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Analysisoffilament-woundfiber-reinforcedsandwichpipeundercombinedinternalpressureandthermomechanicalloading

ARTICLEinCOMPOSITESTRUCTURES·MARCH2001

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Analysis of ®lament-wound ®ber-reinforced sandwich pipe undercombined internal pressure and thermomechanical loading

M. Xia *, K. Kemmochi, H. Takayanagi

Department of Composite Materials, National Institute of Materials and Chemical Research, Agency of Industrial Science and Technology (AIST),

Ministry of International Trade and Industry (MITI), 1-1, Higashi, Tsukuba, Ibaraki 305-8565, Japan

Abstract

This is a presentation based on the classical laminated-plate theory of an elastic solution for the thermal stress and strain in a

®lament-wound ®ber-reinforced sandwich pipe subjected to internal pressure and temperature change. The sandwich pipe is created

using resin material for the core layer and reinforced materials with an alternate-ply for the skin layers. Considering the complicated

material properties of the skin layers reinforced by alternate-ply composites, the thermal stress analysis is based on treating typical

sandwich pipes that are three-dimensional, cylindrical, and orthotropic. A computer program was developed to conduct stress and

deformation analyses of sandwich pipe with di�erent winding angles. Moreover, an optimum winding angle of the ®lament-wound

®ber-reinforced materials was designed by using a netting approach analysis. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Orthotropic analysis; Alternate-ply composite structure; Sandwich cylindrical pipe; Internal pressure; Thermal stress

1. Introduction

The development of sandwich-type pipes for manyindustrial products utilizing composites has already re-ceived considerable attention. Among their applications,a common feature of such products is that they mustsafely undergo a certain working pressure. As ®lament-wound pipes made of ®ber-reinforced plastics havemany potential advantages over pipes made from con-ventional materials, a number of researchers have in-vestigated failure mechanisms of ®lament-wound pipes.For thin-walled cylindrical-pressure vessels with a ratioof applied hoop-to-axial stress of two to one, an opti-mum winding angle of 55° was noted, and many ex-perimental failure analyses were conducted for ®lament-wound pipe with a 55°-winding angle [1±4]. Rosenow [5]used the classical laminated-plate theory to predict thestress and strain response of pipes with winding anglesvarying from 15° to 85°, and he compared them to ex-perimental results. In thin-walled ®lament-wound shells,a 55°-winding angle was shown to be optimum for thehoop-to-axial stress ratio of two, but the optimum anglehad to be about 75° in the case of pressure without axialloading. Spencer and Hull [6] and Uemura and Fuku-

naga [7] have investigated, respectively, the failuremechanism in carbon ®ber-reinforced plastics (CFRP)and glass ®ber-reinforced plastics (GFRP) pipes woundat di�erent winding angles. The maximum weepagestress was found to be around 55°, and negative axialstrains were observed within a range of 35° to 50°. Wildand Vickers [8] have developed an analytical procedurebased on the theory of orthotropic cylindrical sheets andmodeled both plane-stress and plane-strain states ofcylindrical sheets comprising a number of cylindricalsublayers, each of which is cylindrically orthotropic. Theoptimum winding angle was shown to play an importantpart in the design of ®lament-wound cylindrical shells.

Most previous studies on cylindrical ®ber-reinforcedcomposite structures have focused on thin-walled cy-lindrical shells. However, only limited studies have beenpublished dealing with thick-walled cylindrical pipe be-haviors [9±13]. Roy [9] presented a thermal stress anal-ysis of a thick laminated ring assumed to be cylindricallyorthotropic. The analysis was based on treating the ringwith orthotropic materials in the state of plane stress inthe hoop and axial �hÿ r� plane. Ben [10] has reportedan accurate, ®nite-cylindrical element method to obtainthermal stresses and the deformation for thick-walledcylindrical pipes. In his work, the e�ects of thermal re-sidual stresses on the design of thick-walled FRP cy-lindrical pipes were discussed. Ben did not consider axialloading of cylinders with closed ends in his mechanical

www.elsevier.com/locate/compstruct

Composite Structures 51 (2001) 273±283

* Corresponding author. Tel.: +81-0298-6798; fax: +81-0298-61-

6798.

E-mail address: xia@home.nimc.go.jp (M. Xia).

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 2 3 ( 0 0 ) 0 0 1 3 7 - 9

analyses of cylindrical pipes under internal pressure. Inrecent years, a ®nite element method has becomeavailable for analyzing mechanical behaviors of cylin-drical ®ber-reinforced composite structures [11±13].Kitao and Akiyama [14] have analyzed and evaluatedthe progress of failure in thick-walled, ®lament-wound(FW) pipes with di�erent winding-angles. Using the®nite Hankel and the Laplace transforms, the elasto-dynamic solution for the thermal shock stresses in anorthotropic thick cylindrical shell has been reported byCho et al. [15]. The concept of an elastic-plastic stress®eld was predicted in a coated-continuous ®ber com-posite subjected to thermomechanical loading by Youet al. [16].

