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Free vibration and buckling analysis of composite cylindrical shells conveying hot fluid
Composite Structures 60 (2003) 19–32 www.elsevier.com/locate/compstruct
Free vibration and buckling analysis of composite cylindrical shells conveying hot fluid Ravikiran Kadoli, N. Ganesan
*
Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology, Chennai 600 036, India
Abstract A coupled fluid structure interaction problem is analyzed using semi-analytical finite element method involving composite cylindrical shells conveying hot fluid for free vibration and buckling behavior. The system under study is assumed to have a steady flow of hot fluid and the temperature variation is axi-symmetric. First order shear deformation theory is used to model the elastic shells of revolution. Geometric stiffness matrix is evaluated to consider the effects of axi-symmetric temperature variation through the shell continuum due to flow of hot fluid. The fluid domain is modeled using the wave equation. Numerical results of the studies on composite cylindrical shells made of HS-Graphite/Epoxy with two different length to radius ratios and clamped–clamped boundary condition conveying hot fluid are presented. The variation of the natural frequency of the coupled system is evaluated with the steady flow of the hot fluid. The influence of the temperature on the mean axial flow velocity through the shell is critically examined. The critical velocity of the hot fluid and cold fluid which leads to shell instability is compared thus establishing the fact that the lowest critical velocity of the hot fluid coincides with the mode corresponding to the lowest critical thermal buckling temperature. Various fibre angles are also considered in the study and its influence is also examined. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Hot fluid; Axi-symmetric temperature; Semi-analytical finite element method; Fluid-structure interaction; Critical velocity of fluid
1. Introduction Petrochemical industries, chemical processing plants and power generating industries require vessels and pipes for storage and transportation of fluids at high pressure and temperature. There has been ever-growing demand for the manufacture of composite pipes/shells for hot water transportation, gas transportation (compressed air, natural gas, oxygen) and oil industry (production risers). NASAÕs Marshall Space Flight Center [1] has established fabrication techniques for the manufacture composite material pipes/shells and storage vessels of conventional and unique shapes which possess characteristics like lightweight, corrosion resistance, capable of handling diverse fluids, large diameter pipes for high pressure applications, capable of withstanding high temperature etc. These composite pipes/shells and storage vessels are multi-layered in construction. The authors feel that studies related to the dynamics of composite cylindrical shells through which hot fluid is *
Corresponding author. Tel.: +91-44-235-1365; fax: +91-44-2350509. E-mail address: [email protected] (N. Ganesan).
flowing at steady velocity can be an appropriate contribution especially regarding the magnitude of the critical flow velocity of the hot fluid and hence flow induced instability. There is available abundant literature on isotropic shells dealing with the thermal buckling subjected to constant temperature, temperature varying in circumferential direction and temperature varying along the generator of the shell. As pointed by Earl Thornton [2] in his recent review paper on thermal buckling of plates and shells, the availability of studies on thermal buckling in composite shells is scarce. Thangaratnam et al. [3] have performed linear thermal buckling analysis of laminated composite cylindrical and conical shells using finite element method and examined the nature of buckling under thermal load and mechanical load with respect to different fiber orientation. Radhamohan and Venkataramana [4] have made a complete study of thermal buckling of composite cylindrical shell made of fiberglass reinforced plastics. A uniform temperature rise throughout the shell was considered for buckling studies. Very recently the authors [5] have also contributed to the research literature on the thermal buckling studies and the influence of axi-symmetric temperature variation on the natural
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 3 1 3 - 6
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frequencies of the composite cylindrical shells. Birman and Bert [6] considered studies on the buckling and post buckling response of composite shells subjected to high temperature using the equilibrium equations for shells under the simultaneous action of thermal and axial load. Similar to thermal buckling studies of isotropic shells, studies on pipes/shells conveying fluids are abundant in literature. Paidoussis and Li [7] have reviewed and compiled the exhaustive literature available on the studies related with the dynamics of pipes conveying fluids. Contributions by Paidoussis [8] in the area of dynamics of pipes conveying fluid is enormous and one can find a host of problems and solutions associated with fluid flowing through slender structures in the book written by Professor M.P. Paidoussis. Bert and Chen [9] analyzed the vibration modes in piping conveying fluid, pipes made of orthotropic as well as isotropic materials were considered. Chang and Chiou [10] used Mindlin type first order transverse shear deformable cylindrical shell theory and analytical method for the fluid to study the natural frequencies and critical velocities of laminated circular cylindrical shells with fixed ends conveying fluids. Toorani and Lakis [11] have developed a hybrid method based on a combination of refined shell theory, the finite element method and linear fluid theory which enables to account for the influence of shear deformation effect on the natural frequency response of multi-layered anisotropic open or closed cylindrical shells partially filled with or subjected to a non-viscous incompressible flowing fluid. Recently Jayaraj et al. [12] have carried out elaborate parametric studies on various composite cylindrical shells conveying fluids. For thin composite shells they could predict the divergence as well as coupled mode flutter instabilities, and more importantly, established that the buckling mode due to lowest critical velocity of the fluid coincides with the lowest natural frequency mode. Limited studies are available in literature related to shells conveying fluid, which considers the initial tension in the structure. Zhang et al. [13,14] have investigated the vibration characteristics of orthotropic cylindrical shells and tubes with initial tension and conveying fluid. The finite element formulation for the initially tensioned cylindrical shell is based on the three dimensional non-linear theory of elasticity and Eulerian equations. In case of initially tensioned cylindrical tubes, SandersÕ non-linear theory of thin shells and classical potential flow theory is used. Hale et al. [15] have investigated the biaxial stress-strain behavior in filament wound glass fibre reinforced composite pipes exposed to high temperature water. From the literature survey it is found that, to the authors knowledge, there are no studies that deal with the dynamic behavior of composite shells conveying hot fluid. In general the temperature distribution in shells conveying hot fluid leads to compressive stresses and the free vibratory and buckling behavior can be different
when compared to shells subjected to initial tension and conveying fluid. Hence in the present study an attempt is made to analyze a coupled elasto–thermo-fluid interaction problem using semi-analytical finite element method. The structural semi-analytical finite element formulation for axi-symmetric shells of revolution by Rao et al. [16], the fluid domain semi-analytical finite element formulation by Jayaraj et al. [12] are used and extended to include the geometric stiffness matrix which accounts for the effects of initial stresses due to hot fluid flowing through the composite shell. Numerical studies are carried out on composite shells with length to radius ratio equal to 1.048 and 2.08, radius to thickness ratio 292 and clamped–clamped boundary condition. Fibre angles equal to 0°, 15°, 30° and 60° in case of l=r ¼ 1:048 and 0° and 60° in case of l=r ¼ 2:08 are also considered in the study. Static thermal buckling analysis results are presented for the first axial mode associated with first 20 circumferential modes. The fluid considered for study is water at atmospheric pressure and atmospheric temperature (assumed as 20 °C) as well as water at temperatures below saturation temperature at atmospheric pressure. Numerical studies pertaining to variation of natural frequency of the composite shell conveying water at ambient conditions as well as hot water at different temperatures are conducted. The critical flow velocities of cold water as well as hot water at different temperatures are examined for instability of the composite shells. Influence of the fibre angle of the composite shell lamina is examined on the critical velocities of hot water.
2. The physical model The physical model considered for the study is illustrated in Fig. 1. The control volume also referred to as ÔsystemÕ comprises of the composite cylindrical shell and the fluid flowing at uniform velocity and constant temperature which is axi-symmetric in nature. Steady state conditions prevail over the control volume provided the heat influx equals the heat dissipated from the control volume. It is assumed that the temperature across the thickness of shell is constant and equal to that of the hot fluid. The length of the shell is such that temperature throughout the length of the shell does not vary. The fluid enters the control volume at velocity uf and temperature T , and leaves the control volume with the same operating conditions. Since the inlet and outlet mean velocity of fluid is same, it is essential to assume that the pressure drop of the flowing fluid through the shell is neglected. The flow of the fluid is potential, inviscid, irrotational, compressible and isentropic. When the hot fluid flows we assume that two phase of the fluid never exists and there is no flow separation from the wall of the shell. The material properties of the shell and fluid are assumed to be independent of temperature.
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
21
Heat qin Cylindrical shell lamina
Length of shell l meters
Radius of shell r meters
Cold/Hot fluid IN velocity uf Temperature T
Cold/Hot fluid OUT velocity uf Temperature T
Thickness of shell h meters
Control Volume Heat qout
Meridian of the shell
Fig. 1. Illustration of the control volume to the study effects of hot fluid flowing through cylindrical shell on the free vibration and buckling behavior.
panded by Fourier series in the h direction and expressed as
3. Semi-analytical finite element formulation 3.1. Structural domain The (s, h, z) coordinate system for general shells of revolution is illustrated in Fig. 2. By setting the value of the principal radius of curvature R/ equal to infinity (or a large magnitude) and the other principal radius of curvature Rh equal to a finite radius equal to mean radius r, the geometry resembles a cylindrical shell as illustrated in Fig. 3. The mid-surface displacements according to the first order shear deformation theory are expressed as uðs; h; z; tÞ ¼ u0 ðs; h; tÞ þ zws ðs; h; zÞ
ð1aÞ
vðs; h; z; tÞ ¼ v0 ðs; h; tÞ þ zwh ðs; h; zÞ
ð1bÞ
wðs; h; z; tÞ ¼ w0 ðs; h; tÞ
ð1cÞ
where u0 , v0 , w0 are displacement of mid-surface along the s, h and z direction and ws and wh are rotations of the normal to the mid-surface along s and h axes respectively. In the semi-analytical method the generalized displacement field is assumed to depend in the circumferential direction hence these quantities can be ex-
8 9 2 u0 > cos mh 0 0 > > > > > > > 0 sin mh 0 1 6 < v0 = X 6 6 0 0 cos mh w0 ¼ 6 > > m¼0 4 > 0 0 0 > ws > > > : > ; wh 0 0 0
ð2Þ
m stands for the mth circumferential harmonic.
Center line of object
β =s/l 3
Rθ
Rφ
O1
O2 2
1
Parent element
Fig. 3. Schematics of the discretisation of the cylindrical shell using three noded quadratic line element.
Center line of structure rr
φ O1A : Rφ O 2A : R θ rr = Rθ /sinφ
9 38 0 0 u0m > > > > > > > 7 0 0 7< v0m > = 0 0 7 w 7> 0m > cos mh 0 5> w > > > > : sm > ; whm 0 sin mh
A
dφ
B
z (w) θ (v) s (u)
O2 mid-surface of structure O1
Fig. 2. Tri-orthogonal curvilinear coordinate system for shells of revolution.
