stability analysis and experiment of large...

1 Copyright © 2014 by ASME

Proceedings of the ASME 2014 33rd

International Conference on Ocean, Offshore and Arctic Engineering

OMAE2014

June 8-13, 2014, San Francisco, California, USA

OMAE2014-24056

STABILITY ANALYSIS AND EXPERIMENT OF LARGE-SCALE SPHERICAL

MODELS BUILT BY HIGH STRENGTH STEEL

Qinghai Du*

1Hadal Science and Technology Research Center,

Shanghai Ocean University 201306, Shanghai, China

2Southampton Marine and Maritime Institute (SMMI),

University of Southampton, SO17 1BJ, Southampton, England, UK

Email: [email protected]

Weicheng Cui

1Hadal Science and Technology Research Center,

Shanghai Ocean University 201306, Shanghai, China

3Nonlinear Analysis and Applied Mathematics

(NAAM) Research Group, Faculty of Science, King Abdulaziz University

Jeddah 21589, Saudi Arabia

Email: [email protected]

* Ph. D., Senior Engineer and author of correspondence

ABSTRACT The spherical shell is a typical form of revolution shells

which are used widely in engineering especially as underwater

pressure hull. To disclose structural characteristics of the whole

spherical shell with some initial imperfections and residual

stresses, both material nonlinear and geometric nonlinear Finite

Element Analysis (FEA) has been carried out in this paper. In

the FE analysis, the elastic-plasticity stress-strain relations have

been adopted and the initial deflection of spherical shell caused

in manufacture was also taken into account in creating model. It

is also shown that the nonlinear structural characteristics of the

spherical shell vary from its different dimension parameters for

initial imperfection. Compared with the exiting different rule’s

methods, nonlinear FEM may exactly show sphere’s stability

varying by initial imperfections. Then two groups of

experiments of four spherical models, made by a high strength

steel and with two kinds of same main sizes but with different

initial deflection on them for manufacture, have been finished in

model pressure tests. The experiment has been analyzed by

comparing in different method while these results eventually

indicate that the buckling stability of a spherical shell model

varies by its initial imperfection and such materials sphere

critical load could not be accounted for by current rules except

nonlinear FEM or modified theoretical elastic-plasticity analysis

method. Therefore it is essential to obtain the new analysis and

design method for spherical shell made by high strength steel

used to deep-sea vehicle or other ocean engineering.

KEYWORDS

Spherical Shell, Ultimate Strength, Stability, Nonlinear Finite

Element Method, Experiment

INTRODUCTION Spherical shell is widely used in many kinds of practiced

engineering including pressure vessel industry, nuclear and

powder industry, ocean engineering and so on. Especially in

underwater engineering the spherical shell is generally used to

be as the main hull against the strong external pressure, for

example the existing deep manned submersibles of ALVIN of

USA, Nautile of France, MIR I&II of Russia, Shinkai 6500 of

Japan and Jiaolong of China (Fig.1), adopt pressure spherical

hull as human operation cabin. However based on lots of

comparative studies and research, scientists and engineer have

found that all these submersibles could not comply with many

of the existing design rules [1]

. For new category material or new

developed submersibles appearing, it is generally necessary for

engineers to update or modify the design method of spherical

shells hull to optimize its design.


Fig. -1 Chinese HOV “Jiaolong” for diving 7km

seabed In history many researchers in the field of engineering and

mechanics have took great efforts to solve this problem in order

to obtain the analysis and design optimization method of the

spherical shell hull. Among all these works, Zoelly[2]

firstly

derived the elastic buckling pressure formula of a complete

perfect sphere under external pressure in 1915. Until 1960s by

comparing with a series of model tests for more than 200 small

spherical models with different imperfections, Krenzke and

Kiernan [3]

found that the actual buckling load was only 70% of

the perfect one. Then they have considered the effect of

localized imperfections and other effects such as the residual

stresses and defective workmanship. In Krylov Ship-building

Research Institute, Paliy[4]

and his colleagues had developed the

calculation method for the collapse strength of titanium alloy

spherical pressure hulls based on theoretical analyses and model

tests. Galletly [5]

had investigated plastic collapse load for

externally pressurized imperfect hemispherical shells, then

cooperated with Blachut[6]

to make realistic assumption on the

magnitude of initial geometric imperfections in a sphere to be

designed and pointed out that more model tests should be

carried out to verify and extend ranges of parameters. Based on

the optimization method of pressure vessels, Blachut[7,8]

also

developed strength and stability of shells components used in

submersibles. Cui and his colleagues [9,10]

had tried to obtain the

ultimate strength analysis method of titanium alloy spherical

pressure hulls and also to compare with different kinds of rules

for developing new deep manned submersibles in China.