At present, the National Institute of Materials andChemical Research in the Agency of Industrial Scienceand Technology (AIST) of the Ministry of Inter-national Trade and Industry (MITI) is carrying out astudy on FW sandwich pipes used for geothermal de-velopment, such as transportation of undergroundwater with high temperature. Sandwich compositepipes, composed of two skin layers and a core layer,can be suitably tailored with optimum material prop-erties by making e�ective use of each material propertyin the moldings. There are few investigations on themechanical properties of sandwich pipes, and there isalmost no literature reporting on the stress and defor-mation analysis of sandwich composite pipes. We areattempting to provide an analytical foundation for theinvestigation of stress and deformation in a ®lament-wound sandwich pipe under combined internal pressureand temperature change. Considering the especiallycomplicated material properties of the skin layers re-inforced by alternate-ply composites, our analysis isbased on treating typical sandwich pipes that are three-dimensional, cylindrical and orthotropic. With an al-ready developed computer model, we conducted stressand deformation analysis of the sandwich pipe usingdi�erent winding angles.

2. Analysis procedure

The sandwich pipe is created using non-reinforcedmaterials for the core layer and alternate-ply materialsfor the skin layers. The alternate-ply skin layers arethose in which the principal material directions of theadjacent layer have an opposite ®ber orientation (�/)with respect to the axial direction. The adjacent two lay-ups are assumed here to behave together as an ortho-tropic unit. In this paper, the orthotropic unit of thelay-up angle (�/) is referred to as an orthotropic layerof angle /. Figs. 1 and 2 show the cylindrical coordi-nates and con®guration notations for the sandwichcomposite pipe.

2.1. Stress analysis

Subjected to axisymmetric thermomechanical loadingand internal pressure, the circumferential displacementu�k�h is zero. The stresses and strains are independent of/, and there is no shear±extension coupling. Therefore,the equilibrium equation for the current axisymmetricproblem can be expressed as

dr�k�r

dr� r�k�r ÿ r�k�h

r� 0: �1�

The radial and hoop strains, e�k�r and e�k�h , can be given inthe radial displacement u�k�r , we obtain

e�k�r �du�k�r

drand e�k�h �

u�k�r

r: �2�

Fig. 1. Filament-wound sandwich pipe in cylindrical coordinates.

Fig. 2. Cross section of a sandwich pipe.

274 M. Xia et al. / Composite Structures 51 (2001) 273±283

The axial strains e�k�z of all layers are equal to a constant,e0.

Using the cylindrical coordinate system shown inFig. 1, the stress and strain transformation of the kth layerwith the orthotropic alternate-ply material is given by

r�k�r � C�k�11 e�k�r � C

�k�12 e�k�h � C

�k�13 e0 ÿ n�k�r DT ;

r�k�h � C�k�12 e�k�r � C

�k�22 e�k�h � C

�k�23 e0 ÿ n�k�h DT ;

r�k�z � C�k�13 e�k�r � C

�k�23 e�k�h � C

�k�33 e0 ÿ n�k�z DT

�3a�

and

n�k�r � a�k�r C�k�11 � a�k�h C

�k�12 � a�k�z C

�k�13 ;

n�k�h � a�k�r C�k�12 � a�k�h C

�k�22 � a�k�z C

�k�23 ;

n�k�z � a�k�r C�k�13 � a�k�h C

�k�23 � a�k�z C

�k�33 ;

�3b�

where C�k�ij are the sti�ness constants and a�k�i are the

thermal expansion coe�cients (1, 2 and 3 represent r, hand z directions, respectively).

Substituting the expressions for the stress in Eq. (1)and using Eq. (2), we get

d2u�k�r

dr2� 1

rdu�k�r

drÿ C

�k�22 =C

�k�11

r2u�k�r �

a�k�e0

r� g�k�DT

r; �4a�

where

a�k� � C�k�23

�ÿ C

�k�13

�=C�k�11 ; �4b�

and

g�k� � n�k�r

�ÿ n�k�h

�=C�k�11 : �4c�

When C�k�22 =C

�k�11 > 0, if b�k� �

������������������C�k�22 =C

�k�11

q, the solution

for Eqs. (4a)±(4c) can be obtained under the followingtwo conditions:

