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Design of fiber-reinforced composite pressure vessels under various loading conditions
Composite Structures 58 (2002) 83–95 www.elsevier.com/locate/compstruct
Design of fiber-reinforced composite pressure vessels under various loading conditions Levend Parnas a
a,*
, Nuran Katırcı
b
Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey b 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-based composites have been increasing. Various numbers of applications have also been flourishing with this development. Fuel tanks, rocket motor cases, pipes are some examples of pressure vessels made of composite materials. Ever increasing use of this new class of materials in conventional applications is coupled with problems that are intrinsic to the material itself. Difficulties are many folded. Determination of material properties, mechanical analysis and design, failure of the structure are some examples which all require a non-conventional approach. Numerous applications concurrently are accompanied by various researches in the related field. Majority of the studies in the analysis of composite pressure vessels finds their origins in Lethnitskii’s approach [1]. The application of the theory given in this book is later applied to laminated composite structures in tubular form Tsai [2]. The studies followed consider also different loading and environmental conditions. Recently, there are some studies involved directly with tubes under internal pressure [3,4]. In the study by Xia et al. [4], the *
Corresponding author. E-mail addresses: [email protected] (L. Parnas), katirci@ mgeo.aselsan.com.tr (N. Katırcı).
combined effect of thermomechanical loading in addition to internal pressure is considered. In this study, an analytical procedure is developed to design and predict the behavior of fiber-reinforced composite pressure vessels under combined mechanical and hygrothermal loading. The mechanical part of the analysis is similar to the study given in Ref. [5]. The procedure is based on the classical laminated plate theory. A cylindrical shell having a number of sub-layers, each of which is cylindrically orthotropic, is treated as in the state of plane strain. Internal pressure, axial force, body force due to rotation in addition to temperature and moisture variation throughout the body are considered as loading. In the study of Katırcı [6], these parameters are compared with the experimental results.
2. Formulation of problem A thick-walled multi-layered filament wound cylindrical shell is considered in the analysis based on linear elasticity solution. The following assumptions are made for 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 and orthotropic unit,
0263-8223/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 3 7 - 5
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• the vessel is in state of plane strain and only small strains are considered through the analysis, • the length of the vessel is such that the longitudinal bending deformation due to the end closures of the vessel are limited to only small end portions of the pressure vessel. 2.1. Effective elastic properties A laminated composite with its own effective elastic properties, contains a number of anisotropic plates. When these effective elastic properties of the laminate are used, the body is considered to responding to the applied loads as a single unit. The effective elastic properties of the laminate can be determined using the theory of the laminated plates. The filament wound structures, which is the subject of this study, is assumed to be made of angle-ply laminates. An angle-ply laminate has alternating lamina having þa and a winding angles. Therefore, a filament-wound cylindrical shell, having a wind angle a can be treated as an angle-ply laminate. For multi-layered cylinders, each layer is an angle-ply laminate with its own wind angle. Neglecting the effect of curvature, the effective elastic properties of each of these layers can be formulated as follows. For an angle-ply lamina where fibers are oriented at an angle a with the positive x-axis as shown in Fig. 1, the effective elastic properties are given by 1 cos4 a sin4 a ¼ þ þ Ex0 x0 E11 E22
mx0 y 0
1 m12 2 E11 G12
cos2 a sin2 a
1 m12 2 cos2 a sin2 a E11 G12 1 m12 1 1 m12 1 1 ¼ þ2 þ þ þ2 cos2 2a E11 E22 E11 E22 G12 E11 E11 m12 1 m12 1 1 ¼ Ex0 x0 þ þ2 cos2 a sin2 a E11 E11 E22 G12 E11
1 sin4 a cos4 a ¼ þ þ Ey 0 y 0 E11 E22 1 Gx0 y 0
my 0 x0 ¼
Ey 0 y 0 m x0 y 0 Ex0 x0
ð1Þ
In this study, the elastic constants related to the thickness coordinate however, are assumed as Ez0 z0 ¼ E33 ¼ E22 and Gy 0 z0 ¼ Gx0 z0 ¼ Gx0 y 0 . In a filament-wound pressure vessel, the structure is made-up of several angle lay-ups, each of which acts as an orthotropic unit. The elastic constants of each layer is assumed as equal to effective elastic constants of a balanced and symmetric laminate which has two layers of winding angles (þa) and (a) with equal thicknesses. The generalized Hooke’s law in cylindrical coordinates can be written as fegr;h;z ¼ ½a frgr;h;z
For an angle-ply lamina, due to the (a) configuration, the shear coupling terms are zero. Then the compliance matrix [a] can be represented in cylindrical coordinates as 3 2 1 mrh mrz 0 0 0 7 6 Err Err Err 7 6 7 6 m 1 mhz 6 rh 0 0 0 7 7 6 Ehh Ehh 7 6 Err 7 6 7 6 mrz mhz 1 7 6 0 0 0 7 6 Err Ezz Ehh 7 6 ð3Þ ½a ¼ 6 7 1 7 6 0 0 0 0 0 7 6 Ghz 7 6 7 6 1 7 6 6 0 0 7 0 0 0 7 6 Grz 7 6 4 1 5 0 0 0 0 0 Grh The material properties in cylindrical coordinates can be 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 cylindrical anisotropy Lethnitskii [1] started the formulation with the plane stress condition then the problem is converted to the generalized plane strain problem where axial strain of the system is equal to a constant rather than being zero. The equilibrium equations, disregarding rzz , and the equations of generalized Hooke’s Law for a body in cylindrical anisotropy in cylindrical coordinates are given by the following equations. orrr 1 orrh rrr rhh þ þ þR¼0 r oh or r orrh 1 orhh rrh þ þ2 þH¼0 r oh or r err ¼ a11 rrr þ a12 rhh þ a16 rrh ehh ¼ a12 rrr þ a22 rhh þ a26 rrh
Fig. 1. Global, local and material coordinates.
ð2Þ
ezz ¼ a13 rrr þ a23 rhh þ a36 rrh crh ¼ a16 rrr þ a22 rhh þ a66 rrh
ð4Þ ð5Þ
ð6Þ
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
where R and H are the projections of the body forces along r and h directions, respectively. The axial stress, rzz , of the generalized plane strain problem will be obtained by using generalized Hooke’s Law. The strain– displacement relations for the same body are: our or 1 ouh ur ehh ¼ þ r oh r 1 our ouh uh þ crh ¼ r oh or r err ¼
ð7Þ
By eliminating displacements from Eq. (7), the equation of compatibility is obtained, which is: o2 err o2 ðrehh Þ o2 ðrcrh Þ oerr r ¼0 þ r 2 2 or oroh or oh
ð8Þ
The equilibrium equations given in Eq. (5) are satisfied with the following definition of the stress function, F ðr; hÞ: 2
1 oF 1 oF þ þU r or r2 oh2 o2 F rhh ¼ 2 þ U or o2 F rrh ¼ or oh r rrr ¼
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 case which is satisfied by the stress function F ðr; hÞ, is obtained a22
o4 F o4 F 1 o4 F þ 2a 2a ð þ a Þ 26 12 66 or4 or3 oh r2 or2 oh2 4 4 1 oF 1 oF 1 o3 F 2a16 3 þ a þ 2a 11 22 r or oh3 r4 oh4 r or3 3 3 1 oF 1 oF ð2a13 þ a66 Þ 3 þ 2a16 4 3 r or oh2 r oh 1 o2 F 1 o2 F a11 2 2 þ 2ða16 þ a26 Þ 3 r or r or oh 1 o2 F 1 oF þ 2ða11 þ 2a12 þ a66 Þ 4 2 þ a11 3 r oh r or 1 oF þ 2ða16 þ a26 Þ 4 r oh o2 U 1 o2 U ¼ ða12 þ a22 Þ 2 þ ða16 þ a26 Þ or r or oh 1 o2 U 1 oU ða11 a12 Þ 2 þ ða11 2a22 a12 Þ 2 r oh r or 1 oU ð10Þ þ ða16 þ a26 Þ 2 r oh
where a16 and a26 vanish for a body having a angle-ply layers. After introducing the material properties for the
85
compliances and substituting them into Eq. (10), the following non-homogeneous, fourth order differential equation is obtained for an orthotropic cylindrical body for the state of plane stress. 1 o4 F 1 2mrh 1 o4 F 1 1 o4 F þ þ 2 4 4 2 Ehh or Grh Err r or oh Err r4 oh4 2 1 o3 F 1 mrh 1 o2 F 1 1 o2 F þ 2 3 3 Ehh r or Grh Err r or oh Err r2 or2 1 mr 1 1 o2 F 1 1 oF þ 2 þ þ Grh r4 oh2 Err r3 or Err 1 mhr o2 U 1 mrh 1 o2 U ¼ þ Ehh or2 Err r2 oh2 2 1 þ mrh 1 oU : ð11Þ Ehh r or Err
2.3. Stresses and displacements for a rotating anisotropic cylinder At this point, it is easy to obtain the stress distribution for an anisotropic rotating cylinder. It is assumed that the cylinder is orthotropic, so that any radial plane is an elastic symmetry plane. For a rotating cylinder, the body force potential is given by: U ¼
qx2 2 r 2
ð12Þ
where x is the angular speed, q is the density of the material and r is the radial position. Since the problem is axisymmetric, the stress function F depends only on r. Using this fact and Eq. (12), Eq. (11) can be rearranged for the kth layer as follows: 1 4 d4 F 2 3 d3 F 1 2 d2 F 1 dF r þ r r þ k r k k 4 3 k 2 Err dr Err dr Ehh dr Ehh dr 3 2mr þ 1 ¼ qx2 r4 k Ehh Errk
ð13Þ
Eq. (13) is in the form of Euler’s equation and its solution yields the following expression for the stress function F, F ðrÞ ¼ Ak þ Bk r2 þ C k r1þg1k þ Dk r1g1k q x2 Errk 3 2mkrh e2k þ k r4 k 2 36Errk 4Ehh 1=2
ð14Þ
k =Errk Þ . where ek ¼ ðEhh Using the stress function F ðrÞ and the body force potential U in Eq. (9), the stresses can be obtained as:
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L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
rkrr ¼ 2Bk þ C k 1 þ ek rek 1 þ Dk 1 ek rek 1 qk x2 rkhh
k
¼ 2B þ C
k
3 þ mkhr 9 e2k
1 þ ek ek rek 1 Dk 1 ek ek rek 1 qk x2
! r2
3mkhr e2k 9 e2k
!
