design of fiber-reinforced composite pressure vessels under

Design of fiber-reinforced composite pressure vessels undervarious loading conditions

Levend Parnas a,*, Nuran Katırcı b

a Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkeyb ASELSAN Inc., P.O. Box 30, 06011 Etlik, Ankara, Turkey

Abstract

An analytical procedure is developed to design and predict the behavior of fiber reinforced composite pressure vessels. The

classical lamination theory and generalized plane strain model is used in the formulation of the elasticity problem. Internal pressure,

axial force and body force due to rotation in addition to temperature and moisture variation throughout the body are considered.

Some 3D failure theories are applied to obtain the optimum values for the winding angle, burst pressure, maximum axial force and

the maximum angular speed of the pressure vessel. These parameters are also investigated considering hygrothermal effects.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Composite pressure vessels; Filament winding; Generalized plane strain problem; Hygrothermal effects; Burst pressure; Angular speed

1. Introduction

The use of fiber reinforced and polymer-basedcomposites have been increasing. Various numbers ofapplications have also been flourishing with this deve-lopment. Fuel tanks, rocket motor cases, pipes are someexamples of pressure vessels made of composite mate-rials. Ever increasing use of this new class of materials inconventional applications is coupled with problems thatare intrinsic to the material itself. Difficulties are manyfolded. Determination of material properties, mechani-cal analysis and design, failure of the structure are someexamples which all require a non-conventional ap-proach.

Numerous applications concurrently are accompa-nied by various researches in the related field. Majorityof the studies in the analysis of composite pressurevessels finds their origins in Lethnitskii’s approach [1].The application of the theory given in this book is laterapplied to laminated composite structures in tubularform Tsai [2]. The studies followed consider also dif-ferent loading and environmental conditions. Recently,there are some studies involved directly with tubes underinternal pressure [3,4]. In the study by Xia et al. [4], the

combined effect of thermomechanical loading in addi-tion to internal pressure is considered.

In this study, an analytical procedure is developedto design and predict the behavior of fiber-reinforcedcomposite pressure vessels under combined mechanicaland hygrothermal loading. The mechanical part ofthe analysis is similar to the study given in Ref. [5]. Theprocedure is based on the classical laminated plate the-ory. A cylindrical shell having a number of sub-layers,each of which is cylindrically orthotropic, is treated as inthe state of plane strain. Internal pressure, axial force,body force due to rotation in addition to temperatureand moisture variation throughout the body are con-sidered as loading. In the study of Katırcı [6], theseparameters are compared with the experimental results.

2. Formulation of problem

A thick-walled multi-layered filament wound cylin-drical shell is considered in the analysis based on linearelasticity solution. The following assumptions are madefor the formulation of the problem.

• The pressure vessel is cylindrically orthotropic,• the pressure vessel has adjacent �a angle lay-ups and

the adjacent �a lay-ups act as a homogeneous andorthotropic unit,

* Corresponding author.

E-mail addresses: [email protected] (L. Parnas), katirci@

mgeo.aselsan.com.tr (N. Katırcı).

0263-8223/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0263-8223 (02 )00037-5

Composite Structures 58 (2002) 83–95

www.elsevier.com/locate/compstruct

mail to: [email protected]

• the vessel is in state of plane strain and only smallstrains are considered through the analysis,

• the length of the vessel is such that the longitudinalbending deformation due to the end closures of thevessel are limited to only small end portions of thepressure vessel.

2.1. Effective elastic properties

A laminated composite with its own effective elasticproperties, contains a number of anisotropic plates.When these effective elastic properties of the laminateare used, the body is considered to responding to theapplied loads as a single unit. The effective elasticproperties of the laminate can be determined using thetheory of the laminated plates.

The filament wound structures, which is the subject ofthis study, is assumed to be made of angle-ply laminates.An angle-ply laminate has alternating lamina having þaand �a winding angles. Therefore, a filament-woundcylindrical shell, having a wind angle �a can be treatedas an angle-ply laminate. For multi-layered cylinders,each layer is an angle-ply laminate with its own windangle. Neglecting the effect of curvature, the effectiveelastic properties of each of these layers can be formu-lated as follows.

For an angle-ply lamina where fibers are oriented atan angle a with the positive x-axis as shown in Fig. 1, theeffective elastic properties are given by

1

Ex0x0¼ cos4 a

E11

þ sin4 aE22

þ 1

G12

�� 2

m12

E11

�cos2 a sin2 a

1

Ey0y0¼ sin4 a

E11

þ cos4 aE22

þ 1

G12

�� 2

m12

E11

�cos2 a sin2 a

1

Gx0y0¼ 1

E11

þ 2m12

E11

þ 1

E22

� 1

E11

�þ 2

m12

E11

þ 1

E22

� 1

G12

�cos2 2a

mx0y0 ¼ Ex0x0m12

E11

�� 1

E11

�þ 2

m12

E11

þ 1

E22

� 1

G12

�cos2 a sin2 a

�

my0x0 ¼Ey0y0

Ex0x0mx0y0 ð1Þ

In this study, the elastic constants related to thethickness coordinate however, are assumed as Ez0z0 ¼E33 ¼ E22 and Gy0z0 ¼ Gx0z0 ¼ Gx0y0 .