(a) If b�k� 6� 1, which is anisotropic

u�k�r � A�k�rb�k� � B�k�rÿb�k� � a�k�e0 � g�k�DTÿ �

r

1ÿ b�k�� �2

: �5�

(b) If b�k� � 1, which is isotropic or isotropic in(r ÿ h) planer

u�k�r �a�k�e0 � g�k�DTÿ �

r2

ln r � A�k�r � B�k�=r: �6�

When C�k�22 =C

�k�11 < 0, if b�k� �

����������������������ÿC

�k�22 =C

�k�11

q, the solu-

tion for Eq. (4a)±(4c) is

u�k�r � A�k� cos�b�k� ln r� � B�k� sin�b�k� ln r�

� a�k�e0 � g�k�DTÿ �

r

1� b�k�� �2

: �7�

where A�k� and B�k� are unknown constants of integra-tion, and to be determined from the boundary condi-tions and the contact conditions at each interfacebetween the core and skin layers.

2.2. Transformation from ply to laminate properties

The o�-axis sti�ness constants in Eq. (1), fC�k�ij g, canbe calculated from the on-axis sti�ness constants, fC�k�ij g,by using a sti�ness transformation matrix Aij

� �, written

as

C�k�ij

n o� Akl� � C�k�ij

n o; �8a�

where

C�k�ij

n o� C

�k�33 ;C

�k�23 ;C

�k�13 ;C

�k�22 ;C

�k�12 ;C

�k�11

n oT

; �8b�

C�k�ij

n o� C�k�xx ;C

�k�yy ;C

�k�zz ;C

�k�xy ;C

�k�xz ;C

�k�yz ;G

�k�zz

n oT

: �8c�Based upon the classical laminated-plate theory, thesti�ness transformation matrix Aij

� �for the coordinate

system between the on-axis and the cylindrical axisshown in Fig. 3 is given by

Akl� � �

m4 n4 0 2m2n2 0 0 4m2n2

m2n2 m2n2 0 m4� n4 0 0 ÿ4m2n2

0 0 0 0 m2 n2 0n4 m4 0 2m2n2 0 0 4m2n2

0 0 0 0 n2 m2 00 0 1 0 0 0 0

26666664

37777775

�k�

;

�9�where m � cos / and n � sin /, and / is the cylindricalangle of the ®laments from the pipe axis.

To de®ne the three-dimensional alternate-ply mate-rial properties, the material modulus matrix elementsCij �i;j � x; y; z� and Gzz in Eq. (8a)±(8c) are needed.Their values can be calculated from engineering con-stants, de®ned by

Ex;Ey ;Ez; txy ; tzx; tzy ;Gxx;Gyy ;Gzz:

Fig. 3. Relation of coordinate system between principal material axis

and cylindrical axes.

M. Xia et al. / Composite Structures 51 (2001) 273±283 275

For the most general case of orthotropic ply-orientedproperties, these values would have to be experimentallymeasured or estimated using micromechanics. For uni-directional orientation ®ber composites, the ®ber dis-tributions are very similar in the y and z directions.Therefore, assuming transverse isotropy, and based onequivalent properties in the y±z plane for unidirectionalmaterial, we get:

Ey � Ez;

Gyy � Gzz;

tzx � tyx;

where x and y refer to material principal axes along ®berand transverse directions, respectively.

The conversion of engineering constants to modulusmatrix elements are obtained from

D � SxxSyySzz � 2SxySyzSxz ÿ SyyS2xz ÿ SxxS2

yz ÿ SzzS2xy ;

�10a�

Cxx � �SyySzz ÿ S2yz�=D; Cxy � �SxzSyz ÿ SxySzz�D;

Cyy � �SxxSzz ÿ S2xz�=D; Cxz � �SxySyz ÿ SxzSyy�D;

Czz � �SxxSyy ÿ S2xy�=D; Cyz � �SxySxz ÿ SxxSyz�=D;

�10b�where

Sxx � 1=Ex; Sxy � ÿtyx=Ex;

Syy � 1=Ey ; Sxz � ÿtyx=Ex;

Szz � 1=Ez; Syz � ÿtzy=Ey :

�10c�

The laminate-oriented coe�cients of thermal expansion,which represent non-mechanical strains, can be given by

ar

ah

az

8<:9=;�k�

�m2 n2 0n2 m2 00 0 1

24 35 ax

ay

az

8<:9=;�k�

; �11a�

where m � cos / and n � sin /.Considering that the ®ber distributions are very

similar in the y and z directions, we get

az � ay : �11b�

2.3. Boundary conditions

Assuming that the interfaces between the core andskin layers are perfectly bound, the continuance of dis-placements and tractions along the interfaces and trac-tion-free boundary conditions provides a homogeneousequation.

The traction condition (pressure p0) at the innersurface and the traction-free condition at the outersurface are written as

r�1�r �r0� � ÿp0;

r�n�r �ra� � 0:�12�

where r0 and ra are the inner and outer radii, as shown inFig. 2, respectively.