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Rao et al. [16] used first order shear deformation theory and other higher order theories to study the interlaminar stresses in laminated composite shells of revolution. The strain components for a doubly curved shells of revolution based on FSDT are as follows: ess ¼
1 0 1 ðe þ zj1s Þ chz ¼ c0hz A1 ss A2
ð3a; bÞ
ehh ¼
1 0 ðe þ zj1h Þ A2 hh
ð3c; dÞ
csh ¼
1 1 0 ðc þ zj1sh Þ A1 A2 sh
csz ¼
1 0 c A1 sz
ð3eÞ
½ke ¼
Z
T
½B ½D ½B r ds dh
ð9Þ
A
where ½B is the strain-displacement matrix of the shell at stress free temperature T0 . Due to orthogonality principle the stiffness matrix becomes decoupled for each circumferential harmonic. A three node isoparametric element is used in the axial direction s. Fig. 3 depicts the discretization of the cylindrical shell using this element. Each node has five degrees of freedom. The displacement parameters associated with the element are fde g ¼ fu1 ; v1 ; w1; ws1 ; wh1 ; u2 ; v2 ; w2; ws2 ; wh2 ; u3 ; v3 ; w3 ; ws3 ; wh3 g
ð10Þ
where
the subscripts 1, 2, and 3 stand for the three nodes of the element. The shape functions Ni in terms of the isoparametric axial coordinate b ¼ s=l are given by
1 1 ¼ A1 ð1 þ z=R/ Þ and 1 1 ¼ A2 ð1 þ z=Rh Þ in the above equations R/ and Rh are the principle radii of curvature of the shell as illustrated in Fig. 2 and z is the thickness in z-direction. The total strains are denoted as ess , ehh , chz , csz and csh ; which comprises of e0ss , e0hh , c0hz , c0sz and c0sh the normal strains and the shear strains referred to mid-surface, and j1s , j1h and j1sh the change in curvature. The total strain energy in the shell continuum is given by U ¼ U1 þ U2 where U1 is the strain energy due to vibratory stresses and U2 is the strain energy contribution from the initial stresses due to steady state axi-symmetric temperature. Strain energy U1 is given by Z 1 fess rss þ ehh rhh þ chz shz þ csz ssz þ csh ssh g dV U1 ¼ 2 V Z 1 T feg frg dA ð4Þ ¼ 2 A where frg and feg are the generalized stress and strain vectors respectively and dA is the infinitesimal area element on the shell mid-surface. The foregoing vectors are defined as T
feg ¼ fe0ss
e0hh
c0sh
T
j1s
j1h
j1sh
c0sz
c0hz g
frg ¼ fNss Nhh Nsh Mss Mhh Msh Qs Qh g
ð5Þ ð6Þ
The generalized stress resultant vector can be expressed as frg ¼ ½D feg T
ðb2 bÞ ; N2 ¼ ð1 b2 Þ and 2 ðb2 þ bÞ ð11a–cÞ N3 ¼ 2 The constitutive matrix ½D in Eq. (7) or (9) is for a laminate consisting of various integrated shell stiffnesses Aij , B ij , Dij , and F ij . Various integrated shell stiffnesses are defined as follows: Z þh=2 Qij ð1; z; z2 ; z3 Þ dz ð12Þ ðAij ; Bij ; Dij ; Fij Þ ¼ N1 ¼
U1 ¼ 12fde g ½ke fde g
h=2
Qij are the reduced stiffness coefficients, other details of this constitutive matrix can be referred to Jones [17] and Ganesan et al. [5]. 3.2. Geometric stiffness matrix evaluation Neglecting the strain energy due to initial transverse shear stresses, strain energy due to initial stresses U2 is given in [18] as Z 1 2 2 2 U2 ¼ fðei Þ r þ ðeihh Þ r hh þ 2ðcish Þ s sh g dV ð13Þ 2 V ss ss in which r ss , r hh and s sh are the initial stresses and the corresponding strains, eiss , eihh and cish are given as follows: i 1 ow ui i ess ¼ ð14aÞ os R/ 1 þ z=R/ 1 1 owi vi i ð14bÞ ehh ¼ sin / ð1 þ z=Rh Þ r oh r cish ¼ eiss eihh
ð7Þ
Eq. (13) can be put in the following form:
ð8Þ
U2 ¼ 12fdie gT ½kre fdie g
where ½ke is the element stiffness matrix corresponding to the mth harmonic and is computed as follows:
ð14cÞ
ð15Þ
The matrix ½kre in the foregoing equation is the element geometric stiffness matrix
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
½kre ¼
Z
T
½Bi fr g dV
ð16Þ
V
fdie g
represents the elemental displacement vector due to initial stress, ½Bi is the strain displacement matrix based on Eqs. (14a)–(14c) and fr g is the vector comprising of the stress resultants due to thermal load which is described in next section. 3.3. Thermal load and initial stress evaluation It is well known that under thermal environment the expression for the total potential for a finite element of the cylindrical shell is Z Z 1 T T P¼ feg ½D feg dV feg fkgT dV ð17Þ 2 V V The temperature is assumed to be constant over the thickness and the temperature field for the structural continuum is T ðs; h; z; tÞ ¼ T ðs; h; tÞ Steady state temperature is assumed to be dependent in the circumferential direction, and using Fourier series the temperature in the circumferential direction is 1 X T ¼ Tm cos mh ð18Þ m¼0
Minimisation of P with respect to the displacement vector fde g leads to a standard equation for a finite element as ½ke fde g
fFth e g
¼0
ð19Þ
½ke is the structural element stiffness matrix same as Eq. (9). The thermal load evaluation involves Z fFth g ¼ ½e T fkgfT ðzÞg dV ð20Þ e
23
# is the lamina orientation angle, T ðzÞ is the temperature ðkÞ rise from stress free temperature T0 , fag is the vector of coefficient of linear thermal expansion in the kth layer, a1 and a2 are the coefficients of thermal expansion in the directions parallel and perpendicular to the coordinate axis for kth lamina. Using Eq. (20) the thermal load is computed for a steady state axi-symmetric temperature, above the stress free temperature T0 which is assumed to be 20 °C. Solving Eq. (19) in global sense gives the displacement field due to the thermal loading. Using the element displacement vector the stresses and moment resultants are evaluated as follows: fr g ¼ ½D fe0 g fkgT
ð22Þ
fe0 g represents the strain developed in the structural continuum due thermal load, [D] is matrix comprising of various integrated shell stiffnesses in the transformed coordinate, and fkg represents the temperature stress coefficients and is similar to Eq. (21). The stress resultants are found for each finite element and are used in Eq. (16) to compute the element geometric stiffness matrix. 3.4. Mass matrix The mass matrix is obtained from the kinetic energy of the shell continuum, the kinetic energy is Z Z q q KE ¼ ðu_ 2 þ v_ 2 þ w_ 2 Þ dV ¼ fd_ gT fd_ g dV ð23Þ 2 V 2 V Using Eqs. (10) and (11a–c) the kinetic energy will be 1 T KE ¼ fde g ½me fde g 2 where ½me is the element mass matrix given by Z me ¼ q N T N dV
ð24Þ
V
where fkg is the temperature stress coefficients computed for the laminate as nlay Z hk X ðkÞ ðkÞ ½Qij fag ð1; zÞ dA ð21Þ fkg ¼ k¼1
hk 1
ðkÞ
½Qij represents the generally orthotropic lamina properties of the kth layer with proper coordinate transformation and comprises of the stiffness coefficients Aij , and Bij . The coefficient of linear expansion for the kth layer in the shell coordinate system is fagðkÞ ¼ fas
ah
ash
0
where as ¼ a1 cos2 # þ a2 sin2 # as ¼ a1 sin2 # þ a2 cos2 # ash ¼ ða1 a2 Þ cos # sin #
0 0
0
0g
T
3.5. Finite element formulation of the fluid domain Jayaraj et al. [12] very recently have conducted exhaustive studies on the dynamics of composite shells conveying fluid. The finite element formulation for the fluid domain is based on the wave equation 2 1 o o r2 / 2 /¼0 ð25Þ þ Ux c ot ox where / is the velocity potential, c is the velocity of sound, Ux is the mean axial flow velocity of fluid. An eight node isoparametric quadrilateral element is used for discretising the fluid domain as shown in Fig. 4. Velocity potential is the nodal degree of freedom. The velocity potential should satisfy the wave equation. The velocity components in different directions are expressed in terms of velocity potential as
24
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32 Fluid domain mesh using 8 node isoparametric element mid-surface of shell
Elastic shell discretised using 3 node isoparametric element
X
θ Center line of shell
Z
mid-surface displacement of structure, velocity potential of fluid flow domain and temperature distribution are expanded in Fourier series in circumferential direction θ.
Fig. 4. Finite element model of the cylindrical shell conveying fluid.
o/ 1 o/ o/ ; Vh ¼ ; Vr ¼ ð26Þ ox R oh or To satisfy the dynamic boundary condition it is necessary that the radial velocity of the fluid must be equal to the instantaneous velocity of the shell, that is o/ ow ow Vr ¼ þ Ux ð27Þ ¼ or ot ox
Vx ¼ U x þ
r¼R
BernoulliÕs principle is used to compute the pressure exerted by the fluid on shell wall. For unsteady flow o/ 1 2 P Ps þ V þ ¼ ot 2 q q
ð28Þ
where V 2 ¼ Vx2 þ Vh2 þ Vr2 and Ps is the stagnation pressure. P is the sum of a mean pressure P and the perturbation pressure p P ¼P þp
ð29Þ
For small deformation the non-linear terms in V 2 are neglected, we get o/ V 2 ffi Ux2 þ 2Ux ox o/ o/ p ¼ q þ Ux ot ox
ð30Þ
The finite element form of the fluid domain and hence the interaction between the fluid and the structure is obtained by using the GalerkinÕs weighted residual method. The weighted residual form over the domain takes the form
Z
NfT V
1 r / 2 c 2
o o þ Ux ot ox
2 ! / dV ¼ 0
ð31Þ
where the weighting function Nf is the fluid shape functions given in Ross [19]. The variation of velocity potential is expressed in Fourier series in the h direction. Z Z Z 1 T T Nf r/:n dS rNf r/ dV 2 N T /€ dV c V f S V Z Z 2 2Ux Ux2 o2 / T o / dV 2 2 Nf NfT 2 dV ¼ 0 ð32Þ oxot ox c c V V The first term of Eq. (32) describes the interaction of the fluid and the shell wall. Integration of the other terms will yield the kinetic energy of the fluid, compression energy of the fluid, Coriolis energy of the fluid and centrifugal energy of the fluid. Making use of Eq. (27) i.e. the fluid shell interface boundary condition along with the first term of Eq. (32) we have Z Z Z oN T T _ dSfUe g Nf r/:n dS ¼ Nf N dsfUe g þ Ux NfT ox S ð33Þ where N is the w component of the shell shape function and n is the unit normal vector to the structure. The dynamic pressure acting on the shell wall given by Eq. (30) also describes the interaction between fluid and shell wall, hence
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
T
o/ o/ þ Ux dS ot ox
where
N qf S Z Z T T oNf _ dSf/e g N Nf dSf/e g þ qf Ux N ¼ qf ox S S
"
ð34Þ
The various finite element matrices obtained from the above procedure are listed below: Compression energy of the fluid Z X 1 // NfT Nf dV G// ¼ G// Ge ¼ 2 e c V Fluid structure interaction coupling terms due to structural and fluid damping Z X T Cu/ ¼ q N Nf dS Cu/ ¼ Cu/ f e e S Z X C/u NfT N dS C/u ¼ C/u e ¼ e S
Coriolis energy of fluid Z 2 oNf C// dV ¼ NfT e 2 Ux c V ox
M ¼
K ¼ "
C ¼
Muu
0
0
G//
; #
Kuu þ Kuu r
Ux Ku/
Ux K/u
H// Ux2 I// #
0 C/u
;
Cu/ Ux C//
Since the isothermal conditions are assumed to exist, and the changes in density and viscosity of the fluid are not accounted due to changes in temperature, it is apparent from the above Eq. (35) that the off diagonal terms of the damping and stiffness matrices couples the shell and fluid motion. Eq. (36) in the state space form of Eq. (35) is solved for the eigenvalues using LAPACK routine, DGEGV [20]. 4. Validation of the semi-analytical finite element formulation for analyzing hot fluid flowing through shells
C// ¼
X
C// e
Stiffness coupling due to flow Z X T oNf Ku/ dS Ku/ ¼ ¼ q N Ku/ f e e ox S Z X oN dS K/u ¼ K/u NfT K/u e ¼ e ox S Kinetic energy of fluid Z X ¼ rNfT rNf dV H// ¼ H// H// e e V
Centrifugal energy of fluid Ux2 Z X 1 oNfT oNf dV I// ¼ I// ¼ I// e e 2 c V ox ox Considering the structural global mass matrix Muu , global stiffness matrix Kuu and global geometric stiffness matrix Kuu r the complete fluid-structure finite element equation is uu € M 0 u u_ 0 Cu/ þ 0 G// /€ /_ C/u Ux C// uu u Ux Ku/ K þ Kuu r ¼0 ð35Þ þ / Ux K/u H// Ux2 I// The above equation is expressed in the state-space form by letting fag ¼ fu / u_ /_ gT
0 K C M
fag ¼ k ð36Þ
fag 0 M M 0
The present formulation developed for analysis of shells conveying hot fluid has been validated with the results reported by Jayaraj et al. [12] for a Graphite/ Epoxy composite cylindrical shell with l=r ¼ 1:0; r=h ¼ 100 and clamped–clamped boundary conditions. Jayaraj et al. have validated their formulation with published results of Chang and Chiou [10]. The variation of the dimensionless frequency versus dimensionless velocity has been compared in Fig. 5 for the first two axial modes associated with circumferential harmonics 4. Even though the fluid domain modeling is same as that developed by Jayaraj et al., the difference lies in the structural finite element. In the present case the structural finite element has five degrees of freedom per node whereas the structural finite element used by Jayaraj et al. has viscoelastic layer and hence the nodal degrees
Dimensionless frequency
Z
25
8.25 7.50 6.75 6.00 5.25 4.50 3.75 3.00 2.25 1.50 0.75 0.00
m=4, n=1: m=4,n=2:
present present
Jayaraj et al.[12] Jayaraj et al.[12]
0.002 0.004 0.006 0.008 0.010 0.012 0.014
Dimensionless velocity Fig. 5. Validation of the results obtained from the present study with that of Jayaraj et al. [12]. Composite shell is made of Graphite/Epoxy, l=r ¼ 1:0, r=h ¼ 100 and clamped–clamped boundary condition.
