Buckling analysis of thin-walled metal liner of cylindrical ...

Research Article

Guo Zhang, Haiyang Zhu, Qi Wang, Xiaowen Zhang, Mingfa Ren*, Shunan Xue and Gang Li

Buckling analysis of thin-walled metal liner ofcylindrical composite overwrapped pressurevessels with depressions after autofrettageprocessing

https://doi.org/10.1515/secm-2021-0051received May 08, 2021; accepted September 02, 2021

Abstract: The cylindrical filament wound composite over-wrapped pressure vessels (COPV) with metal liner hasbeen widely used in spaceflight due to their high strengthand low weight. After the autofrettage process, the plasticdeformation of the metal liner is constrained by compo-site winding layers, which introduce depressions to themetal liner that causes local buckling. To predict the localbuckling of the inner liner with depressions of the pres-sure vessel after the autofrettage process, a local bucklinganalysis method for the metal liner of COPV was devel-oped in this article. The finite element method is used tocalculate the overall stress distribution in the pressurevessel before and after the autofrettage process, and theinfluence of local depressions on the buckling is

evaluated. The axial buckling of the pressure vessel underexternal pressure is analyzed. The control equation of themetal liner with depressions is developed, consider-ing the changes in the pressure and the bending momentof the liner depressions and its vicinity during the loadingand unloading process. Taking the cylindrical COPV (38 L)with aluminum alloy liner as an example, the effects of linerthickness, liner radius, the thickness-to-diameter ratio, auto-frettage pressure, and the length of straight section on theautofrettage process are discussed. The results show that thethickness of the inner liner has themost significant influenceon the buckling of the liner, followed by the length of thestraight section and the radius of the inner liner, while theautofrettage pressure has the least influence.

Keywords: autofrettage processing, buckling analysis,composite overwrapped pressure vessels

1 Introduction

As a core component of the space reentry vehicle, pres-sure vessels occupy the largest proportion of weight andvolume in the propulsion system, which is the most sig-nificant part of the carrier structures [1,2]. Due to theanisotropy, designability, lightweight, and high strengthof the composite materials, the weight of composite over-wrapped pressure vessels (COPV) can be reduced byabout 25% compared to metal pressure vessels. There-fore, COPV have been widely applied in aerospace struc-tures and process equipment in recent years [3–10].COPV are mainly divided into cylindrical pressure ves-sels, spherical pressure vessels, annular pressure vesselsby shape, of which the cylindrical pressure vessels aremost widely used for their advantages of simple structureand high filling coefficient [11]. By the type of structure,the cylindrical COPV can be divided into circumferen-tially wound COPV with metal liner, metal liner COPV,

Guo Zhang: Department of Engineering Mechanics, DalianUniversity of Technology, Dalian 116024, China; Beijing ChinatankIndustry Co. Ltd, Beijing 100024, ChinaHaiyang Zhu: Department of Engineering Mechanics, DalianUniversity of Technology, Dalian 116024, China; National KeyLaboratory of Combustion, Thermal Structure and Flow,Xi’an Aerospace Propulsion Institute, Xi’an 710125, ChinaQi Wang, Shunan Xue: Department of Engineering Mechanics,Dalian University of Technology, Dalian 116024, ChinaXiaowen Zhang: School of Materials Science and Engineering,Dalian University of Technology, Dalian 116024, China

* Corresponding author: Mingfa Ren, Department of EngineeringMechanics, Dalian University of Technology, Dalian 116024, China;Department of Engineering Mechanics, The State Key Laboratory ofStructural Analysis for Industrial Equipment, Dalian University ofTechnology, Dalian 116024, China, e-mail: [email protected], tel:+86-411-8470-9161,fax: +86-411-8470-9161

Gang Li: Department of Engineering Mechanics, Dalian University ofTechnology, Dalian 116024, China; Department of EngineeringMechanics, The State Key Laboratory of Structural Analysis forIndustrial Equipment, Dalian University of Technology,Dalian 116024, China

Science and Engineering of Composite Materials 2021; 28: 540–554

Open Access. © 2021 Guo Zhang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0International License.

https://doi.org/10.1515/secm-2021-0051

mailto:[email protected]

liner-less COPV, and so on. To make full use of the highstrength of composites and the plasticity of the metal mate-rial [12], the wound layer should bear most of the load undertheworking pressure, while the inner liner only acts as a seal.Therefore, the thin-walled liner structure is adopted, andthe autofrettage process that exceeds its working pressureis applied to the pressure vessel before it is put into use[13,14]. Then, after the autofrettage pressure is completelyunloaded, the residual compressive stresses remain in themetal liner. Under working pressure, this pre-stressed statecan make the stress distribution COPV more reasonable,effectively reduce the stress level of the liner [15–19] andsignificantly improve the overall capacity of COPV [20–23].

After the autofrettage process of the COPV, the plasticstate of the metal liner is under compression. Thereforethe liner is prone to buckling deformation, which willgreatly reduce the load-bearing capacity and service lifeof the COPV [24]. Aggarwal and Cooper [25] conducted49 sets of experiments to study the liner buckling underdifferent boundary conditions. The results indicate that,for the same inner liner, when the outer side is rigidlyrestrained, the critical buckling load is 7 times larger thanthat of the unconstrained one. Yang et al. [26] developeda 3D finite element analysis (FEA) of the COPV by equatingthe wound layer to composite laminates. With the FEAmethod, the nonlinear buckling analysis of the COPVwith metal liner was carried out, and the multiphase buck-ling modes and critical buckling loads were obtained. Fuet al. [27] used the FEA of nonlinear buckling analysis toobtain the buckling mode shapes of the cylinder liner. Atthe same time, the relationship between the liner displace-ment and the external pressure was established, and thecritical buckling external pressure of the inner liner wasobtained. Liang et al. [28] used the eigenvalue analysismethod to calculate the critical buckling pressure of thecylinder liner of COPV. It is found that the critical buck-ing pressure becomes more sensitive to the geometricdepression by increasing the length-to-diameter ratio. Thebuckling of the metal liner of COPV is solved as a globalbuckling problem under external pressure in these studies.Therefore, the finite element model of the eigenvalue ana-lysis or nonlinear analysis is established to study the stabi-lity of the metal liner after the autofrettage process.

