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A study of stress concentration effect around

penetrations on curved shell and failure modes for

deep-diving submersible vehicle

Ching-Yu Hsua,*, Cho-Chung Liangb, Sheau-Wen Shiahc,Chan-Yung Jend

aDepartment of Marine Mechanical Engineering, Chinese Naval Academy, 669 Chun Hsiao Road,

Tsoying, Kaohsiung 813 Taiwan, R.O.C.bDepartment of Mechanical and Automation Engineering, Da-Yeh University, 112. Shan-Jiau Road,

Da-Tsuen, Changhua 515, Taiwan, R.O.C.cDepartment of Naval Architecture and Marine Engineering, University of National Defense Chung Cheng

Institute of Technology, Ta-Shi, Tao-Yuan 335, Taiwan, R.O.C.dInstitute of System Engineering, University of National Defense Chung Cheng Institute of Technology,

Ta-Shi, Tao-Yuan 335, Taiwan, R.O.C.

Received 5 December 2003; accepted 12 March 2004

Available online 18 January 2005

Abstract

For deep-diving submersibles, penetrations in the pressure hull induced by discontinuities in the

shell can generate stress concentration effects that represent one of the most interesting and

challenging areas of structural design. However, only the slightest trace of rational theory, collateral

test results or application techniques have been reported in open literature. Therefore, in this paper,

an opened shallow cylindrical shell and deep-diving submersible vehicle GUPPY is adopted to

investigate. In this study, the finite element procedure based on the Hibbitt and Karlsson’s

methodology is used to analyze the curvature effects and failure modes influence on stress

concentrations around opening. These results can hopefully provide valuable insight for the future

the designer of all underwater vehicles.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Stress concentrations; Curvature effects; Penetrations; Deep-diving submersible vehicle

Ocean Engineering 32 (2005) 1098–1121

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2004.05.011

* Corresponding author. Tel.: C886 7 5834700x1702; fax: C886 7 5834861.

E-mail address: [email protected] (C.-Y. Hsu).

http://www.elsevier.com/locate/oceaneng

C.-Y. Hsu et al. / Ocean Engineering 32 (2005) 1098–1121 1099

1. Introduction

Penetrations are necessary either for access or for the passage of piping, ventilation

ducts, uptakes, and electrical systems. A hole can be found in a missile skin, aircraft

fuselage, a ship hatch, boilers and submersible pressure hulls, amongst other numerous

situations. For instance, a modern submarine has more than 300–400 openings in its

pressure hull.

On the deep-diving submersible vehicles penetrations are also discovered. During the

past three decades a number of deep-diving submersible vehicles (DSV) has been

developed for many diverse fields, e.g. the oil industry, scientific deepwater marine

research, underwater cable and piping laying, deep ocean mining, as well as submarine

rescues. In most cases, deep-diving submersible vehicles must dive to a depth of 1000–

4000 m. Most of DSV pressure hulls are single or multiple sphere configurations for

external hydrostatic pressure loading. Some of the most used forms of pressure hulls are

shown in Fig. 1 (Allmendinger, 1990; Jackson, 1992; Watson, 1971; Gorman and Louie,

1991).

Penetrations are a major design consideration. For deep-diving submersibles, major

penetrations are limited to access hatches and viewports. There are local stress variations

around the penetrations that affect the design’s depth limitations, as well as the cyclic life

Fig. 1. The deep-diving submersible vehicle (Allmendinger, 1990; Jackson, 1992; Watson, 1971; Gorman, 1991).

C.-Y. Hsu et al. / Ocean Engineering 32 (2005) 1098–11211100

of the pressure vessel. Small penetrations affect only local stresses, but large penetrations

could affect the design’s collapse depth. Hence, the problem of stress concentrations

around holes in the shell is very important and cannot be neglected by a study in deep-

diving submersible vehicle.

While the structure designer proceeds with stress analysis for the opened shell, the

holes in the shell are neglected to simplify the problem. For instance, the submersible

pressure hull that is fabricated with great axisymmetric precision is subsequently

punctured randomly. The designer used the ideal structure model to calculate the

structure’s strength. Thus, the analysis results could not reflect the actual situation.