Continuity conditions for the displacements andstresses in the interfaces lead to

u�k�r �rk� � u�k�1�r �rk� k � 1; 2; . . . ; n;

r�k�r �rk� � r�k�1�r �rk� k � 1; 2; . . . ; n:

�13�

For a cylinder with closed ends, the axial equilibrium issatis®ed by the following relation:

2pXn

k�1

Z rk

rkÿ1

r�k�z �r�r dr � pr20p0: �14�

Eqs. (12)±(14) can give a set of equations to determineunknown constants A�k�, B�k�, and axial strain e0 inEqs. (5)±(7). The simultaneous equation, for the sand-wich pipe �n � 3� shown in Fig. 2, can be written asfollows:

k11 k12 k13 k14 k15 k16 k17

k21 k22 k23 k24 k25 k26 k27

k31 k32 k33 k34 k35 k36 k37

k41 k42 k43 k44 k45 k46 k47

k51 k52 k53 k54 k55 k56 k57

k61 k62 k63 k64 k65 k66 k67

k71 k72 k73 k74 k75 k76 k77

26666664

37777775A�1�A�2�A�3�B�1�B�2�B�3�e0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

�

d1

d2

d3

d4

d5

d6

d7

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; �15�

where kij and dj �i; j � 1; . . . ; 7� in two conditions of

C�k�22 =C

�k�11 > 0 and C

�k�22 =C

�k�11 < 0 are given in Appendix A.

Once values of A�k�, B�k� �k � 1; 2; 3�, and e0 obtainedfrom Eq. (15) are known, the strains, stresses anddisplacements are thus determined from Eqs. (2), (3a),(3b), (5)±(7).

3. Numerical results and discussion

A computer procedure based on the above analysishas been incorporated into a FORTRAN program thatallows user input of geometric parameters and materialproperties for the core and skin layers, and of the in-ternal pressure load and temperature changes. Theprogram can calculate stress, strain, and deformation of®lament-wound sandwich pipes.

The procedure is applied to an example of a com-posite sandwich pipe with an isotropic-core layer andorthotropic-skin layers. The con®guration notation ofthe sandwich pipe is shown in Fig. 2, which has an innerradius �r0� of 50 mm, a core-layer thickness �tc� of 20

276 M. Xia et al. / Composite Structures 51 (2001) 273±283

mm, and a 2 mm skin-layer thickness �tf�. In the presentstudy, the ®rst and third skin layers (inner layer andouter layer) of the sandwich pipe are made of the samematerial that is based on carbon ®ber/epoxy (T300/934)and E-glass/epoxy [17]. The material properties used inthis study are given in Table 1.

Netting analysis is a simpli®ed approach to the designof cylindrical ®lament-wound structures under externalapplied loading [18]. Netting analysis assumes that allstrength and sti�ness properties are derived from the®bers alone and that no forces are transmitted by resin.The analysis gives the optimum winding angle written as

aopt � tanÿ1

�������rh

rz:

r�16�

For a thin-walled pressure cylinder with closed ends, thehoop stress is twice the axial stress. Using Eq. (16), theoptimal winding angle aopt is equal to 54:7°. Hull [4]described the deformation and failure modes in glass-reinforced polyester pipe wound at 55°. He has testedpipes wound at 55° and has shown that this angle was anoptimum winding angle.

With the exception of a thin-walled laminate-ply cyl-inder, the stress distribution through the wall of a ®la-ment-wound cylinder will not be uniform. The ratio ofhoop-to-axial stress will also vary with the winding angle.

Fig. 4 shows the curve of aopt � tanÿ1�rh=rz�0:5 on theinner layer of the composite pipe with T300/934. Thepipe is subjected to the combination of a 100 K-tem-perature increase and an internal pressure of 0.1 GPa.The optimal winding angle, which is on the crossings ofa curve and a straight line, can be obtained from Fig. 4.The value of the optimum winding angle for the com-posite pipe is about 58°. It can be certi®ed that the de-sign of an optimal winding angle is independent of aninternal pressure or a thermomechanical loading sub-jected alone.

Table 1

Material properties of skin layers and resin

Properties T300/934 E-glass/epoxy Resin (core)

Ex (GPa) 141.6 43.4 1.2

Ey (GPa) 10.7 15.2 1.2

Gzz(GPa) 3.88 6.14 0.46

myx 0.268 0.29 0.30

mzy 0.495 0.38 0.30

ax�10ÿ6 Kÿ1� 0.006 2.32 110

ay�10ÿ6 Kÿ1� 30.04 35.19 110

Fig. 4. Estimation of optimum winding angle under the loading con-

ditions: a 100 K increase superimposed on an internal pressure of

0.1 GPa.

Fig. 5. Stress distributions within a sandwich pipe at DP � 0:1 GPa:

(a) radial, (b) hoop, and (c) axial.