However, the buckling deformation of the COPV linerafter the autofrettage process is the unilateral bucklingconstrained by the outer wound layer, which is generallyexpressed as the local buckling of the metal liner. Huet al. [29] used acoustic emission technology to study thebuckling deformation of the liner of COPV in the unloadingprocess of internal pressure and found that the deformationbetweenthecomposite layersandthe linerwas incompatible,

which resulted in the buckling deformation and the failureof the liner. High-amplitude acoustic signals were detectedin theunloadingprocess,and localbucklingof the linerdueto the release of stress waves from structural buckling wasobserved. The results suggest that the local buckling willgenerate in the unloading of the autofrettage processrather than the global buckling.

In addition, from the perspective of the manufac-turing process of COPV, foreign objects may be mixedduring the winding process and the inner liner may behit by hard objects occasionally. These factors induceinitial depressions into the inner liner, which will furtherincrease the possibility of local buckling around thedepressions [30]. This kind of depressions can be avoidedthrough strict production management. However, duringthe autofrettage process of the filament wound thin-walled metal-lined COPV, at the overlap of the first layerand the second layer of the fiber tape, due to the unevendistribution, the matrix material will be compacted toincrease the rigidity under the action of autofrettage pres-sure, which will inevitably cause the thin-walled metalliner to produce a certain depth of depression after theautofrettage processing [31]. In view of the research onthe local buckling problem of the COPV liner with depres-sions, based on the assumption of plane strain, scholarssimplified the cylindrical inner liner into a cylindricalring, carried out an analysis of the circumferential buck-ling of the cylindrical ring with depression. Zhou [32]carried out external pressure buckling tests on thin-walled stainless steel liners and obtained the criticalbuckling loads of different sizes of liners. Lo et al. [33]equated the buckling part of the ring to a rod with initialcurvature and used the elliptic integral function to derivethe post-buckling topography. Sun et al. [34] conductedresearch on the shrinkage buckling of the inner linerthrough theoretical derivation and experimental researchand found that the buckling load is very sensitive to thesmall depressions of the rigid boundary. Based on thetheory of finite deflection, Yamamoto and Matsubara[35,36] studied the buckling of the cylindrical ring underexternal pressure and found that the initial depressioncan decrease the critical buckling pressure of the cylind-rical ring. Based on the finite element method (FEM),Wang et al. [37] established a simplified ring model ofthe ultra-thin metal liner of COPV with depressions onthe liner and realized the numerical simulation of thelocal buckling of the liner. With the nonlinear FEM, Vasi-likis [38,39] established the 2D model to conduct thenumerical simulation of the cylinder and its surroundingmedium. The results showed that the critical bucklingpressure is sensitive to the initial buckling depression.

Buckling analysis of thin-walled metal liner of COPVs 541

The cylindrical COPV’s liner is simplified to be acylindrical ring for buckling analysis, and the analyticaland numerical solutions of critical buckling can be obtained.The influence of axial stress and deformation on buckling isneglected. However, studies have shown that after autofret-tage processes, the liner is simultaneously constrained bythe winding layer in circumferential and axial directions,and the axial stress and deformation of the liner have a greatinfluence on buckling [28]. The cylindrical pressure vessel ismore likely to have axial buckling after the autofrettageprocess. It is also found that the liner has a damaged mor-phology caused by local buckling in the axial direction (asshown in Figure 1).

According to the buckling analysis on the metal lineof COPV with the initial depression, Phoenix and Kezirian[31] believed that the elastic support of the compositematerial winding layer on the liner limits the deformationof the liner. When the axial in-plane pressure N of theliner is greater than the axial critical pressure Ncr of theliner, the liner will not buckle but the initial depressionwill cause the interface pressure between the liner andthe winding layer to change. If the interface pressureis less than 0 and the length of the residual bendingmoment of the inner liner distributed in the axial

direction is greater than the half-wavelength of the in-plane axial buckling, the liner will buckle. The bucklinganalysis of a metal line of spherical COPV with depres-sion is carried out, and the problem is simplified to asquare thin plate model on an elastic foundation sub-jected to a bidirectional load. And in the process of sol-ving, using the symmetry of the square thin plate tofurther solve the elastic surface differential equation ofthe rectangular thin plate by the ordinary differentialequation, and the analytical solution of the inner bladderinterface pressure and residual bending moment isobtained. This study mainly used the equal stress anddeformation of the spherical shell in all directions. It isbelieved that the critical buckling load of the square thinplate with side length as half the wavelength of sphericalshell buckling in both directions is equal to the criticalbuckling load of the spherical shell. However, the stressand deformation of the cylindrical pressure vessel in thehoop and axial directions are inconsistent, which makesthe listed three-dimensional elastoplastic equationsunable to be simplified accordingly, and it is difficult toobtain the required analytical solutions for the in-planetension N and the interface pressure. If a three-dimen-sional finite element model for the buckling analysis of

Figure 1: Damaged morphology of COPV with metal liner caused by local buckling in the axial direction. (a) Schematic diagram of the axiallocal buckling of the metal liner and (b) damage appearance of the metal liner resulting from axial buckling after the autofrettage process.