ð15Þ
r2
rkrh ¼ 0
The structure is mechanically subjected to a radial body force due to rotation, internal pressure and axial force as shown in Fig. 2. Boundary conditions for the given geometry and loading can be represented as follows, For r ¼ rint ) r1rr ¼ Pint For r ¼ rext ) rnrr ¼ 0
and
and
r1rh ¼ 0
rnrh ¼ 0
At the interface of adjacent layers, the following boundary conditions are applied, r ¼ bk1
k rk1 rr ¼ rrr
)
uk1 ¼ ukr r
and
When the boundary conditions are applied, two unknown coefficients of the stress function, Ak and Bk are obtained as zero, and other coefficients as ( k 1 bk bg k k1 C ¼ gk gk qk 1 þ gk ck ck gk 12gk k b b þ b1gk ðcgk cg k Þ k k1 gk k1 gkk qk1 ck ck " #) gk 32gk k 3 gk 3gk 2 3 þ mhr bk bk1 bk bk1 þ qk x þ bk1 k cgk k cg 9 gk2 k
Dk ¼
1 1 gk
"
ð16Þ gk 1 qk1 bkgk 1 qk bk1 k cgk k cg k
bk bk1
3 þ mkhr 2 gk 1 gk 1 b2k bk1 bk bk1 qk x2 b b 9 gk2 k1 k
# ð17Þ
where ck ¼ bk1 =bk , gk ¼ ðnk11 =nk22 Þ1=2 and k denotes the layer number. The reduced strain coefficients, nkij , extend the plane stress problem into the generalized plane strain problem as proposed by Lekhnitskii [7]. They can be defined for a multi-layered cylinder as, nkij ¼ akij
aki3 akj3 ak33
i; j ¼ 1; 2
where akij are the components of the compliance tensor in cylindrical coordinates for the kth layer. Using the stress function F ðrÞ, layer stresses can be derived as: ! " g 1 k 1 cgk k þ3 r k k 2 2 3 þ mhr rrr ¼ qk x bk k bk 9 gk2 1 c2g k ! gk þ1 2 # gk 3 1 ck bk r þ ckgk þ3 k b r 1 c2g k k !" # g 1 g þ1 qk1 ckgk þ1 r k bk k þ k bk r 1 c2g k !" gk 1 # g þ1 qk bk k r 2gk þ ck 2gk bk r 1 ck ð18Þ rkhh
( 2
"
!
1 ckgk þ3 r gk ¼ 3þ 2gk b 1 ck k ! gk þ1 # gk 3 1 ck bk ckgk þ3 2gk r 1 ck 2 ) r gk2 þ 3mkhr bk !" gk þ1 # g 1 qk1 ckgk 1 gk r k bk þ 2gk bk r 1 ck ! " g þ1 # g 1 qk gk r k bk k 2gk þ ck k bk r 1 c2g k qk x2 bk 9 gk2
mkhr
gk 1
ð19Þ
In Eqs. (18) and (19), symbols qk1 and qk denote the internal and external forces in radial direction acting on the kth layer as given in Fig. 3, and mkhr ¼ nk12 =nk22 . Since the pressure vessel is assumed to be in the state of generalized plane strain, axial strains of all layers is equal to the constant, e0zz . Then the axial stress can be obtained as: rkzz ¼
e0zz 1 k ak13 rkrr þ ak23 rkhh k a33 a33
ð20Þ
The displacements are obtained as follows: ukr ¼ r nk12 rkrr þ nk22 rkhh mkzh e0zz ; ukh ¼ 0 Fig. 2. Mechanical loading on a closed end cylindrical pressure vessel.
and
ukz ¼ ze0zz
where mkzh ¼ ak23 =ak33 .
ð21Þ
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
87
At each layer interface, radial displacements of adjacent layers must be continuous, which follows that: ukr ðbk Þ ¼ ukþ1 r ðbk Þ
ð24Þ
Using Eqs. (23) and (24), a set of simultaneous equations in terms of qk , one for each interface, is determined as:
Fig. 3. Multi-layered cylinder showing layer notation.
uk qkþ1 þ vk qk þ gk qk1 þ x2 kk ! k X n mzh mkþ1 2 zh þ qi1 di þ qi li þ x wi D i¼1 k FA mzh mkþ1 2 2 zh ¼ Pext rext þ Pint rint D p
bk1
k¼1
where rint is the internal radius of the cylinder. Substituting Eqs. (18) and (19) for rkrr and rkhh into Eq. (20) for rkzz and evaluating the integral in Eq. (22), the relation for e0zz is determined as: " FA 1 2 2 Pint rint þ e0zz ¼ Pext rext D p # n X 2 qk1 dk þ qk lk þ x wk ð23Þ
gk ¼
2gk nk12 ckgk þ1 k 1 c2g k
kk ¼
b2k
(
k¼1
where, D ¼ dk ¼
ak33
Pn
k¼1
2 k 1 c2g k
b2k b2k1 ak33
(
and
bk ckgk þ1 1 þ gk
ak13 þ gk ak23 ðbk cgk k bk1 Þ
) bk1 k g a gk ak23 ðbk ck k bk1 Þ 1 gk 13 2 bk k lk ¼ k a13 þ gk ak23 ðbk cgk k bk1 Þ 2gk 1 þ g a33 1 ck k gk bk c k þ ak13 gk ak23 ðbk cgk k bk1 Þ 1 gk ( 2qk b2k 3 þ mkhr ak13 ak23 wk ¼ gk þ 9 gk2 ak33 ak33 ! gk þ1 gk þ1 1 ckgk þ3 bk1 b2k
bk 2gk 1 c k ð 1 þ gk Þ ! k 1 ckgk 3 ckgk þ3 a13 ak23 gk þ k ak33 ak33 1 c2g ð 1 gk Þ k k gk þ1 gk þ1 a ak23 gk2 þ 3mkhr 2 13
bk bk1 bk þ þ ak33 ak33 ð3 þ mkhr Þ ) 1 4
bk b4k1 2 4bk
1
kþ1 k þ1 Þ=ð1 c2g where, uk ¼ ð2gkþ1 nkþ1 kþ1 Þ 22 ckþ1 k gk bk22 1 þ c2g k k uk ¼ b12 þ bkþ1 12 2gk 1 ck 2g gkþ1 bkþ1 1 þ ckþ1kþ1 22 2g 1 ckþ1kþ1
g
For a cylinder with closed ends, the axial equilibrium is satisfied by the following relation, Z bk n X 2 2p rkzz r dr ¼ print ðPint Pext Þ þ FA ð22Þ
ð25Þ
" ! k 1 2ckgk þ3 þ c2g qk nk22 k k gk 3 þ mhr k 9 gk2 1 c2g k # q nkþ1 gkþ1 3 þ mkþ1 gk2 þ 3mkhr b2kþ1 kþ1 22 hr 2 9 gkþ1 ! ) gkþ1 1 2g þ2 2 2ckþ1 ckþ1kþ1 c2kþ1 kþ1 2 gkþ1 þ 3mhr ckþ1
2g 1 ckþ1kþ1
Therefore, the unknown interface pressures, qk , are solved by using Eq. (25), which eventually leads to the complete solution of the elasticity problem. 2.4. Analysis of pressure vessels using thin wall theory The thickness ratio is defined as the ratio between external and internal radii of the pressure vessel. For pressure vessels of thickness ratios less than 1.1, the thin wall analysis can satisfactorily be used. In this theory, the radial stress is assumed to be zero in addition to hoop and axial stresses to be constant through the thickness. The hoop and the axial stresses of a pressure vessel subjected to internal and external pressure, and an axial force can be calculated, respectively, as follows: ðPint Pext Þrint and t ðPint Pext Þrint FA rzz ¼ þ 2t 2print t rhh ¼
where t is the wall thickness of the vessel.