In a filament-wound pressure vessel, the structure ismade-up of several angle lay-ups, each of which acts asan orthotropic unit. The elastic constants of each layer isassumed as equal to effective elastic constants of a bal-anced and symmetric laminate which has two layers ofwinding angles (þa) and (�a) with equal thicknesses.

The generalized Hooke’s law in cylindrical coordi-nates can be written as

fegr;h;z ¼ ½a�frgr;h;z ð2Þ

For an angle-ply lamina, due to the (�a) configura-tion, the shear coupling terms are zero. Then the com-pliance matrix [a] can be represented in cylindricalcoordinates as

½a� ¼

1

Err� mrh

Err� mrz

Err0 0 0

� mrh

Err

1

Ehh� mhz

Ehh0 0 0

� mrz

Err� mhz

Ehh

1

Ezz0 0 0

0 0 01

Ghz0 0

0 0 0 01

Grz0

0 0 0 0 01

Grh

2666666666666666666664

3777777777777777777775

ð3Þ

The material properties in cylindrical coordinates canbe obtained by simply replacing cartesian coordinates,x, y and z, with r, h and z, respectively (Fig. 1).

2.2. Plane stress problem for a body in cylindricalanisotropy

Lethnitskii [1] started the formulation with the planestress condition then the problem is converted to thegeneralized plane strain problem where axial strain ofthe system is equal to a constant rather than being zero.The equilibrium equations, disregarding rzz, and theequations of generalized Hooke’s Law for a body incylindrical anisotropy in cylindrical coordinates aregiven by the following equations.

orrr

orþ 1

rorrh

ohþ rrr � rhh

rþ R ¼ 0 ð4Þ

orrh

orþ 1

rorhh

ohþ 2

rrh

rþ H ¼ 0 ð5Þ

err ¼ a11rrr þ a12rhh þ a16rrh

ehh ¼ a12rrr þ a22rhh þ a26rrh

ezz ¼ a13rrr þ a23rhh þ a36rrh

crh ¼ a16rrr þ a22rhh þ a66rrh

ð6Þ

Fig. 1. Global, local and material coordinates.

84 L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95

where R and H are the projections of the body forcesalong r and h directions, respectively. The axial stress,rzz, of the generalized plane strain problem will be ob-tained by using generalized Hooke’s Law. The strain–displacement relations for the same body are:

err ¼our

or

ehh ¼1

rouh

ohþ ur

r

crh ¼1

rour

ohþ ouh

or� uh

r

ð7Þ

By eliminating displacements from Eq. (7), theequation of compatibility is obtained, which is:

o2erroh2

þ ro2 rehhð Þor2

� o2 rcrhð Þoroh

� roerror

¼ 0 ð8Þ

The equilibrium equations given in Eq. (5) are satis-fied with the following definition of the stress function,F ðr; hÞ:

rrr ¼1

roFor

þ 1

r2

o2F

oh2þ U

rhh ¼o2For2

þ U

rrh ¼ � o2

orohFr

� � ð9Þ

where U is the body force potential.On the basis of equations of compatibility, stress–

strain relations and equilibrium equations given above,the following differential equation for plane stress casewhich is satisfied by the stress function F ðr; hÞ, is ob-tained

a22

o4For4

� 2a26

o4For3 oh

þ 2a12ð þ a66Þ1

r2

o4F

or2 oh2

� 2a16

1

r3

o4F

oroh3þ a11

1

r4

o4F

oh4þ 2a22

1

ro3For3

� ð2a13 þ a66Þ1

r3

o3F

oroh2þ 2a16

1

r4

o3F

oh3

� a11

1

r2

o2For2

þ 2 a16ð þ a26Þ1

r3

o2Foroh

þ 2 a11ð þ 2a12 þ a66Þ1

r4

o2F

oh2þ a11

1

r3

oFor

þ 2 a16ð þ a26Þ1

r4

oFoh

¼ � a12ð þ a22Þo2Uor2

þ a16ð þ a26Þ1

ro2Uoroh

� a11ð � a12Þ1

r2

o2U

oh2þ a11ð � 2a22 � a12Þ

1

roUor

þ a16ð þ a26Þ1

r2

oUoh

ð10Þ

where a16 and a26 vanish for a body having �a angle-plylayers. After introducing the material properties for the

compliances and substituting them into Eq. (10), thefollowing non-homogeneous, fourth order differentialequation is obtained for an orthotropic cylindrical bodyfor the state of plane stress.