542 Guo Zhang et al.

the inner liner with initial depressions is established forcylindrical COPV, the contact between the liner and thewinding layer must be defined, and the different contactstates of the depression and the winding layer before andafter the autofrettage need to be considered. Meanwhile,the contact area has strict requirements on the qualityand accuracy of the mesh [40], which often makes itdifficult to obtain the calculation results of convergence.In addition, there is obvious stress concentration nearthe depression, and the liner will enter the plastic flowstate earlier in the vicinity of the depression [41],resulting in an ill-conditioned stiffness matrix duringthe solution process, which will further cause the calcu-lation results of the FEM not to converge, and therefore,it is impossible to obtain accurate calculation results[41,42].

In this article, a novel buckling analysis method forthe metal liner of cylindrical COPV with depressions isproposed. Firstly, the stress distributions of pressure ves-sels before and after autofrettage are calculated by FEAto obtain the in-plane tension and interface pressure ofthe metal liner; Secondly, an analytical model of theeffect of depression on the interface pressure and residualbending moment of the liner is established to calculatethe change in the depression profile before and after theautofrettage to determine whether the metal liner will belocally buckled. Then, the effect of the changes of dimen-sional parameters on local buckling of cylindrical COPVwith initial depressions is further discussed, such as theautofrettage pressure, liner thickness, radius, and thick-ness-to-diameter ratio.

2 Establishment of the analysismethod

2.1 Buckling criterion of metal liners withdepression

For the thin-walled liner of cylindrical COPV, the straightsection is a cylindrical shell. Under the pressure of theautofrettage process, the liner will be plastically deformed,and the composite winding layer is always in an elasticstate [22]. After the pressure is unloaded, the liner will bein a compressed state. The critical buckling pressure FNcrof the cylindrical shell under the axial load is [42]

F E tR μ3 1

,Ncrl l

2

l2( )

=

−

(1)

where El is the elastic modulus of liners, tl is the thicknessof liners, Rl is the radius of liners, and μ is the Poissonratio of the liners.

The buckling wavelength λ corresponding to the cri-tical in-plane buckling pressure of a cylindrical shellunder pressure is [42]

λ mπ R tμ

212 1

,l2

l2

2

14

⎜ ⎟⎛

⎝ ( )⎞

⎠=

−

(2)

where m is the order. Taking m = 1 and dividing thewavelength by 2, the half-wavelength of the buckling isobtained as

λ π R tμ2 12 1

.l2

l2

2

14

⎜ ⎟⎛

⎝ ( )⎞

⎠=

−

(3)

The distance between the zero points of the residualbending moment of the liners near the depressions isgreater than π R t μ12 1l

2l2 2 1 4( ( ))/ −

/ . The interface pressurebetween the liner and the winding layer is less than 0. Ifthe above three conditions are met at the same time, localbuckling of the liner occurs.

2.2 Overall calculation process

The metal liners of the cylindrical COPV usually thickenthe sealing section and the transition regions from thestraight section to the sealing section under the spinningprocess, and the middle part of the straight section is thethinnest. This results in the straight section being morelikely to produce an initial depression under the autofret-tage pressure and the local buckling after unloading [22];so, the metal liner in the straight section with initialdepressions is studied in this article.

The commercial software ANSYS is utilized for thesimulation calculation of the FEM. A geometrical modelof the 1/8 cross-section of the cylindrical COPV withmetal liner is established, where the type of SOLID 46element is employed for the metal liner and type ofSOLID 45 element is employed for composite layers.The contact elements CONTA174 are added between themetal liner and the composite layers for the calculationsof the stress distribution of the liner before and afterautofrettage pressure. The symmetrical boundary condi-tions are carried out by the edges of the FEM. The axialstress of each node in the middle of the straight section ofthe FEM is extracted along with the thickness orientationof the liner, and integrated with the thickness direction toobtain the in-plane pressure of the thin-walled metal liner.


With the deduction, an analytical model was establishedfor the contour changes of the straight cylinder sectionwith depressions before and after the autofrettage process,and the analytical model of the effect of the depression onthe liner interface pressure and residual bending moment.

If the deformation distribution of the cylindrical shellin the axial and circumferential directions is consideredat the same time, a partial differential equation systemneedsto be established, which cannot be resolved. Therefore, onlythe displacement distribution in one direction is considered,the ordinary differential equation is listed, and the influenceof the other direction is corrected in the equation to solveit. Brush [43] has used a simply supported beam model onthe elastic foundation to describe the deformation ofthe cylindrical shell under confining pressure. The specificmethod was to extract the beammodel along the axial direc-tion of the cylindrical shell to obtain the displacement dis-tribution of the cylindrical shell along the axis. Since thebeam model is in one dimension, in order to describe theinfluence of the circumferential stiffness, the displacementof the beam restricted by the circumferential direction of thecylindrical shell is taken as equivalent to the stiffness treat-ment of the elastic foundation.

In this way, when analyzing the change of displace-ment along the axial direction of the cylindrical shell, thetwo-dimensional problem is simplified to a one-dimen-sional problem, and the partial differential equation issimplified into the ordinary differential equation as fol-lows [43]

D wx

F wx

E tR

wdd

dd

0,4

4 N2

2l l

l2⎜ ⎟

⎛

⎝

⎞

⎠− + = (4)

where w is the radial displacement of the liner, Nl is theaxial in-plane tension of the liner, FN is the axial in-planetension of the liner, D is the lateral stiffness of the liner, Elis the elastic modulus of the liner, tl is the thickness of theliner, μl is the Poisson ratio, and Rl is the radius of theliner.

