Analytical solution for thermal stresses of laminated hollow cylinders under transient nonuniform thermal loading/Laminuoto tusciavidurio cilindro apkrovimo nestacionaria nevienalyte silumine apkrova analitinis skaiciavimas.1. Introduction
Filament-wound composite pipes and vessels are widely used in commercial industries such as fuel tanks, portable oxygen storage, and compressed natural gas (CNG) pressure vessel transportation. Using of these materials under a high temperature environment is increasing. One of the causes for damage in these laminated composite materials includes delimitation. In order to evaluate this phenomenon, the thermal stress analysis taking into account the transverse shearing stresses and the normal stress in the thickness direction is necessary. In addition, a transient thermal stress analysis as well as a steady thermal stress analysis becomes important, because maximum thermal stress distribution occurs in a transient state.
Lee  preformed the analysis of thermal stresses within multilayered cylinder under axial symmetry periodic boundary conditions. Ootao and Tanigawa  considered an angle-ply laminated cylindrical panel with simply supported edges due to a nonuniform heat supply in the circumferential direction. Shahani and Nabavi  solved transiented thermoelasticity problem in an isotropic thick-walled cylinder analytically by using the finite Hankel transform. Talor and Radu  considered the sinusoidal transient temperature for an isotropic hollow cylinder in an axis symmetric condition for the analyses of the thermal fatigue in the pipe lines. Hocine  investigated on thermo-mechanical behaviour of multilayer tubular composite in axis metric steady state conditions. His results show that thermal effect has the greatest influence on radial stresses and strains. Zamani Nejad and Rahimi developed a complete and consistent 3-D set of field equa-tions by tensor analysis to characterize the behavior of FGM thick shells of revolution with arbi-trary curvature and variable thickness along the meridional direction .
Bakaiyan and Hosseini  studied the stress and deformation of the filament-wound pipes under combined internal pressure and temperature variations with axisymmetric and steady-state consideration.
In the previous works thermal conduction and elasticity equations was solved only for the steady or the symmetric condition.
This investigation is concerned with the theoretical treatment of the transient thermal stress problem involving a two layered anisotropic hollow cylinder with non uniform temperature in the circumferential direction.
An infinitely long angle-ply laminated hollow cylinder composed of N layers is shown in Fig. 1. The cylinder's inner and outer radii are denoted as a and b respectively. It is assumed that each layer has the orthotropic material properties and the fiber direction in the ith layer is alternated with ply angle p to the z-axis.
[FIGURE 1 OMITTED]
2.1 Heat conduction problem
It is assumed that initially the laminated hollow cylinder is at constant temperature and the temperature of outer surface changes by an arbitrary function of time and angle of the form [bar.f] ([theta])[bar.g]([tau]). The temperature distribution is assumed to be a two-dimensional distribution in r-[theta] plane. The transient heat conduction of each layer and the initial and thermal boundary conditions in a dimensionless form are given in the following forms 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[[bar.K].sub.r] = [[bar.K].sub.T] (4)
[[bar.K].sub.[theta] = [K.sub.L] [sin.sup.2] ([PHI]) + [[bar.K].sub.T] [co[s.sup.2] ([PHI])}
The following dimensionless values are introduced
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where T is the temperature, [K.sub.r] and [[K.sub.[theta]] are the thermal diffusivities in the r and 6 directions, respectively, [[lambda].sub.r] is the thermal conductivity in the r direction, t is time, [T.sub.0], [K.sub.0] and [[lambda].sub.0] are the typical values of temperature, thermal diffusivity and thermal conductivity, respectively. In Eq. (4) the subscripts L and T denote the fiber and transverse directions, respectively. [bar.g] ([tau]) and [bar.f] ([theta]) are the arbitrary functions of time and angle respectively. Introducing the finite sine transformation with respect to the variable [theta], the solution of Eq. (1) can be obtained to satisfy Eq. (3). This solution is shown as follows
[bar.T] = [[infinity.summation over (n=1)][[bar.T].sub.n] ([rho], [tau])([C.sub.l]sin(n[theta]) + [C.sub.2] cos(n[theta])) (6)
where [C.sub.1] and [C.sub.2] are constants and can be found from Eq. (2). By using Fourier transform for function [bar.f] ([theta]), Eq. (2) can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
As a result, it is obvious that
[C.sub.1] = [A.sub.n] ,[C.sub.2] = [B.sub.n], [[bar.T].sub.n] (1, [tau]) = [bar.g]([tau]) (8)
Since, the function [[bar.T].sub.n] ([rho], [tau]) is still unknown, accordingly, substituting [bar.T] from Eq. (6) into Eq. (1) results in
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
The solution of the Eq. (9) may be accomplished by using the finite Hankel transform as described in .