Stress concentration around holes in shells is a relevant issue at the design stage.

Therefore, precisely analyzing the stress distributed around the holes of the shell is crucial

for the shell structure design. The stress around the holes on flat plates has received

extensive interest (Patel et al., 1969; Abdul-Mihsein et al., 1979; Mangalgiri, 1984;

Pinchas and Jacob, 1985; Gao, 1996). Those investigations focus primarily on the plate

loaded for in-plane force, internal pressure and other conditions. Nevertheless, stress

around the hole on the curved shell has seldom been mentioned. Table 1 summarizes more

significant investigrations.

For a military submarine, the most appropriate pressure hull consists of a cylinder, cone

and dom shell. However, for deep-diving research spherical pressure hulls, which can bear

a high hydrostatic pressure, are frequently preferred. McKee (1959) thoroughly described

the displacement, weight, external shape and structure of US and German submarines used

in World War II. According to that investigation, the spherical pressure hull offers a better

strength/weight ratio than a cylinder and cone. Garland (1968) described the design

requirement and fabrication procedures used to construct the pressure hull of a deep

submergence vehicle (DSV). Watson (1971) presented the design, construction and

operational experiences of the deep-diving submersible vehicle GUPPY. According to his

result, the structural failure attributed to buckling of a shell may occur at nominal stress

levels far below the yield strength of the material. Gifford and Jones (1971) discussed the

feasibility of applying FEM (Finite Element Method) to analyze current and future deep

submersible vehicles. That investigation also examined the structural response of the

dicontinuous member of a deep submergence pressure hull, e.g. plug hatches, juncture

rings and viewport. Winker and Nordseewerke (1983) reviewed the design trends of

conventional submarines, including extension of the submerged cruising range between

two snorkelling periods, reduction of noise radiation, and introduction of an improved

rescue system. Faulker (1983) studied the underlying structural mechanics and design

philosophy with simple strength formulations deemed appropriate for manual calcu-

lations. Gorman and Louie (1991) assessed the merits of advanced hull shapes,

architectures and materials via a novel methodology. In 1993, in addition to reviewing

manned and unmanned submersible vehicles systems, Garzke et al. (1993) highlighted the

importance of the systems in the exploration of ocean bottom geology and underwater

wrecks, e.g. the Titanic and Bismarck. Those contented that developing deep-diving

manned and unmanned submersibles could lead to new architectural fields. Liang et al.

(1997) proposed a minimum weight design of a submarine pressure hull under hydrostatic

pressure with constraints on factors, e.g. general instability, buckling of shell between

frames, plate yielding and frame yielding. Liang et al. (1998) investigated the non-linear

Table 1

The synopsis of the study for the hole in the shell

Author Synopsis Notes

Bonde and Rao

(1980)

Model: two equal circular elastic inclusions in a pressurized

cylindrical shell

Loading: pressure loading

Method: analytical method

Result: the interaction between the two inclusions is more

pronounced at qZ08 than at any other angle. Maximum

shell stress increases as the inclusions come closer. The

effect of inter-inclusion distance on bending stresses is more

pronounced than on membrane stresses

Bull (1982) Model: cylindrical shell with large circular cut-out

Loading: axial compression, torsion and three point bending

Method: experimental and finite element analysis

Result: stress concentration factors around the hole are

obtained from the experimental and finite element analysis

Deans (1990) Model: ring-reinforced shell with opening

Loading: end loading, external pressure

Method: finite element method

Result: a series of tables of the stress concentration factors

(SCF) in areas adjacent to the opening are obtained and,

using these SCF, modeling rules for utilization of part

models are developed

Dyke (1965) Model: an infinite thin shallow cylindrical shell

Loading: axial tension, internal pressure, and torsion

Method: analytic method

Result: three ranges of curvature parameter are defined by

the dimensionless parameter b

Foo (1992) Model: cylindrical shells with a single cutout

Loading: tensile and torsion loading

Method: non-linear mathematical programming techniques

Result: approximate lower bounds to the limit load and

torque of cylindrical shells with a single cutout is obtained

Hooke and

Demunshi

(1977)