In the middle of the cylindrical shell, assuming theinitial depth of the depression is δ, then the coordinatesof the center position of the depression are taken asw x δ0( )= = . The boundary condition of the equation isdefined as w xlim 0x ( ) =

∣ ∣→∞， w x xlim d d 0x 0 ( )/ =

∣ ∣→, in

combinationwith the boundary condition, the above ordinarydifferential equation can be solved to obtain the radialdirection of the cylindrical shell. Then, the derivative ofthe equation is taken to obtain the shear force and thebending moment. In addition, for the cylindrical COPV,the influence of the lateral stiffness of the outer windinglayers on the deformation of the inner liners shouldbe taken into account. Therefore, the lateral stiffness

correction term should be added to the linear term of thegoverning equation (the stiffness term of the elasticfoundation) to reflect this effect, and obtain the gov-erning equation,

D wx

F wx

E tR

Et

wdd

dd

0.4

4 N2

2l l

l2

O

O⎜ ⎟⎛

⎝

⎞

⎠− + + = (5)

Considering the change of the in-plane tension of thepressure vessel before and after the autofrettage processand the contour change of the depression on the surfaceof the cylindrical shell in the autofrettage process, equa-tion (5) is solved. The interface pressure and residualbending moment of the metal liner at the depressioncan be obtained. The calculation flow chart is shown inFigure 2.

2.3 Solution of depression profile, interfacepressure, the and residual bendingmoment

2.3.1 Establishment of the control equation on thedepression profile

In order to facilitate the solution of the governing equa-tion (5), dimensionless processing is performed to obtain

w xx

F w xx

w xdd

2 dd

Φ 0,4

4 l2

2( ) ( )

( )− ⋅ + ⋅ = (6)

where

FF μ

E t6 1

,lN l

2

l l3

( )=

− (7)

Figure 2: Calculation flow chart.


E tR

Et

E tμ

Φ12 1

.1l l

l2

O

o

l l3

l2⎜ ⎟⎜ ⎟

⎛

⎝

⎞

⎠

⎛

⎝ ( )

⎞

⎠

= +

−

(8)

Here, EO,t is the elastic modulus of the windinglayer, tO is the thickness of the winding layer, Rl is theradius of the inner liner, and pin is the autofrettagepressure.

In order to obtain the interface pressure and the resi-dual bending moment of the inner liner depressionduring the loading phase, the ordinary differential equa-tion (6) is transformed: F F μ

E tl16 1N1 l

2

l l3

( )=

−

, where FN1 is the

axial in-plane tension of the liner when the autofrettage

pressure reaches its peak, Φ E tR

Et

E tμ1 12 1

l l

l2

O

o

l l3

l2⎛

⎝⎞⎠( )( )

= +

−

. The

transverse stiffness of the winding layer is the same asthat before compaction and it is 1/10 of the transverserigidity after compaction.

Define the initial depth of the depression δb and theboundary conditions w x δ01 b1( )= = − , w xlim 0x 1( ) =

∣ ∣→∞,

lim 0xw x

x0d

d1( )

=∣ ∣→

, the solution of the equation can be

obtained:

w x δ e φ ω x ω xsin cos ,ω φ x1 b1 1 1 11 1( ) [ ( ∣ ∣) ( )]= − ⋅ ⋅ + ⋅

− ⋅ ∣ ∣ (9)

where the values of each parameter are ω Φ1 14= ⋅

FΦ2 Φ

1 l1

1

− and φ Φ FF1 1

ΦΦ

4 1 l1

1 l1= ⋅

+

−

.

Taking the derivative of the solution 4 times, andsubstituting it into the governing equation (6), and sortingout the governing equation with the initial depressionprofile,

w xx

F w xx

w x

ω φ φ δ δ x

dd

2 dd

Φ

4 1 0.

41

4 l12

12 1 1

13

12

1 b1

( ) ( )( )

( ) ( )

− ⋅ + ⋅

+ ⋅ + ⋅ ⋅ ⋅ =

(10)

The fourth derivative is

w xx

δ e φ ω x

ω x ω φ φ δ δ x

ω φ φ δ e

φ ω x ω x

dd

Φ sin

cos 4 1

2 1 1

sin cos .

ω φ x

ω φ x

41

4 1 b1 1 1

1 13

12

1 b1

14

12

12

b1

1 1 1

1 1

1 1

( )[ ( ∣ ∣)

( )] ( ) ( )

( ) ( )

[ ( ∣ ∣) ( )]

= ⋅ ⋅ ⋅ ⋅ ⋅

+ ⋅ − ⋅ + ⋅ ⋅ ⋅

+ ⋅ + ⋅ − + ⋅ ⋅

⋅ − ⋅ ⋅ + ⋅

− ⋅ ∣ ∣

− ⋅ ∣ ∣

(11)

From the first two terms of equitation (11), the expres-sion of the interface pressure is obtained as

p x C E I ω φ δ e

φ ω x ω xC E I ω φ φ δ x

1

sin cos4 1 ,

l l lω φ x

l l

14

12 2

b1

1 1 1

13

12

1

1 1( ) ( )

[ ( ∣ ∣) ( )]

( ) ( )

= ⋅ ⋅ ⋅ + ⋅ ⋅

⋅ ⋅ ⋅ + ⋅

− ⋅ ⋅ ⋅ + ⋅ ⋅

− ⋅ ∣ ∣

(12)

where C is the coefficient of the residual bending moment,and the bending stiffness E Il l of the liner is

E I E tμ12 1

.l ll l

3

l2( )

=

−

(13)

The expressionp xp( ) for the local pressure distribu-tion at the center of the depression is [31]

p xF φ ω

φe φ ω x

ω x F δ x

14

sin

cos ,

ω φ xp

pc 12

1

11 1

1 pc

1 1( )( )

[ ( ∣ ∣)

( )] ( )

=

⋅ + ⋅

⋅ ⋅ ⋅ ⋅

+ ⋅ − ⋅

− ⋅ ∣ ∣

(14)

where

F x C E I ω φ φ4 1 .pc l l 13

12

1( ) ( )= − ⋅ ⋅ ⋅ + ⋅ (15)

As equation (14) involves the function δ, it is not easyto understand their effects intuitively. Therefore, the lineload F δ xpc ( )⋅ in the region is processed with interpola-tion, which does not affect the overall analysis of thechange in the depression profile [31]. Assuming that

p x p A xx

B xx

1 ,pc maxc

4

c

2⎜ ⎟ ⎜ ⎟( )