Finally, the temperature distribution in the cylinder can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and [[lambda].sub.j] are the positive roots of Kernel([bar.a], [[lambda].sub.j]) = 0. Also from 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
2.2. Thermal stress analyses
In this section, the transient thermal stress of a laminated hollow cylinder is analyzed as a generalized plane strain problem. In each layer, the fiber direction, the in-plane transverse direction and the radial direction are denoted by L, T and R, respectively. Each layer has orthotropic material properties between the fiber-reinforced direction and its orthogonal direction.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Stress-strain relations in the dimensionless form for the ith layer are given in Eq. (14) .Applying the coordinate transformation rule to Eq.(14), stress-strain relations for the global coordinate system (r, [theta], z) are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
In the above equations, the following dimensionless values are introduced
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where [[sigma].sub.kli] are the stress components, [[epsilon].sub.kli] are the strain tensor, [[alpha].sub.ki] and [[alpha].sub.[theta]zi] are the coefficients of linear thermal expansion, [Q.sub.kli] and [Q.sup.*.sub.kli] are the elastic stiffness constants, and [[alpha].sub.0] = [[alpha].sub.r] and [E.sub.0] = [E.sub.r] are the typical values of the coefficient of linear thermal expansion and Young's modulus of elasticity, respectively. The strain-displacement equations can be displayed as flowing equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
Using stress--strain Eqs. (18), the displacement equations can be displayed in the following form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
In the above equations, comma denotes partial differentiation with respect to variables that follows. The boundary conditions in the surface and interfaces are represented accordingly
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
Here, we deal with nonuniform thermal loading. However, mechanical loading can be considered with equality of radial stresses with this kind of loading in the boundary conditions.
2.3 Solving equations
In order to satisfy periodic condition in the angular direction, the solutions of Eq. (14) are assumed in the following form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
Substituting Eq. (22) into Eq. (19) and considering p = es for the left hand side of Eq. (19), it can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
where [bar.D] = [partial derivative]/[partial derivative]s. This is a system of the second order differential equations which can be explained as the first order equations as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
Substituting Eq. (24) into Eq. (23), the system of equations can be written in a matrix form of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
The double bar notation is used for displaying matrix essence of parameters and [??], [??] and [[??].sub.T] were defined in Eq. (26).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
Finally, the solution of (23) can be presented in the following form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
where the matrix e is diagonal matrix and the diagonal kith element and can be displayed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Also, [Y.sub.nk], is the eigenvalue and [??], is the matrix of eigenvectors of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In the above equation, the unknown parameters [c.sup.1], ..., [c.sub.6] are found from the boundary conditions Eq. (17) for each layer. Noteworthy is that adding of mechanical loading in boundary conditions, only changes the constant parameter [c.sub.i]. Substituting displacement from Eq. (27) into strain-displacement equations, Eq. (16), and using stress-strain equation, Eq. (15), the stresses in each layer are obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
where [[??].sub.n](i) is ith member of the matrix [[??].sub.n].
3. Numerical results
For validation of the proposed solution, the results were compared with the stress function method solution in isotropic case (in reference ) which is a special type of our solution. It is seen that the results are in close agreement and the average error in each case is less than 0.5%. Further two-layered pipe with different boundary contions are investigated. The temperature and stresses in all figures of this paper are on the maximum value in the angular direction. According to Eq. (28) and thermal boundary conditions in these cases, the maximum peripheral angles, [theta], for the temperature, [[bar.[sigma]].sub.r[theta] and [[bar.[sigma]].sub.rz] is 90[degrees] and for other stresses is 0[degrees]. Because of using series method, an oscillating zone in the outer boundary can be seen in the most of the figures.
To illustrate the foregoing analyses, a typical composite pipe which composed of E-Glass/Epoxy with the material properties as stated in Table 1 is considered.
The thermal conductivity and thermal expansion coefficients are considered as 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
It is assumed that each layer consists of the same orthotropic material. A 2-layered antisymmetric angle-ply laminated hollow cylindrical with the fibber orientation ([phi]/-[phi]) with the same thickness is assumed. The inner and outer dimensionless radii are taken as 0.9 and 1 respectively. Following cases are investigated
1 [bar.T](1, [theta],[tau] = sin([theta]), [phi] = +85[degrees] [bar.T] (a,[theta],[tau]) = 0
2 - T(1, [theta][tau]) = sin([theta])sin(2[pi][tau], [phi] = [+ or -] 85[degrees] T (a,[theta],[tau]) = 0
3 - [bar.T](1,[theta],[tau] = sin ([theta]), [phi] = [+ or -] 55[degrees] T (a,[theta],[tau]) = 0
For the sake of brevity, the results of case 2 and 3 are not drawn and only the maximum stresses of these conditions are compared with case 1.