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R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
Table 1 Validation of critical buckling temperature for symmetric cross-ply cylindrical shell r=h
Critical temperature (°C) Ref. [21]
200 300 400 500 a
quencies and critical velocity of the hot fluid corresponding to instability of shell. The numerical study hence involves the following objectives:
Results from present study a
1304.298 (11) 912.434 (13) 659.610 (18) 514.745 (19)
1258.4 (11), 1250 (12), 1268.4 (13) 851.3 (13), 835.8 (14), 836.6 (15) 632.2 (16), 631 (17), 637.8 (18) 507.3 (18), 507 (19), 511.74 (20)
Numbers in bracket indicate circumferential mode.
of freedom are seven. The results obtained by the present formulation are in close agreement with those reported by [12]. The above validation is for shell conveying cold fluid. Due to non-availability of literature on hot fluid flowing through pipes/shells, it was felt that the formulation could be validated for critical thermal buckling temperature in case of composite shells operating under axi-symmetric temperature without flow of fluid. For this purpose Eq. (35) is solved as an uncoupled problem. Thangarathman et al. [21] have carried out finite element analysis of buckling of composite cylindrical shells under the influence of mechanical and thermal loads using semi-loof finite element. They have presented the results of their studies on the critical buckling temperature of composite cylindrical shell under uniform temperature rise with simply supported boundary condition. The length to radius ratio (l=r) was 0.5 and different ratios of shell radius to shell thickness (r=h) were considered. The laminate properties used were as follows: Ell =Ett ¼ 10; at =al ¼ 2:0
Glt =Ett ¼ 0:5; and
llt ¼ 0:25;
al ¼ 0:1 10 5
These data were used to obtain the critical buckling temperatures using the present code. Table 1 lists the comparison of the results obtained by the present code and those reported by Thangaratnam et al. The trend in the results tallies very well. Also it is seen that the circumferential mode at which buckling occurs nearly tally. These two validations are sufficient to indicate the adaptability of the formulation and the computer code developed for the studies on composite shells conveying hot fluid.
5. Numerical simulations––results and discussion Analysis of the composite cylindrical shell conveying hot fluid is carried out in order to determine the effect of temperature of the hot fluid flowing on the natural fre-
(a) Variation of the natural frequencies of the composite shell corresponding to the first axial mode of the coupled system involving the flow of hot fluid through the shell. (b) Influence of the temperature of water on the magnitude of the mean axial velocity and hence the critical velocity of hot water which cause instability of the system. (c) Comparison of the behavior of instability of the shell with flow of cold and hot water. (d) Simultaneously bring out the influence of fiber orientation on the above stated objectives. It is essential to understand the nature of the variation of the free vibration frequencies of the composite shell as well of the magnitude of the thermal buckling temperatures for different modes. Hence the study starts with the evaluation of (i) the critical buckling temperature of the first axial mode (n ¼ 1) associated with first 25 circumferential modes (m ¼ 1 to 25) for the composite shells exposed to axi-symmetric temperature variation without fluid and (ii) the free vibration frequencies for the composite shells at ambient temperature and without fluid corresponding to first axial mode associated with first 25 circumferential modes. The geometric details of the composite cylindrical shells considered for the study are given in Table 2. The properties of HS-Graphite/Epoxy material are listed in Table 3. 5.1. Evaluation of static thermal buckling eigenvalues Static thermal buckling analysis is carried out using the following equation: Table 3 Material properties of HS-Graphite/Epoxy HS-Graphite/ Epoxy YoungÕs Modulus (GPa)
E11 , E22 , E33
Shear Modulus (GPa) PoissonÕs ratio Density (kg/m3 ) Coefficient of thermal expansion (°C) Environment temperature (°C)
G12 , G31 , G23 m12 , m13 , m23 q a11 , a22 T0
181.0, 10.34, 10.34 7.2, 7.2, 7.2 0.28, 0.28, 0.28 1389.23 11:34 10 6 , 36:9 10 6 20.0
Table 2 Details of cylindrical shell and boundary conditions Cylindrical shell
Length (l m)
Radius (r m)
l=r ratio
Thickness of cylindrical shell (h m)
Boundary condition
1 2
0.914 1.8288
0.876 0.876
1.048 2.08
0.0015 0.0015
Clamped–clamped Clamped–clamped
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
ðKuu þ vKuu r Þxb ¼ 0 In the above equation Kuu is the global structural stiffness matrix and Kuu r is the global geometric stiffness matrix, contribution from the thermal load due to steady state axi-symmetric temperature. v is the buckling eigenvalues and xb is the corresponding buckling eigenvector. The buckling parameters of the first axial mode (n ¼ 1) associated with first 25 circumferential modes (m) are computed at arbitrary temperature of 10 °C. The critical buckling temperatures of composite cylindrical shell with l=r ¼ 1:048 and for different fibre orientations are plotted with respect to circumferential harmonic, refer to Figs. 6 and 7. In Figs. 6 and 7 the lowest critical buckling temperature modes are also in-
80
fibre angle 0o o fibre angle 15 o fibre angle 30
70 60 50 40 30
m=14,n=1 m=13,n=1
20
m=11,n=1
10 0
2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 6. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 1:048 and clamped–clamped boundary condition.
70
Critical buckling temperature oC
600
o
fibre angle 60
500 400 300 200
dicated. For composite shells with fibre angle 0°, 15° and 30° the magnitude of the critical buckling temperature is minimum for circumferential modes m ¼ 14, 11 and 12. One can notice that the critical buckling temperatures are well within the saturation temperature of water at atmospheric pressure (1 atmp. or 1.0 bar) for the circumferential modes considered. The critical buckling temperatures for composite cylindrical shell tailored to fibre angle 60° as seen in Fig. 7 exhibit increasing temperature as the number of circumferential harmonics increase. The critical buckling temperatures are considerably high compared to the other fibre angles. These temperatures are above 296.77 °C, which is much high compared to saturation temperature of water at atmospheric pressure. Similarly critical buckling temperatures are evaluated for composite cylindrical shell with l=r ¼ 2:08 and for fibre angles 0° and 60°. Typical results are presented in Fig. 8 for composite shell with fibre angle 0°. The variation of the buckling temperature corresponding to first axial mode (n ¼ 1) associated with 25 circumferential modes m is similar to that noticed in case of l=r ¼ 1:048, see Fig. 6. Again notice that the critical buckling temperatures are well within the saturation temperature of water at atmospheric pressure. The critical buckling temperature are considerably higher in case of composite shell with l=r ¼ 2:08 and fibre angle 60° (these results are not presented here). For example, the critical buckling temperature is 296.4° for mode (11,1). In the present work an attempt is made to study the buckling behavior of shells conveying hot fluids. For the purpose of numerical computation and simulation, water at atmospheric pressure is considered as the working fluid. It is assumed that the density variation of water is
m=1,n=1
100 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 7. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with fibre angle 60°, l=r ¼ 1:048 and clamped–clamped boundary condition.
Critical buckling temperature o C
Critical buckling temperature oC
90
27
60 50 40 30
m=14,n=1 o
fibre angle 0
20 10 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26
Circumferential modes m Fig. 8. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with fibre angle 0°, l=r ¼ 2:08 and clamped–clamped boundary condition.
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R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
lowest natural frequency and the lowest critical buckling temperature are not the same.
small at different operating temperatures. Even though at lower circumferential modes the buckling temperatures are high, especially in case of composite cylindrical shell with fibre angle 60° and l=r ¼ 1:048, the studies are limited to temperature of water equal to 90 °C for all circumferential modes.
5.3. Studies on variation of natural frequency of the shell conveying hot fluid In order to determine the influence of the hot fluid flowing through the composite shell on the natural frequency of the first axial mode numerical studies are carried out on clamped–clamped composite cylindrical shell with l=r ratio equal to 1.048 and 2.08. Further the influence of the fibre orientation is also considered simultaneously in the study. Water at ambient temperature as well as water at various temperatures is considered in the study. In the present study however the maximum magnitude of the water temperature is limited either by the critical buckling temperature of the shell or 90 °C which is close to the saturation temperature of water at atmospheric pressure. Considering the flow of
5.2. Natural frequencies of composite cylindrical shells The free vibration eigenvalues of the composite shell with l=r ¼ 1:048, for fibre orientations of 0°, 15°, 30° and 60° and l=r ¼ 2:08 for fibre orientations of 0° and 60° are illustrated in Figs. 9 and 10. It is also indicated in the figures the mode which has the lowest natural frequency. The lowest natural frequencies, lowest critical buckling temperatures and their modes are listed in Tables 4 and 5. Note that the mode corresponding to
fibre angle 0o o fibre angle 60
Natural frequency (Hz)
1000 800 600 400
m =9, n=1
200 0
m=6,n=1
0 2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 9. Plots of the variation of the first axial mode natural frequencies for 25 circumferential modes. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 1:048 and clamped–clamped boundary condition.
Fig. 10. Plot of the variation of the first axial mode natural frequencies for first 25 circumferential modes. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 2:08 and clamped–clamped boundary condition.
Table 4 Lowest natural frequency and lowest critical buckling temperature for composite cylindrical shell with l=r ¼ 1:048 Sl. no.
Fibre angle
Lowest natural frequency (Hz)
Mode ðm; nÞ
Lowest critical thermal buckling temperature (°C)
Mode ðm; nÞ
1 2 3 4
0° 15° 30° 60°
137.4 153.0 167.9 175.2
(12,1) (12,1) (10,1) (8,1)
47.98 37.62 50.50 296.77
(14,1) (13,1) (11,1) (1,1)
Table 5 Lowest natural frequency and lowest critical buckling temperature for composite cylindrical shell with l=r ¼ 2:08 Sl. no.
Fibre angle
Lowest natural frequency (Hz)
Mode ðm; nÞ
Lowest critical thermal buckling temperature (°C)
Mode ðm; nÞ
1 2
0° 60°
77.01 90.23
(9,1) (6,1)
46.51 296.42
(14,1) (11,1)
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20oC 30oC o 40 C 45oC o 50 C o 60 C
Dimensionless frequencies
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0.001 0.002 0.003 0.004 0.005 0.006
Dimensionless velocity
(a)
1.8 m=12, n=1
Dimensionless frequencies
5.3.1. Composite shell with l=r ¼ 1:048 and fibre angle 0° conveying fluid The variation of dimensionless frequency of the first axial mode with respect to dimensionless velocity are illustrated in Fig. 11(a)–(c) for circumferential modes m ¼ 1, 12 and 14. The mode (12,1) corresponds to the lowest natural frequency of the composite shell and mode (14,1) has the lowest critical buckling temperature. In Fig. 11(a) the variation of the dimensionless frequency versus dimensionless velocity for mode (1,1) is illustrated. The critical buckling temperature for this mode is 67.67 °C. Hence one can consider the flow of water at temperatures like 20, 30, 40, 45, and 60 °C. The velocity of water is varied from 0.0 m/s to critical velocity that leads to shell instability. The natural frequency of the coupled system decreases but not appreciably as the velocity of water increases. The fall in frequency is very rapid as the velocity of water flowing approaches the critical velocity. This trend is observed irrespective of the temperature of the water. In contrast for higher modes (12,1) and (14,1) the drop in the magnitude of natural frequencies smooth and continuous with respect to flow as seen in Fig. 11(b) and (c). Apart from the decrease in the natural frequencies, one can also observe the decrease in the magnitude of the maximum velocity that is permissible for conveying through the shell as the temperature of the water is increased. As seen in the Fig. 11(b), the maximum velocity of water that can be conveyed through the shell at 20 °C is greater than that of the maximum velocity of water passing through the shell at 40 °C. Consider the flow of hot water at 45 °C in the case of mode (12,1) in Fig. 11(b). The critical thermal buckling temperature for this mode is 49.39 °C. Hence we have a situation where the hot water temperature is close to the thermal buckling temperature. It is clear from the curve that the natural frequency as well as the maximum velocity of water
m=1,n=1
1.8
1.6
o
20 C 30oC o 40 C o 45 C
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.001
0.002
0.003
0.004
0.005
Dimensionless velocity
(b)
2.0
m = 14, n = 1
1.8
Dimensionless frequencies
hot water at constant temperature, the velocity of water is varied from 0.0 m/s to a velocity that causes the system to buckle, in increments of 0.5 m/s. The eigenvalues are computed by solving the coupled equation of motion, Eq. (36), using the DGEVG subroutine, for the shell conveying water at steady velocity and constant temperature. Eigenvalues obtained are expressed in dimensionless quantities as X ¼ ðx=x0 Þ 100, where x0 ¼ U0 =r, x is the real natural frequency and r is the radius of the shell. The flow velocity of the water is also expressed inffi dimensionless form as Ux =U0 where pffiffiffiffiffiffiffiffiffiffiffi U0 ¼ E0 =qs , in which E11 =E0 ¼ 21, E11 is YoungÕs modulus of composite shell and qs is the density of shell material. The dimensionless quantities are similar to Chiou et al. [10] for a single layer in its material principal axes. Typical results of the study are presented and discussed in the next section.
