Elastic analysis of pressurized thick hollow cylindrical shells with clamped-clamped ends/Plonasieniu hermetiniu cilindru su uzsandarintais galais tamprioji analize.

Ghannad, M. ; Nejad, M. Zamani

1. Introduction

There are many engineering applications of commonly used structures, such as rods, annular disks, cylindrical and spherical shells when subjected to different loading and boundary conditions. Deformation and stress analysis of thick-walled cylinders subjected to either internal or external pressure is an important topic in engineering because of their rigorous applications in industry as well as in daily life. For this reason, the classical problem of a pressurized thick hollow cylinder has been the topic of a variety of theoretical investigations.

Naghdi and Cooper [1], assuming the cross shear effect, formulated the shear deformation theory (SDT). Mirsky and Hermann [2], derived the solution of thick cylindrical shells of homogenous and isotropic materials, using the first shear deformation theory (FSDT). Greenspon [3], opted to make a comparison between the findings regarding the different solutions obtained for cylindrical shells. Making use of Mirsky-Hermann theory and the finite difference method (FDM), Ziv and Perl [4] obtained the vibration response for semi-long cylindrical shells. Using SDT and Frobenius series, Suzuki et. al. [5], obtained the solution of the free vibration of cylindrical shells with variable thickness, and Takashaki et. al. [6] obtained the same solution for conical shells. A paper was published by Kang and Leissa [7] where equations of motion and energy functional were derived for a three-dimensional coordinate system. The field equations are utilized to express such energy functional in terms of displacement components. The stress state of two-layer hollow bars in which they are exposed to axial load is analyzed [8]. The layers are made of isotropic, homogeneous, linearly elastic material, and they are considered as concentric cylinders.

Assuming that the material properties vary nonlinearly in the radial direction and the Poisson's ratio is constant, Zamani Nejad and Rahimi [9] obtained closed form solutions for one-dimensional steady-state thermal stresses in a rotating functionally graded pressurized thick-walled hollow circular cylinder. A complete and consistent 3D set of field equations has been developed by tensor analysis to characterize the behavior of FGM thick shells of revolution with arbitrary curvature and variable thickness along the meridional direction [10]. Using the analytical method for stress strain state of two-layer mechanically inhomogeneous pipe subjected to internal pressure at elastic plastic loading are analyzed by Brazenas and Vaiciulis [11].

This article presents the general method of derivation and the analysis of an internally pressurized thick-walled cylinder shell with clamped-clamped ends, taking into account the effect of shear stresses and strains.

2. Classical theory

The plane elasticity theory (PET) or classical theory is based on the assumption that the straight lines perpendicular to the central axis of the cylinder remain unchanged after loading and deformation. According to this theory, the deformations are axisymmetric and do not change along the longitudinal cylinder. In other words, the elements do not have any rotation, and the shear strain is assumed to be zero. Thus, equilibrium equations are independent of one another, and the coupling of the equations is deleted. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The differential equation based on the Navier Solution is

[d.sup.2][u.sub.r]/d[r.sup.2] + 1/r d[u.sub.r]/dr - 1/[r.sup.2] [u.sub.r] = 0 (3)

The solution of the Eq. (3) is

[u.sub.r](r) = [C.sub.1]r + [C.sub.2]/r (4)

This method is applicable in problems in which shear stresses and strains are considered zero. However, to solve the problems such as the following it is not possible to use the PET

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. Shear deformation theory (SDT)

In SDT, the straight lines perpendicular to the central axis of the cylinder do not necessarily remain unchanged after loading and deformation, suggesting that the deformations are axial axisymmetric and change along the longitudinal cylinder. In other words, the elements have rotation, and the shear strain is not zero.

In Fig. 1, the location of a typical point m (r), within the shell element may be determined by R and z, as

r = R(x) + z (6)

where R represents the distance of middle surface from the axial direction, and z is the distance of typical point from the middle surface.

In Eq. (6), x and z must be as follows

- h/2 [less than or equal to] z [less than or equal to] h/2, 0 [less than or equal to] x [less than or equal to] L (7)

where h and L are the thickness and the length of the cylinder.