ð26Þ
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L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
2.5. Environmental effects on composite materials
2.6. Hygrothermal degradation
The influence of environmental factors, such as elevated temperature, humidity and corrosive fluids must be taken into consideration since they affect mechanical and physical properties of composite materials resulting in a change of the mechanical performance. The effect of the elevated temperature can be seen in the composite material properties with a decrease in the modulus and strength because of thermal softening. Especially in polymer-based composites, the matrix-dominated properties are more affected then the fiber-dominated properties. For example, the longitudinal strength and modulus of a unidirectional composite specimen remain almost constant but off-axis properties of the same specimen are significantly reduced as the temperature approaches the glass transition temperature of the polymer. When exposed to humid air or water environment, many polymeric matrix composites absorb moisture by instantaneous surface absorption followed by diffusion through the matrix. Analysis of moisture absorption shows that for epoxy and polyester matrix composites, the moisture concentration increases initially with time and approaches an equilibrium (saturation) level after several days of exposure to humid environments [2]. The analysis of composites due to elevated temperature and moisture absorption is called as ‘‘hygrothermal problem’’. It can be solved mainly in three steps: First, the temperature distribution and the moisture content inside the material are calculated. Then from known temperature and moisture distribution, the hygrothermal deformations and stresses are calculated. Finally, the changes in performance due to both affects are determined. The assumptions used through these steps are:
In addition to creating stresses, temperature and moisture degrade the material properties as well. By following the method given by Tsai [2], the non-dimensional temperature T can be defined as,
• temperature and moisture content inside the material vary only in the thickness direction, • thermal conductivity of the material is independent of temperature and moisture level, • the environmental conditions (temperature and moisture level) are constant. The temperature distribution is obtained by using the one dimensional steady-state heat conduction analysis throughout the body. So for the kth layer, one can write: Tk ¼ Tk1 qRk
ð27Þ Pn
where q ¼ ðTint Text Þ= i¼1 Ri and Rk ¼ Kk =hk . Here Tk temperature, hk thickness and Kk are the thermal conductivity of the kth layer, respectively. In this study, the moisture content is taken as constant and equal to the saturated moisture level throughout the material.
T ¼
ðTg Topr Þ ðTg Trm Þ
ð28Þ
where Tg is the glass-transition temperature, Topr is the operation temperature and Trm is the room temperature. It is also assumed that the moisture suppresses the glass transition temperature by a simple moisture shift as, Tg ¼ Tg0 gc
ð29Þ
where Tg0 is the glass-transition temperature at the dry state, g is the temperature shift per unit moisture absorbed and c is the moisture absorption of the structure. The term T is used to empirically fit the fiber and matrix stiffness and strength data as functions of both moisture and temperature. Using empirical fiber and matrix properties, the ply stiffness and strength properties are given here [2], first in terms of stiffness ratios as vf f 0 ðT Þ E11 v0f h ih i 1 a 0 b 0 1 f 0 1 þ ðT Þ E þ ðT Þ g 1 ðT Þ E 1 g0y 0 m y vf f E22 vf h i h i ¼ 0 b 1 E22 0 þ Em 1 g0y Ef0 1 þ ðTvf Þ ðT Þb ðT Þa ðT Þf v0f i h 0 b ih g0 ð1v0 Þ gs ðT Þ ð1vf Þ þ 1 G10 þ sv0 G0 f vf Es f f m ¼ Es0 g0s v0m ðT Þb g0s ð1vf Þ 1 1 þ v0 f 0 þ b 0 ðT Þ G v ðT Þ G
E11 ¼
f
f
f
m
ð30Þ and strength ratios as X vf h ¼ ðT Þ X 0 v0f e X0 vf h E s ð31Þ ¼ ðT Þ X 00 v0f Es0 Y Y0 S d ¼ ¼ ¼ ðT Þ Y 0 Y 00 S 0 where g is the mutual influence coefficient and subscripts f and m denote fiber and matrix, respectively. The constants a, b, d, f and h are determined empirically and the exponent for example on X 0 denotes the values of the corresponding property X obtained at room temperature with 0.5% moisture content. 2.7. Hygrothermal stresses The hygrothermal and mechanical strains can be superposed in strain level to obtain total strains as, mech etot þ ehygr ij ij ¼ eij
ð32Þ
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
or
89
4. Discussion of results
mech etot þ aij DT þ bij c ij ¼ eij
where aij and bij are thermal and moisture expansion coefficients, respectively. Total stresses however, can be obtained using anisotropic stress–strain relations. Total stresses due to hygrothermal and mechanical loads can be written as, 1
tot rtot ij ¼ ½a eij
ð33Þ
2.8. Failure analysis The main reason for performing the stress analysis is to determine the failure behavior of the pressure vessel. Design of a structure or a component is performed by comparing stresses (or strains) created by applied loads with the allowable strength (or strain capacity) of the material [2]. Tsai–Hill, Tsai–Wu [8], Hoffman [9] and 3D-Quadratic Failure Theories [2] are used in this study for comparison and it is seen that 3D-Quadratic Failure Criteria gives the most conservative results for strength.
3. Numerical solution In order to see how structures behave, the numerical results are necessary for a given material, geometry and loading combination. A preliminary design package program is developed using the derived formulation of stresses. In order to determine the burst pressure, the maximum axial force and the maximum angular speed, the performance (load carrying capacity) of the specified composite pressure vessel is taken as the only limiting value. The strength ratio is the ratio between the maximum or ultimate strength and the applied stress. It must be slightly larger than one because of the safety reason. Burst pressure and maximum angular speed are determined by using the first-ply failure criterion and maximum axial force is determined by using the last-ply failure criterion. The winding angle obtained by developed computer program is called as optimum without using any optimization procedure. This is not wrong because only one constraint is taken into account, which has to be maximized in this case and all possible solutions are checked every time to get the winding angle satisfying the constraints. Since the winding angle varies between 0 and 90, layer stresses are obtained for each angle with a step size of 0.1. The strength ratios of the worst layers are compared with each other. Then the angle having the highest strength ratio is taken as the optimum winding angle for the specified loading and geometry conditions.