29
1.6
20oC (ambient temperature) 30oC 40oC 45oC
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.001
(c)
0.002
0.003
0.004
0.005
0.006
Dimensionless velocity
Fig. 11. Variation of the dimensionless frequency with respect to dimensionless velocity for water flowing through composite cylindrical shell with l=r ¼ 1:048, fibre angle 0° and clamped–clamped boundary condition.
considerably reduces when the temperature of the water conveyed through the shell nears the critical thermal buckling temperature of the shell. This study reveals that the maximum velocity of water that can be maintained through the composite shell is highly
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5.4. Influence of water temperature on the critical velocity of fluid conveyed through the composite shell In the previous section based on the variation of dimensionless natural frequency with respect to dimensionless flow velocities for different temperatures of water it was pointed out that the mean axial critical velocity of water flowing through the composite shell, with l=r ¼ 1:048 and 0° fibre orientation, decreases as the temperature of water increases. In this section the details of the influence of water temperature on critical flow velocity is examined for different fibre orientations viz. 0°, 15°, 30° and 60°. Figs. 12–14 illustrates the variations of mean axial flow velocity with respect to varying temperature of water for composite cylindrical shell with l=r ¼ 1:048 and fibre angles 0°, 30° and 60°. Various circumferential harmonics are also considered in the study. In addition Table 6 lists the critical velocity of water at various temperature for various modes ðm; nÞ for cylindrical composite shell with fibre angle 15°. Referring to Fig. 12, at ambient temperature (¼20 °C), the critical velocity of water reduces as the number of circumferential harmonics increases. The critical velocity attains a lowest value of 55.0 and 56.5 m/s for circumferential modes m ¼ 10 and 12 respectively. Further as the number of circumferential harmonics increase the permissible mean critical velocity increases. The lowest critical velocity of water at ambient temperature will nearly coincide with the circumferential mode which corresponds to a lower magnitude of natural frequency of the shell, these findings are already reported by l/r =1.048; fibre angle 0
mean axial velocity (m/sec)
90
o
m= 1 m= 4 m= 7 m = 10 m = 12 m = 14 m = 17 m = 20
80 70 60 50 40 30 20 10 0 0
10
20
30
40
50
60 o
temperature of flowing water C Fig. 12. Variation of the mean axial critical velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
l/r=1.048; fibre angle 30o
mean axial velocity (m/sec)
interdependent on the temperature of water as well as the critical thermal buckling temperature of the shell.
m= 1 m = 10 m = 11 m = 17 m = 20
120 100 80 60 40 20 0 0
10
20
30
40
50
60
70
o
temperature of flowing water C Fig. 13. Variation of the mean axial flow velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
o
l/r=1.048; fibre angle 60
mean axial velocity (m/sec)
30
110 100 90 80 70 60 50 40 30 20 10 0
m= 1 m= 4 m= 8 m = 10 m = 12 m = 14 m = 17
0
10
20
30
40
50
60
70
80
temperature of flowing water oC Fig. 14. Variation of the mean axial flow velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
Jayaraj et al. [12]. In Fig. 8, the natural frequency for mode (10,1) is 142.59 Hz and the critical buckling velocity is 55.0 m/s. For the same mode assuming water flows at 45 °C, the permissible lowest critical velocity is 28.5 m/s. In the case of mode (14,1) the velocity at ambient temperature is 59.5 m/s which is high compared to mode (10,1). But when flow of hot water at 45 °C is considered for mode (14,1) the permissible lowest critical velocity is 19.0 m/s. It is found that even though at ambient temperature the mode (14,1) has higher critical velocity compared to mode (10,1), but the critical
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
31
Table 6 Mean axial critical velocity of water through composite cylindrical shell with l=r ¼ 1:048 and fibre angle 15° Mode ðm; nÞ
Temperature °C 20
25
30
35
36
37
38
39
40
42
44
45.0 43.0 35.5 27.0 23.0 23.5 30.0 45.0
43.0 40.5 32.5 24.0 18.0 18.5 25.5 41.0
40.5 38.0 29.5 20.0 11.5 11.0 19.5 37.0
38.0 35.5 27.0 15.5
35.5 32.5 22.5 7.0
33.0 29.5 18.5
25.0 21.5
13.5 6.5
11.5 32.0
26.0
19.0
Mean critical axial velocity (m/s) (1,1) (4,1) (8,1) (10,1) (12,1) (13,1) (15,1) (18,1)
68.0 65.5 59.0 57.0 59.0 61.5 68.0 83.5
61.5 60.0 52.5 49.5 50.0 52.0 58.5 73.5
54.0 52.5 44.5 40.0 39.0 40.5 46.5 61.5
velocity of water flowing at 45 °C for mode (10,1) is higher when compared to mode (14,1) which is due to the fact that the critical buckling temperature of mode (10,1) is higher compared to mode (14,1). The critical thermal buckling temperature of the shell for mode (14,1) is 47.98 °C and that of mode (10,1) is 53.36 °C. From the above discussion it is clear that as the temperature of the water increases the magnitude of the lowest critical velocity decreases apart from this the associated mode for lowest critical velocity of the hot water shifts to that mode which has the lowest critical buckling temperature. From Table 6 the lowest critical velocity of water at ambient temperature (57.0 m/s) coincides with the mode (10,1) having a lower magnitude of natural frequency (158.3 Hz), whereas the lowest critical velocity of hot water (11.0 m/s) will coincide with the mode (13,1) corresponding to the lowest critical thermal buckling temperature (37.62 °C). Thus the lowest critical velocity of water at ambient temperature and the lowest natural frequency have the same mode, whereas the mode which is associated with lowest critical velocity of the hot water coincides with mode associated with the lowest critical thermal buckling temperature. Similar observation can be concluded from Figs. 13 and 14, the results of the numerical studies on composite cylindrical shell with l=r ¼ 1:048 and fibre angles 30°, and 60°. Further studies are repeated for composite cylindrical shell with l=r ¼ 2:08 and fibre angles 0° and 60°. The results pertaining to this are not reported, further the findings from these studies also revealed the above stated facts. 5.5. Influence of fibre angle of composite lamina on the mean axial critical velocity From the plots of the critical thermal buckling temperatures with respect to the circumferential modes for various fibre angles (Figs. 6 and 7) and from the plots of the variation of mean axial critical velocity versus tem-
perature (Figs. 12–14 and Table 6) an indirect of influence of the fibre angle on the mean axial critical velocity of water flowing through the shell can be deduced. Studies on the thermal buckling behavior of composite cylindrical shells by Ganesan et al. [5] indicate that the fibre angle of the composite lamina strongly influences the critical thermal buckling temperature. The critical buckling temperatures are very high in magnitude in case of composite cylindrical shell with fibre angle 60° whereas the critical thermal buckling temperatures are very low in magnitude for composite shell with 15° fibre orientation, see Figs. 7 and 6. Composite shell tailored with 60° fibre orientation will be well suited for high flow velocities of water at ambient conditions and also for conveying water at higher temperatures (which also means higher saturation pressure).
6. Conclusions Composites pipes/shells have found their place in specific engineering application requiring the transportation of fluids at high pressure and temperature. To study the dynamics of composite pipes/shells conveying hot fluid a semi-analytical finite element formulation has been presented. The structural finite element and the fluid finite element domain are coupled by appropriate fluid structure interface boundary conditions. Initial stresses produced by the thermal loading were employed to derive the geometric stiffness matrix. Thus the fluid structure interaction problem accounts for the temperature of the fluid flowing through the composite shell under isothermal conditions. Validation of the code for flow of water at ambient temperature and for the thermal buckling temperature gives an indication that the code is suitable for studies on the buckling behavior of composite shells conveying hot fluid. Two composite cylindrical shells with different length to radius ratios each with r=h ¼ 292 and for clamped–clamped boundary conditions were considered for numerical studies
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R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
and simulations. Composite shells with various fibre orientations are also considered in the study. The following are the conclusions from the numerical studies:
[2] [3]
1. Natural frequency variation of the shell conveying cold water and hot water have been computed for flow velocities ranging from 0.0 m/s to a velocity which leads to the buckling of the system. The nature of variation of the natural frequency of the coupled system involving the flow of water at ambient conditions and hot water is same. 2. The influence of water temperature on the critical flow velocities show that the maximum mean axial velocity of water that can be conveyed through the shell is high in the case of the water at ambient temperature when compared to high temperature water and in both cases it is limited by the critical buckling flow velocity. 3. The lowest critical velocity of water at ambient condition is associated with the mode that corresponds to the lowest natural frequency of the shell, whereas the lowest critical velocity of the hot water is associated with the mode corresponding to lowest thermal buckling temperature. 4. The influence of fibre angle on the critical mean axial velocities is considerable. Based on the fiber angles considered in the study it is found that 60° fibre angle for composite shells/pipes permits flow of water at high velocities in case of ambient conditions. The same composite shell also permits transportation of water at higher temperature since the critical buckling temperatures are high compared to 0°, 15° and 30° fibre angle. The studies presented in this paper bring out a methodology for estimating the maximum safe velocity of hot water for conveying through the composite shell based on the shell design parameters. This data estimated at the design stage can be used to quote the operating specifications for the pipes/shells.
[4] [5]
[6]
[7] [8] [9]
[10]
[11]
[12]
[13]
[14]
[15]
[16] [17] [18]
[19]
References
[20] [21]
[1] Nabors SA. Technology opportunity composite technology: strong, lightweight composite tanks and pipes for demanding
applications. NASAÕs Marshall Space Flight Center. www.nasasolutions.com. Thornton E. Thermal buckling of plates and shells. Appl Mech Rev 1993;46(10):485–506. Thangartnam RK, Palaninathan R, Ramachandran J. Thermal buckling of laminated composite shells. J AIAA 1990;28(5):859– 60. Radhamohan SK, Venkatramana J. Thermal buckling of orthotropic cylindrical shells. J AIAA 1975;13(3):397–9. Ganesan N, Kadoli R. Buckling and dynamic analysis of piezothermoelastic composite cylindrical shell. Compos Struct 2003; 59(1):45–60. Birman V, Bert CW. Buckling and post-buckling of composite plates and shells subjected to elevated temperature. ASME J Appl Mech 1993;60:514–9. Paidoussis MP, Li GX. Pipes conveying fluid: a model dynamic problem. J Fluids Struct 1993;7:137–204. Paidoussis MP. Fluid-structure interactions: slender structures and axial flow. London: Academic Press; 1998. vol. I. Bert CW, Chen TLC. Wave propagation in fluid conveying piping constructed of composite material. J Pressure Vessel Technol Trans ASME 1975;178:178–84. Chang JS, Chiou WJ. Natural frequencies and critical velocities of fixed-laminated circular cylindrical shells conveying fluids. Comput Struct 1995;57(5):929–39. Toorani MH, Lakis AA. Dynamic analysis of anisotropic cylindrical shells containing flowing fluid. J Pressure Vessel Technol Trans ASME 2001;123:454–60. Jayaraj K, Ganesan N, Chandramouli P. A semi-analytical coupled finite element formulation for composite shells conveying fluids. J Sound Vib 2002;258(2):287–307. Zhang YL, Reese JM, Gorman DG. Initially tensioned orthotropic cylindrical shells conveying fluid: A vibration analysis. J Fluids Struct 2002;16(1):53–70. Zhang YL, Gorman DG, Reese JM. A finite element method for modelling the vibration of initially tensioned thin-walled orthotropic cylindrical tubes conveying fluid. J Sound Vib 2001;245(1):93–112. Hale JM, Shaw BA, Speake SD, Gibson AG. High temperature failure envelopes for thermosetting composite pipes in water. J Plastics, Rubber Compos 2000;29(10):539–48. Rao SR, Ganesan N. Interlaminar stresses in shells of revolution. Mech Compos Mater Struct 1996;3:321–39. Jones RM. Mechanics of composite materials. New York: Hemisphere Publishing Corporation; 1998. Rao SR. Static and dynamic problems in laminated beams and axi-symmetric shells. Ph.D. Thesis, Indian Institute of Technology, Madras, India, 1997. Ross CTF. Pressure vessels under external pressure: static and dynamics. London: Elsevier Applied Science; 1994. Golub GH, Van Loan CF. Matrix computations. Third ed. Baltimore: Johns Hopkins University Press; 1996. Thangaratnam RK, Palaninathan, Ramachandran J. Buckling of composite cylindrical shells. J Aero Soc India 1989;41(1):47– 54.