M. Xia et al. / Composite Structures 51 (2001) 273±283 277

The analysis of the sandwich pipe was carried outunder the internal pressure of 0.1 GPa and with a 100K-temperature increase, respectively. The stress distri-butions through the wall of the sandwich pipe with a 60°-winding angle is observed in the axial, hoop, and radialdirections when the internal pressure is 0.1 GPa, as shownin Fig. 5. Fig. 5(a) shows that the radial stress through thewall of the sandwich pipe is subjected to a compressivestress from a given internal pressure of 0.1 GPa to zero. InFig. 5(b) and (c), the stresses in the hoop and axial di-rections are tensile stresses. The skin layers are subjectedto much higher stresses than the core layer. The values of

stress distributions are larger at the inner layer than at theouter layer. Fig. 6 gives stress distributions through thewall of the sandwich pipe under the 100 K-temperatureincrease. Compared with the internal pressure loading,the hoop and axial stresses acting on the core and innerlayers are the compressive thermal stresses.

Analyses of the hoop and axial stresses in the skinlayers are obtained from Figs. 7±10. The stresses actingon the pipe vary with the design of the winding angle.The e�ect of the winding angle on stress variation ismuch larger for the carbon ®ber (T300/934) than for theglass ®ber (E-Glass/Epoxy) because the carbon ®bermaterial has larger anisotropic properties.

Figs. 7 and 8 show the hoop stress curves varying withthe winding angles under the internal pressure and thethermomechanical loading, respectively. In case of theinternal pressure loading, as shown in Fig. 7(a), the hoopstresses in the inner layer will increase when the windingangle increases. Fig. 7(b) shows that the variations of thehoop stresses in the outer layer are quite small. This isbecause the outer layer is subjected to low stress, whichis transmitted from the internal pressure. Both inner andouter skin layers of the pipe are subjected to tensilestresses. For the thermomechanical loading, as shown inFig. 8(a), the stresses of the inner layer decrease whenthe winding angle is increased and can be subjected from

Fig. 6. Stress distributions within a sandwich pipe at DT � 100 K:

(a) radial, (b) hoop, and (c) axial.

Fig. 7. In¯uence of winding angle on hoop stress of (a) inner and

(b) outer layers at DP � 0:1 GPa.

278 M. Xia et al. / Composite Structures 51 (2001) 273±283

the tensile to the compressive thermal stresses. As shownin Fig. 8(b), the stresses of the outer layers increase whenthe winding angle is increased.

The axial stress curves of pipe under the internalpressure and the thermomechanical loading are shownin Figs. 9 and 10, respectively. In Fig. 9(a), the axialstresses in the inner layer increase when the windingangle is increased and have the largest values at about35°, whereas variations of the stresses in the outer layersare reversed, as shown in Fig. 9(b). Fig. 10(a) shows thatthe thermal stresses in the inner layer decrease with thewinding angle and obtain the smallest value at about50°. The thermal stresses in the outer layer have amaximum value at around 45°, as shown in Fig. 10(b).

The strain curve in the axial direction varying withthe winding angle is shown in Fig. 11. For the pipesubjected to the internal pressure, the axial strain of thecylinder must be greater than zero for the isotropicsandwich pipe or the pipe with low anisotropic property.As shown in Fig. 11(a), the axial strain for T300/934 isnegative within a 15° to 55° range of winding anglesbecause of the e�ect of the anisotropic elasticity on theaxial strain. This result has also been obtained in otherexperiments [5±7]. Fig. 11(b) shows that the thermalstrains increase when the winding angle is increased.

Fig. 12 shows the in¯uence of the core material on thehoop stress of the pipe with T300/934 when the internal

pressure is 0.1 GPa. The hoop stress decreases whileincreasing the modulus of the core layer. In the mean-while, the di�erence of the hoop stress between the innerand outer layers becomes smaller when the modulus ofthe core layer is increased. This is because the core layermaterial with a high degree of sti�ness intensi®es thetransfer of the force due to the internal pressure fromthe inner to outer layers. For the core material with verylow sti�ness, the internal pressure is applied primarily tothe inner layer so that the outer layer is subjected tosmaller loading. The outer layer undergoes the maxi-mum stress value when the modulus of the core materialis about 2 GPa.

4. Conclusions

This research presents a method to analyze thestresses and strains of a ®lament-wound sandwich pipesubjected to internal pressure and thermomechanicalloading. This procedure is based on the classical lami-nated-plate theory. The sandwich pipe is considered in3D analysis and in an orthotropic-material model. Thecalculating method developed here provides a basis forpredicting the elastic behavior of the ®lament-woundsandwich pipe.