1

Ehh

o4For4

þ 1

Grh

�� 2mrh

Err

�1

r4

o4F

or2 oh2þ 1

Err

1

r4

o4F

oh4

þ 2

Ehh

1

ro3For3

� 1

Grh

�� 2

mrh

Err

�1

r3

o2Foroh

� 1

Err

1

r2

o2For2

þ 21 � mr

Err

�þ 1

Grh

�1

r4

o2F

oh2þ 1

Err

1

r3

oFor

¼ � 1 � mhr

Ehh

o2Uor2

�þ 1 � mrh

Err

1

r2

o2U

oh2

�

� 2

Ehh

�� 1 þ mrh

Err

�1

roUor

: ð11Þ

2.3. Stresses and displacements for a rotating anisotropiccylinder

At this point, it is easy to obtain the stress distribu-tion for an anisotropic rotating cylinder. It is assumedthat the cylinder is orthotropic, so that any radial planeis an elastic symmetry plane. For a rotating cylinder, thebody force potential is given by:

U ¼ � qx2

2r2 ð12Þ

where x is the angular speed, q is the density of thematerial and r is the radial position.

Since the problem is axisymmetric, the stress functionF depends only on r. Using this fact and Eq. (12), Eq.(11) can be rearranged for the kth layer as follows:

1

Ekhh

r4 d4Fdr4

þ 2

Ekhh

r3 d3Fdr3

� 1

Ekrr

r2 d2Fdr2

þ 1

Ekrr

rdFdr

¼ 3

Ekhh

�� 2mr þ 1

Ekrr

�qx2r4 ð13Þ

Eq. (13) is in the form of Euler’s equation and itssolution yields the following expression for the stressfunction F,

F ðrÞ ¼ Ak þ Bkr2 þ Ckr1þg1k þ Dkr1�g1k

þ qkx2

23�

� 2mkrh � �ee2

k

� Ekrr

36Ekrr � 4Ek

hh

r4 ð14Þ

where �eek ¼ ðEkhh=E

krrÞ

1=2.

Using the stress function F ðrÞ and the body forcepotential U in Eq. (9), the stresses can be obtained as:

L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95 85

The structure is mechanically subjected to a radialbody force due to rotation, internal pressure and axialforce as shown in Fig. 2. Boundary conditions for thegiven geometry and loading can be represented as fol-lows,

For r ¼ rint ) r1rr ¼ �Pint and r1

rh ¼ 0

For r ¼ rext ) rnrr ¼ 0 and rn

rh ¼ 0

At the interface of adjacent layers, the followingboundary conditions are applied,

r ¼ bk�1 ) rk�1rr ¼ rk

rr and uk�1r ¼ uk

r

When the boundary conditions are applied, two un-known coefficients of the stress function, Ak and Bk areobtained as zero, and other coefficients as

Ck ¼ 1

1 þ gk

bkb�gkk�1

cgkk � c�gk

k

qk

(

�bgk

k b1�2gkk�1 þ b1�gk

k�1 cgkk � c�gk

kð Þ� �

cgkk � c�gk

k

qk�1

þ qkx2 3 þ mk

hr

9 � g2k

bgkk b3�2gk

k�1 � b3kb

�gkk�1

cgkk � c�gk

k

"þ b3�gk

k�1

#)

ð16Þ

Dk ¼ 1

1 � gk

qk�1bgk�1k � qkb

gk�1k�1

� �bkbk�1

cgkk � c�gk

k

"

� qkx2 3 þ mk

hr

9 � g2k

b2k�1b

gk�1k

�� b2

kbgk�1k�1

�bkbk�1

#ð17Þ

where ck ¼ bk�1=bk, gk ¼ ðnk11=n

k22Þ

1=2and k denotes the

layer number. The reduced strain coefficients, nkij, extend

the plane stress problem into the generalized plane strainproblem as proposed by Lekhnitskii [7]. They can bedefined for a multi-layered cylinder as,

nkij ¼ ak

ij �ak

i3akj3

ak33

i; j ¼ 1; 2

where akij are the components of the compliance tensor in

cylindrical coordinates for the kth layer. Using the stressfunction F ðrÞ, layer stresses can be derived as:

rkrr ¼ qkx

2b2k

3 þ mkhr

9 � g2k

� �1 � cgkþ3

k

1 � c2gkk

!rbk

� �gk�1"

þ 1 � cgk�3k

1 � c2gkk

!cgkþ3k

bk

r

� �gkþ1

� rbk

� �2#

þ qk�1cgkþ1k

1 � c2gkk

!rbk

� �gk�1"

� bk

r

� �gkþ1#

þ qk

1 � c2gkk

!c2gkk

bk

r

� �gkþ1"

� rbk

� �gk�1#

ð18Þ

rkhh ¼

qkx2b2

k

9 � g2k

� �3�(

þ mkhr

�gk

1 � cgkþ3k

1 � c2gkk

!rbk

� �gk�1"