Model: two cylindrical holes intersecting at right angles in

an infinite homogenous, isotropic, elastic body

Loading: uniform tension at an infinite distance from holes

Method: the point matching or boundary value least

squares’ technique

Result: the effect of the intersection on stress distribution is

very much localized in nature and decreases rapidly as the

point is moved away from the intersection along each of

holes. When the load is perpendicular to both holes, the stress

concentration factor has been found to be increased by

decreasing the ratio of the hole radii from 5.43 for equal sized

holes to 7.28 with one hole much larger than the other. Where

the holes are under hydrostatic pressure the stress concen-

tration factor varies from 2.99 with equal sized holes to 3.53

with one hole much larger than the other

(continued on next page)


Table 1 (continued)


Huang (1972) Model: an infinite elastic–plastic plate with a circular hole

or inclusions

Loading: pure tension and shear at infinity

Method: semianalytical method based on fourier series and

finite difference

Result: the validity of Neuber’s relationship between stress

and strain concentration factors for the plane stress

problems is examined and a generalized Stowell formula for

the stress concentration factor is proposed for problems in

which the applied loading may be pure shear as well as pure

tension

Meyer and

Dharani (1991)

Model: a buffer strip laminate containing a circular cutout

Method: the classical shear-lag stress–displacement

relations

Result: lower modulus and higher failure strain buffer strips

are most efficient in increasing the notched strength of

laminates with circular cutouts

Palazotto

(1977)

Model: cylindrical shell with cutout

Loading: compressive loading

Method: the finite difference method

Result: the bifurcation load is within 11% of the

collapse force for each configuration

Pattabiraman

and Ramamerti

(1977)

Model: a cylindrical shell with a cutout

Loading: asymmetric bending

Method: superposed method

Result: the cutouts located at the bottom experience

greater stresses than those at the top and stresses

increase with load angle

Paul and Rao

(1995)

Model: thick FRP finite laminated plates containing two

circular holes

Loading: transverse uniform loading

Method: The finite element displacement method, along

with the Lo-Christensen-Wu high order bending theory

Result: the results obtained by this theory compare

favourably with the exact elasticity solutions, which

demonstrates the validity of the method. The variation of

the stress concentration factor, with respect to plate

thickness, hole size and distance between holes, is

presented in graphical form and discussed

(continued on next page)


Table 1 (continued)


Sanders (1970) Model: an arbitrary shape cutout of shallow shells

Method: Betti–Rayleigh principle

Result: obtained integral representations of general

solutions

Starnes (1972) Model: thin cylindrical shells with a circular hole

Loading: axial compression

Method: experimental method and analytical method based

on a simplified Rayleigh–Ritz type approximation

Result: The experimental results indicated that the character

of the shell buckling was depended on a parameter which is

proportional to the hole radius divided by the square root of

the product of the shell radius and thickness. The analytical

method provided an upper bound for the buckling stresses

of the cylinders tested with hole radii less than 10% of the

shell radii, and verified the dependence of the shell buckling

characteristics on the parameter used to correlate the

experimental results


transient dynamic response of a submersible pressure hull which sustains the hydrostatic

pressure and/or shock wave.

Although the above investigations have provided numerous preliminary results related

to shells with holes and deep-diving submarine pressure hulls, to our knowledge, the effect

of a curved angle on the shell and the failure modes of the deep-diving spherical vehicle

under external pressure have not been systematically studied.

In light of the above developments, this work elucidates the curvature effects on stress

concentrations around a circular hole in an opened shallow cylindrical shell and the failure

modes of the deep-diving spherical vehicle under external pressure using the finite element

method. Design data are also developed for stress distribution around the hole for different

curved angles of the shell and for different failure modes for the deep-diving spherical

vehicle. Results presented herein provide a valuable reference for the designer of a shell

structure.

2. Theoretical background

The finite element procedure based on Hibbitt and Karlsson (1979, 1996) methodology

is adopted herein to investigate the curvature effects and failure modes’ influence on stress

concentrations around penetrations on an opened shallow cylindrical shell and Guppy type

shell under external pressure. The procedure uses the Newmark time implicit integration

scheme and Newton’s method, which includes dynamic equilibrium interaction

considering the half-step residual convergence tolerance. The plasticity relations, based

on von Mises’ yield criterion, assume the isotropic hardening rule for the elastroplastic

behavior of the material under study. The governing equation, solution method and

convergence tolerance are described as follows.