R(x) and inner and outer radii ([r.sub.i], [r.sub.o]) of the cylinder are as follows

[r.sub.i] = R - h/2 = const., [r.sub.o] = R + h/2 = const. (8)

Based on PET, the radial displacement of the cylinder is

[u.sub.r](r) = [C.sub.1](R + z) + [C.sub.2]/R + z (9)

Using the Taylor's expansion for [absolute value of z/R] < 1,

[u.sub.r](r) = [C.sub.1] (R + z) + [C.sub.2]/R(1 - z/R + [z.sup.2]/[R.sup.2] +...) (10)

Thus,

[u.sub.r](r) = [u.sub.0] + [u.sub.1]z + [u.sub.2][z.sup.2] +.... (11)

According to Eq. (11), the radial displacement is written in the form of a polynomial function of z. When z = 0, it shows the displacement of the mid-plane.

The general axisymmetric displacement field ([U.sub.x], [U.sub.z]), in the first-order Mirsky-Hermann's theory could be expressed on the basis of axial and radial displacements, as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where u(x) and w(x) are the displacement components of the middle surface. Also, [phi](x) and [psi](x) are the functions used to determine the displacement field.

[FIGURE 1 OMITTED]

The strain-displacement relations in the cylindrical coordinates system are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In addition, the stresses on the basis of constitutive equations for homogenous and isotropic materials are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where [[sigma]].sub.i] and [[epsilon].sub.i] are the stresses and strains in the axial (x), circumferential ([theta]), and radial (z) directions; [upsilon] and E are Poisson's ratio and Young's modulus, respectively.

The normal forces ([N.sub.x],[N.sub.[theta]],[N.sub.z]),shear force ([Q.sub.x]), bending moments ([M.sub.x], [M.sub.[theta]], [M.sub.z]), and the torsional moment ([M.sub.xz]) in terms of stress resultants are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

On the basis of the principle of virtual work, the variations of strain energy are equal to the variations of the external work as follows

[delta]U = [delta]W (19)

where U is the total strain energy of the elastic body and W is the total external work due to internal pressure. The strain energy is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

and the external work is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where P is internal pressure.

The variation of the strain energy is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The resulting Eq. (22) will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

and the variation of the external work is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

The resulting Eq. (24) will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Substituting Eqs. (13) and (14) into Eq. (19), and drawing upon calculus of variation and the virtual work principle, we will have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

and the boundary conditions are

R[[[N.sub.x][delta]u + [M.sub.x][delta][phi] + [Q.sub.x][delta]w + [M.sub.xz][delta][psi]].sup.L.sub.O] = 0 (27)

Eq. (27) states the boundary conditions which must exist at the two ends of cylinder. In order to solve the set of differential equations (26), forces and moments need to be expressed in terms of the components of displacement field, using Eqs. (15) to (18). Thus, set of differential equations (26) could be derived as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

The set of equations (28) is a set of linear nonhomogenous equations with constant coefficients. The coefficients matrices [[[[bar.A].sub.i]].sub.4x4], and force vector {[bar.F]} are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

{[bar.F]} = P/[lambda]E(R - h/2)[{0 0 -1 h/2}.sup.T] (32)

The parameters u and a are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where K is the shear correction factor that is embedded in the shear stress term.

It is assumed that in the static state, for cylindrical shells K = 5/6 [12].

[[[bar.A].sub.3]] is irreversible and its reverse is needed in the next calculations. In order to make [[[[bar.A].sub.3]].sup.-1], the first equation in the set of Eqs. (26) is integrated.

R[N.sub.x] = [C.sub.0] (34)

In Eq. (28), it is apparent that does not exist, but du/dx does. Taking du/dx as v,

u = [integral] vdx + [C.sub.7] (35)

Thus, set of differential Eqs. (28) could be deived as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

where, the coefficients matrices [[[A.sub.i]].sub.4x4], and force vector {F} are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

The equations (36) are the set of nonhomogenous linear differential equations with constant coefficients.

4. Analytical solution

Defining the differential operator P(D), Eq. (36) is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

Thus

P(D){y} = {F} (42)

The differential Eq. (42) has the general solution including general solution for homogeneous case [{y}.sub.g] and particular solution [{y}.sub.p], as follows

{y} = [{y}.sub.g] + [{y}.sub.p] (43)

For the general solution for homogeneous case, [{y}.sub.g] = {V}[e.sup.mx] is substituted in P(D){y} = 0.

[e.sup.mx] [[m.sup.2][[A.sub.1]] + m[[A.sub.2]] + [[A.sub.3]]]{V} = {0} (44)

Given that [e.sup.mx] [not equal to] 0, the following eigenvalue problem is created.

[[m.sup.2][[A.sub.1]] + m[[A.sub.2]] + [[A.sub.3]]]{V} = {0} (45)

To obtain the eigenvalues, the determinant of coefficients must be considered zero.