The design outputs of the computer program are optimum winding angle, burst pressure and maximum angular speed of the vessel for a given material, geometry and loading combination. Also the affects of axial force and hygrothermal forces on burst pressure and angular speed are studied. In each of these analysis, the material used is a graphite-epoxy composite (T300/ N5208). The properties of the unidirectional laminate of this material are given in Table 1. Note that the residual stresses due to material itself are not considered in this study. 4.1. Optimum winding angle In literature [5], the optimum winding angle for filament wound composite pressure vessels is given as 54.74 by netting analysis. Using the current procedure for the internal pressure loading, the optimum winding angle is obtained as ranging between 52.1 and 54.2 depending on geometry and failure criteria used. 3DQuadratic Failure Criterion always gives greater optimum winding angle than other theories, because the circumferential stress or strain is more effective in this criterion. If angular speed is applied at the same time with the internal pressure, the optimum winding angle values obtained for the pure internal pressure case are increased and shifted to 90. 4.2. Stress distribution The stress distribution through the thickness of a filament wound vessel is not uniform but varying depending on the geometry and loading. The stress graphs for pure internal pressure and pure angular speed cases with a constant winding angle of 53, are given in Figs. 4 and 5, respectively. The symbols a and b represent
Table 1 Properties of unidirectional laminate (T300/N5208) Elastic modulus in fiber direction (GPa) Elastic modulus in matrix direction (GPa) In plane shear modulus (GPa) Major Poisson’s ratio Ultimate tensile strength in fiber direction (MPa) Ultimate compressive strength in fiber direction (MPa) Ultimate tensile strength in matrix direction (MPa) Ultimate compressive strength in matrix direction (MPa) Ultimate in-plane shear strength (MPa) Thermal expansion coefficient in fiber direction (106 /C) Thermal expansion coefficient in matrix direction (106 /C) Moisture expansion coefficient in fiber direction Moisture expansion coefficient in matrix direction Thermal conductivity normal to the thickness direction (W/m per C)
181 10.3 7.17 0.28 1500 1500 40 146 68 0.02 22.5 0 0.6 0.865
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Fig. 4. Radial, axial and circumferential stress distributions for pure internal pressure.
internal and external radii of the tube, respectively. The stresses are normalized with the values of thin-wall solution. In Fig. 4, the results for radial stresses and to some degree with hoop stresses are similar with the ones of Ref. [5]. However, the axial stress distributions show a considerable difference. The errors in the formulation of Ref. [5] would be a reason for the difference; another reason might be originated from the fact that a variable winding angle (54–56) was considered in Ref. [5], contrary to the constant winding angle used in the current study. As it can be seen in Fig. 5, the thin-wall analysis gives only an average result. When thickness increases, the thick-wall analysis has to be used, instead. Radial stress is zero at the inside and outside of the pressure vessel and positive through the thickness. Its maximum value is approximately at the mid-point of the thickness.
When the wall thickness is increased, the layer having maximum radial stress becomes closer to the inner boundary of the pressure vessel. Larger circumferential and axial stresses are obtained at the inner layers of the vessel. In Fig. 5, it is seen that axial stress always changes sign at the point where the radial stress reaches to a maximum. 4.3. Thick- and thin-walled solutions for burst pressure The corresponding burst pressure values are obtained using an iterative procedure where the loading is increased until the failure of a single layer. In Fig. 6, the burst pressure, Pburst , by using both thin and thick-walled solution techniques are plotted versus winding angle. The burst pressure is normalized with that of 0 winding angle. For the thin-wall tube, both thick and thin-wall
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91
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].
solutions predict almost the same burst pressure. Actually, since the thin-wall solution neglects the radial stress, burst pressure values obtained with the thin-
walled analysis are slightly higher than those obtained with thick-walled analysis. The agreement between these two solutions is satisfied except for the values near the
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Fig. 7. Variation of burst pressure with increasing winding angle [rext =rint ¼ 1:4].
Fig. 8. Burst pressure for increasing wall thickness.
optimum winding angle where they differ for thick and thin-wall solutions. For a thick pressure vessel (Fig. 7), there is a significant difference between thin-wall and thick-wall solutions especially near the optimum winding angle. In this wall thickness value (rext =rint ¼ 1:40), the thick-wall solution gives higher burst pressure than the thin-wall solution between angles 48 and 64. To check the limiting value of the thin-wall solution on the wall thickness, burst pressures are calculated for increasing wall thickness Thin-wall and thick-wall solutions yield very similar burst pressure values up to the thickness ratio of rext =rint ¼ 1:1. For thicknesses with rext =rint P 1:1, the deviation between thick and thin-wall solutions becomes larger (Fig. 8). 4.4. Effect of angular speed and axial force on burst pressure The effect of angular speed and axial force on the burst pressure can be seen in Fig. 9. The burst pressure decreases with angular speed, when the winding angle is less than its optimum value. It is not an expected result, since the burst pressure increases with the speed for angles greater than the optimum winding angle. The
burst pressure increases in negligible amount for increasing axial tensile force for angles smaller than the optimum winding angle. If the winding angle of the structure is larger than its optimum value, the burst pressure always decreases with increasing axial force. 4.5. Maximum angular speed The effect of wall thickness and winding angle on the maximum angular speed can be seen in Fig. 10. If only angular speed is applied, the optimum winding angle is obtained as ranging between 81 and 83 depending on the wall thickness of the structure. For small winding angles up to 30, the thin and thick-wall constructions give almost the same maximum angular speed. As the wall thickness increases, the maximum angular speed decreases opposite to the case of burst pressure. It is an expected result, since an increasing wall thickness means more inertia that affects the speed in the negative sense. 4.6. Hygrothermal stresses and strains Hygrothermal stresses in the macro-mechanical level calculated by using the laminated plate theory. In order to assess the effects of residual stresses on the failure of
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
93
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.
composite materials, the hygrothermal expansion coefficients have to be determined correctly. As a sample calculation of hygrothermal stresses, the data given in Table 2 for T300/N5208 [2] is used where DF is the degradation factor.
Table 2 Hygrothermal effects on burst pressure [rext =rint ¼ 1:05, a ¼ 53] Topr (C) c (%) DF (%) Pburst (MPa)
22 0 0 18.3
22 0 0 20.3
122 0 0 16.3
122 0 10 14.4
122 1 30 12.1
Since the curing temperature is the stress free state for composite materials, the operation temperature affects the failure of the composite depending on whether it is below or above the curing temperature. If the operation temperature is less than zero or if it is less than the curing temperature, the burst pressure is increased since the thermal strains and mechanical strains for pure internal pressure case work in opposite senses. It should be pointed out that, the negative temperature for constant moisture content also cause an increase in the mechanical properties of the composite material. It can be concluded that if the operation temperature is less than the curing temperature, burst pressure is increased. If operation temperature is greater than the curing
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Fig. 11. Effect of temperature on burst pressure for different operating temperatures.
temperature, however, both thermal and mechanical strains have a cumulative effect. This can be seen as a decrease in burst pressure values in Fig. 11. Although, the performance of the composite material is negatively influenced by the presence of moisture, it creates less residual strains compared with the thermal ones and does not change the burst pressure, significantly.
5. Conclusion An analytical procedure is developed to assess the behavior of a cylindrical composite structure under loading conditions particular to a rocket motor case. Available loading conditions are internal pressure, axial force and body force due to rotation. Additionally, temperature and moisture variations throughout the body are considered in the analysis. The procedure is based on the classical laminated plate theory. It models the plane strain state of the cylindrical body, which consists of a number of cylindrical sub-layers. The cylindrical pressure vessel is analyzed using two approaches, which are thin wall and thick wall solutions. It is shown that for composite pressure vessels with a ratio of outer to inner radius, up to 1.1, two approaches give similar results in terms of the optimum winding angle, the burst pressure, etc. As the ratio increases, the thick wall analysis is required. The optimum winding angle for the thick-wall pressure vessel analysis with the pure internal pressure loading case is obtained as ranging between 52.1 and 54.1 degrees depending on the material type. If the angular speed is applied at the same time with the internal pressure, optimum winding angle values obtained for the pure internal pressure case are shifted towards 90. The influence of the axial force is, however, opposite to the one of the angular speed. The addition of the axial force has a decreasing effect on the winding angle.
The burst pressure value is greatly depends on the analysis type used. The deviation between thin and thick-wall solutions is quite large especially near the optimum winding angle. As the wall thickness is increased, the thick-wall solution gives almost 30% higher burst pressure values. Therefore, the thin-wall analysis is said to be an average but a safe analysis. If angular speed is applied, the maximum stress occurs in the hoop direction. The optimum winding angle of the analyzed body for this type of loading is obtained as ranging between 81 and 83. The value of the maximum angular speed that the system can be rotated is greatly affected by the thickness of the pressure vessel. Hygrothermal effects are analyzed in this study in two levels. The effect of temperature and moisture to the performance of the materials is determined by using the micromechanics of the composite materials. By taking a linear variation of temperature and constant value for moisture content throughout the body, hygrothermal stresses and strains are determined. Since the thermal and moisture expansion coefficients of the materials have to be determined experimentally, always some amount of error is expected in these calculations. If the material has a tendency of expanding due to a positive temperature difference, the increasing operating temperature is shown to reduce the mechanical performance of the system.
References [1] Lekhnitskii SG. Anisotropic plates. New York, NY: Gordon and 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.
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95 [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 vessels. M.S. Thesis, Middle East Technical University, Turkey, 1998.
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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 and design. New York, NY: Marcel Dekker, Inc; 1988. [9] Lee SM. Reference book for composite technology-2. Technomic Publishing Company; 1989.
