compressive Force

(a) F/Fcr-01.719, Mcr-/ Mcr-01.560

(b) F/Fcr-01.719, Mcr-/ Mcr-0-1.638

Fig.20 Buckling Mode of Pontoon with One Ring Stiffener due to both Bending Moment and Compressive Force

(a) F/Fcr-03.768, Mcr-/ Mcr-03.434

(b) F/Fcr-03.768, Mcr-/ Mcr-0-3.591

Fig.21 Buckling Mode of Pontoon with Two Ring Stiffeners due to both Bending Moment and Compressive Force

CONCLUSION The buckling characteristics of the pontoons without ring stiffeners subjected to circumferential compressive load and those with and without ring stiffeners subjected to both circumferential bending and compressive load are investigated using the linear elastic axisymmetric shell finite element analysis.



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(8)S.Yoshida, et.al., 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, Vol.370, 1998, pp.121-128.


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Buckling 2

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