3.1. Results for case 1
Fig. 2 shows that the temperature distribution changes quickly in the time from [tau] = [K.sub.0]t / [b.sup.2] =
= 0.001-0.003 and reaches finally to the steady state on [tau] = 0.005
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
The distributions of the radial stress, hoop stress and longitudinal stress across the thickness of the cylinder at different instants of time were shown in Fig. 3. The high values of hoop and longitudinal stresses in the external boundary of the cylinder can be noticed. The maximum value of radial stress changes with time duration and its position approaches to the centre of cylinder.
This stress is small in comparison with the hoop and longitudinal stresses, but has a significant effect on the destruction of layers . In addition, the maximum values of normal stresses decrease as the time proceeds.
[FIGURE 4 OMITTED]
The variation of the shearing stresses indicated in the Fig. 4. It can be noticed that the shearing stresses [[sigma].sub.rz], [[sigma].sub.z[theta]] and [[sigma].sub.r[theta]] show the maximum value on the interface of layers.
The shearing stress [[bar.[sigma]].sub.r[theta]] shows the maximum value near [rho] = 0.96 in a transient state. The value of the shearing stress [[sigma].sub.rz], rises as the time proceeds and has a maximum value in the steady state.
The maximum values of [[bar.[sigma]].sub.z[theta] are in the interface of layers which does not change with time. The maximum of stresses for different boundary conditions and angles are compared and shown in Table 2.
The high value of longitudinal stress in sinusoidal loading compare to sudden loading can be noticed. In addition, the maximum stress is on the hoop stress for [phi] = [+ or -]85[degrees] for sudden loading. Increasing in angle of orientation result in increasing all stresses except of [bar.[sigma]].sub.z] and [bar.[sigma]].sub.rz].
The results of numerical analyses in summary can be classified as follows:
1. by increasing angle of orientation; the amount of stress discontinuity in layer interface reduces;
2. the most discontinuity in the stresses is seen in [bar.sigma]].sub.[theta]z];
3. the shearing stresses [bar.sub.rz] [bar.[sigma]].sub.r[theta]] and [bar.[sigma]].sub.r[theta]] show the maximum value on the interface of the layers;
4. in an anisotropic cylinder for two dimensional temperature fields, the strain in the axial direction is available and varies with time, but in the isotropic cylinder this strain is zero;
5. the longitudinal stress in sinusoidal loading comparing to a sudden loading has great value;
6. the increasing in the angle of orientation in sudden loading, results in all stresses increasing. Further for the sudden loading, the maximum stress is the hoop stress;
7. in the sinusoidal loading the hoop stress is less than the sudden loading and the maximum stress is longitudinal stress.
Received August 18, 2010
Accepted January 17, 2011
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M. A. Ehteram, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15875-4413, Iran, E-mail: firstname.lastname@example.org
M. Sadighi, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15875-4413, Iran, E-mail: email@example.com
H. Basirat Tabrizi, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15875-4413, Iran, E-mail: firstname.lastname@example.org
Table 1 Material properties for CNG pressure Vessels  Properties Value [E.sub.11], kg/[mm.sup.2] 5483.946 [E.sub.22], kg/[mm.sup.2] 1827.982 [E.sub.33], kg/[mm.sup.2] 1927.982 [v.sub.12], [v.sub.13] 0.25 [G.sub.12], kg/[mm.sup.2] 913.991 Table 2 Maximum absolute stresses for different boundary conditions Case: 1 (Sudden) 2 (Sinusoidal) Orientation: [phi] = [phi] = [+ or -] [+ or -] 85[degrees] 85[degrees] [bar.[[sigma].sub.zz]] 0.7 0.7 [bar.[[sigma].sub.[theta][theta]]] 0.8254 0.58 [bar.[[sigma].sub.rr]] 0.014 0.0136 [bar.[[sigma].sub.z[theta]]] 0.01923 0.0199 [bar.[[sigma].sub.rz]] 0.00053 0.00053 [bar.[[sigma].sub.r[theta]]] 0.014 0.0136 Case: 3 (Sudden) Orientation: [phi] = [+ or -] 55[degrees] [bar.[[sigma].sub.zz]] 0.8 [bar.[[sigma].sub.[theta][theta]]] 0.70 [bar.[[sigma].sub.rr]] 0.0114 [bar.[[sigma].sub.z[theta]]] 0.0986 [bar.[[sigma].sub.rz]] 0.0023 [bar.[[sigma].sub.r[theta]]] 0.0114