Design of fiber-reinforced composite pressure vessels under various loading conditions Levend Parnas a
a,*
, Nuran Katırcı
b
Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey b 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-based composites have been increasing. Various numbers of applications have also been flourishing with this development. Fuel tanks, rocket motor cases, pipes are some examples of pressure vessels made of composite materials. Ever increasing use of this new class of materials in conventional applications is coupled with problems that are intrinsic to the material itself. Difficulties are many folded. Determination of material properties, mechanical analysis and design, failure of the structure are some examples which all require a non-conventional approach. Numerous applications concurrently are accompanied by various researches in the related field. Majority of the studies in the analysis of composite pressure vessels finds their origins in Lethnitskii’s approach [1]. The application of the theory given in this book is later applied to laminated composite structures in tubular form Tsai [2]. The studies followed consider also different loading and environmental conditions. Recently, there are some studies involved directly with tubes under internal pressure [3,4]. In the study by Xia et al. [4], the *
Corresponding author. E-mail addresses: [email protected] (L. Parnas), katirci@ mgeo.aselsan.com.tr (N. Katırcı).
combined effect of thermomechanical loading in addition to internal pressure is considered. In this study, an analytical procedure is developed to design and predict the behavior of fiber-reinforced composite pressure vessels under combined mechanical and hygrothermal loading. The mechanical part of the analysis is similar to the study given in Ref. [5]. The procedure is based on the classical laminated plate theory. A cylindrical shell having a number of sub-layers, each of which is cylindrically orthotropic, is treated as in the state of plane strain. Internal pressure, axial force, body force due to rotation in addition to temperature and moisture variation throughout the body are considered as loading. In the study of Katırcı [6], these parameters are compared with the experimental results.
2. Formulation of problem A thick-walled multi-layered filament wound cylindrical shell is considered in the analysis based on linear elasticity solution. The following assumptions are made for 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 and orthotropic unit,
0263-8223/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 3 7 - 5
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• the vessel is in state of plane strain and only small strains are considered through the analysis, • the length of the vessel is such that the longitudinal bending deformation due to the end closures of the vessel are limited to only small end portions of the pressure vessel. 2.1. Effective elastic properties A laminated composite with its own effective elastic properties, contains a number of anisotropic plates. When these effective elastic properties of the laminate are used, the body is considered to responding to the applied loads as a single unit. The effective elastic properties of the laminate can be determined using the theory of the laminated plates. The filament wound structures, which is the subject of this study, is assumed to be made of angle-ply laminates. An angle-ply laminate has alternating lamina having þa and a winding angles. Therefore, a filament-wound cylindrical shell, having a wind angle a can be treated as an angle-ply laminate. For multi-layered cylinders, each layer is an angle-ply laminate with its own wind angle. Neglecting the effect of curvature, the effective elastic properties of each of these layers can be formulated as follows. For an angle-ply lamina where fibers are oriented at an angle a with the positive x-axis as shown in Fig. 1, the effective elastic properties are given by 1 cos4 a sin4 a ¼ þ þ Ex0 x0 E11 E22
mx0 y 0
1 m12 2 E11 G12
cos2 a sin2 a
1 m12 2 cos2 a sin2 a E11 G12 1 m12 1 1 m12 1 1 ¼ þ2 þ þ þ2 cos2 2a E11 E22 E11 E22 G12 E11 E11 m12 1 m12 1 1 ¼ Ex0 x0 þ þ2 cos2 a sin2 a E11 E11 E22 G12 E11
1 sin4 a cos4 a ¼ þ þ Ey 0 y 0 E11 E22 1 Gx0 y 0
my 0 x0 ¼
Ey 0 y 0 m x0 y 0 Ex0 x0
ð1Þ
In this study, the elastic constants related to the thickness coordinate however, are assumed as Ez0 z0 ¼ E33 ¼ E22 and Gy 0 z0 ¼ Gx0 z0 ¼ Gx0 y 0 . In a filament-wound pressure vessel, the structure is made-up of several angle lay-ups, each of which acts as an orthotropic unit. The elastic constants of each layer is assumed as equal to effective elastic constants of a balanced and symmetric laminate which has two layers of winding angles (þa) and (a) with equal thicknesses. The generalized Hooke’s law in cylindrical coordinates can be written as fegr;h;z ¼ ½a frgr;h;z
For an angle-ply lamina, due to the (a) configuration, the shear coupling terms are zero. Then the compliance matrix [a] can be represented in cylindrical coordinates as 3 2 1 mrh mrz 0 0 0 7 6 Err Err Err 7 6 7 6 m 1 mhz 6 rh 0 0 0 7 7 6 Ehh Ehh 7 6 Err 7 6 7 6 mrz mhz 1 7 6 0 0 0 7 6 Err Ezz Ehh 7 6 ð3Þ ½a ¼ 6 7 1 7 6 0 0 0 0 0 7 6 Ghz 7 6 7 6 1 7 6 6 0 0 7 0 0 0 7 6 Grz 7 6 4 1 5 0 0 0 0 0 Grh The material properties in cylindrical coordinates can be 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 cylindrical anisotropy Lethnitskii [1] started the formulation with the plane stress condition then the problem is converted to the generalized plane strain problem where axial strain of the system is equal to a constant rather than being zero. The equilibrium equations, disregarding rzz , and the equations of generalized Hooke’s Law for a body in cylindrical anisotropy in cylindrical coordinates are given by the following equations. orrr 1 orrh rrr rhh þ þ þR¼0 r oh or r orrh 1 orhh rrh þ þ2 þH¼0 r oh or r err ¼ a11 rrr þ a12 rhh þ a16 rrh ehh ¼ a12 rrr þ a22 rhh þ a26 rrh
Fig. 1. Global, local and material coordinates.
ð2Þ
ezz ¼ a13 rrr þ a23 rhh þ a36 rrh crh ¼ a16 rrr þ a22 rhh þ a66 rrh
ð4Þ ð5Þ
ð6Þ
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
where R and H are the projections of the body forces along r and h directions, respectively. The axial stress, rzz , of the generalized plane strain problem will be obtained by using generalized Hooke’s Law. The strain– displacement relations for the same body are: our or 1 ouh ur ehh ¼ þ r oh r 1 our ouh uh þ crh ¼ r oh or r err ¼
ð7Þ
By eliminating displacements from Eq. (7), the equation of compatibility is obtained, which is: o2 err o2 ðrehh Þ o2 ðrcrh Þ oerr r ¼0 þ r 2 2 or oroh or oh
ð8Þ
The equilibrium equations given in Eq. (5) are satisfied with the following definition of the stress function, F ðr; hÞ: 2
1 oF 1 oF þ þU r or r2 oh2 o2 F rhh ¼ 2 þ U or o2 F rrh ¼ or oh r rrr ¼
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 case which is satisfied by the stress function F ðr; hÞ, is obtained a22
o4 F o4 F 1 o4 F þ 2a 2a ð þ a Þ 26 12 66 or4 or3 oh r2 or2 oh2 4 4 1 oF 1 oF 1 o3 F 2a16 3 þ a þ 2a 11 22 r or oh3 r4 oh4 r or3 3 3 1 oF 1 oF ð2a13 þ a66 Þ 3 þ 2a16 4 3 r or oh2 r oh 1 o2 F 1 o2 F a11 2 2 þ 2ða16 þ a26 Þ 3 r or r or oh 1 o2 F 1 oF þ 2ða11 þ 2a12 þ a66 Þ 4 2 þ a11 3 r oh r or 1 oF þ 2ða16 þ a26 Þ 4 r oh o2 U 1 o2 U ¼ ða12 þ a22 Þ 2 þ ða16 þ a26 Þ or r or oh 1 o2 U 1 oU ða11 a12 Þ 2 þ ða11 2a22 a12 Þ 2 r oh r or 1 oU ð10Þ þ ða16 þ a26 Þ 2 r oh
where a16 and a26 vanish for a body having a angle-ply layers. After introducing the material properties for the
85
compliances and substituting them into Eq. (10), the following non-homogeneous, fourth order differential equation is obtained for an orthotropic cylindrical body for the state of plane stress. 1 o4 F 1 2mrh 1 o4 F 1 1 o4 F þ þ 2 4 4 2 Ehh or Grh Err r or oh Err r4 oh4 2 1 o3 F 1 mrh 1 o2 F 1 1 o2 F þ 2 3 3 Ehh r or Grh Err r or oh Err r2 or2 1 mr 1 1 o2 F 1 1 oF þ 2 þ þ Grh r4 oh2 Err r3 or Err 1 mhr o2 U 1 mrh 1 o2 U ¼ þ Ehh or2 Err r2 oh2 2 1 þ mrh 1 oU : ð11Þ Ehh r or Err
2.3. Stresses and displacements for a rotating anisotropic cylinder At this point, it is easy to obtain the stress distribution for an anisotropic rotating cylinder. It is assumed that the cylinder is orthotropic, so that any radial plane is an elastic symmetry plane. For a rotating cylinder, the body force potential is given by: U ¼
qx2 2 r 2
ð12Þ
where x is the angular speed, q is the density of the material and r is the radial position. Since the problem is axisymmetric, the stress function F depends only on r. Using this fact and Eq. (12), Eq. (11) can be rearranged for the kth layer as follows: 1 4 d4 F 2 3 d3 F 1 2 d2 F 1 dF r þ r r þ k r k k 4 3 k 2 Err dr Err dr Ehh dr Ehh dr 3 2mr þ 1 ¼ qx2 r4 k Ehh Errk
ð13Þ
Eq. (13) is in the form of Euler’s equation and its solution yields the following expression for the stress function F, F ðrÞ ¼ Ak þ Bk r2 þ C k r1þg1k þ Dk r1g1k q x2 Errk 3 2mkrh e2k þ k r4 k 2 36Errk 4Ehh 1=2
ð14Þ
k =Errk Þ . where ek ¼ ðEhh Using the stress function F ðrÞ and the body force potential U in Eq. (9), the stresses can be obtained as:
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rkrr ¼ 2Bk þ C k 1 þ ek rek 1 þ Dk 1 ek rek 1 qk x2 rkhh
k
¼ 2B þ C
k
3 þ mkhr 9 e2k
1 þ ek ek rek 1 Dk 1 ek ek rek 1 qk x2
! r2
3mkhr e2k 9 e2k
!