⎡

⎣⎢

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎤

⎦⎥

= − ⋅ ⋅ − ⋅ + (16)

where x x xc c− < < . To ensure the continuity of the solu-tion, the boundary condition is defined as the value andthe sloppiness of the pressure distribution at the transi-tion line, x xc= , are 0, and then the two undeterminedcoefficients of the above equation can be determined.

p x p xx

xx

2 1 .pc maxc

4

c

2⎜ ⎟ ⎜ ⎟( )

⎡

⎣⎢

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎤

⎦⎥

= − ⋅ − + (17)

In order to show the equivalent effect, the integral ofthe interface pressure in the dispersion region is equalbefore and after the numerical processing [31], and it canbe determined as

pFx

1516

.maxpc

c= (18)

The dispersed local interface pressure can be expressedas

p x

xx

xx

Fx

x x x

x x or x x

2 11516

,

0, ,

pc

c

4

c

2pc

cc c

c c

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

( )

⎧

⎨

⎪

⎩⎪

⎡

⎣⎢

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎤

⎦⎥

⎛

⎝

⎞

⎠=

− + ⋅ − − < <

< − >

(19)The analytical solution of the interfacial pressure of

the whole field is


The distribution of the bending moments is

where

Θ F x x xx

x xx

x xx

x xx

132

532

1532

12

.

pc cc6 6

c6

c4 4

c4

c2 2

c2

c

c

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎡

⎣⎢

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

∣ ∣⎞

⎠

⎤

⎦⎥

= ⋅ ⋅

−

−

−

+

−

−

−

(22)

Substituting x = 0, the residual bending moment dis-tribution at the center of the depression can be obtainedas

MFω φ

F x04

532

.ppc

1 1pc c( )= − ⋅ (23)

If M M0p pf,max( ) = , where Mpf,max is the maximumplastic moment, we can obtain

xω φ

MF

85

325

.c1 1

pf, max

pc= − (24)

2.3.2 Deformation analysis of depression in theunloading stage

Starting from the governing equation (6) for the depressionprofile, the displacement solution of the initial depressionprofile is defined as

w x δ e φ ω x

ω x

sin

cos .b

ω φ x1 ,0 1 1

1

1 1( ) [ ( ∣ ∣)

( )]

= − ⋅ ⋅ ⋅

+ ⋅

− ⋅ ∣ ∣

(25)

Define the lateral displacement in the unloading stagewf as the sum of the displacement increment (caused bythe axial load FN applied at each phase) and the initialdisplacement of the liner depression, then

w x w x w x .f 1( ) ( ) ( )= − (26)

In the unloading stage, the lateral displacement of thelining differential equation shall meet the followingequitation:

w xx

F w xx

w xd

d2 d

dΦ 0.f

l f

4

4

2

2 2( ) ( )

( )− ⋅ + ⋅ =(27)

In the second term on the left of the equation, theapplied load depends on the total displacement and so wis the total displacement.

The general solution of the inhomogeneous equation(28) is

w x e A ω x B ω xsin cos ,fω φ x

1 11 1( ) [ ( ) ( )]= ⋅ ⋅ ⋅ + ⋅ ⋅

− ⋅ (28)

where A and B are undetermined coefficients, which canbe obtained by using the method of undefined coefficient:

A H IF G

B J KF G

, ,=

+

+

=

+

+

(29)

where

F ω φ ω φ ω ω

F ω φ F ω

G ω φ ω ω φ ω

F ω φ ω

H ω φ ω φ ω ω

F ω φ F ω φ

I ω φ ω ω φ ω

F ω φ ω

J ω φ ω ω φ ω

F ω φ ω φ

K ω φ ω φ ω ω

F ω φ F ω

6

2 2 Φ

4 4

4

6

2 2 Φ

4 4

4

4 4

4

6

2 2 Φ ,

l

1 14

1 12

12

14

1 12

l 12

22

1 13

1 1 1 13

l 1 1 12

1 14

1 12

12

14

l 1 12

l 12

2 1

1 13

1 1 1 13

l 1 1 1

1 13

1 1 1 13

l 1 1 1 1

1 14

1 12

12

14

l 1 12

l 12

2

[( ) ( )

( ) ]

[ ( ) ( )

( ) ]

[( ) ( )

( ) ]

( ) ( )

( )

[ ( ) ( )

( ) ]

( ) ( )

( )

= ⋅ − ⋅ ⋅ +

− ⋅ ⋅ + ⋅ +

= − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅

+ ⋅ ⋅ ⋅

= − ⋅ − ⋅ ⋅ +

− ⋅ ⋅ + ⋅ + ⋅

= − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅

+ ⋅ ⋅ ⋅

= ⋅ ⋅ − ⋅ ⋅

− ⋅ ⋅ ⋅

= ⋅ − ⋅ ⋅ +

− ⋅ ⋅ + ⋅ +

(30)

p x

F φ ωφ

e φ ω x ω x p x x x

F φ ωφ

e φ ω x ω x x x x x

14

sin cos ,

14

sin cos , or .