Fig. 8. In¯uence of winding angle on hoop stress of (a) inner and

(b) outer layers at DT � 100 K. Fig. 9. In¯uence of winding angle on axial stress of (a) inner and

(b) outer layers at DP � 0:1 GPa.

M. Xia et al. / Composite Structures 51 (2001) 273±283 279

The design of the optimum winding angle can beobtained from netting analysis, which depends upongeometry and construction materials. For a thick-walledlaminate-ply sandwich pipe, a 55°-winding angle is nolonger an ideal arrangement.

Under internal pressure, the axial strain of a pipewith T300/934 changes from positive to negative withrespect to the winding angle. Because the core layer witha high degree of sti�ness intensi®es the connection be-tween the inner and outer layers, both hoop stress andthe di�erence of stress between the inner and outerlayers decrease while increasing the modulus of the corelayer. For the sandwich composite pipe constructionwith low core sti�ness, the in¯uence of the core materialon the strength of the pipe is quite large.

Appendix A

(1) When C�k�22 =C

�k�11 > 0:

k11 � b�1�C�1�11

�� C

�1�12

�rb�1�ÿ1

0 ;

k14 ��ÿ b�1�C

�1�11 � C

�1�12

�rÿb�1�ÿ1

0 ;

k17 � a�1�

1ÿ �b�1��2 C�1�11

h� C

�1�12

i� C

�1�13 ;

k12 � k13 � k15 � k16 � 0;

d1 � ÿp0 ÿ g�1�

1ÿ �b�1��2 C�1�11

h(� C

�1�12

iÿ n�1�r

)DT ;

k21 � rb�1�1 ; k22 � ÿr1; k23 � 0;

k24 � rÿb�1�1 ; k25 � ÿ1=r1; k26 � 0;

Fig. 10. In¯uence of winding angle on axial stress of (a) inner and

(b) outer layers at DT � 100 K.

Fig. 11. In¯uence of winding angle on axial strain under the condition

of (a) DP � 0:1 GPa and (b) DT � 100 K.

Fig. 12. In¯uence of core sti�ness on hoop stress behavior.

280 M. Xia et al. / Composite Structures 51 (2001) 273±283

k27 � a�1�r1

1ÿ �b�1��2 ÿa�2�r1

2ln r1;

d2 � g�2�r1

2ln r1

(ÿ g�1�r1

1ÿ �b�1��2)

DT ;

k31 � 0; k32 � r2; k33 � ÿrb�3�2 ;

k34 � 0; k35 � 1=r2; k36 � ÿrÿb�3�2 ;

k37 � a�2�r2

2ln r2 ÿ a�3�r2

1ÿ �b�3��2 ;

d3 � g�3�r2

1ÿ �b�3��2(

ÿ g�2�r2

2ln r2

)DT ;

k41 � b�1�C�1�11

�� C

�1�12

�rb�1�ÿ1

1 ;

k42 � ÿ C�2�11

�� C

�2�12

�;

k44 ��ÿ b�1�C

�1�11 � C

�1�12

�rÿb�1�ÿ1

1 ;

k45 � ÿ C�2�12

�ÿ C

�2�11

�.r2

1;

k47 � a�1�

1ÿ �b�1��2 C�1�11

h� C

�1�12

i� C

�1�13

ÿa�2� ln r1 C

�2�11 � C

�2�12

h i� a�2�C

�2�11 � 2C

�2�13

2;

k43 � k46 � 0;

d4 �g�2� ln r1 C

�2�11 � C

�2�12

h i� g�2�C

�2�11

2

8<:ÿ g�1�

1ÿ �b�1��2 C�1�11

h� C

�1�12

i� n�1�r

�ÿ n�2�r

�9=;DT ;

k51 � k54 � 0;

k52 � C�2�11 � C

�2�12 ;

k53 � ÿ b�3�C�3�11

�� C

�3�12

�rb�3�ÿ1

2 ;

k55 � C�2�12

�ÿ C

�2�11

�=r2

2;

k56 � ÿ�ÿ b�3�C

�3�11 � C

�3�12

�rÿb�3�ÿ1

2 ;

k57 �a�2� ln r2 C

�2�11 � C

�2�12

h i� a�2�C

�2�11 � 2C

�2�13

2

ÿ a�3�

1ÿ �b�3��2 C�3�11

h� C

�3�12

i� C

�3�13 ;

d5 � g�3�

1ÿ �b�3��2 C�3�11

h(� C

�3�12

i

ÿg�2� ln r2 C

�2�11 � C

�2�12

h i� g�2�C

�2�11

2� n�2�r

�ÿ n�3�r

�)DT ;