Free vibration and buckling analysis of composite cylindrical shells conveying hot fluid Ravikiran Kadoli, N. Ganesan
*
Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology, Chennai 600 036, India
Abstract A coupled fluid structure interaction problem is analyzed using semi-analytical finite element method involving composite cylindrical shells conveying hot fluid for free vibration and buckling behavior. The system under study is assumed to have a steady flow of hot fluid and the temperature variation is axi-symmetric. First order shear deformation theory is used to model the elastic shells of revolution. Geometric stiffness matrix is evaluated to consider the effects of axi-symmetric temperature variation through the shell continuum due to flow of hot fluid. The fluid domain is modeled using the wave equation. Numerical results of the studies on composite cylindrical shells made of HS-Graphite/Epoxy with two different length to radius ratios and clamped–clamped boundary condition conveying hot fluid are presented. The variation of the natural frequency of the coupled system is evaluated with the steady flow of the hot fluid. The influence of the temperature on the mean axial flow velocity through the shell is critically examined. The critical velocity of the hot fluid and cold fluid which leads to shell instability is compared thus establishing the fact that the lowest critical velocity of the hot fluid coincides with the mode corresponding to the lowest critical thermal buckling temperature. Various fibre angles are also considered in the study and its influence is also examined. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Hot fluid; Axi-symmetric temperature; Semi-analytical finite element method; Fluid-structure interaction; Critical velocity of fluid
1. Introduction Petrochemical industries, chemical processing plants and power generating industries require vessels and pipes for storage and transportation of fluids at high pressure and temperature. There has been ever-growing demand for the manufacture of composite pipes/shells for hot water transportation, gas transportation (compressed air, natural gas, oxygen) and oil industry (production risers). NASAÕs Marshall Space Flight Center [1] has established fabrication techniques for the manufacture composite material pipes/shells and storage vessels of conventional and unique shapes which possess characteristics like lightweight, corrosion resistance, capable of handling diverse fluids, large diameter pipes for high pressure applications, capable of withstanding high temperature etc. These composite pipes/shells and storage vessels are multi-layered in construction. The authors feel that studies related to the dynamics of composite cylindrical shells through which hot fluid is *
Corresponding author. Tel.: +91-44-235-1365; fax: +91-44-2350509. E-mail address: [email protected] (N. Ganesan).
flowing at steady velocity can be an appropriate contribution especially regarding the magnitude of the critical flow velocity of the hot fluid and hence flow induced instability. There is available abundant literature on isotropic shells dealing with the thermal buckling subjected to constant temperature, temperature varying in circumferential direction and temperature varying along the generator of the shell. As pointed by Earl Thornton [2] in his recent review paper on thermal buckling of plates and shells, the availability of studies on thermal buckling in composite shells is scarce. Thangaratnam et al. [3] have performed linear thermal buckling analysis of laminated composite cylindrical and conical shells using finite element method and examined the nature of buckling under thermal load and mechanical load with respect to different fiber orientation. Radhamohan and Venkataramana [4] have made a complete study of thermal buckling of composite cylindrical shell made of fiberglass reinforced plastics. A uniform temperature rise throughout the shell was considered for buckling studies. Very recently the authors [5] have also contributed to the research literature on the thermal buckling studies and the influence of axi-symmetric temperature variation on the natural
0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 3 1 3 - 6
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frequencies of the composite cylindrical shells. Birman and Bert [6] considered studies on the buckling and post buckling response of composite shells subjected to high temperature using the equilibrium equations for shells under the simultaneous action of thermal and axial load. Similar to thermal buckling studies of isotropic shells, studies on pipes/shells conveying fluids are abundant in literature. Paidoussis and Li [7] have reviewed and compiled the exhaustive literature available on the studies related with the dynamics of pipes conveying fluids. Contributions by Paidoussis [8] in the area of dynamics of pipes conveying fluid is enormous and one can find a host of problems and solutions associated with fluid flowing through slender structures in the book written by Professor M.P. Paidoussis. Bert and Chen [9] analyzed the vibration modes in piping conveying fluid, pipes made of orthotropic as well as isotropic materials were considered. Chang and Chiou [10] used Mindlin type first order transverse shear deformable cylindrical shell theory and analytical method for the fluid to study the natural frequencies and critical velocities of laminated circular cylindrical shells with fixed ends conveying fluids. Toorani and Lakis [11] have developed a hybrid method based on a combination of refined shell theory, the finite element method and linear fluid theory which enables to account for the influence of shear deformation effect on the natural frequency response of multi-layered anisotropic open or closed cylindrical shells partially filled with or subjected to a non-viscous incompressible flowing fluid. Recently Jayaraj et al. [12] have carried out elaborate parametric studies on various composite cylindrical shells conveying fluids. For thin composite shells they could predict the divergence as well as coupled mode flutter instabilities, and more importantly, established that the buckling mode due to lowest critical velocity of the fluid coincides with the lowest natural frequency mode. Limited studies are available in literature related to shells conveying fluid, which considers the initial tension in the structure. Zhang et al. [13,14] have investigated the vibration characteristics of orthotropic cylindrical shells and tubes with initial tension and conveying fluid. The finite element formulation for the initially tensioned cylindrical shell is based on the three dimensional non-linear theory of elasticity and Eulerian equations. In case of initially tensioned cylindrical tubes, SandersÕ non-linear theory of thin shells and classical potential flow theory is used. Hale et al. [15] have investigated the biaxial stress-strain behavior in filament wound glass fibre reinforced composite pipes exposed to high temperature water. From the literature survey it is found that, to the authors knowledge, there are no studies that deal with the dynamic behavior of composite shells conveying hot fluid. In general the temperature distribution in shells conveying hot fluid leads to compressive stresses and the free vibratory and buckling behavior can be different
when compared to shells subjected to initial tension and conveying fluid. Hence in the present study an attempt is made to analyze a coupled elasto–thermo-fluid interaction problem using semi-analytical finite element method. The structural semi-analytical finite element formulation for axi-symmetric shells of revolution by Rao et al. [16], the fluid domain semi-analytical finite element formulation by Jayaraj et al. [12] are used and extended to include the geometric stiffness matrix which accounts for the effects of initial stresses due to hot fluid flowing through the composite shell. Numerical studies are carried out on composite shells with length to radius ratio equal to 1.048 and 2.08, radius to thickness ratio 292 and clamped–clamped boundary condition. Fibre angles equal to 0°, 15°, 30° and 60° in case of l=r ¼ 1:048 and 0° and 60° in case of l=r ¼ 2:08 are also considered in the study. Static thermal buckling analysis results are presented for the first axial mode associated with first 20 circumferential modes. The fluid considered for study is water at atmospheric pressure and atmospheric temperature (assumed as 20 °C) as well as water at temperatures below saturation temperature at atmospheric pressure. Numerical studies pertaining to variation of natural frequency of the composite shell conveying water at ambient conditions as well as hot water at different temperatures are conducted. The critical flow velocities of cold water as well as hot water at different temperatures are examined for instability of the composite shells. Influence of the fibre angle of the composite shell lamina is examined on the critical velocities of hot water.
2. The physical model The physical model considered for the study is illustrated in Fig. 1. The control volume also referred to as ÔsystemÕ comprises of the composite cylindrical shell and the fluid flowing at uniform velocity and constant temperature which is axi-symmetric in nature. Steady state conditions prevail over the control volume provided the heat influx equals the heat dissipated from the control volume. It is assumed that the temperature across the thickness of shell is constant and equal to that of the hot fluid. The length of the shell is such that temperature throughout the length of the shell does not vary. The fluid enters the control volume at velocity uf and temperature T , and leaves the control volume with the same operating conditions. Since the inlet and outlet mean velocity of fluid is same, it is essential to assume that the pressure drop of the flowing fluid through the shell is neglected. The flow of the fluid is potential, inviscid, irrotational, compressible and isentropic. When the hot fluid flows we assume that two phase of the fluid never exists and there is no flow separation from the wall of the shell. The material properties of the shell and fluid are assumed to be independent of temperature.
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
21
Heat qin Cylindrical shell lamina
Length of shell l meters
Radius of shell r meters
Cold/Hot fluid IN velocity uf Temperature T
Cold/Hot fluid OUT velocity uf Temperature T
Thickness of shell h meters
Control Volume Heat qout
Meridian of the shell
Fig. 1. Illustration of the control volume to the study effects of hot fluid flowing through cylindrical shell on the free vibration and buckling behavior.
panded by Fourier series in the h direction and expressed as
3. Semi-analytical finite element formulation 3.1. Structural domain The (s, h, z) coordinate system for general shells of revolution is illustrated in Fig. 2. By setting the value of the principal radius of curvature R/ equal to infinity (or a large magnitude) and the other principal radius of curvature Rh equal to a finite radius equal to mean radius r, the geometry resembles a cylindrical shell as illustrated in Fig. 3. The mid-surface displacements according to the first order shear deformation theory are expressed as uðs; h; z; tÞ ¼ u0 ðs; h; tÞ þ zws ðs; h; zÞ
ð1aÞ
vðs; h; z; tÞ ¼ v0 ðs; h; tÞ þ zwh ðs; h; zÞ
ð1bÞ
wðs; h; z; tÞ ¼ w0 ðs; h; tÞ
ð1cÞ
where u0 , v0 , w0 are displacement of mid-surface along the s, h and z direction and ws and wh are rotations of the normal to the mid-surface along s and h axes respectively. In the semi-analytical method the generalized displacement field is assumed to depend in the circumferential direction hence these quantities can be ex-
8 9 2 u0 > cos mh 0 0 > > > > > > > 0 sin mh 0 1 6 < v0 = X 6 6 0 0 cos mh w0 ¼ 6 > > m¼0 4 > 0 0 0 > ws > > > : > ; wh 0 0 0
ð2Þ
m stands for the mth circumferential harmonic.
Center line of object
β =s/l 3
Rθ
Rφ
O1
O2 2
1
Parent element
Fig. 3. Schematics of the discretisation of the cylindrical shell using three noded quadratic line element.
Center line of structure rr
φ O1A : Rφ O 2A : R θ rr = Rθ /sinφ
9 38 0 0 u0m > > > > > > > 7 0 0 7< v0m > = 0 0 7 w 7> 0m > cos mh 0 5> w > > > > : sm > ; whm 0 sin mh
A
dφ
B
z (w) θ (v) s (u)
O2 mid-surface of structure O1
Fig. 2. Tri-orthogonal curvilinear coordinate system for shells of revolution.