ω φ x

ω φ xl

pc 12

1

11 1 1 pc c c

pc 12

1

11 1 1 c c

1 1

1 1

( )

⎧

⎨

⎪⎪

⎩

⎪⎪

( )[ ( ∣ ∣) ( )]

( )[ ( ∣ ∣) ( )]

=

⋅ + ⋅

⋅ ⋅ ⋅ ⋅ + ⋅ + − < <

⋅ + ⋅

⋅ ⋅ ⋅ ⋅ + ⋅ > < −

− ⋅ ∣ ∣

− ⋅ ∣ ∣

(20)

M x

Fω φ

ω x φ ω x e Θ x x x

Fω φ

ω x φ ω x e x x x x

4cos sin ,

4cos sin , or ,

ω φ x

ω φ xp

pc

1 11 1 1 c c

pc

1 11 1 1 c c

1 1

1 1

( )

⎧

⎨

⎪

⎩⎪

[ ( ) ( ∣ ∣)]

[ ( ) ( ∣ ∣)]

=

⋅ ⋅ − ⋅ ⋅ ⋅ + − < <

⋅ ⋅ − ⋅ ⋅ ⋅ > < −

− ⋅ ∣ ∣

− ⋅ ∣ ∣

(21)


in which ω Φ F1 1

Φ2 Φ

4 1 l1

1= ⋅

− , φ Φ FF1 1

ΦΦ

4 1 l1

1 l1= ⋅

+

−

,

Φ E tR

Et

E tμ1 12 1

l l

l2

O

o

l l3

l2⎛

⎝⎞⎠( )( )

= +

−

, Φ E tR

Et

E tμ2 12 1

l l

l2

O1

o

l l3

l2⎛

⎝⎞⎠( )( )

= +

−

.

Within the interval x > 0, the general solution ofthe inhomogeneous equation is the general solution of thehomogeneous equation plus a special solution of the inho-mogeneous equation:

w x P e ω x Q e

ω x R e ω xS e ω xe A ω x B ω x

sin

cos sincos

sin cos .

fω φ x ω φ x

ω φ x

ω φ x

ω φ x

2

2 2

2

1 1

2 2 2 2

2 2

2 2

1 1

( ) ( )

( ) ( )

( )

[ ( ) ( )]

= ⋅ ⋅ ⋅ + ⋅

⋅ ⋅ + ⋅ ⋅ ⋅

+ ⋅ ⋅ ⋅

+ ⋅ ⋅ + ⋅ ⋅

− ⋅ − ⋅

− ⋅

− ⋅

− ⋅

(31)

Substituting the boundary conditions w xlim 0x f ( ) =→∞

,we obtain

R S 0,= = (32)

w x w x w x ,f 1( ) ( ) ( )= − (33)

where ω Φ F1 1

Φ2 Φ

4 1 l1

1= ⋅

− , φ Φ FF1 1

ΦΦ

4 1 l1

1 l1= ⋅

+

−

, Φ1 =

E tR

Et

E tμ12 1

l l

l2

O

o

l l3

l2⎛

⎝⎞⎠( )( )

+

−

, ω Φ F2 2

Φ2 Φ

4 2 l2

2= ⋅

− , φ Φ2 24= ⋅

FF

ΦΦ

2 l2

2 l2

+

−

, Φ E tR

Et

E tμ2 12 1

O

o

l l

l2

1 l l3

l2⎛

⎝⎞⎠( )( )

= +

−

.

Expanding the solution to x−∞ < < +∞, we obtain

w x P e ω x Q e

ω x e A ω xB ω x

sin

cos sincos ,

fωφ x ωφ x

ω φ x1

1

1 1

( ) ( ∣ ∣)

( ) [ ( ∣ ∣)

( )]

= ⋅ ⋅ ⋅ + ⋅

⋅ ⋅ + ⋅ ⋅

+ ⋅ ⋅

− ⋅ ∣ ∣ − ⋅ ∣ ∣

− ⋅ ∣ ∣ (36)

w xx

P e ω x Q e

ω x e A ω xB ω x

dd

sin

cos sincos .

f ωφ x ωφ x

ω φ x

2

2

1

1

1 1

( )( ∣ ∣)

( ) [ ( ∣ ∣)

( )]

= ″⋅ ⋅ ⋅ + ″⋅

⋅ ⋅ + ″⋅ ⋅

+ ″⋅ ⋅

− ⋅ ∣ ∣ − ⋅ ∣ ∣

− ⋅ ∣ ∣

(37)

When calculating the displacement of the liner loadedby the autofrettage pressure, EO1 = EO, the value of Fl2 isthe maximum value of in-plane tension, the displacementwf obtained by formula (36) is the total radial displacementof the liner during the loading process, which is wf 1. Whencalculating the displacement of the liner unloaded bythe autofrettage pressure, EO1 = 10EO, the value of Fl2 isthe maximum value of the in-plane pressure, and the

displacementwf obtained by formula (36) is the total radialdisplacement of the liner during the unloading process,which is wf 2.

The ultimate form of moment distribution, pressuredistribution, and displacement near the depression afterunloading is obtained [31] as

M x C E Iwx

wx

dd

dd

,f fun l l

22

2

21

2⎜ ⎟( ) ⎛

⎝

⎞

⎠= ⋅ ⋅ − (38)

p x C E I E μ w w t1 .f fun l l O l2

2 1 O( ) ( ) ( )= ⋅ ⋅ ⋅ − ⋅ − / (39)

The specific calculation process is shown in Figure 3.In detail, an FE model is established for the COPV with ametal liner, the autofrettage pressure is applied, the in-plane tension when the autofrettage pressure reaches thepeak is calculated, and the internal tension and interfacepressure after the autofrettage pressure are completelyunloaded. Establishing the governing equation of the

depression profile and solving the ordinary differentialequation by substituting the above coefficients, we obtainthe distribution of the interface pressure and residualbending moment near the depression after the autofret-tage pressure is completely unloaded. Combined with thebuckling criterion, one can judge whether the depressionwill cause the buckling of the liner.

3 Model validation

To validate the analysis methods applied in this article,the analytical results are compared with the results ofnumerical examples in the literature [31]. The resultsare shown in Table 1, and the initial depth of the depres-sion is set to 0.51 and 2.04mm, respectively, to calculatethe interface pressure distribution. The results are shownin Figure 3.

The calculation results in Table 1 and Figure 4 showthat the deviation between the results of the critical depthof the depression in the literature and that in this articleis 7.9%, and the minimum value of the interface pressurenear the depression is in good agreement.