ð15Þ
r2
rkrh ¼ 0
The structure is mechanically subjected to a radial body force due to rotation, internal pressure and axial force as shown in Fig. 2. Boundary conditions for the given geometry and loading can be represented as follows, For r ¼ rint ) r1rr ¼ Pint For r ¼ rext ) rnrr ¼ 0
and
and
r1rh ¼ 0
rnrh ¼ 0
At the interface of adjacent layers, the following boundary conditions are applied, r ¼ bk1
k rk1 rr ¼ rrr
)
uk1 ¼ ukr r
and
When the boundary conditions are applied, two unknown coefficients of the stress function, Ak and Bk are obtained as zero, and other coefficients as ( k 1 bk bg k k1 C ¼ gk gk qk 1 þ gk ck ck gk 12gk k b b þ b1gk ðcgk cg k Þ k k1 gk k1 gkk qk1 ck ck " #) gk 32gk k 3 gk 3gk 2 3 þ mhr bk bk1 bk bk1 þ qk x þ bk1 k cgk k cg 9 gk2 k
Dk ¼
1 1 gk
"
ð16Þ gk 1 qk1 bkgk 1 qk bk1 k cgk k cg k
bk bk1
3 þ mkhr 2 gk 1 gk 1 b2k bk1 bk bk1 qk x2 b b 9 gk2 k1 k
# ð17Þ
where ck ¼ bk1 =bk , gk ¼ ðnk11 =nk22 Þ1=2 and k denotes the layer number. The reduced strain coefficients, nkij , extend the plane stress problem into the generalized plane strain problem as proposed by Lekhnitskii [7]. They can be defined for a multi-layered cylinder as, nkij ¼ akij
aki3 akj3 ak33
i; j ¼ 1; 2
where akij are the components of the compliance tensor in cylindrical coordinates for the kth layer. Using the stress function F ðrÞ, layer stresses can be derived as: ! " g 1 k 1 cgk k þ3 r k k 2 2 3 þ mhr rrr ¼ qk x bk k bk 9 gk2 1 c2g k ! gk þ1 2 # gk 3 1 ck bk r þ ckgk þ3 k b r 1 c2g k k !" # g 1 g þ1 qk1 ckgk þ1 r k bk k þ k bk r 1 c2g k !" gk 1 # g þ1 qk bk k r 2gk þ ck 2gk bk r 1 ck ð18Þ rkhh
( 2
"
!
1 ckgk þ3 r gk ¼ 3þ 2gk b 1 ck k ! gk þ1 # gk 3 1 ck bk ckgk þ3 2gk r 1 ck 2 ) r gk2 þ 3mkhr bk !" gk þ1 # g 1 qk1 ckgk 1 gk r k bk þ 2gk bk r 1 ck ! " g þ1 # g 1 qk gk r k bk k 2gk þ ck k bk r 1 c2g k qk x2 bk 9 gk2
mkhr
gk 1
ð19Þ
In Eqs. (18) and (19), symbols qk1 and qk denote the internal and external forces in radial direction acting on the kth layer as given in Fig. 3, and mkhr ¼ nk12 =nk22 . Since the pressure vessel is assumed to be in the state of generalized plane strain, axial strains of all layers is equal to the constant, e0zz . Then the axial stress can be obtained as: rkzz ¼
e0zz 1 k ak13 rkrr þ ak23 rkhh k a33 a33
ð20Þ
The displacements are obtained as follows: ukr ¼ r nk12 rkrr þ nk22 rkhh mkzh e0zz ; ukh ¼ 0 Fig. 2. Mechanical loading on a closed end cylindrical pressure vessel.
and
ukz ¼ ze0zz
where mkzh ¼ ak23 =ak33 .
ð21Þ
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95
87
At each layer interface, radial displacements of adjacent layers must be continuous, which follows that: ukr ðbk Þ ¼ ukþ1 r ðbk Þ
ð24Þ
Using Eqs. (23) and (24), a set of simultaneous equations in terms of qk , one for each interface, is determined as:
Fig. 3. Multi-layered cylinder showing layer notation.
uk qkþ1 þ vk qk þ gk qk1 þ x2 kk ! k X n mzh mkþ1 2 zh þ qi1 di þ qi li þ x wi D i¼1 k FA mzh mkþ1 2 2 zh ¼ Pext rext þ Pint rint D p
bk1
k¼1
where rint is the internal radius of the cylinder. Substituting Eqs. (18) and (19) for rkrr and rkhh into Eq. (20) for rkzz and evaluating the integral in Eq. (22), the relation for e0zz is determined as: " FA 1 2 2 Pint rint þ e0zz ¼ Pext rext D p # n X 2 qk1 dk þ qk lk þ x wk ð23Þ
gk ¼
2gk nk12 ckgk þ1 k 1 c2g k
kk ¼
b2k
(
k¼1
where, D ¼ dk ¼
ak33
Pn
k¼1
2 k 1 c2g k
b2k b2k1 ak33
(
and
bk ckgk þ1 1 þ gk
ak13 þ gk ak23 ðbk cgk k bk1 Þ
) bk1 k g a gk ak23 ðbk ck k bk1 Þ 1 gk 13 2 bk k lk ¼ k a13 þ gk ak23 ðbk cgk k bk1 Þ 2gk 1 þ g a33 1 ck k gk bk c k þ ak13 gk ak23 ðbk cgk k bk1 Þ 1 gk ( 2qk b2k 3 þ mkhr ak13 ak23 wk ¼ gk þ 9 gk2 ak33 ak33 ! gk þ1 gk þ1 1 ckgk þ3 bk1 b2k
bk 2gk 1 c k ð 1 þ gk Þ ! k 1 ckgk 3 ckgk þ3 a13 ak23 gk þ k ak33 ak33 1 c2g ð 1 gk Þ k k gk þ1 gk þ1 a ak23 gk2 þ 3mkhr 2 13
bk bk1 bk þ þ ak33 ak33 ð3 þ mkhr Þ ) 1 4
bk b4k1 2 4bk
1
kþ1 k þ1 Þ=ð1 c2g where, uk ¼ ð2gkþ1 nkþ1 kþ1 Þ 22 ckþ1 k gk bk22 1 þ c2g k k uk ¼ b12 þ bkþ1 12 2gk 1 ck 2g gkþ1 bkþ1 1 þ ckþ1kþ1 22 2g 1 ckþ1kþ1
g
For a cylinder with closed ends, the axial equilibrium is satisfied by the following relation, Z bk n X 2 2p rkzz r dr ¼ print ðPint Pext Þ þ FA ð22Þ
ð25Þ
" ! k 1 2ckgk þ3 þ c2g qk nk22 k k gk 3 þ mhr k 9 gk2 1 c2g k # q nkþ1 gkþ1 3 þ mkþ1 gk2 þ 3mkhr b2kþ1 kþ1 22 hr 2 9 gkþ1 ! ) gkþ1 1 2g þ2 2 2ckþ1 ckþ1kþ1 c2kþ1 kþ1 2 gkþ1 þ 3mhr ckþ1
2g 1 ckþ1kþ1
Therefore, the unknown interface pressures, qk , are solved by using Eq. (25), which eventually leads to the complete solution of the elasticity problem. 2.4. Analysis of pressure vessels using thin wall theory The thickness ratio is defined as the ratio between external and internal radii of the pressure vessel. For pressure vessels of thickness ratios less than 1.1, the thin wall analysis can satisfactorily be used. In this theory, the radial stress is assumed to be zero in addition to hoop and axial stresses to be constant through the thickness. The hoop and the axial stresses of a pressure vessel subjected to internal and external pressure, and an axial force can be calculated, respectively, as follows: ðPint Pext Þrint and t ðPint Pext Þrint FA rzz ¼ þ 2t 2print t rhh ¼
where t is the wall thickness of the vessel.