22
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
Rao et al. [16] used first order shear deformation theory and other higher order theories to study the interlaminar stresses in laminated composite shells of revolution. The strain components for a doubly curved shells of revolution based on FSDT are as follows: ess ¼
1 0 1 ðe þ zj1s Þ chz ¼ c0hz A1 ss A2
ð3a; bÞ
ehh ¼
1 0 ðe þ zj1h Þ A2 hh
ð3c; dÞ
csh ¼
1 1 0 ðc þ zj1sh Þ A1 A2 sh
csz ¼
1 0 c A1 sz
ð3eÞ
½ke ¼
Z
T
½B ½D ½B r ds dh
ð9Þ
A
where ½B is the strain-displacement matrix of the shell at stress free temperature T0 . Due to orthogonality principle the stiffness matrix becomes decoupled for each circumferential harmonic. A three node isoparametric element is used in the axial direction s. Fig. 3 depicts the discretization of the cylindrical shell using this element. Each node has five degrees of freedom. The displacement parameters associated with the element are fde g ¼ fu1 ; v1 ; w1; ws1 ; wh1 ; u2 ; v2 ; w2; ws2 ; wh2 ; u3 ; v3 ; w3 ; ws3 ; wh3 g
ð10Þ
where
the subscripts 1, 2, and 3 stand for the three nodes of the element. The shape functions Ni in terms of the isoparametric axial coordinate b ¼ s=l are given by
1 1 ¼ A1 ð1 þ z=R/ Þ and 1 1 ¼ A2 ð1 þ z=Rh Þ in the above equations R/ and Rh are the principle radii of curvature of the shell as illustrated in Fig. 2 and z is the thickness in z-direction. The total strains are denoted as ess , ehh , chz , csz and csh ; which comprises of e0ss , e0hh , c0hz , c0sz and c0sh the normal strains and the shear strains referred to mid-surface, and j1s , j1h and j1sh the change in curvature. The total strain energy in the shell continuum is given by U ¼ U1 þ U2 where U1 is the strain energy due to vibratory stresses and U2 is the strain energy contribution from the initial stresses due to steady state axi-symmetric temperature. Strain energy U1 is given by Z 1 fess rss þ ehh rhh þ chz shz þ csz ssz þ csh ssh g dV U1 ¼ 2 V Z 1 T feg frg dA ð4Þ ¼ 2 A where frg and feg are the generalized stress and strain vectors respectively and dA is the infinitesimal area element on the shell mid-surface. The foregoing vectors are defined as T
feg ¼ fe0ss
e0hh
c0sh
T
j1s
j1h
j1sh
c0sz
c0hz g
frg ¼ fNss Nhh Nsh Mss Mhh Msh Qs Qh g
ð5Þ ð6Þ
The generalized stress resultant vector can be expressed as frg ¼ ½D feg T
ðb2 bÞ ; N2 ¼ ð1 b2 Þ and 2 ðb2 þ bÞ ð11a–cÞ N3 ¼ 2 The constitutive matrix ½D in Eq. (7) or (9) is for a laminate consisting of various integrated shell stiffnesses Aij , B ij , Dij , and F ij . Various integrated shell stiffnesses are defined as follows: Z þh=2 Qij ð1; z; z2 ; z3 Þ dz ð12Þ ðAij ; Bij ; Dij ; Fij Þ ¼ N1 ¼
U1 ¼ 12fde g ½ke fde g
h=2
Qij are the reduced stiffness coefficients, other details of this constitutive matrix can be referred to Jones [17] and Ganesan et al. [5]. 3.2. Geometric stiffness matrix evaluation Neglecting the strain energy due to initial transverse shear stresses, strain energy due to initial stresses U2 is given in [18] as Z 1 2 2 2 U2 ¼ fðei Þ r þ ðeihh Þ r hh þ 2ðcish Þ s sh g dV ð13Þ 2 V ss ss in which r ss , r hh and s sh are the initial stresses and the corresponding strains, eiss , eihh and cish are given as follows: i 1 ow ui i ess ¼ ð14aÞ os R/ 1 þ z=R/ 1 1 owi vi i ð14bÞ ehh ¼ sin / ð1 þ z=Rh Þ r oh r cish ¼ eiss eihh
ð7Þ
Eq. (13) can be put in the following form:
ð8Þ
U2 ¼ 12fdie gT ½kre fdie g
where ½ke is the element stiffness matrix corresponding to the mth harmonic and is computed as follows:
ð14cÞ
ð15Þ
The matrix ½kre in the foregoing equation is the element geometric stiffness matrix
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
½kre ¼
Z
T
½Bi fr g dV
ð16Þ
V
fdie g
represents the elemental displacement vector due to initial stress, ½Bi is the strain displacement matrix based on Eqs. (14a)–(14c) and fr g is the vector comprising of the stress resultants due to thermal load which is described in next section. 3.3. Thermal load and initial stress evaluation It is well known that under thermal environment the expression for the total potential for a finite element of the cylindrical shell is Z Z 1 T T P¼ feg ½D feg dV feg fkgT dV ð17Þ 2 V V The temperature is assumed to be constant over the thickness and the temperature field for the structural continuum is T ðs; h; z; tÞ ¼ T ðs; h; tÞ Steady state temperature is assumed to be dependent in the circumferential direction, and using Fourier series the temperature in the circumferential direction is 1 X T ¼ Tm cos mh ð18Þ m¼0
Minimisation of P with respect to the displacement vector fde g leads to a standard equation for a finite element as ½ke fde g
fFth e g
¼0
ð19Þ
½ke is the structural element stiffness matrix same as Eq. (9). The thermal load evaluation involves Z fFth g ¼ ½e T fkgfT ðzÞg dV ð20Þ e
23
# is the lamina orientation angle, T ðzÞ is the temperature ðkÞ rise from stress free temperature T0 , fag is the vector of coefficient of linear thermal expansion in the kth layer, a1 and a2 are the coefficients of thermal expansion in the directions parallel and perpendicular to the coordinate axis for kth lamina. Using Eq. (20) the thermal load is computed for a steady state axi-symmetric temperature, above the stress free temperature T0 which is assumed to be 20 °C. Solving Eq. (19) in global sense gives the displacement field due to the thermal loading. Using the element displacement vector the stresses and moment resultants are evaluated as follows: fr g ¼ ½D fe0 g fkgT
ð22Þ
fe0 g represents the strain developed in the structural continuum due thermal load, [D] is matrix comprising of various integrated shell stiffnesses in the transformed coordinate, and fkg represents the temperature stress coefficients and is similar to Eq. (21). The stress resultants are found for each finite element and are used in Eq. (16) to compute the element geometric stiffness matrix. 3.4. Mass matrix The mass matrix is obtained from the kinetic energy of the shell continuum, the kinetic energy is Z Z q q KE ¼ ðu_ 2 þ v_ 2 þ w_ 2 Þ dV ¼ fd_ gT fd_ g dV ð23Þ 2 V 2 V Using Eqs. (10) and (11a–c) the kinetic energy will be 1 T KE ¼ fde g ½me fde g 2 where ½me is the element mass matrix given by Z me ¼ q N T N dV
ð24Þ
V
where fkg is the temperature stress coefficients computed for the laminate as nlay Z hk X ðkÞ ðkÞ ½Qij fag ð1; zÞ dA ð21Þ fkg ¼ k¼1
hk 1
ðkÞ
½Qij represents the generally orthotropic lamina properties of the kth layer with proper coordinate transformation and comprises of the stiffness coefficients Aij , and Bij . The coefficient of linear expansion for the kth layer in the shell coordinate system is fagðkÞ ¼ fas
ah
ash
0
where as ¼ a1 cos2 # þ a2 sin2 # as ¼ a1 sin2 # þ a2 cos2 # ash ¼ ða1 a2 Þ cos # sin #
0 0
0
0g
T
3.5. Finite element formulation of the fluid domain Jayaraj et al. [12] very recently have conducted exhaustive studies on the dynamics of composite shells conveying fluid. The finite element formulation for the fluid domain is based on the wave equation 2 1 o o r2 / 2 /¼0 ð25Þ þ Ux c ot ox where / is the velocity potential, c is the velocity of sound, Ux is the mean axial flow velocity of fluid. An eight node isoparametric quadrilateral element is used for discretising the fluid domain as shown in Fig. 4. Velocity potential is the nodal degree of freedom. The velocity potential should satisfy the wave equation. The velocity components in different directions are expressed in terms of velocity potential as
24
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32 Fluid domain mesh using 8 node isoparametric element mid-surface of shell
Elastic shell discretised using 3 node isoparametric element
X
θ Center line of shell
Z
mid-surface displacement of structure, velocity potential of fluid flow domain and temperature distribution are expanded in Fourier series in circumferential direction θ.
Fig. 4. Finite element model of the cylindrical shell conveying fluid.
o/ 1 o/ o/ ; Vh ¼ ; Vr ¼ ð26Þ ox R oh or To satisfy the dynamic boundary condition it is necessary that the radial velocity of the fluid must be equal to the instantaneous velocity of the shell, that is o/ ow ow Vr ¼ þ Ux ð27Þ ¼ or ot ox
Vx ¼ U x þ
r¼R
BernoulliÕs principle is used to compute the pressure exerted by the fluid on shell wall. For unsteady flow o/ 1 2 P Ps þ V þ ¼ ot 2 q q
ð28Þ
where V 2 ¼ Vx2 þ Vh2 þ Vr2 and Ps is the stagnation pressure. P is the sum of a mean pressure P and the perturbation pressure p P ¼P þp
ð29Þ
For small deformation the non-linear terms in V 2 are neglected, we get o/ V 2 ffi Ux2 þ 2Ux ox o/ o/ p ¼ q þ Ux ot ox
ð30Þ
The finite element form of the fluid domain and hence the interaction between the fluid and the structure is obtained by using the GalerkinÕs weighted residual method. The weighted residual form over the domain takes the form
Z
NfT V
1 r / 2 c 2
o o þ Ux ot ox
2 ! / dV ¼ 0
ð31Þ
where the weighting function Nf is the fluid shape functions given in Ross [19]. The variation of velocity potential is expressed in Fourier series in the h direction. Z Z Z 1 T T Nf r/:n dS rNf r/ dV 2 N T /€ dV c V f S V Z Z 2 2Ux Ux2 o2 / T o / dV 2 2 Nf NfT 2 dV ¼ 0 ð32Þ oxot ox c c V V The first term of Eq. (32) describes the interaction of the fluid and the shell wall. Integration of the other terms will yield the kinetic energy of the fluid, compression energy of the fluid, Coriolis energy of the fluid and centrifugal energy of the fluid. Making use of Eq. (27) i.e. the fluid shell interface boundary condition along with the first term of Eq. (32) we have Z Z Z oN T T _ dSfUe g Nf r/:n dS ¼ Nf N dsfUe g þ Ux NfT ox S ð33Þ where N is the w component of the shell shape function and n is the unit normal vector to the structure. The dynamic pressure acting on the shell wall given by Eq. (30) also describes the interaction between fluid and shell wall, hence
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
T
o/ o/ þ Ux dS ot ox
where
N qf S Z Z T T oNf _ dSf/e g N Nf dSf/e g þ qf Ux N ¼ qf ox S S
"
ð34Þ
The various finite element matrices obtained from the above procedure are listed below: Compression energy of the fluid Z X 1 // NfT Nf dV G// ¼ G// Ge ¼ 2 e c V Fluid structure interaction coupling terms due to structural and fluid damping Z X T Cu/ ¼ q N Nf dS Cu/ ¼ Cu/ f e e S Z X C/u NfT N dS C/u ¼ C/u e ¼ e S
Coriolis energy of fluid Z 2 oNf C// dV ¼ NfT e 2 Ux c V ox
M ¼
K ¼ "
C ¼
Muu
0
0
G//
; #
Kuu þ Kuu r
Ux Ku/
Ux K/u
H// Ux2 I// #
0 C/u
;
Cu/ Ux C//
Since the isothermal conditions are assumed to exist, and the changes in density and viscosity of the fluid are not accounted due to changes in temperature, it is apparent from the above Eq. (35) that the off diagonal terms of the damping and stiffness matrices couples the shell and fluid motion. Eq. (36) in the state space form of Eq. (35) is solved for the eigenvalues using LAPACK routine, DGEGV [20]. 4. Validation of the semi-analytical finite element formulation for analyzing hot fluid flowing through shells
C// ¼
X
C// e
Stiffness coupling due to flow Z X T oNf Ku/ dS Ku/ ¼ ¼ q N Ku/ f e e ox S Z X oN dS K/u ¼ K/u NfT K/u e ¼ e ox S Kinetic energy of fluid Z X ¼ rNfT rNf dV H// ¼ H// H// e e V
Centrifugal energy of fluid Ux2 Z X 1 oNfT oNf dV I// ¼ I// ¼ I// e e 2 c V ox ox Considering the structural global mass matrix Muu , global stiffness matrix Kuu and global geometric stiffness matrix Kuu r the complete fluid-structure finite element equation is uu € M 0 u u_ 0 Cu/ þ 0 G// /€ /_ C/u Ux C// uu u Ux Ku/ K þ Kuu r ¼0 ð35Þ þ / Ux K/u H// Ux2 I// The above equation is expressed in the state-space form by letting fag ¼ fu / u_ /_ gT
0 K C M
fag ¼ k ð36Þ
fag 0 M M 0
The present formulation developed for analysis of shells conveying hot fluid has been validated with the results reported by Jayaraj et al. [12] for a Graphite/ Epoxy composite cylindrical shell with l=r ¼ 1:0; r=h ¼ 100 and clamped–clamped boundary conditions. Jayaraj et al. have validated their formulation with published results of Chang and Chiou [10]. The variation of the dimensionless frequency versus dimensionless velocity has been compared in Fig. 5 for the first two axial modes associated with circumferential harmonics 4. Even though the fluid domain modeling is same as that developed by Jayaraj et al., the difference lies in the structural finite element. In the present case the structural finite element has five degrees of freedom per node whereas the structural finite element used by Jayaraj et al. has viscoelastic layer and hence the nodal degrees
Dimensionless frequency
Z
25
8.25 7.50 6.75 6.00 5.25 4.50 3.75 3.00 2.25 1.50 0.75 0.00
m=4, n=1: m=4,n=2:
present present
Jayaraj et al.[12] Jayaraj et al.[12]
0.002 0.004 0.006 0.008 0.010 0.012 0.014
Dimensionless velocity Fig. 5. Validation of the results obtained from the present study with that of Jayaraj et al. [12]. Composite shell is made of Graphite/Epoxy, l=r ¼ 1:0, r=h ¼ 100 and clamped–clamped boundary condition.