� 1 � cgk�3k

1 � c2gkk

!cgkþ3k

bk

r

� �gkþ1#

� g2k

�þ 3mk

hr

� rbk

� �2)

þ qk�1cgk�1k gk

1 � c2gkk

!rbk

� �gk�1"

� bk

r

� �gkþ1#

� qkgk

1 � c2gkk

!rbk

� �gk�1"

þ c2gkk

bk

r

� �gkþ1#

ð19Þ

In Eqs. (18) and (19), symbols qk�1 and qk denote theinternal and external forces in radial direction acting onthe kth layer as given in Fig. 3, and mk

hr ¼ �nk12=n

k22. Since

the pressure vessel is assumed to be in the state of gen-eralized plane strain, axial strains of all layers is equal tothe constant, e0

zz. Then the axial stress can be obtained as:

rkzz ¼

e0zz

ak33

� 1

ak33

ak13r

krr

�þ ak

23rkhh

�ð20Þ

The displacements are obtained as follows:

ukr ¼ r nk

12rkrr

�þ nk

22rkhh � mk

zhe0zz

�;

ukh ¼ 0 and uk

z ¼ ze0zz ð21Þ

where mkzh ¼ �ak

23=ak33.Fig. 2. Mechanical loading on a closed end cylindrical pressure vessel.

rkrr ¼ 2Bk þ Ck 1

�þ �eek

�r�eek�1 þ Dk 1

�� eek

�r��eek�1 � qkx

2 3 þ mkhr

9 � �ee2k

!r2

rkhh ¼ 2Bk þ Ck 1

�þ �eek

��eekr�eek�1 � Dk 1

�� eek

��eekr��eek�1 � qkx

2 3mkhr � �ee2

k

9 � �ee2k

!r2

rkrh ¼ 0

ð15Þ


For a cylinder with closed ends, the axial equilibriumis satisfied by the following relation,Xn

k¼1

2pZ bk

bk�1

rkzzrdr ¼ pr2

int Pintð � PextÞ þ FA ð22Þ

where rint is the internal radius of the cylinder. Substi-tuting Eqs. (18) and (19) for rk

rr and rkhh into Eq. (20) for

rkzz and evaluating the integral in Eq. (22), the relation

for e0zz is determined as:

e0zz ¼

1

DPintr2

int

�"� Pextr2

ext

�þ FA

p

�Xn

k¼1

qk�1dk

�þ qklk þ x2wk

�#ð23Þ

where, D ¼Pn

k¼1

b2k�b2

k�1

ak33

and

dk ¼�2

ak33 1 � c2gk

k

� � bkcgkþ1k

1 þ gkak

13

�(þ gkak

23

�bkð � cgk

k bk�1Þ

� bk�1

1 � gkak

13

�� gkak

23

�bkc

gkkð � bk�1Þ

)

lk ¼�2

ak33 1 � c2gk

k

� � �bk

1 þ gkak

13

��þ gkak

23

�bkð � cgk

k bk�1Þ

þ bkcgkk

1 � gkak

13

�� gkak

23

�bkc

gkkð � bk�1Þ

�

wk ¼2qkb

2k 3 þ mk

hr

� �9 � g2

k

ak13

ak33

�(þ ak

23

ak33

gk

�

1 � cgkþ3k

1 � c2gkk

� �1 þ gkð Þ

!b�gkþ1

k bgkþ1k�1

�� b2

k

�

þ ak13

ak33

�� ak

23

ak33

gk

�1 � cgk�3

k

� �cgkþ3k

1 � c2gkk

� �1 � gkð Þ

!

bgkþ1k bgkþ1

k�1

�� b2

k

�þ ak

13

ak33

�þ ak

23

ak33

g2k þ 3mk

hr

� �3 þ mk

hrð Þ

�

1

4b2k

� �b4

k

�� b4

k�1

�)

At each layer interface, radial displacements of ad-jacent layers must be continuous, which follows that:

ukr ðbkÞ ¼ ukþ1

r ðbkÞ ð24Þ

Using Eqs. (23) and (24), a set of simultaneousequations in terms of qk, one for each interface, is de-termined as:

ukqkþ1 þ vkqk þ gkqk�1 þ x2kk

þ mkzh � mkþ1

zh

D

� � Xn

i¼1

qi�1di

� þ qili þ x2wi

�!

¼ mkzh � mkþ1

zh

D

� �Pintr2

int

�� Pextr2

ext

�þ FA

p

�ð25Þ

where, uk ¼ ð2gkþ1nkþ122 cgkþ1�1

kþ1 Þ=ð1 � c2gkþ1kþ1 Þ

uk ¼ �bk12 �

gkbk22 1 þ c2gk

k

� �1 � c2gk

k

þ bkþ112

�gkþ1b

kþ122 1 þ c2gkþ1

kþ1

� �1 � c2gkþ1

kþ1

gk ¼2gkn

k12c

gkþ1k

1 � c2gkk

kk ¼ b2k

qknk22

9 � g2k

gk 3�"(

þ mkhr

� 1 � 2cgkþ3k þ c2gk

k

1 � c2gkk

!