ð26Þ
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2.5. Environmental effects on composite materials
2.6. Hygrothermal degradation
The influence of environmental factors, such as elevated temperature, humidity and corrosive fluids must be taken into consideration since they affect mechanical and physical properties of composite materials resulting in a change of the mechanical performance. The effect of the elevated temperature can be seen in the composite material properties with a decrease in the modulus and strength because of thermal softening. Especially in polymer-based composites, the matrix-dominated properties are more affected then the fiber-dominated properties. For example, the longitudinal strength and modulus of a unidirectional composite specimen remain almost constant but off-axis properties of the same specimen are significantly reduced as the temperature approaches the glass transition temperature of the polymer. When exposed to humid air or water environment, many polymeric matrix composites absorb moisture by instantaneous surface absorption followed by diffusion through the matrix. Analysis of moisture absorption shows that for epoxy and polyester matrix composites, the moisture concentration increases initially with time and approaches an equilibrium (saturation) level after several days of exposure to humid environments [2]. The analysis of composites due to elevated temperature and moisture absorption is called as ‘‘hygrothermal problem’’. It can be solved mainly in three steps: First, the temperature distribution and the moisture content inside the material are calculated. Then from known temperature and moisture distribution, the hygrothermal deformations and stresses are calculated. Finally, the changes in performance due to both affects are determined. The assumptions used through these steps are:
In addition to creating stresses, temperature and moisture degrade the material properties as well. By following the method given by Tsai [2], the non-dimensional temperature T can be defined as,
• temperature and moisture content inside the material vary only in the thickness direction, • thermal conductivity of the material is independent of temperature and moisture level, • the environmental conditions (temperature and moisture level) are constant. The temperature distribution is obtained by using the one dimensional steady-state heat conduction analysis throughout the body. So for the kth layer, one can write: Tk ¼ Tk1 qRk
ð27Þ Pn
where q ¼ ðTint Text Þ= i¼1 Ri and Rk ¼ Kk =hk . Here Tk temperature, hk thickness and Kk are the thermal conductivity of the kth layer, respectively. In this study, the moisture content is taken as constant and equal to the saturated moisture level throughout the material.
T ¼
ðTg Topr Þ ðTg Trm Þ
ð28Þ
where Tg is the glass-transition temperature, Topr is the operation temperature and Trm is the room temperature. It is also assumed that the moisture suppresses the glass transition temperature by a simple moisture shift as, Tg ¼ Tg0 gc
ð29Þ
where Tg0 is the glass-transition temperature at the dry state, g is the temperature shift per unit moisture absorbed and c is the moisture absorption of the structure. The term T is used to empirically fit the fiber and matrix stiffness and strength data as functions of both moisture and temperature. Using empirical fiber and matrix properties, the ply stiffness and strength properties are given here [2], first in terms of stiffness ratios as vf f 0 ðT Þ E11 v0f h ih i 1 a 0 b 0 1 f 0 1 þ ðT Þ E þ ðT Þ g 1 ðT Þ E 1 g0y 0 m y vf f E22 vf h i h i ¼ 0 b 1 E22 0 þ Em 1 g0y Ef0 1 þ ðTvf Þ ðT Þb ðT Þa ðT Þf v0f i h 0 b ih g0 ð1v0 Þ gs ðT Þ ð1vf Þ þ 1 G10 þ sv0 G0 f vf Es f f m ¼ Es0 g0s v0m ðT Þb g0s ð1vf Þ 1 1 þ v0 f 0 þ b 0 ðT Þ G v ðT Þ G
E11 ¼
f
f
f
m
ð30Þ and strength ratios as X vf h ¼ ðT Þ X 0 v0f e X0 vf h E s ð31Þ ¼ ðT Þ X 00 v0f Es0 Y Y0 S d ¼ ¼ ¼ ðT Þ Y 0 Y 00 S 0 where g is the mutual influence coefficient and subscripts f and m denote fiber and matrix, respectively. The constants a, b, d, f and h are determined empirically and the exponent for example on X 0 denotes the values of the corresponding property X obtained at room temperature with 0.5% moisture content. 2.7. Hygrothermal stresses The hygrothermal and mechanical strains can be superposed in strain level to obtain total strains as, mech etot þ ehygr ij ij ¼ eij
ð32Þ
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or
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4. Discussion of results
mech etot þ aij DT þ bij c ij ¼ eij
where aij and bij are thermal and moisture expansion coefficients, respectively. Total stresses however, can be obtained using anisotropic stress–strain relations. Total stresses due to hygrothermal and mechanical loads can be written as, 1
tot rtot ij ¼ ½a eij
ð33Þ
2.8. Failure analysis The main reason for performing the stress analysis is to determine the failure behavior of the pressure vessel. Design of a structure or a component is performed by comparing stresses (or strains) created by applied loads with the allowable strength (or strain capacity) of the material [2]. Tsai–Hill, Tsai–Wu [8], Hoffman [9] and 3D-Quadratic Failure Theories [2] are used in this study for comparison and it is seen that 3D-Quadratic Failure Criteria gives the most conservative results for strength.
3. Numerical solution In order to see how structures behave, the numerical results are necessary for a given material, geometry and loading combination. A preliminary design package program is developed using the derived formulation of stresses. In order to determine the burst pressure, the maximum axial force and the maximum angular speed, the performance (load carrying capacity) of the specified composite pressure vessel is taken as the only limiting value. The strength ratio is the ratio between the maximum or ultimate strength and the applied stress. It must be slightly larger than one because of the safety reason. Burst pressure and maximum angular speed are determined by using the first-ply failure criterion and maximum axial force is determined by using the last-ply failure criterion. The winding angle obtained by developed computer program is called as optimum without using any optimization procedure. This is not wrong because only one constraint is taken into account, which has to be maximized in this case and all possible solutions are checked every time to get the winding angle satisfying the constraints. Since the winding angle varies between 0 and 90, layer stresses are obtained for each angle with a step size of 0.1. The strength ratios of the worst layers are compared with each other. Then the angle having the highest strength ratio is taken as the optimum winding angle for the specified loading and geometry conditions.