Pω Bω φ φ Bω φ Aω φ Aω φ Bω φ Bω φ Aω Aω

ω φ3 3 3 3

2 1,1 2

222

1 12

13

22

22

12

12

22

1 12

1 22

12

23

22

( )

( )=

⋅ − − + − + + −

− +

(34)

Qω Bω φ φ Bω φ Aω φ Aω φ Bω φ Bω φ Aω Aω

ω φ φ3 3 3 3

2 1,1 2

222

1 12

13

22

22

12

12

22

1 12

1 22

12

23

2 22

( )

( )=

⋅ − − + − + + −

− +

(35)


4 Example discussion

The critical depth of the depression is defined as theminimum value of the depression depth, which is relatedto the autofrettage pressure, material parameters, anddimension parameters. In this article, for the cylindrical

COPV with an initial depression, the calculation modelfor buckling analysis is established. The influence of fac-tors such as the autofrettage pressure, the radius of theliner, the thickness of the liner, the thickness-to-diameterratio of the liner, and the length of the straight section onthe critical depth of the depression is studied.

Figure 3: Calculation process.


4.1 Calculation results of the interfacepressure and residual bending moment

The FEM of the cylindrical COPV is established, and thecross-sectional view of the pressure vessel is shown inFigure 5. The liner of the pressure vessel is of aluminumalloy, Young’s modulus is 69 GPa, the winding layer ofthe composite material is T700/epoxy, the length of thestraight section is 270mm, the radius of the inner liner is170mm, the thickness of the inner liner is 0.8 mm, andthe thickness of the winding layer is 10.8 mm. The auto-frettage pressure is 45 MPa. The stiffness properties ofT700/epoxy are shown in Table 2.

From equation (2), the critical in-plane pressure ofthe liner is calculated to be 149.52 N/m. The in-plane pres-sure in the middle of the straight section is 192.88 N/mafter autofrettage, which is larger than the critical in-planepressure of the inner liners. Therefore, the metal liner of theexample meets the first condition of the buckling criterion.

The analytical model of the metal liner of the cylind-rical pressure vessel with depression was adopted forcalculations. The depths of the depression are set to be

0.2, 0.4, and 0.6 mm, respectively. The change of inter-face pressure and residual bending moment near thedepression are shown in Figure 6.

It can be seen from Figure 6 that the fluctuation rangeof the residual bending moment corresponding to thethree depression depths is about 40mm (the depressionis about 20 mm on the left and right sides of the lineraxis). From equation (3), the axial buckling half-wave-length of the liner is 20.1mm, and the fluctuation range ofthe residual bending moment is larger than the axial buck-ling half-wavelength of the liner, which meets the secondcondition of the buckling criterion; When the initial depthof the depression is 0.2mm, the interface pressure of thedepression is greater than 0, which does not meet the thirdcondition of the buckling criterion, so the initial depressionwith a depth of 0.2mm will not cause local buckling of theliner. When the initial depth of the depression is 0.4 or0.6mm, the interface pressure has a distribution less than0, which meets the third condition of the buckling criterionand the inner liner has the local buckling.

Table 1: Comparison between the calculation results of the literature [31] and results of verification examples in this article

Item Literature [31] results Verification examples in this article Deviation (%)

Material property and dimension parameter Winding material: Kevlar 49/epoxy; —Lining material: titanium alloy;Lining thickness: 2.7 mmThickness of the winding layer: 18.8mmContainer radius: 480mm

In-plane tension with full loading 2206.2 N/m 2196.5 N/m 0.4In-plane tension with unloading completely 1491.5 N/m 1250.1 N/m 16.2Critical depth of depression 1.65 mm 1.78mm 7.9

Figure 4: Comparison of interface pressure distribution calculations.

Figure 5: Section view of the pressure vessel.


4.2 Parameter discussion

4.2.1 Relationship between the critical depth ofdepression and thickness of the inner liner

Keeping the radius of the liner at 170mm, the thicknessesof the liner are 0.7, 0.8, and 0.9 mm. The relationshipbetween the critical depth of depression and autofrettagepressure is shown in Figure 7. The relationship betweenthe ratio of the critical depth of the depression to thethickness of the liner (hereinafter referred to as “the

depression-to-thickness ratio”) and the autofrettage pres-sure are shown in Figure 8.

From Figures 7 and 8, it can be seen that as theautofrettage pressure increases, the critical depth andthe depression-to-thickness ratio decrease, and the cri-tical depth level and the depression-to-thickness ratiodecrease with the decrease of the liner’s thickness. Thecritical depth corresponding to the thickness of the innerliner of 0.9 mm is significantly greater than that

Table 2: Stiffness properties of T700/epoxy

E1 (GPa E2 (GPa) E3 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) μ12 μ13 μ23

150 12 12 7 5 5 0.3 0.3 0.3

Figure 6: Distribution of (a) interface pressure and (b) residualbending moment near depressions after autofrettage.

Figure 7: Relationship between the critical depth of depression andautofrettage pressure considering the variation of liner thickness.

Figure 8: Relationship between the depression-thickness ratio andautofrettage pressure considering the variation of liner thickness.


corresponding to the thickness inner liner of 0.8 and0.7mm. As the autofrettage pressure increases, local buck-ling will occur in the metal liner regardless of the thick-ness. The thinner the inner liner, the smaller the criticaldepth of the depression, which means that the inner liner

is more prone to local buckling and is more sensitive to thesize of initial depression after autofrettage.

4.2.2 Relationship between the critical depth ofdepression and radius of the inner liner

Keeping the thickness of the liner at 0.8 mm, the radii ofthe liner are 165, 170, and 175 mm, respectively. The rela-tionship between the critical depth of the depression andthe autofrettage pressure is shown in Figure 9 and thatbetween the depression-to-thickness ratio and the auto-frettage pressure is shown in Figure 10.