� g2k

�þ 3mk

hr

�#� b2

kþ1

qkþ1nkþ122

9 � g2kþ1

gkþ1 3�

þ mkþ1hr

�

2cgkþ1�1kþ1 � c2gkþ1þ2

kþ1 � c2kþ1

1 � c2gkþ1

kþ1

!� g2

kþ1

�þ 3mkþ1

hr

�c2kþ1

)

Therefore, the unknown interface pressures, qk, aresolved by using Eq. (25), which eventually leads to thecomplete solution of the elasticity problem.

2.4. Analysis of pressure vessels using thin wall theory

The thickness ratio is defined as the ratio betweenexternal and internal radii of the pressure vessel. Forpressure vessels of thickness ratios less than 1.1, the thinwall analysis can satisfactorily be used. In this theory,the radial stress is assumed to be zero in addition tohoop and axial stresses to be constant through thethickness. The hoop and the axial stresses of a pressurevessel subjected to internal and external pressure, and anaxial force can be calculated, respectively, as follows:

rhh ¼ðPint � PextÞrint

tand

rzz ¼ðPint � PextÞrint

2tþ FA

2printtð26Þ

where t is the wall thickness of the vessel.

Fig. 3. Multi-layered cylinder showing layer notation.


2.5. Environmental effects on composite materials

The influence of environmental factors, such as ele-vated temperature, humidity and corrosive fluids mustbe taken into consideration since they affect mechanicaland physical properties of composite materials resultingin a change of the mechanical performance. The effect ofthe elevated temperature can be seen in the compositematerial properties with a decrease in the modulusand strength because of thermal softening. Especiallyin polymer-based composites, the matrix-dominatedproperties are more affected then the fiber-dominatedproperties. For example, the longitudinal strength andmodulus of a unidirectional composite specimen remainalmost constant but off-axis properties of the samespecimen are significantly reduced as the temperatureapproaches the glass transition temperature of thepolymer. When exposed to humid air or water envi-ronment, many polymeric matrix composites absorbmoisture by instantaneous surface absorption followedby diffusion through the matrix. Analysis of moistureabsorption shows that for epoxy and polyester matrixcomposites, the moisture concentration increases ini-tially with time and approaches an equilibrium (satu-ration) level after several days of exposure to humidenvironments [2].

The analysis of composites due to elevated tempera-ture and moisture absorption is called as ‘‘hygrothermalproblem’’. It can be solved mainly in three steps: First,the temperature distribution and the moisture contentinside the material are calculated. Then from knowntemperature and moisture distribution, the hygrother-mal deformations and stresses are calculated. Finally,the changes in performance due to both affects aredetermined. The assumptions used through these stepsare:

• temperature and moisture content inside the materialvary only in the thickness direction,

• thermal conductivity of the material is independent oftemperature and moisture level,

• the environmental conditions (temperature and mois-ture level) are constant.

The temperature distribution is obtained by using theone dimensional steady-state heat conduction analy-sis throughout the body. So for the kth layer, one canwrite:

Tk ¼ Tk�1 � qRk ð27Þ

where q ¼ ðTint � TextÞ=Pn

i¼1 Ri and Rk ¼ Kk=hk.Here Tk temperature, hk thickness and Kk are the

thermal conductivity of the kth layer, respectively. Inthis study, the moisture content is taken as constant andequal to the saturated moisture level throughout thematerial.

2.6. Hygrothermal degradation

In addition to creating stresses, temperature andmoisture degrade the material properties as well. By fol-lowing the method given by Tsai [2], the non-dimen-sional temperature T � can be defined as,

T � ¼ ðTg � ToprÞðTg � TrmÞ

ð28Þ

where Tg is the glass-transition temperature, Topr is theoperation temperature and Trm is the room temperature.It is also assumed that the moisture suppresses the glasstransition temperature by a simple moisture shift as,

Tg ¼ T 0g � gc ð29Þ

where T 0g is the glass-transition temperature at the dry

state, g is the temperature shift per unit moisture ab-sorbed and c is the moisture absorption of the structure.The term T � is used to empirically fit the fiber and ma-trix stiffness and strength data as functions of bothmoisture and temperature.