2.1. Governing equation

The basis of the displacement-based finite element procedure is the principle of virtual

displacement. The surface traction at any point on surface S is the force fs per unit of

current area. Also, the body force at any point within the volume V of the material under

consideration is f b, per unit of volume.

Then the force equilibrium for the volume is:ðS

f sdA C

ðS

f bdV Z 0 (1)

By using the Gauss theorem and the definition of Cauchy stress, the virtual work

equation can be obtained as:ðV

tdedV Z

ðS

duf SdS C

ðV

duf bdV (2)

where t, e, and u denote the stress, strain and displacement, respectively. In addition, t, e,

and u are an equilibrium set:

f S Z nt;vdu

vx

� �t C f b Z 0; t Z tT (3)

and de and du are compatible:

de Z1

2

vdu

vxC

vdu

vx

� �T� �(4)

and du is compatible with all kinematics constraints.

The left-hand side of Eq. (2) is replaced with the integral over the reference volume of

the virtual work rate per reference volume, as defined by any conjugate pairing of stress s

and strain 3:ðV

sd3dV Z

ðS

ff SgTdudA C

ðV

ff bgdudV (5)

and since

du Z ½N�due; d3 Z ½B�due; s Z ½D�3 Z D½B�due;

where [N] represents interpolation functions which depend on some material coordinate

system, and [B] defines the strain variation from the variations of the kinematic variables

which are derivable immediately from the interpolation functions once the particular strain

measure to be used is defined. In addition, [D] denotes the relation between stress and

strain. Moreover, ue expresses the vector of nodal displacement of an element. The non-

linear equilibrium equations can be derived to the following from:ðVe

½B�T½D�½B�uedVe Z

ðSe

½N�Tff sgdSe C

ðV

½N�Tff bgdVe (6)


This system of equations form the basis for the displacement-based finite element

analysis procedure, and has of the following form:

FnðumÞ Z 0 (7)

where Fn is the force component of the nth variable in the problem, and um denotes the

displacement of the mth variable.

2.2. Solution method

Herein, Newton’s method is used as a numerical technique for solving the non-linear

equilibrium equations. Allow cmiC1 to be the difference between the approximate solution

and exact solution to the discrite equilibrium equation. Eq. (7) can be denoted as:

Fnðumi Ccm

i Þ Z 0 (8)

Expanding the left-hand side of this equation in a Taylor series about the approximate

solution umi leads to:

Fnðumi ÞC

vFn

vupðun

i ÞcpiC1 C

v2Fn

vupvuqðum

i ÞcpiC1c

qiC1 C/Z 0 (9)

If the ith iteration solution umi is a close approximation to the exact solution, the

magnitude of the (iC1)th iteration solution umiC1 is small. Then, the first two terms

mentioned above can be neglected. In addition, the non-linear system of equations can be

expressed as follows:

Jnpi c

piC1 ZKFn

i (10)

where Jnpi is the Jacobian matrix:

Jnpi Z

vFn

vupðum

i Þ (11)

Then the next approximation to the solution is:

umiC1 Z um

i CcmiC1 (12)

2.3. Convergence tolerance

The convergence tolerance x of Eq. (12), which is an iteration process, is measured by

comparing the consecutive quantities of cmiC1 to see if they are sufficient small. Restated,

jumiC1 Kum

i j Z cmiC1%x (13)

3. Problem description

An opened shallow cylindrical shell and a Guppy type pressure shell which is

constructed of a spherical shell subjected to external pressure, are modeled to elucidate

Fig. 2. The opened shallow cylindrical shell profiles.


how curvature and failure modes influence stress concentrations around a circular hole.

Their geometrical configuration, boundary condition, material property and element mesh

are described as follows.

3.1. Geometrical configuration and boundary condition

3.1.1. Opened shallow cylindrical shell

Fig. 2 presents the profiles of an opened shallow cylindrical shell and Table 2 lists the

detailed dimensions, where some symbols are defined as follows:

L:
the side of the shell and LZRap/180Zconstant
P:
external pressure
r:
the radius of the hole
R:
the radius of curvature of the shells
t:
thickness of the shell
a:
the curved angle of the shell
b:
the location angle of the hole in the shell
The upper surface that comes into contact with the external pressure loading is defined

as the outer surface. The lower surface is the inner surface.