k61 � k62 � k64 � k65 � 0;

k63 � b�3�C�3�11

�� C

�3�12

�rb�3�ÿ1

a ;

k66 ��ÿ b�3�C

�3�11 � C

�3�12

�rÿb�3�ÿ1

a ;

k67 � a�3�

1ÿ �b�3��2 C�3�11

h� C

�3�12

i� C

�3�13 ;

d6 � ÿ g�3�

1ÿ �b�3��2 C�3�11

h(� C

�3�12

iÿ n�3�r

)DT ;

k71 � 2�b�1�C�1�13 � C�1�23 �

1� b�1�rb�1��1

1

hÿ rb�1��1

0

i;

k72 � C�2�13

�� C

�2�23

�r2

2

ÿ ÿ r21

�;

k73 �2 b�3�C

�3�13 � C

�3�23

� �1� b�3�

rb�3��1a

hÿ rb�3��1

2

i;

k74 �2 ÿ b�1�C

�1�13 � C

�1�23

� �1ÿ b�1�

rÿb�1��11

hÿ rÿb�1��1

0

i;

k75 � 2 C�2�23

�ÿ C

�2�13

�ln�r2=r1�;

k76 �2 ÿ b�3�C

�3�13 � C

�3�23

� �1ÿ b�3�

rÿb�3��1a

hÿ rÿb�3��1

2

i;

k77 �a�1� C

�1�13 � C

�1�23

� �1ÿ �b�1��2

24 � C�1�33

35�r21 ÿ r2

0�

� a�2� C�2�13

�� C

�2�23

��r2

2 ln r2

� ÿ r21 ln r1�=2ÿ �r2

2 ÿ r21�=4

�� a�2�C

�2�13

.2

h� C

�2�33

ir2

2

ÿ ÿ r21

�� a�3��C�3�13 � C

�3�23 �

1ÿ �b�3��2"

� C�3�33

#�r2

a ÿ r22�;

d7 � p0r20 ÿ

g�1� C�1�13 � C

�1�23

� �1ÿ �b�1��2

248<: ÿ n�1�z

35�r21 ÿ r2

0�

� g�2� C�2�13

�� C

�2�23

��r2

2 ln r2

� ÿ r21 ln r1�=2ÿ �r2

2 ÿ r21�=4

�� g�2�C

�2�13 =2

hÿ n�2�z

ir2

2

ÿ ÿ r21

�� g�3��C�3�13 � C

�3�23 �

1ÿ �b�3��2"

ÿ n�3�z

#�r2

a ÿ r22�9=;DT ;

(2) When C�k�22 =C

�k�11 < 0:

M. Xia et al. / Composite Structures 51 (2001) 273±283 281

k11 �hÿ b�1�C

�1�11 sin�b�1� ln r0� � C

�1�12 cos�b�1� ln r0�

i.r0;

k14 � b�1�C�1�11 cos�b�1� ln r0�

h� C

�1�12 sin�b�1� ln r0�

i.r0;

k17 � a�1�

1� �b�1��2 C�1�11

h� C

�1�12

i� C

�1�13 ;

k12 � k13 � k15 � k16;

d1 � ÿp0 ÿ g�1�

1� �b�1��2 C�1�11

h(� C

�1�12

iÿ n�1�r

)DT ;

k21 � cos�b�1� ln r1�;k22 � ÿr1;

k24 � sin�b�1� ln r1�;k25 � ÿ1=r1;

k27 � a�1�r1

1� �b�1��2 ÿa�2�r1

2ln r1;

k23 � k26 � 0;

d2 � g�2�r1

2ln r1

(ÿ g�1�r1

1� �b�1��2)

DT ;

k31 � k34 � 0;

k32 � r2;

k33 � ÿ cos�b�3� ln r2�;k35 � 1=r2;

k36 � ÿ sin�b�3� ln r2�;

k37 � a�2�r2

2ln r2 ÿ a�3�r2

1� �b�3��2 ;

d3 � g�3�r2

1� �b�3��2(

ÿ g�2�r2

2ln r2

)DT ;

k41 �hÿ b�1�C

�1�11 sin�b�1� ln r1� � C

�1�12 cos�b�1� ln r1�

i.r1;

k42 � ÿ C�2�11

�� C

�2�12

�;

k44 � b�1�C�1�11 cos�b�1� ln r1�

h� C

�1�12 sin�b�1� ln r1�

i.r1;

k45 � ÿ C�2�12

�ÿ C

�2�11

�.r2

1;

k47 � a�1�

1� �b�1��2 C�1�11

h� C

�1�12

i� C

�1�13

ÿa�2� ln r1 C

�2�11 � C

�2�12

h i� a�2�C

�2�11 � 2C

�2�13

2;