In this paper some of main rule methods and nonlinear

FEM for spherical shell design are investigated and compared

respectively. The characteristics of spherical shell under

external pressure will be discussed. And then experiments of

four spherical models built by high strength steel will be

presented and it is also compared and analyzed with different

methods for them. Based on the results it is confirmed that it is

necessary to update or modify the design method of spherical

pressure shell for new developing deep manned submersibles.

THE RULE METHODS FOR SPHERICAL SHELL UNDE-

R EXTERNAL PRESSURE

In the world there are some accordingly rules for pressure

spherical shell to different countries in underwater engineering.

The spherical shell under external pressure faces two questions

including strength and buckling.

The strength of spherical shell is generally checked by the

following formula:

[ ]0 ,2

s b

s b

pR

t n n

σ σσ σ

= ≤ =

(1)

Here p is the working pressure. R and t are mean radius and

mean thickness of spherical shell. σs and σb are yield strength

and ultimate strength of materials constructed. While ns and nb

are safety factors to elastic or plastic failure controlled state.

The solutions of the limit buckling pressure of spherical

shell are varying for different countries because of affecting

factors of manufacture and material. Therefore three rules and

their design check methods are mainly discussed in this section.

(1) RMRS [11]

rule method

Russian Maritime Register of Shipping (RMRS) has speci-

fic rules for the classification and construction of manned

submersibles, ship’s diving systems and passenger submer-

sibles. In the rule spherical hulls, semispherical and near-

semispherical ends of cylindrical and conical hulls, as well as

semispherical caps are calculated by the following formulae for

checking:

cp

B

Pp

n≤ (2)

cp s EP Pη= (3)

Where

( )2

2

2

3 1E

E tP

Rµ = −

( ) 2

1 1/ 1 1s s sfη η η δ = + + , 2

E

s

P R

tδ

σ=

( ) 2 / 3

1 1/ 1 2.8s f fη = + + , /f f t=

Here E and µ are material modulus of elasticity and Poisson’s

ratio. While f is the maximum deviation of spherical surface

from the regular round forms. nB is safety factor for stability.

(2) ABS [12]

rule method

American Bureau of Shipping (ABS) presents special rules

for building and classing to underwater vehicles, systems and

hyperbaric facilities. The maximum allowable working pressure

and limit pressure for spherical shells are to be obtained from

the following equation:

cpp Pη≤ (4)


21

0.7391 / 1 0.3

0.2124

Ecp

E E

p for P pP

P for P p

δ

+ > = ≤

(5)

Where

2 s

o

tp

R

σ=

However the geometry of spherical shell is determined by outer

dimension of spherical shell in ABS rule here. Therefore η is

usage factor and Ro is outer radius of spherical shell. And PE

must be calculated by Ro.

(3) CCS [13]

rule method

China Classification Society (CCS) has its special rules for

the construction and classification of diving system and

submersibles. In this rule the design method and calculation of

spherical shell is expressly determined as followings:

cp s z eP C C P= (6)

Where

20.84eP EC= ,c

tC F

R

=

Here Cs and Cz are material performance factor and geometrical

or manufacture imperfection factor. And C is the function of t to

R radio. The geometry of spherical shell is determined by mean

dimension of spherical shell.

THE NONLINEAR FINITE ELEMENT METHOD FOR

ANALYSIS OF STABILITY The buckling analysis should include not only large

deformation and nonlinear strain-displacement relations, but

also material nonlinearity, because that the material of

somewhere in the pressure hull has become into plastic stage

when the buckling approached. And in this paper the stress-

strain relations have been adopted just as Fig.6 in Ref.[14] to

simulate the true material behavior.

The nonlinear buckling analysis had been presented in

detail by Wang [15]

. And in this paper Von-Mises yield criterion

was adopted to judge whether the local buckling has happened.