26
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
Table 1 Validation of critical buckling temperature for symmetric cross-ply cylindrical shell r=h
Critical temperature (°C) Ref. [21]
200 300 400 500 a
quencies and critical velocity of the hot fluid corresponding to instability of shell. The numerical study hence involves the following objectives:
Results from present study a
1304.298 (11) 912.434 (13) 659.610 (18) 514.745 (19)
1258.4 (11), 1250 (12), 1268.4 (13) 851.3 (13), 835.8 (14), 836.6 (15) 632.2 (16), 631 (17), 637.8 (18) 507.3 (18), 507 (19), 511.74 (20)
Numbers in bracket indicate circumferential mode.
of freedom are seven. The results obtained by the present formulation are in close agreement with those reported by [12]. The above validation is for shell conveying cold fluid. Due to non-availability of literature on hot fluid flowing through pipes/shells, it was felt that the formulation could be validated for critical thermal buckling temperature in case of composite shells operating under axi-symmetric temperature without flow of fluid. For this purpose Eq. (35) is solved as an uncoupled problem. Thangarathman et al. [21] have carried out finite element analysis of buckling of composite cylindrical shells under the influence of mechanical and thermal loads using semi-loof finite element. They have presented the results of their studies on the critical buckling temperature of composite cylindrical shell under uniform temperature rise with simply supported boundary condition. The length to radius ratio (l=r) was 0.5 and different ratios of shell radius to shell thickness (r=h) were considered. The laminate properties used were as follows: Ell =Ett ¼ 10; at =al ¼ 2:0
Glt =Ett ¼ 0:5; and
llt ¼ 0:25;
al ¼ 0:1 10 5
These data were used to obtain the critical buckling temperatures using the present code. Table 1 lists the comparison of the results obtained by the present code and those reported by Thangaratnam et al. The trend in the results tallies very well. Also it is seen that the circumferential mode at which buckling occurs nearly tally. These two validations are sufficient to indicate the adaptability of the formulation and the computer code developed for the studies on composite shells conveying hot fluid.
5. Numerical simulations––results and discussion Analysis of the composite cylindrical shell conveying hot fluid is carried out in order to determine the effect of temperature of the hot fluid flowing on the natural fre-
(a) Variation of the natural frequencies of the composite shell corresponding to the first axial mode of the coupled system involving the flow of hot fluid through the shell. (b) Influence of the temperature of water on the magnitude of the mean axial velocity and hence the critical velocity of hot water which cause instability of the system. (c) Comparison of the behavior of instability of the shell with flow of cold and hot water. (d) Simultaneously bring out the influence of fiber orientation on the above stated objectives. It is essential to understand the nature of the variation of the free vibration frequencies of the composite shell as well of the magnitude of the thermal buckling temperatures for different modes. Hence the study starts with the evaluation of (i) the critical buckling temperature of the first axial mode (n ¼ 1) associated with first 25 circumferential modes (m ¼ 1 to 25) for the composite shells exposed to axi-symmetric temperature variation without fluid and (ii) the free vibration frequencies for the composite shells at ambient temperature and without fluid corresponding to first axial mode associated with first 25 circumferential modes. The geometric details of the composite cylindrical shells considered for the study are given in Table 2. The properties of HS-Graphite/Epoxy material are listed in Table 3. 5.1. Evaluation of static thermal buckling eigenvalues Static thermal buckling analysis is carried out using the following equation: Table 3 Material properties of HS-Graphite/Epoxy HS-Graphite/ Epoxy YoungÕs Modulus (GPa)
E11 , E22 , E33
Shear Modulus (GPa) PoissonÕs ratio Density (kg/m3 ) Coefficient of thermal expansion (°C) Environment temperature (°C)
G12 , G31 , G23 m12 , m13 , m23 q a11 , a22 T0
181.0, 10.34, 10.34 7.2, 7.2, 7.2 0.28, 0.28, 0.28 1389.23 11:34 10 6 , 36:9 10 6 20.0
Table 2 Details of cylindrical shell and boundary conditions Cylindrical shell
Length (l m)
Radius (r m)
l=r ratio
Thickness of cylindrical shell (h m)
Boundary condition
1 2
0.914 1.8288
0.876 0.876
1.048 2.08
0.0015 0.0015
Clamped–clamped Clamped–clamped
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
ðKuu þ vKuu r Þxb ¼ 0 In the above equation Kuu is the global structural stiffness matrix and Kuu r is the global geometric stiffness matrix, contribution from the thermal load due to steady state axi-symmetric temperature. v is the buckling eigenvalues and xb is the corresponding buckling eigenvector. The buckling parameters of the first axial mode (n ¼ 1) associated with first 25 circumferential modes (m) are computed at arbitrary temperature of 10 °C. The critical buckling temperatures of composite cylindrical shell with l=r ¼ 1:048 and for different fibre orientations are plotted with respect to circumferential harmonic, refer to Figs. 6 and 7. In Figs. 6 and 7 the lowest critical buckling temperature modes are also in-
80
fibre angle 0o o fibre angle 15 o fibre angle 30
70 60 50 40 30
m=14,n=1 m=13,n=1
20
m=11,n=1
10 0
2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 6. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 1:048 and clamped–clamped boundary condition.
70
Critical buckling temperature oC
600
o
fibre angle 60
500 400 300 200
dicated. For composite shells with fibre angle 0°, 15° and 30° the magnitude of the critical buckling temperature is minimum for circumferential modes m ¼ 14, 11 and 12. One can notice that the critical buckling temperatures are well within the saturation temperature of water at atmospheric pressure (1 atmp. or 1.0 bar) for the circumferential modes considered. The critical buckling temperatures for composite cylindrical shell tailored to fibre angle 60° as seen in Fig. 7 exhibit increasing temperature as the number of circumferential harmonics increase. The critical buckling temperatures are considerably high compared to the other fibre angles. These temperatures are above 296.77 °C, which is much high compared to saturation temperature of water at atmospheric pressure. Similarly critical buckling temperatures are evaluated for composite cylindrical shell with l=r ¼ 2:08 and for fibre angles 0° and 60°. Typical results are presented in Fig. 8 for composite shell with fibre angle 0°. The variation of the buckling temperature corresponding to first axial mode (n ¼ 1) associated with 25 circumferential modes m is similar to that noticed in case of l=r ¼ 1:048, see Fig. 6. Again notice that the critical buckling temperatures are well within the saturation temperature of water at atmospheric pressure. The critical buckling temperature are considerably higher in case of composite shell with l=r ¼ 2:08 and fibre angle 60° (these results are not presented here). For example, the critical buckling temperature is 296.4° for mode (11,1). In the present work an attempt is made to study the buckling behavior of shells conveying hot fluids. For the purpose of numerical computation and simulation, water at atmospheric pressure is considered as the working fluid. It is assumed that the density variation of water is
m=1,n=1
100 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 7. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with fibre angle 60°, l=r ¼ 1:048 and clamped–clamped boundary condition.
Critical buckling temperature o C
Critical buckling temperature oC
90
27
60 50 40 30
m=14,n=1 o
fibre angle 0
20 10 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26
Circumferential modes m Fig. 8. Critical buckling temperatures of the first axial mode associated with 25 harmonics. HS-Graphite/Epoxy cylindrical shell with fibre angle 0°, l=r ¼ 2:08 and clamped–clamped boundary condition.
28
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
lowest natural frequency and the lowest critical buckling temperature are not the same.
small at different operating temperatures. Even though at lower circumferential modes the buckling temperatures are high, especially in case of composite cylindrical shell with fibre angle 60° and l=r ¼ 1:048, the studies are limited to temperature of water equal to 90 °C for all circumferential modes.
5.3. Studies on variation of natural frequency of the shell conveying hot fluid In order to determine the influence of the hot fluid flowing through the composite shell on the natural frequency of the first axial mode numerical studies are carried out on clamped–clamped composite cylindrical shell with l=r ratio equal to 1.048 and 2.08. Further the influence of the fibre orientation is also considered simultaneously in the study. Water at ambient temperature as well as water at various temperatures is considered in the study. In the present study however the maximum magnitude of the water temperature is limited either by the critical buckling temperature of the shell or 90 °C which is close to the saturation temperature of water at atmospheric pressure. Considering the flow of
5.2. Natural frequencies of composite cylindrical shells The free vibration eigenvalues of the composite shell with l=r ¼ 1:048, for fibre orientations of 0°, 15°, 30° and 60° and l=r ¼ 2:08 for fibre orientations of 0° and 60° are illustrated in Figs. 9 and 10. It is also indicated in the figures the mode which has the lowest natural frequency. The lowest natural frequencies, lowest critical buckling temperatures and their modes are listed in Tables 4 and 5. Note that the mode corresponding to
fibre angle 0o o fibre angle 60
Natural frequency (Hz)
1000 800 600 400
m =9, n=1
200 0
m=6,n=1
0 2 4 6 8 10 12 14 16 18 20 22 24 26
circumferential modes m Fig. 9. Plots of the variation of the first axial mode natural frequencies for 25 circumferential modes. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 1:048 and clamped–clamped boundary condition.
Fig. 10. Plot of the variation of the first axial mode natural frequencies for first 25 circumferential modes. HS-Graphite/Epoxy cylindrical shell with l=r ¼ 2:08 and clamped–clamped boundary condition.
Table 4 Lowest natural frequency and lowest critical buckling temperature for composite cylindrical shell with l=r ¼ 1:048 Sl. no.
Fibre angle
Lowest natural frequency (Hz)
Mode ðm; nÞ
Lowest critical thermal buckling temperature (°C)
Mode ðm; nÞ
1 2 3 4
0° 15° 30° 60°
137.4 153.0 167.9 175.2
(12,1) (12,1) (10,1) (8,1)
47.98 37.62 50.50 296.77
(14,1) (13,1) (11,1) (1,1)
Table 5 Lowest natural frequency and lowest critical buckling temperature for composite cylindrical shell with l=r ¼ 2:08 Sl. no.