Eighteen curved angles (a) are considered in this study. Four sides of the shell are

considered as fully fixed.

Table 2

Detailed dimension of the opened shallow cylindrical shell

Curved angle, a

(8)

Radius of curva-

ture, R (m)

The side of the shell, LZ0.254 m; the radius of the hole, rZ0.0254 m; the external pressure, PZ6895 Pa; the location

angle of hole of the shell, b; the thickness of the shell, t.

Note: LZRap/180

0 Infinity

1 145.53

2 72.77

3 48.51

4 36.38

5 29.11

6 24.26

7 20.79

8 18.19

9 16.17

10 14.55

15 9.70

20 7.28

30 4.85

40 3.64

45 3.23

50 2.91

60 2.43


3.1.2. Guppy type pressure hull

The Guppy type pressure hull is shown in Fig. 1 and her model is a single spherical

configuration with two hemispherical viewports forward and one hemispherical access

hatch overhead. Fig. 3 presents the geometrical configuration of a spherical shell and

Table 3 lists the detailed dimensions.

Herein, the boundary conditions are described as follows:

ux Z4x Z4y Z4zZ0, uys0, uzs0 at the nodes on the symmetric (Y–Z) plane, and

uy ZuzZ4x Z4y Z4zZ0, uys0 at the node A, where ux, uy, and uz denotes

displacements in the global X-, Y- and Z-direction; fx, fy, and fz denote rotation about

the global X-, Y- and Z-direction, respectively.

The upper surface that comes into contact with the external pressure loading is defined

as the outer surface. The lower surface is the inner surface.

3.2. Material property


The model is assumed herein to be constructed of high-tension steel (HTS). The

material properties are described as follows:

Young’s modulus: EZ2.04!1011 Pa

Poisson’s ratio: nZ0.3

Yield stress: syZ3.24!108 Pa

Fig. 3. The geometrical configuration of a Guppy type pressure hull.


Table 3

The geometrical dimensions of a spherical shell

Spherical shell Manhole Viewport

The center of circle

(sphere) (R, q, f)

OZ(0, 08, 08) O1Z(0.8201 m, 08, 08) O2Z(0.8382 m, 678, 188)

Radius (m) RZ0.84455 r1Z0.2032 r2Z0.1016

Thickness (m) 0.0127 0.0508 0.0508



The model is assumed here to be constructed of HY100 high-tension steel. The material

properties are described as follows:

Young’s modulus: EZ2.10!1011 Pa

Poisson’s ratio: nZ0.29

Ultimate stress: suZ7.935!108 Pa

Yielding stress: syZ6.90!108 Pa

Mass density: rZ7.828!103 kg/m3

3.3. Loading


This study considers only the external pressure loading 6.8948!103 Pa (1.0 psi).


There is a diving depth: for every 100 m, water pressure increases 9.795!105 Pa in

standard sea water. This study considers only the external hydrostatic pressure loading.

3.4. Meshing

In this study, the opened shallow cylindrical shell with the circular hole was modeled

by 9-node doubly curved thin shell elements and the Gyppy type shell was modeled using

two curved thin shell elements. The one is a 3-node triangular curved thin shell element.

The other hull is a 4-node doubly curved thin shell element

The 9-node doubly curved thin shell element has five degrees of freedom per node

(three displacement components and two in-surface rotation components)—ux, uy, uz, fx,

fy, and has four Gauss integration points in this element (Fig. 4). The 3-node triangular

curved thin shell element and 4-node doubly curved thin shell element also has five

degrees of freedom per node—ux, uy, uz, fx, fy, (Figs. 5 and 6).

The positive normal of the element is given by a right-hand rule going around the nodes

of the element in the order that they are given on the element data line.


There are 18 opened shallow cylindrical shell models, and each is modelled with

specified curved angle (a). At the specified curved angle (a), the relative finite element

model is illustrated in Fig. 7. The finite element model is meshed by 212 doubly curved

thin shell elements.