It can be seen from Figures 9 and 10 that, regardlessof the diameter, increasing the autofrettage pressure canlead to local buckling of the metal lining. As the radiusof the liner increases, both the critical depth and the

Figure 9: Relationship between the critical depth of depression andautofrettage pressure considering the variation of liner radius.

Figure 10: Relationship between the depression-thickness ratio andautofrettage pressure considering the variation of liner radius.

Table 3: Table of factor level

Level Factor

Autofrettage pressure (MPa) Liner radius (mm) Liner thickness (mm) Length of the straight section (mm)

1 40 160 0.6 1502 45 170 0.7 2003 50 180 0.8 2504 55 190 0.9 300

Table 4: Parameter table for orthogonal case

Casenumber

Level(autofrettagepressure)(MPa)

Level(linerradius)(mm)

Level (linerthickness)(mm)

Level(length ofthestraightsection)(mm)

1 1(40.0) 1(160.0) 1(0.6) 1(150.0)2 1(40.0) 2(170.0) 2(0.7) 2(200.0)3 1(40.0) 3(180.0) 3(0.8) 3(250.0)4 1(40.0) 4(190.0) 4(0.9) 4(300.0)5 2(45.0) 1(160.0) 2(0.7) 3(250.0)6 2(45.0) 2(170.0) 1(0.6) 4(300.0)7 2(45.0) 3(180.0) 4(0.9) 1(150.0)8 2(45.0) 4(190.0) 3(0.8) 2(200.0)9 3(50.0) 1(160.0) 3(0.8) 4(300.0)10 3(50.0) 2(170.0) 4(0.9) 3(250.0)11 3(50.0) 3(180.0) 1(0.6) 2(200.0)12 3(50.0) 4(190.0) 2(0.7) 1(150.0)13 4(55.0) 1(160.0) 4(0.9) 2(200.0)14 4(55.0) 2(170.0) 3(0.8) 1(150.0)15 4(55.0) 3(180.0) 2(0.7) 4(300.0)16 4(55.0) 4(190.0) 1(0.6) 3(250.0)


depression-to-thickness ratio decrease under the sameautofrettage pressure, indicating that the larger the innerliner radius, the easier the local buckling of the liner tooccur after autofrettage, and is more sensitive to the sizeof the initial depression.

4.3 Parametric sensitivity analysis

In order to quantitatively study the influence of the auto-frettage pressure, vessel surface area, liner thickness,and the local buckling of cylindrical COPV with initialdepression, and the sensitivity analysis of the abovefour factors, it is necessary to conduct the sensitivityanalysis of the four factors above.

4.3.1 Orthogonal case design

Orthogonal experimental design is an efficient designmethod to study multiple factors and multiple levels.Based on orthogonality, some representative points fromthe overall points are selected for testing. These represen-tative points have the characteristics of “evenly dispersed,neat, and comparable,”which not only reduces the numberof necessary sample points but also comprehensively ana-lyzes the sensitivity of each factor [44–46]. Therefore, theorthogonal test is adopted in this article to analyze theorthogonal calculation examples and sort out the most sen-sitive factors that affect the buckling of the pressure vessel[47]. The design factor levels are shown in Table 3.

According to the feature of the orthogonal test table[40], the parameters for the orthogonal case are shown inTable 4.

4.3.2 Parametric sensitivity analysis

The critical depth of the depression is analyzed accordingto the numerical example shown in Table 4, and the var-iance analysis is performed for the critical depth of thedepression to judge the influence of the autofrettage pres-sure, the radius of vessels, thickness of liners, and thestraight length on the critical depth of the depression.The results are shown in Table 5.

The variance of each factor is compared. It is foundthat the influence of four factors on the critical depth ofthe depressions, from large to small, is as follows: thethickness of the liner, the length of the straight section,the radius of liners, and the autofrettage pressure.

The variance analysis of the depression-to-thick-ness ratio is carried out to determine the influence ofthe autofrettage pressure, the radius of the vessel, thethickness of liners, and the length of the straight sectionon the depression-to-thickness ratio. The results are inTable 6.

The variance of each factor is compared by the ana-lysis of variance. It is found that the influence of the fourfactors on the depression-thickness ratio, from large tosmall, is as follows: the length of the straight section, theradius of the liners, the thickness of the liners, and theautofrettage pressure.

Table 5: Analysis of variance

Influencing factor Autofrettage pressure Liner radius Liner thickness Length of the straight section

Sum of squared deviations of a factor 0.002877 0.017081 0.0833 0.02141Sum of squared deviation errors 0.000177Freedom of each factor 3 3 3 3Factor-based statistics 16.22 96.27 469.51 120.67

Table 6: Analysis of variance table

Influencing factor Autofrettage pressure Liner radius Liner thickness Length of the straight section

Sum of squared deviations of a factor 0.004854 0.032071 0.024679 0.035591Sum of squared deviation errors 0.00029Freedom of each factor 3 3 3 3Factor-based statistics 16.72 110.51 85.04 122.64


5 Conclusions

(1) The numerical and analytical method proposed inthis article is an effective strategy to analyze the metalliner buckling behavior of the cylindrical COPV withdepression. By comparison with the results in ref. [31],the proposed method can predict the effect of theinitial depression size on the local buckling of themetal liner accurately.

(2) For the constant initial depression depth, the innerliner of the pressure vessel is more likely to bucklelocally near the depression after autofrettage whenthe autofrettage pressure is greater. Therefore, a sui-table autofrettage pressure should be set to ensurethe COPV have a satisfying autofrettage effect andwill not enter local buckling.

(3) In the local buckling of the metal liner in the cylind-rical composite pressure vessel, the thickness of themetal liner is the most sensitive factor, followed bythe length of the straight section and the radius of theliner, while the autofrettage pressure is the least sen-sitive factor.

The method established in this article can providereferences for the engineering design and autofrettageprocessing of COPV.

Conflict of interest: Authors state no conflict of interest.

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