Fibre angle
Lowest natural frequency (Hz)
Mode ðm; nÞ
Lowest critical thermal buckling temperature (°C)
Mode ðm; nÞ
1 2
0° 60°
77.01 90.23
(9,1) (6,1)
46.51 296.42
(14,1) (11,1)
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
20oC 30oC o 40 C 45oC o 50 C o 60 C
Dimensionless frequencies
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
0.001 0.002 0.003 0.004 0.005 0.006
Dimensionless velocity
(a)
1.8 m=12, n=1
Dimensionless frequencies
5.3.1. Composite shell with l=r ¼ 1:048 and fibre angle 0° conveying fluid The variation of dimensionless frequency of the first axial mode with respect to dimensionless velocity are illustrated in Fig. 11(a)–(c) for circumferential modes m ¼ 1, 12 and 14. The mode (12,1) corresponds to the lowest natural frequency of the composite shell and mode (14,1) has the lowest critical buckling temperature. In Fig. 11(a) the variation of the dimensionless frequency versus dimensionless velocity for mode (1,1) is illustrated. The critical buckling temperature for this mode is 67.67 °C. Hence one can consider the flow of water at temperatures like 20, 30, 40, 45, and 60 °C. The velocity of water is varied from 0.0 m/s to critical velocity that leads to shell instability. The natural frequency of the coupled system decreases but not appreciably as the velocity of water increases. The fall in frequency is very rapid as the velocity of water flowing approaches the critical velocity. This trend is observed irrespective of the temperature of the water. In contrast for higher modes (12,1) and (14,1) the drop in the magnitude of natural frequencies smooth and continuous with respect to flow as seen in Fig. 11(b) and (c). Apart from the decrease in the natural frequencies, one can also observe the decrease in the magnitude of the maximum velocity that is permissible for conveying through the shell as the temperature of the water is increased. As seen in the Fig. 11(b), the maximum velocity of water that can be conveyed through the shell at 20 °C is greater than that of the maximum velocity of water passing through the shell at 40 °C. Consider the flow of hot water at 45 °C in the case of mode (12,1) in Fig. 11(b). The critical thermal buckling temperature for this mode is 49.39 °C. Hence we have a situation where the hot water temperature is close to the thermal buckling temperature. It is clear from the curve that the natural frequency as well as the maximum velocity of water
m=1,n=1
1.8
1.6
o
20 C 30oC o 40 C o 45 C
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.001
0.002
0.003
0.004
0.005
Dimensionless velocity
(b)
2.0
m = 14, n = 1
1.8
Dimensionless frequencies
hot water at constant temperature, the velocity of water is varied from 0.0 m/s to a velocity that causes the system to buckle, in increments of 0.5 m/s. The eigenvalues are computed by solving the coupled equation of motion, Eq. (36), using the DGEVG subroutine, for the shell conveying water at steady velocity and constant temperature. Eigenvalues obtained are expressed in dimensionless quantities as X ¼ ðx=x0 Þ 100, where x0 ¼ U0 =r, x is the real natural frequency and r is the radius of the shell. The flow velocity of the water is also expressed inffi dimensionless form as Ux =U0 where pffiffiffiffiffiffiffiffiffiffiffi U0 ¼ E0 =qs , in which E11 =E0 ¼ 21, E11 is YoungÕs modulus of composite shell and qs is the density of shell material. The dimensionless quantities are similar to Chiou et al. [10] for a single layer in its material principal axes. Typical results of the study are presented and discussed in the next section.
29
1.6
20oC (ambient temperature) 30oC 40oC 45oC
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.001
(c)
0.002
0.003
0.004
0.005
0.006
Dimensionless velocity
Fig. 11. Variation of the dimensionless frequency with respect to dimensionless velocity for water flowing through composite cylindrical shell with l=r ¼ 1:048, fibre angle 0° and clamped–clamped boundary condition.
considerably reduces when the temperature of the water conveyed through the shell nears the critical thermal buckling temperature of the shell. This study reveals that the maximum velocity of water that can be maintained through the composite shell is highly
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
5.4. Influence of water temperature on the critical velocity of fluid conveyed through the composite shell In the previous section based on the variation of dimensionless natural frequency with respect to dimensionless flow velocities for different temperatures of water it was pointed out that the mean axial critical velocity of water flowing through the composite shell, with l=r ¼ 1:048 and 0° fibre orientation, decreases as the temperature of water increases. In this section the details of the influence of water temperature on critical flow velocity is examined for different fibre orientations viz. 0°, 15°, 30° and 60°. Figs. 12–14 illustrates the variations of mean axial flow velocity with respect to varying temperature of water for composite cylindrical shell with l=r ¼ 1:048 and fibre angles 0°, 30° and 60°. Various circumferential harmonics are also considered in the study. In addition Table 6 lists the critical velocity of water at various temperature for various modes ðm; nÞ for cylindrical composite shell with fibre angle 15°. Referring to Fig. 12, at ambient temperature (¼20 °C), the critical velocity of water reduces as the number of circumferential harmonics increases. The critical velocity attains a lowest value of 55.0 and 56.5 m/s for circumferential modes m ¼ 10 and 12 respectively. Further as the number of circumferential harmonics increase the permissible mean critical velocity increases. The lowest critical velocity of water at ambient temperature will nearly coincide with the circumferential mode which corresponds to a lower magnitude of natural frequency of the shell, these findings are already reported by l/r =1.048; fibre angle 0
mean axial velocity (m/sec)
90
o
m= 1 m= 4 m= 7 m = 10 m = 12 m = 14 m = 17 m = 20
80 70 60 50 40 30 20 10 0 0
10
20
30
40
50
60 o
temperature of flowing water C Fig. 12. Variation of the mean axial critical velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
l/r=1.048; fibre angle 30o
mean axial velocity (m/sec)
interdependent on the temperature of water as well as the critical thermal buckling temperature of the shell.
m= 1 m = 10 m = 11 m = 17 m = 20
120 100 80 60 40 20 0 0
10
20
30
40
50
60
70
o
temperature of flowing water C Fig. 13. Variation of the mean axial flow velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
o
l/r=1.048; fibre angle 60
mean axial velocity (m/sec)
30
110 100 90 80 70 60 50 40 30 20 10 0
m= 1 m= 4 m= 8 m = 10 m = 12 m = 14 m = 17
0
10
20
30
40
50
60
70
80
temperature of flowing water oC Fig. 14. Variation of the mean axial flow velocity of water with respect to temperature in the first axial mode associated with various circumferential harmonics. HS-Graphite/Epoxy composite cylindrical shell with clamped–clamped boundary condition.
Jayaraj et al. [12]. In Fig. 8, the natural frequency for mode (10,1) is 142.59 Hz and the critical buckling velocity is 55.0 m/s. For the same mode assuming water flows at 45 °C, the permissible lowest critical velocity is 28.5 m/s. In the case of mode (14,1) the velocity at ambient temperature is 59.5 m/s which is high compared to mode (10,1). But when flow of hot water at 45 °C is considered for mode (14,1) the permissible lowest critical velocity is 19.0 m/s. It is found that even though at ambient temperature the mode (14,1) has higher critical velocity compared to mode (10,1), but the critical
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
31
Table 6 Mean axial critical velocity of water through composite cylindrical shell with l=r ¼ 1:048 and fibre angle 15° Mode ðm; nÞ
Temperature °C 20
25
30
35
36
37
38
39
40
42
44
45.0 43.0 35.5 27.0 23.0 23.5 30.0 45.0
43.0 40.5 32.5 24.0 18.0 18.5 25.5 41.0
40.5 38.0 29.5 20.0 11.5 11.0 19.5 37.0
38.0 35.5 27.0 15.5
35.5 32.5 22.5 7.0
33.0 29.5 18.5
25.0 21.5
13.5 6.5
11.5 32.0
26.0
19.0
Mean critical axial velocity (m/s) (1,1) (4,1) (8,1) (10,1) (12,1) (13,1) (15,1) (18,1)
68.0 65.5 59.0 57.0 59.0 61.5 68.0 83.5
61.5 60.0 52.5 49.5 50.0 52.0 58.5 73.5
54.0 52.5 44.5 40.0 39.0 40.5 46.5 61.5
velocity of water flowing at 45 °C for mode (10,1) is higher when compared to mode (14,1) which is due to the fact that the critical buckling temperature of mode (10,1) is higher compared to mode (14,1). The critical thermal buckling temperature of the shell for mode (14,1) is 47.98 °C and that of mode (10,1) is 53.36 °C. From the above discussion it is clear that as the temperature of the water increases the magnitude of the lowest critical velocity decreases apart from this the associated mode for lowest critical velocity of the hot water shifts to that mode which has the lowest critical buckling temperature. From Table 6 the lowest critical velocity of water at ambient temperature (57.0 m/s) coincides with the mode (10,1) having a lower magnitude of natural frequency (158.3 Hz), whereas the lowest critical velocity of hot water (11.0 m/s) will coincide with the mode (13,1) corresponding to the lowest critical thermal buckling temperature (37.62 °C). Thus the lowest critical velocity of water at ambient temperature and the lowest natural frequency have the same mode, whereas the mode which is associated with lowest critical velocity of the hot water coincides with mode associated with the lowest critical thermal buckling temperature. Similar observation can be concluded from Figs. 13 and 14, the results of the numerical studies on composite cylindrical shell with l=r ¼ 1:048 and fibre angles 30°, and 60°. Further studies are repeated for composite cylindrical shell with l=r ¼ 2:08 and fibre angles 0° and 60°. The results pertaining to this are not reported, further the findings from these studies also revealed the above stated facts. 5.5. Influence of fibre angle of composite lamina on the mean axial critical velocity From the plots of the critical thermal buckling temperatures with respect to the circumferential modes for various fibre angles (Figs. 6 and 7) and from the plots of the variation of mean axial critical velocity versus tem-
perature (Figs. 12–14 and Table 6) an indirect of influence of the fibre angle on the mean axial critical velocity of water flowing through the shell can be deduced. Studies on the thermal buckling behavior of composite cylindrical shells by Ganesan et al. [5] indicate that the fibre angle of the composite lamina strongly influences the critical thermal buckling temperature. The critical buckling temperatures are very high in magnitude in case of composite cylindrical shell with fibre angle 60° whereas the critical thermal buckling temperatures are very low in magnitude for composite shell with 15° fibre orientation, see Figs. 7 and 6. Composite shell tailored with 60° fibre orientation will be well suited for high flow velocities of water at ambient conditions and also for conveying water at higher temperatures (which also means higher saturation pressure).
6. Conclusions Composites pipes/shells have found their place in specific engineering application requiring the transportation of fluids at high pressure and temperature. To study the dynamics of composite pipes/shells conveying hot fluid a semi-analytical finite element formulation has been presented. The structural finite element and the fluid finite element domain are coupled by appropriate fluid structure interface boundary conditions. Initial stresses produced by the thermal loading were employed to derive the geometric stiffness matrix. Thus the fluid structure interaction problem accounts for the temperature of the fluid flowing through the composite shell under isothermal conditions. Validation of the code for flow of water at ambient temperature and for the thermal buckling temperature gives an indication that the code is suitable for studies on the buckling behavior of composite shells conveying hot fluid. Two composite cylindrical shells with different length to radius ratios each with r=h ¼ 292 and for clamped–clamped boundary conditions were considered for numerical studies
32
R. Kadoli, N. Ganesan / Composite Structures 60 (2003) 19–32
and simulations. Composite shells with various fibre orientations are also considered in the study. The following are the conclusions from the numerical studies:
[2] [3]
1. Natural frequency variation of the shell conveying cold water and hot water have been computed for flow velocities ranging from 0.0 m/s to a velocity which leads to the buckling of the system. The nature of variation of the natural frequency of the coupled system involving the flow of water at ambient conditions and hot water is same. 2. The influence of water temperature on the critical flow velocities show that the maximum mean axial velocity of water that can be conveyed through the shell is high in the case of the water at ambient temperature when compared to high temperature water and in both cases it is limited by the critical buckling flow velocity. 3. The lowest critical velocity of water at ambient condition is associated with the mode that corresponds to the lowest natural frequency of the shell, whereas the lowest critical velocity of the hot water is associated with the mode corresponding to lowest thermal buckling temperature. 4. The influence of fibre angle on the critical mean axial velocities is considerable. Based on the fiber angles considered in the study it is found that 60° fibre angle for composite shells/pipes permits flow of water at high velocities in case of ambient conditions. The same composite shell also permits transportation of water at higher temperature since the critical buckling temperatures are high compared to 0°, 15° and 30° fibre angle. The studies presented in this paper bring out a methodology for estimating the maximum safe velocity of hot water for conveying through the composite shell based on the shell design parameters. This data estimated at the design stage can be used to quote the operating specifications for the pipes/shells.
[4] [5]
[6]
[7] [8] [9]
[10]
[11]
[12]
[13]
[14]
[15]
[16] [17] [18]
[19]
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