So that is

( ) 02

32 =−=−= s

ijijseq

SSJF σσσ (7)

Generally the equivalent stress strength could be expressed

as follows:

23Jeq =σ (8)

ijij SSJ2

12 =

( ) ( ) ( )[ ] 222222

6

1ZXYZXYXXZZZZYYYYXX σσσσσσσσσ +++−+−+−=

(9)

Here Sij is deviator tensor of stress; σs is yield stress; σXX, σYY,

σZZ are the normal stresses of the X, Y and Z direction; σXY, σYZ,

σZX are the shear stresses of the XY, YZ and ZX plane

coordinate.

The analysis process chart of nonlinear FEA is shown in

Figure 2.

STRUCTURAL CHARACTERISTICS OF SPHERICAL

SHELL UNDER EXTERNAL PRESSURE The main problem to spherical shell under external

pressure is how to calculate its ultimate strength especially for

buckling or damage. As it is all known well that the stability of

spherical shell is sensitive to its initial deformation, material

mechanical property or external disturbance except its geomet-

ic dimension which could be found from equations (3), (5) and

(6). From formulas it is known that these effect factors are

principally R/t, f/t and δ .

(1) Ultimate state and buckling

The spherical shell has good strength and stability for its

Analysis of the linear-elastic and eigenvalue buckling

Set initial deflection to the model

Use elastoplastic stress-strain relation

Apply arc-length method to model Post-buckling

Analysis of the non-linear finite element method

Analysis of the results

Obtain the critical pressure

Get the deformation-pressure curve

Fig. -2 The nonlinear FEM analysis process chart

special structure. Here one model of spherical shell is adopted

to analyze with parameters of R/t=71.5 and δ =3.65. Its stress

and ultimate strength are solved by FEM presented here and it

is assumed that the model material also has ideal elastic-plastic

performance and here the form of initial deflection of model

was set as same as the first elastic buckling modal. Then Fig.3

from FEA results shows the pressure-flexibility curve as into

post-buckling track. From pressure- displace curve it could be

found that the model appears obviously sharply critical state

and perfect post-buckling plastic flow. Fig.4 presents

deformation contour when spherical model is at ultimate state.

While Fig.5 shows out the deformation contour after the

spherical model has being entered plastic flow phase. If the

spherical model even has a large opening in it, this buckling

state would be still appearing such as Fig.6 with a large opening

in it.


Fig.-3 The pressure- flexibility curve as into post-

buckling track

Fig. -4 The deformation contour of spherical model at

ultimate state

Fig. -5 The deformation contour of spherical model in

post-buckling flow

Fig. -6 The deformation contour of spherical model

with large opening in post-buckling flow

(2) Effect to buckling critical pressure

From formulas (3) or based on exiting investigation data [5,

6], it is easily seen that the buckling critical pressure of spherical

shell is the most sensitive to the local geometric initial

imperfection than other effecting factors. So in CCS and ABS

Rule, the maximum allowance imperfection has been

constrained not to exceed one specific size based on the

sphere’s dimension. Fig.7 shows out the results of comparison

between four kinds of methods to the spherical model with

parameters of R/t=71.5 and δ =3.65. Here it is obviously

found that the limit pressure is sharply reducing as initial

imperfection enlarging and the relation between them is not the

simple linearity. While the effect of sphere’s size is relatively

simple and this could be seen from Fig.8.

Fig. -7 Comparison of the different rules for R/t=71.5


Fig. -8 The influence of the initial deflection on critical

pressure by varying R/t In fact the critical pressure would be also affect by the yield

strength of shell material. The yield stressσs and ultimate

strength σb are generally very close to high strength steel or

titanium alloy, which means that the material may be more

fragile than common steel and it is more easily to appear fragile

destroy for the pressure hull. Fig.9 presents the influence of the

initial deflection and material yield strength on critical pressure.

Fig. -9 The influence of the initial deflection and

material yield strength on critical pressure

EXPERIMENTS OF SPHERICAL MODELS According to above research and in the development of

Jiaolong deep manned submersibles, Cui [10]

and his colleagues

have done series pressure tests of titanium alloy models and

their aim to obtain the optimization design of submersible.

Besides titanium alloy, the steel especially high strength steel

is often used as pressure hull in underwater engineering.

Therefore it is necessary to done some model experiments by

high strength steel.

(1) The spherical models

Four steel models have been built here, two small (model I

and II) and two bigger (model III and IV). These four models

are all manufactured by high strength steel, and their dimension

parameters are listed in table 1.

The two small model I and II are consisting of two semi-

spherical shells which is the whole one and two hemispheres are

welded into a whole sphere, see Fig.10. The model geometric

imperfection size has been measured by bridge gauge device as

Fig.11.