Using empirical fiber and matrix properties, the plystiffness and strength properties are given here [2], firstin terms of stiffness ratios as

E11 ¼vf

v0f

ðT �Þf E011

E22

E022

¼ðT �ÞaE0

m þ ðT �Þbg0y

1vf� 1

� �ðT �Þf E0

f

h i1 þ 1

v0f

� 1� �

g0y

h iE0

m þ 1v0

f

� 1� �

g0yE

0f

h i1 þ ðT �Þb

vf� ðT �Þb

h iðT �ÞaðT �Þf

Es

E0s

¼g0s ðT �Þbð1�vf Þ

vfþ 1

h i1G0

f

þ g0s ð1�v0

fÞ

v0fG0

m

h i1 þ g0

s v0m

v0f

� �1

ðT �Þf G0f

þ ðT �Þbg0s ð1�vf Þ

vf ðT �ÞbG0m

� �ð30Þ

and strength ratios as

XX 0

¼ vf

v0f

ðT �Þh

X 0

X 00 ¼vf

v0f

ðT �Þh Es

E0s

� �e

YY 0

¼ Y 0

Y 00 ¼SS0

¼ T �ð Þd

ð31Þ

where g is the mutual influence coefficient and subscriptsf and m denote fiber and matrix, respectively. Theconstants a, b, d, f and h are determined empirically andthe exponent for example on X 0 denotes the values ofthe corresponding property X obtained at room tem-perature with 0.5% moisture content.

2.7. Hygrothermal stresses

The hygrothermal and mechanical strains can be su-perposed in strain level to obtain total strains as,

etotij ¼ emech

ij þ ehygrij ð32Þ


or

etotij ¼ emech

ij þ aijDT þ bijc

where aij and bij are thermal and moisture expansioncoefficients, respectively.

Total stresses however, can be obtained using aniso-tropic stress–strain relations. Total stresses due to hy-grothermal and mechanical loads can be written as,

rtotij ¼ a½ ��1etot

ij ð33Þ

2.8. Failure analysis

The main reason for performing the stress analysis isto determine the failure behavior of the pressure vessel.Design of a structure or a component is performed bycomparing stresses (or strains) created by applied loadswith the allowable strength (or strain capacity) of thematerial [2]. Tsai–Hill, Tsai–Wu [8], Hoffman [9] and3D-Quadratic Failure Theories [2] are used in this studyfor comparison and it is seen that 3D-Quadratic FailureCriteria gives the most conservative results for strength.

3. Numerical solution

In order to see how structures behave, the numericalresults are necessary for a given material, geometry andloading combination. A preliminary design packageprogram is developed using the derived formulation ofstresses. In order to determine the burst pressure, themaximum axial force and the maximum angular speed,the performance (load carrying capacity) of the specifiedcomposite pressure vessel is taken as the only limitingvalue. The strength ratio is the ratio between the maxi-mum or ultimate strength and the applied stress. It mustbe slightly larger than one because of the safety reason.Burst pressure and maximum angular speed are deter-mined by using the first-ply failure criterion and maxi-mum axial force is determined by using the last-plyfailure criterion.

The winding angle obtained by developed computerprogram is called as optimum without using any opti-mization procedure. This is not wrong because only oneconstraint is taken into account, which has to be maxi-mized in this case and all possible solutions are checkedevery time to get the winding angle satisfying the con-straints.

Since the winding angle varies between 0� and 90�,layer stresses are obtained for each angle with a step sizeof 0.1�. The strength ratios of the worst layers arecompared with each other. Then the angle having thehighest strength ratio is taken as the optimum windingangle for the specified loading and geometry conditions.

4. Discussion of results

The design outputs of the computer program areoptimum winding angle, burst pressure and maximumangular speed of the vessel for a given material, geo-metry and loading combination. Also the affects of axialforce and hygrothermal forces on burst pressure andangular speed are studied. In each of these analysis, thematerial used is a graphite-epoxy composite (T300/N5208). The properties of the unidirectional laminate ofthis material are given in Table 1. Note that the residualstresses due to material itself are not considered in thisstudy.

4.1. Optimum winding angle

In literature [5], the optimum winding angle for fila-ment wound composite pressure vessels is given as54.74� by netting analysis. Using the current procedurefor the internal pressure loading, the optimum windingangle is obtained as ranging between 52.1� and 54.2�depending on geometry and failure criteria used. 3D-Quadratic Failure Criterion always gives greater opti-mum winding angle than other theories, because thecircumferential stress or strain is more effective in thiscriterion. If angular speed is applied at the same timewith the internal pressure, the optimum winding anglevalues obtained for the pure internal pressure case areincreased and shifted to 90�.