Fig. 4. Nine-nodes doubly curved thin shell element.


3.4.2. Guppy pressure hull

The spherical submersible pressure hull was modeled using two curved thin shell

elements. The one is a 3-node triangular curved thin shell element (Fig. 5). The other hull

is a 4-node doubly curved thin shell element (Fig. 6). Owing to symmetry, only half of the

model is studied, as shown in Fig. 8. The pole circle of the spherical shell was modeled by

the former element and the others were modeled by the latter element. The 1/2 spherical

shell was meshed by 1584 four-node doubly curved thin shell elements and 40 three-node

triangular curved thin shell elements.

Fig. 5. The 3-node triangular curved thin shell element.

Fig. 6. The 4-node doubly curved thin shell element.


3.5. von Mises stress definition

This study considers the von Mises stress (or equivalent stress) to reveal the stress

distribution phenomenon around the hole. The von Mises stress is defined as follows:

Fig. 7. Mesh of the opened shallow cylindrical shell.

Fig. 8. The finite mesh diagram of 1/2 spherical shell.


von Mises Stress

sM Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2½ðs1 Ks2Þ

2 C ðs2 Ks3Þ2 C ðs3 Ks1Þ

2�

r(14)

where s1, s2, s3 denotes the principal stress

otherwise,

von Mises stress on the outer surface is denoted as so.

von Mises stress on the inner surface is denoted as si.


4. Numerical results

4.1. Curvature effects on stress concentrations around the circular hole in opened shallow

cylindrical shell

In this study, the shell thickness is maintained at a constant value of 22.23 mm. The

curved angle a is expressed in degree form, ranging from 0 to 608. In addition, eighteen

curved angles are considered as follows:

aZ0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 15; 20; 30; 40; 45; 50; 608

Figs. 9–12 summarize the analysis results of curvature effects.

Figs. 9 and 10 display von Mises stress around the circular hole on outer and inner

surface of the opened cylindrical shell for different curved angles aZ0–608, respectively.

These figures reveal the following:

1.
Extreme stress occurred about aZ0–58 for both outer and inner surfaces.
2.
For the outer surface (Fig. 9):
(a) The form of von Mises stress so resembles an exponential decay function.

(b) Maximum stress occurs on curved angle aZ38 and appears at the location angle

around the hole aZ908 where the value is 24.81 MPa.

Fig
. 9. The von Mises stress (Pa) so on outer surface for different curved angles of a (8) of the shell.

Fig. 10. The von Mises stress (Pa) si on inner surface for different curved angles of a (8) of the shell.


3.
For the inner surface (Fig. 10):
(a) Maximum von Mises stress appears at curved angle aZ08. Herein, the location

angle is bZ458. Maximum stress values are 18.48 Mpa.

(b) Two groups of the series line are shown in Fig. 10. The group of the location angles is

less than bZ458, revealing that the curved angle increases and the stress around the

hole decreases as well. For the other groups, minimum stress appears in the interval

in which curved angles range from aZ0 to 98 and the form of stress line resembles a

concave form. In these groups, the observation that the curved angles exceed aZ208

reveals the curved angle increase and the stress around the hole will decrease.

4.
The comparison of the maximum stress so is about 25% higher than that of si. Figs. 11
and 12 summarize the stress distribution configuration around the hole boundary for

different curved angles. These figures reveal the following results:

(a) Extreme stress nearly occurs at around bZ0, 45, 90, 135 180, 225, 270, 3158, for

both outer and inner surfaces.

(b) If the shell is flat (aZ08), maximum stress appears at bZ45, 135, 225, 3158. In

addition, the value is 18.48 MPa.

(c) In the outer surface (Fig. 11):

(i) When the curved angle ranges from aZ0 to 308, maximum stress occurs on

location angle bZ90, 2708.

Fig. 11. The von Mises stress distribution (Pa), so around the hole on inner surface for different curved

angles of a (8).


Fig. 12. The von Mises stress distribution (Pa), si around the hole on inner surface for different curved

angles of a (8).


Fig. 13. The von Mises stress (Pa) around the access hatch on inner and outer surface for pressure hull of actual model.