The design outputs of the computer program are optimum winding angle, burst pressure and maximum angular speed of the vessel for a given material, geometry and loading combination. Also the affects of axial force and hygrothermal forces on burst pressure and angular speed are studied. In each of these analysis, the material used is a graphite-epoxy composite (T300/ N5208). The properties of the unidirectional laminate of this material are given in Table 1. Note that the residual stresses due to material itself are not considered in this study. 4.1. Optimum winding angle In literature [5], the optimum winding angle for filament wound composite pressure vessels is given as 54.74 by netting analysis. Using the current procedure for the internal pressure loading, the optimum winding angle is obtained as ranging between 52.1 and 54.2 depending on geometry and failure criteria used. 3DQuadratic Failure Criterion always gives greater optimum winding angle than other theories, because the circumferential stress or strain is more effective in this criterion. If angular speed is applied at the same time with the internal pressure, the optimum winding angle values obtained for the pure internal pressure case are increased and shifted to 90. 4.2. Stress distribution The stress distribution through the thickness of a filament wound vessel is not uniform but varying depending on the geometry and loading. The stress graphs for pure internal pressure and pure angular speed cases with a constant winding angle of 53, are given in Figs. 4 and 5, respectively. The symbols a and b represent
Table 1 Properties of unidirectional laminate (T300/N5208) Elastic modulus in fiber direction (GPa) Elastic modulus in matrix direction (GPa) In plane shear modulus (GPa) Major Poisson’s ratio Ultimate tensile strength in fiber direction (MPa) Ultimate compressive strength in fiber direction (MPa) Ultimate tensile strength in matrix direction (MPa) Ultimate compressive strength in matrix direction (MPa) Ultimate in-plane shear strength (MPa) Thermal expansion coefficient in fiber direction (106 /C) Thermal expansion coefficient in matrix direction (106 /C) Moisture expansion coefficient in fiber direction Moisture expansion coefficient in matrix direction Thermal conductivity normal to the thickness direction (W/m per C)
181 10.3 7.17 0.28 1500 1500 40 146 68 0.02 22.5 0 0.6 0.865
90
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Fig. 4. Radial, axial and circumferential stress distributions for pure internal pressure.
internal and external radii of the tube, respectively. The stresses are normalized with the values of thin-wall solution. In Fig. 4, the results for radial stresses and to some degree with hoop stresses are similar with the ones of Ref. [5]. However, the axial stress distributions show a considerable difference. The errors in the formulation of Ref. [5] would be a reason for the difference; another reason might be originated from the fact that a variable winding angle (54–56) was considered in Ref. [5], contrary to the constant winding angle used in the current study. As it can be seen in Fig. 5, the thin-wall analysis gives only an average result. When thickness increases, the thick-wall analysis has to be used, instead. Radial stress is zero at the inside and outside of the pressure vessel and positive through the thickness. Its maximum value is approximately at the mid-point of the thickness.
When the wall thickness is increased, the layer having maximum radial stress becomes closer to the inner boundary of the pressure vessel. Larger circumferential and axial stresses are obtained at the inner layers of the vessel. In Fig. 5, it is seen that axial stress always changes sign at the point where the radial stress reaches to a maximum. 4.3. Thick- and thin-walled solutions for burst pressure The corresponding burst pressure values are obtained using an iterative procedure where the loading is increased until the failure of a single layer. In Fig. 6, the burst pressure, Pburst , by using both thin and thick-walled solution techniques are plotted versus winding angle. The burst pressure is normalized with that of 0 winding angle. For the thin-wall tube, both thick and thin-wall
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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].
solutions predict almost the same burst pressure. Actually, since the thin-wall solution neglects the radial stress, burst pressure values obtained with the thin-
walled analysis are slightly higher than those obtained with thick-walled analysis. The agreement between these two solutions is satisfied except for the values near the
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Fig. 7. Variation of burst pressure with increasing winding angle [rext =rint ¼ 1:4].
Fig. 8. Burst pressure for increasing wall thickness.
optimum winding angle where they differ for thick and thin-wall solutions. For a thick pressure vessel (Fig. 7), there is a significant difference between thin-wall and thick-wall solutions especially near the optimum winding angle. In this wall thickness value (rext =rint ¼ 1:40), the thick-wall solution gives higher burst pressure than the thin-wall solution between angles 48 and 64. To check the limiting value of the thin-wall solution on the wall thickness, burst pressures are calculated for increasing wall thickness Thin-wall and thick-wall solutions yield very similar burst pressure values up to the thickness ratio of rext =rint ¼ 1:1. For thicknesses with rext =rint P 1:1, the deviation between thick and thin-wall solutions becomes larger (Fig. 8). 4.4. Effect of angular speed and axial force on burst pressure The effect of angular speed and axial force on the burst pressure can be seen in Fig. 9. The burst pressure decreases with angular speed, when the winding angle is less than its optimum value. It is not an expected result, since the burst pressure increases with the speed for angles greater than the optimum winding angle. The
burst pressure increases in negligible amount for increasing axial tensile force for angles smaller than the optimum winding angle. If the winding angle of the structure is larger than its optimum value, the burst pressure always decreases with increasing axial force. 4.5. Maximum angular speed The effect of wall thickness and winding angle on the maximum angular speed can be seen in Fig. 10. If only angular speed is applied, the optimum winding angle is obtained as ranging between 81 and 83 depending on the wall thickness of the structure. For small winding angles up to 30, the thin and thick-wall constructions give almost the same maximum angular speed. As the wall thickness increases, the maximum angular speed decreases opposite to the case of burst pressure. It is an expected result, since an increasing wall thickness means more inertia that affects the speed in the negative sense. 4.6. Hygrothermal stresses and strains Hygrothermal stresses in the macro-mechanical level calculated by using the laminated plate theory. In order to assess the effects of residual stresses on the failure of
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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.
composite materials, the hygrothermal expansion coefficients have to be determined correctly. As a sample calculation of hygrothermal stresses, the data given in Table 2 for T300/N5208 [2] is used where DF is the degradation factor.
Table 2 Hygrothermal effects on burst pressure [rext =rint ¼ 1:05, a ¼ 53] Topr (C) c (%) DF (%) Pburst (MPa)
22 0 0 18.3
22 0 0 20.3
122 0 0 16.3
122 0 10 14.4
122 1 30 12.1
Since the curing temperature is the stress free state for composite materials, the operation temperature affects the failure of the composite depending on whether it is below or above the curing temperature. If the operation temperature is less than zero or if it is less than the curing temperature, the burst pressure is increased since the thermal strains and mechanical strains for pure internal pressure case work in opposite senses. It should be pointed out that, the negative temperature for constant moisture content also cause an increase in the mechanical properties of the composite material. It can be concluded that if the operation temperature is less than the curing temperature, burst pressure is increased. If operation temperature is greater than the curing
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Fig. 11. Effect of temperature on burst pressure for different operating temperatures.
temperature, however, both thermal and mechanical strains have a cumulative effect. This can be seen as a decrease in burst pressure values in Fig. 11. Although, the performance of the composite material is negatively influenced by the presence of moisture, it creates less residual strains compared with the thermal ones and does not change the burst pressure, significantly.
5. Conclusion An analytical procedure is developed to assess the behavior of a cylindrical composite structure under loading conditions particular to a rocket motor case. Available loading conditions are internal pressure, axial force and body force due to rotation. Additionally, temperature and moisture variations throughout the body are considered in the analysis. The procedure is based on the classical laminated plate theory. It models the plane strain state of the cylindrical body, which consists of a number of cylindrical sub-layers. The cylindrical pressure vessel is analyzed using two approaches, which are thin wall and thick wall solutions. It is shown that for composite pressure vessels with a ratio of outer to inner radius, up to 1.1, two approaches give similar results in terms of the optimum winding angle, the burst pressure, etc. As the ratio increases, the thick wall analysis is required. The optimum winding angle for the thick-wall pressure vessel analysis with the pure internal pressure loading case is obtained as ranging between 52.1 and 54.1 degrees depending on the material type. If the angular speed is applied at the same time with the internal pressure, optimum winding angle values obtained for the pure internal pressure case are shifted towards 90. The influence of the axial force is, however, opposite to the one of the angular speed. The addition of the axial force has a decreasing effect on the winding angle.
The burst pressure value is greatly depends on the analysis type used. The deviation between thin and thick-wall solutions is quite large especially near the optimum winding angle. As the wall thickness is increased, the thick-wall solution gives almost 30% higher burst pressure values. Therefore, the thin-wall analysis is said to be an average but a safe analysis. If angular speed is applied, the maximum stress occurs in the hoop direction. The optimum winding angle of the analyzed body for this type of loading is obtained as ranging between 81 and 83. The value of the maximum angular speed that the system can be rotated is greatly affected by the thickness of the pressure vessel. Hygrothermal effects are analyzed in this study in two levels. The effect of temperature and moisture to the performance of the materials is determined by using the micromechanics of the composite materials. By taking a linear variation of temperature and constant value for moisture content throughout the body, hygrothermal stresses and strains are determined. Since the thermal and moisture expansion coefficients of the materials have to be determined experimentally, always some amount of error is expected in these calculations. If the material has a tendency of expanding due to a positive temperature difference, the increasing operating temperature is shown to reduce the mechanical performance of the system.
References [1] Lekhnitskii SG. Anisotropic plates. New York, NY: Gordon and 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.
L. Parnas, N. Katırcı / Composite Structures 58 (2002) 83–95 [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 vessels. M.S. Thesis, Middle East Technical University, Turkey, 1998.
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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 and design. New York, NY: Marcel Dekker, Inc; 1988. [9] Lee SM. Reference book for composite technology-2. Technomic Publishing Company; 1989.