The two bigger models III and IV are built through by eight

pulps and each pulp was welded and connected into a whole

sphere model, shown as Fig.12. Because of the big size, the

model geometric imperfection size has been measured by 3D

laser detector from inside and be checked from outside.

(a)

(b)


(c)

Fig.-10 Components and model of spherical shell

(a) (b)

Fig.-11 Measuring model external imperfection by bri-

dge gauge device

Fig. -12 Large model of spherical shell with manhole

(2) Model test in pressure tank

The four spherical models had been successively tested in

pressure tank which is similar to underwater environment under

external pressure. And in process of experiment, the strain on

some key or special positions on spherical model had been

measured and being monitored online during all the tests.

Four pictures in Fig.13 accordingly show the damage state

and deformation of spherical models after external pressure test.

It is easily found that three models appear large plastic

deformation flow and stability buckling obviously, except that

model I happened some tore broken state. Meanwhile it would

be simply noticed that the plastic deformation of Fig.5 is very

similar to those of Fig.13.

(I)

(II)


(III)

(IV)

Fig. -13 Deformation of spherical model after external

pressure test

COMPARISON AND ANALYSIS OF SPHERICAL MOD-

ELS TESTS

Based on the real initial deflection and thickness of models,

the four models have been checked by rule methods and even

nonlinear FEM which was set the real initial deflection in them.

In ABS and CCS rules method, the real initial deflections are

still in the range of allowance initial maximum geometric error.

Table 1 shows the comparing results of four spherical

models. From this table, it is confirmed that buckling critical

pressure is sensitive to the imperfection and it is actually

different from results calculated by rule methods to tests ones.

ABS and CCS rules’ methods are relatively conservative

because of their allowance initial maximum geometric error in

manufacture, while RMRS rule method is based on titanium

alloy model tests which material is similar to presented model

material.

Fig.14 has shown the deflection-pressure curve of model II

at its firstly collapse local point as into post-buckling track by

FEA method presented in paper, and other model curves could

be obtained easily such as that.

Fig.-14 The deflection-pressure curve of model II as

into post-buckling track

However although here the FEM results are very close to

those of tests, it must be noticed that calculation result would be

sharply sensitive to inputting initial deformation of model shell

and the choice of nonlinear equation solution method in

presented nonlinear FEM, which mostly depends on experience

of researcher in FE analysis.

SUMMARY

In this paper rules’ design method of spherical pressure

shell have been studied and then characteristics of the whole

spherical shell have been demonstrated by different ways. Two

groups of four spherical models built by high strength steel and

with two kinds of same main sizes but with different initial

deflection on them have been finished in model tests. Those

results eventually indicate that the stability of a spherical shell

varies by its initial imperfection and such materials sphere

stability design could not be identical by current rules designs

method except presented nonlinear FEM. It is essentially

necessary to obtain the new theoretical analysis and design

method for spherical shell made by such steel, with high yield

strength but closely to ultimate strength, used to deep manned

submersibles or other ocean engineering.


Tab.1 Comparing results of four spherical shell models

I II III IV

R/t 53.53 53.53 67.9 66.5

f0/t 0.08 0.075 0.305 0.195

δ 4.46 4.46 3.69 3.90

-- Pcp/PE Error Pcp/PE Error Pcp/PE Error Pcp/PE Error

Test Data 0.326 -- 0.332 -- 0.252 -- 0.285 --

RMRS Rule 0.315 3.4% 0.315 5.1% 0.273 8.3% 0.301 5.7%

ABS Rule 0.173 46.9% 0.173 47.9% 0.187 25.8% 0.184 35.4%

CCS Rule 0.266 18.4% 0.266 19.9% 0.233 7.5% 0.223 21.8%

Nonlinear FEA 0.327 0.31% 0.333 0.3% 0.254 0.8% 0.280 1.7%

ACKNOWLEDGMENTS

The experiments were carried out in China Ship Scientific

Research Center (CSSRC) and many helps from our former

colleagues including Prof. Zhenquan Wan and Prof. Yongjun

Wang, are greatly appreciated.

The authors are thankful to cooperated Prof. R Ajit Shenoi

of University of Southampton for study condition and equipm-

ent, and also thankful to China Scholarship Council (CSC) for

providing financial funds to author (Q. Du) as a visiting scholar

in Southampton Marine and Maritime Institute (SMMI) at Univ-

ersity of Southampton.

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