4.2. Stress distribution

The stress distribution through the thickness of afilament wound vessel is not uniform but varying de-pending on the geometry and loading. The stress graphsfor pure internal pressure and pure angular speed caseswith a constant winding angle of 53�, are given in Figs. 4and 5, respectively. The symbols a and b represent

Table 1

Properties of unidirectional laminate (T300/N5208)

Elastic modulus in fiber direction (GPa) 181

Elastic modulus in matrix direction (GPa) 10.3

In plane shear modulus (GPa) 7.17

Major Poisson’s ratio 0.28

Ultimate tensile strength in fiber direction (MPa) 1500

Ultimate compressive strength in fiber direction (MPa) 1500

Ultimate tensile strength in matrix direction (MPa) 40

Ultimate compressive strength in matrix direction (MPa) 146

Ultimate in-plane shear strength (MPa) 68

Thermal expansion coefficient in fiber direction (10�6/�C) 0.02

Thermal expansion coefficient in matrix direction (10�6/�C) 22.5

Moisture expansion coefficient in fiber direction 0

Moisture expansion coefficient in matrix direction 0.6

Thermal conductivity normal to the thickness direction

(W/m per �C)

0.865


internal and external radii of the tube, respectively. Thestresses are normalized with the values of thin-wall so-lution. In Fig. 4, the results for radial stresses and tosome degree with hoop stresses are similar with the onesof Ref. [5]. However, the axial stress distributions showa considerable difference. The errors in the formulationof Ref. [5] would be a reason for the difference; anotherreason might be originated from the fact that a variablewinding angle (54–56�) was considered in Ref. [5], con-trary to the constant winding angle used in the currentstudy.

As it can be seen in Fig. 5, the thin-wall analysis givesonly an average result. When thickness increases, thethick-wall analysis has to be used, instead. Radial stressis zero at the inside and outside of the pressure vesseland positive through the thickness. Its maximum valueis approximately at the mid-point of the thickness.

When the wall thickness is increased, the layer havingmaximum radial stress becomes closer to the innerboundary of the pressure vessel. Larger circumferentialand axial stresses are obtained at the inner layers of thevessel. In Fig. 5, it is seen that axial stress alwayschanges sign at the point where the radial stress reachesto a maximum.

4.3. Thick- and thin-walled solutions for burst pressure

The corresponding burst pressure values are obtainedusing an iterative procedure where the loading is in-creased until the failure of a single layer. In Fig. 6, theburst pressure, Pburst, by using both thin and thick-wal-led solution techniques are plotted versus winding angle.The burst pressure is normalized with that of 0� windingangle. For the thin-wall tube, both thick and thin-wall

Fig. 4. Radial, axial and circumferential stress distributions for pure internal pressure.


solutions predict almost the same burst pressure. Actu-ally, since the thin-wall solution neglects the radialstress, burst pressure values obtained with the thin-

walled analysis are slightly higher than those obtainedwith thick-walled analysis. The agreement between thesetwo solutions is satisfied except for the values near the

Fig. 5. Radial, axial and circumferential stress distributions for pure angular speed.

Fig. 6. Variation of burst pressure with increasing winding angle [rext=rint ¼ 1:05].


optimum winding angle where they differ for thick andthin-wall solutions.

For a thick pressure vessel (Fig. 7), there is a signif-icant difference between thin-wall and thick-wall solu-tions especially near the optimum winding angle. In thiswall thickness value (rext=rint ¼ 1:40), the thick-wall so-lution gives higher burst pressure than the thin-wallsolution between angles 48� and 64�.

To check the limiting value of the thin-wall solutionon the wall thickness, burst pressures are calculated forincreasing wall thickness Thin-wall and thick-wall so-lutions yield very similar burst pressure values up to thethickness ratio of rext=rint ¼ 1:1. For thicknesses withrext=rint P 1:1, the deviation between thick and thin-wallsolutions becomes larger (Fig. 8).

4.4. Effect of angular speed and axial force on burstpressure

The effect of angular speed and axial force on theburst pressure can be seen in Fig. 9. The burst pressuredecreases with angular speed, when the winding angle isless than its optimum value. It is not an expected result,since the burst pressure increases with the speed forangles greater than the optimum winding angle. The

burst pressure increases in negligible amount for in-creasing axial tensile force for angles smaller than theoptimum winding angle. If the winding angle of thestructure is larger than its optimum value, the burstpressure always decreases with increasing axial force.

4.5. Maximum angular speed

The effect of wall thickness and winding angle on themaximum angular speed can be seen in Fig. 10. If onlyangular speed is applied, the optimum winding angle isobtained as ranging between 81� and 83� depending onthe wall thickness of the structure. For small windingangles up to 30�, the thin and thick-wall constructionsgive almost the same maximum angular speed. As thewall thickness increases, the maximum angular speeddecreases opposite to the case of burst pressure. It is anexpected result, since an increasing wall thickness meansmore inertia that affects the speed in the negative sense.

4.6. Hygrothermal stresses and strains

Hygrothermal stresses in the macro-mechanical levelcalculated by using the laminated plate theory. In orderto assess the effects of residual stresses on the failure of

Fig. 7. Variation of burst pressure with increasing winding angle [rext=rint ¼ 1:4].

Fig. 8. Burst pressure for increasing wall thickness.


composite materials, the hygrothermal expansion coef-ficients have to be determined correctly. As a samplecalculation of hygrothermal stresses, the data given inTable 2 for T300/N5208 [2] is used where DF is thedegradation factor.