k43 � k46 � 0;

d4 �g�2� ln r1 C

�2�11 � C

�2�12

h i� g�2�C

�2�11

2

8<:ÿ g�1�

1� �b�1��2 C�1�11

h� C

�1�12

i� n�1�r

�ÿ n�2�r

�9=;DT ;

k51 � k54 � 0;

k52 � C�2�11 � C

�2�12 ;

k53 � b�3�C�3�11 sin�b�3� ln r2�

hÿ C

�3�12 cos�b�3� ln r2�

i.r2;

k55 � C�2�12

�ÿ C�2�11

�.r2

2;

k56 � ÿ b�3�C�3�11 cos�b�3� ln r2�

h� C

�3�12 sin�b�3� ln r2�

i.r2;

k57 �a�2� ln r2 C

�2�11 � C

�2�12

h i� a�2�C

�2�11 � 2C

�2�13

2

ÿ a�3�

1� �b�3��2 C�3�11

h� C

�3�12

i� C

�3�13 ;

d5 � g�3�

1� �b�3��2 C�3�11

h8<: � C�3�12

i

ÿg�2� ln r2 C

�2�11 � C

�2�12

h i� g�2�C

�2�11

2� n�2�r

�ÿ n�3�r

�9=;DT ;

k61 � k62 � k64 � k65 � 0;

k63 �hÿ b�3�C

�3�11 sin�b�3� ln ra� � C

�3�12 cos�b�3� ln ra�

i.ra;

k66 � b�3�C�3�11 cos�b�3� ln ra�

h� C

�3�12 sin�b�3� ln ra�

i.ra;

k67 � a�3�

1� �b�3��2 C�3�11

h� C

�3�12

i� C

�3�13 ;

d6 � ÿ g�3�

1� �b�3��2 C�3�11

h(� C

�3�12

iÿ n�3�r

)DT ;

k71 � 2

1� �b�1��2 b�1� C�1�23

�nÿ C

�1�13

�� r1 sin�b�1� ln r1�h

ÿ r0 sin�b�1� ln r0�i

� C�1�13 �b�1��2

�� C

�1�23

�� r1 cos�b�1� ln r1�h

ÿ r0 cos�b�1� ln r0�io;

k72 � C�2�13

�� C

�2�23

�r2

2

ÿ ÿ r21

�;

k73 � 2

1� �b�3��2 b�3� C�3�23

�nÿ C

�3�13

�� ra sin�b�3� ln ra�h

ÿ r2 sin�b�3� ln r2�i

� C�3�13 �b�3��2

�� C

�3�23

�� ra cos�b�3� ln ra�h

ÿ r2 cos�b�3� ln r2�io;

k74 � 2

1� �b�1��2 C�1�13 �b�1��2

�n� C

�1�23

�� r1 sin�b�1� ln r1�h

ÿ r0 sin�b�1� ln r0�i

� b�1� C�1�13

�ÿ C

�1�23

�� r1 cos�b�1� ln r1�h

ÿ r0 cos�b�1� ln r0�io;

282 M. Xia et al. / Composite Structures 51 (2001) 273±283

k75 � 2 C�2�23

�ÿ C

�2�13

�ln�r2=r1�;

k76 � 2

1� �b�3��2 C�3�13 �b�3��2

�n� C

�3�23

�� ra sin�b�3� ln ra�h

ÿ r2 sin�b�3� ln r2�i

� b�3� C�3�13

�ÿ C

�3�23

�� ra cos�b�3� ln ra�h

ÿ r2 cos�b�3� ln r2�io;

k77 �a�1� C

�1�13 � C

�1�23

h i1� �b�1��2

24 � C�1�33

35� �r2

1 ÿ r20� � a�2� C

�2�13

�� C

�2�23

�� �r2

2 ln r2

� ÿ r21 ln r1�=2ÿ �r2

2 ÿ r21�=4

�� a�2�C

�2�13 =2

h� C

�2�33

ir2

2

ÿ ÿ r21

�� a�3��C�3�13 � C

�3�23 �

1� �b�3��2"

� C�3�33

#�r2

a ÿ r22�;

d7 � p0r20 ÿ

g�1� C�1�13 � C

�1�23

� �1ÿ �b�1��2

248<: ÿ n�1�z

35� �r2

1 ÿ r20� � g�2� C

�2�13

�� C

�2�23

�� �r2

2 ln r2

� ÿ r21 ln r1�=2ÿ �r2

2 ÿ r21�=4

�� g�2�C

�2�13 =2

hÿ n�2�z

ir2

2

ÿ ÿ r21

�� g�3��C�3�13 � C

�3�23 �

1ÿ �b�3��2"

ÿ n�3�z

#�r2

a ÿ r22�9=;DT ;

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