(ii) Maximum stress gradually increases from aZ0 to 38 and gradually decreases

from aZ4 to 608.

(d) For the inner surface of the shell (Fig. 12):

(i) If the curved angle ranges from aZ0 to 98, maximum stress occurs on location

Fig. 14. The global buckling of pressure hull of actual model from buckling mode shapes (mode1wmode10).



angle bZ0, 1808. When the curved angle ranges from aZ9 to 608, maximum

stress occurs on location angle bZ90, 2708.

(ii) The maximum stress gradually decreases from aZ0 to 608.

4.2. The failure modes of Guppy type pressure hull vessel under uniform external pressure

For the Guppy type pressure hull, shell thickness is maintained at a constant value of

12.70 mm and shell thickness arround the access hatch and viewports linearly increases

from 12.70 to 50.8 mm. Under uniform external pressure, a thin-walled spherical pressure

hull can collapse from external pressure in three different modes, due to axisymmetric

yield, ultimate stress or by buckling manner. The failure modes depend on a number of

factors, including thickness, radius ratio of the vessel and the mechanical properties of its

construction material.

Failure due to yield is perhaps the most important failure mode of vessels under

external pressure. From Fig. 13, we can find that the diving depth of the HY-100 steel at

yielding strength is 800 m, the hydrostatic pressure is 7.836!106 Pa, and the maximum

von Mises stress around the access hatch on inner surface of the spherical shell is 6.87!108 Pa at q1Z08.

The failure mechanism may be divided into two areas—plastic buckling at stress

levels below the proportional limit of the stress–strain curve and inelastic buckling at

stress levels above the proportional limit. The eigenvalue buckling analysis is

generally used to estimate the critical load of stiff structures and can provide useful

estimates of collapse mode shapes.

The critical buckling load, Pcr, is defined by:

Pcr Z P0 Cli !P (15)

where P0 is the dead load, P is caused by application of pressure, and li are the

eigenvalues. Fig. 14 indicates the first ten buckling mode shapes. This finding suggests that

the first buckling mode eigenvalue is 334, critical buckling load is 3.34!107 Pa, and

buckling collapse depth is 3410 m.

5. Conclusion

This study elucidates the curvature effect on stress concentrations around a circular hole

in the curved shell and the failure modes of the Guppy type pressure hull under external

pressure loading using the finite element method.

Based on the results presented herein, we can conclude the following:

5.1. Curvature effect for an opened shallow cylindrical shell

1.
Extreme stress always occurs at about bZ0, 45, 90, 135, 180, 225, 270, 3158 for
different curved angles a.


2.
If the shell is flat, i.e. aZ08, maximum stress appears at bZ45, 135, 225, 3158.
3.
For the outer surface:
(a) When the curved angle ranges from aZ0 to 308, maximum stress occurs on

location angle bZ90, 2708. Stress gradually decreases from aZ0 to 38 and

gradually increases from aZ4 to 308.

(b) If aZ40 and 458, maximum stress appears at the bZ67.5, 112.5, 247.5, 292.58.

(c) If aZ50 and 608, maximum stress appears at the bZ45, 135, 225, 3158.

4.
For the inner surface of the shell,
(a) If the curved angle is aZ0–88, maximum stress occurs on location angle bZ0,

1808.

(b) When the curved angle is aZ9–608, maximum stress occurs on location angle

bZ90, 2708. Stress gradually decreases from aZ0 to 608.

5.
For different curved angles, most maximum stress levels gradually decrease with an
increasingly curved angle. This finding suggests that the shell that has larger curved

angles can more aptly resist to external pressure loading.

5.2. For the Guppy type pressure hull

This study examines the results presented herein. We can conclude the following:

1.
The stress level around holes on the spherical shell is higher than at other sections.
2.
The stress level increases with the increasing radius of the shell penetration.
3.
The stress level of critical sections of the hull structure always occurs around holes.
These phenomena are induced by the discontinuity and stress concentration. Therefore

when the designing, the shell’s thickness must be reinforced around holes.

4.
We can find that the diving depth of the HY-100 steel at yielding strength is 800 m.
5.
The buckling collapse depth of the Guppy type pressure hull is 3410 m.
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