Since the curing temperature is the stress free state forcomposite materials, the operation temperature affectsthe failure of the composite depending on whether it isbelow or above the curing temperature. If the operationtemperature is less than zero or if it is less than thecuring temperature, the burst pressure is increased sincethe thermal strains and mechanical strains for pure in-ternal pressure case work in opposite senses. It should bepointed out that, the negative temperature for constantmoisture content also cause an increase in the mechan-ical properties of the composite material. It can beconcluded that if the operation temperature is lessthan the curing temperature, burst pressure is increased.If operation temperature is greater than the curing

Fig. 9. Effect of angular speed and axial force on burst pressure [rext=rint ¼ 1:05].

Fig. 10. Maximum angular velocity versus winding angle.

Table 2

Hygrothermal effects on burst pressure [rext=rint ¼ 1:05, a ¼ 53�]

Topr (�C) 22 �22 122 122 122

c (%) 0 0 0 0 1

DF (%) 0 0 0 10 30

Pburst (MPa) 18.3 20.3 16.3 �14.4 12.1


temperature, however, both thermal and mechanicalstrains have a cumulative effect. This can be seen as adecrease in burst pressure values in Fig. 11.

Although, the performance of the composite materialis negatively influenced by the presence of moisture, itcreates less residual strains compared with the thermalones and does not change the burst pressure, signifi-cantly.

5. Conclusion

An analytical procedure is developed to assess thebehavior of a cylindrical composite structure underloading conditions particular to a rocket motor case.Available loading conditions are internal pressure, axialforce and body force due to rotation. Additionally,temperature and moisture variations throughout thebody are considered in the analysis. The procedure isbased on the classical laminated plate theory. It modelsthe plane strain state of the cylindrical body, whichconsists of a number of cylindrical sub-layers.

The cylindrical pressure vessel is analyzed using twoapproaches, which are thin wall and thick wall solutions.It is shown that for composite pressure vessels with aratio of outer to inner radius, up to 1.1, two approachesgive similar results in terms of the optimum windingangle, the burst pressure, etc. As the ratio increases, thethick wall analysis is required.

The optimum winding angle for the thick-wall pres-sure vessel analysis with the pure internal pressureloading case is obtained as ranging between 52.1 and54.1 degrees depending on the material type. If the an-gular speed is applied at the same time with the internalpressure, optimum winding angle values obtained forthe pure internal pressure case are shifted towards90. The influence of the axial force is, however, oppositeto the one of the angular speed. The addition of the axialforce has a decreasing effect on the winding angle.

The burst pressure value is greatly depends on theanalysis type used. The deviation between thin andthick-wall solutions is quite large especially near theoptimum winding angle. As the wall thickness is in-creased, the thick-wall solution gives almost 30% higherburst pressure values. Therefore, the thin-wall analysis issaid to be an average but a safe analysis.

If angular speed is applied, the maximum stressoccurs in the hoop direction. The optimum windingangle of the analyzed body for this type of loading isobtained as ranging between 81� and 83�. The value ofthe maximum angular speed that the system can be ro-tated is greatly affected by the thickness of the pressurevessel.

Hygrothermal effects are analyzed in this study in twolevels. The effect of temperature and moisture to theperformance of the materials is determined by using themicromechanics of the composite materials. By taking alinear variation of temperature and constant value formoisture content throughout the body, hygrothermalstresses and strains are determined. Since the thermaland moisture expansion coefficients of the materialshave to be determined experimentally, always someamount of error is expected in these calculations.

If the material has a tendency of expanding due to apositive temperature difference, the increasing operatingtemperature is shown to reduce the mechanical perfor-mance of the system.

References

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Breach Science Publishers; 1968.

[2] Tsai SW. Composites design. USA, Think Composites, 1985.

[3] Xia M, Takayanagi H, Kemmochi K. Analysis of multi-layered

filament-wound composite pipes under internal pressure. Compos

Struct 2001;53(4):483–91.

[4] Xia M, Kemmochi K, Takayanagi H. Analysis of filament-wound

fiber-reinforced sandwich pipe under combined internal pressure

and thermomechanical loading. Compos Struct 2001;51(3):273–83.

Fig. 11. Effect of temperature on burst pressure for different operating temperatures.


[5] Wild PM, Vickers GW. Analysis of filament-wound cylindrical

shells under combined centrifugal, pressure and axial loading.

Compos Part A 1997;28(1):47–55.

[6] Katırcı N. Design of fiber reinforced composite pressure ves-

sels. M.S. Thesis, Middle East Technical University, Turkey,

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[7] Lekhnitskii SG. Theory of elasticity of an anisotropic body. USSR,

Moscow: Mir Publishers; 1981.

[8] Mallick PK. Fiber-reinforced composites, materials, manufacturing

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