[absolute value of [[m.sup.2[[A.sub.1]] + m[[A.sub.2]] + [[A.sub.3]]] = 0 (46)

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

The result of the determinant above is a six-order polynomial which is a function of m, the solution of which is a 6 eigenvalues [m.sub.i]. The eigenvalues are 3 pairs of conjugated root. Substituting the calculated eigenvalues in Eq. (45), the corresponding eigenvectors [{V}.sub.i] are obtained. Therefore, the general solution for homogeneous Eq. (42) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

The constants [C.sub.1] to [C.sub.6] are obtained by applying boundary conditions. Given that {F} is comprised of constant parameters, the particular solution is obtained as follows.

[{y}.sub.P] = [[[A.sub.3]].sup.-1] {F} (50)

Therefore, the general solution for Eq. (42) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

In general, the problem consists of 8 unknown values of [C.sub.i], including [C.sub.0] (Eq. 34), [C.sub.1] to [C.sub.6] (Eq. 51), and [C.sub.7] (Eq. 35). By applying boundary conditions, one can obtain the constants of [C.sub.i].

Given that the two ends of the cylinder are clamped-clamped, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

Based on PET in the plane strain state, radial stress, circumferential stress and radial displacement are as follows [13]:

[[sigma].sub.r] = P/[k.sup.2] - 1 [1 - [k.sup.2]/[([bar.r]).sup.2]] (53)

[[sigma].sub.[theta]] = P/[k.sup.2] - 1 [1 - [k.sup.2]/[([bar.r]).sup.2]] (54)

[[sigma].sub.x] = 2[upsilon]P/[k.sup.2] - 1 (55)

[u.sub.r] = P[r.sub.i][bar.r](1 + [upsilon])/E([k.sup.2] - 1)[1 - 2[upsilon] + [k.sup.2]/[([bar.r]).sup.2]] (56)

where [bar.r] = r/[r.sub.i] and k = [r.sub.o]/[r.sub.i].

5. Results and discussion

In this section, we present the results for a homogeneous and isotropic thick hollow cylindrical shell with [r.sub.i] = 40 mm, h = 20 mm and L = 800 mm. The Young's modulus and Poisson's ratio, respectively, have the values of E = 200 GPa and [upsilon] = 0.3. The applied internal pressure is 80 MPa.

The analytical solution is carried out by writing the program in MAPLE 12. The numerical solution is obtained through finite element method (FEM).

Table presents the results of the different solutions for the middle of the cylinder (x = L/2) and mid-layer (z = 0). The results suggest that in points further away from the boundary it is possible to make use of PET.

Fig. 2 shows the distribution of axial displacement at different layers. At points away from the boundaries, axial displacement does not show significant differences in different layers, while at points near the boundaries, the reverse holds true. The distribution of radial displacement at different layers is plotted in Fig. 3. The radial displacement at points away from the boundaries depends on radius and length. According to Figs. 2 and 3, the greatest axial and radial displacement occurs in the internal surface (z = - h/2). Distribution of circumferential stress in different layers is shown in Fig. 4. The circumferential stress at all points depends on radius and length. The circumferential stress at layers close to the external surface at points near boundary is negative, and at other layers positive. The greatest circumferential stress occurs in the internal surface (z = - h/2).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Fig. 5 shows the distribution of shear stress at different layers. The shear stress at points away from the boundaries at different layers is the same and trivial. However, at points near the boundaries, the stress is significant, especially in the internal surface, which is the greatest.

[FIGURE 5 OMITTED]

In the Figs. 6-10, displacement and stress distributions are obtained using FSDT are compared with the solutions of FEM and are presented in the form of graphs.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

6. Conclusions

In the present study, the advantages as well as the disadvantages of the PET (Lame' solution) for hollow thick-walled cylindrical shells with different boundary conditions at the two ends were indicated. Regarding the problems which could not be solved through PET, the solution based on the FSDT is suggested. At the boundary areas (20 percent of the length of the cylinder) of a thick-walled cylinder with clamped-clamped ends, having constant thickness and uniform pressure, given that displacements and stresses are dependent on radius and length, use cannot be made of PET, and FSDT must be used. In the areas further away from the boundaries (80% of the length of the cylinder), as the displacements and stresses along the cylinder remain constant and dependent on radius, PET ought to be used. The shear stress in boundary areas cannot be ignored, but in areas further away from the boundaries, it can be ignored. Therefore, the PET can be used, provided that the shear strain is zero. The maximum displacements and stresses in all the areas of the cylinder occur on the internal surface. The analytical solutions and the solutions carried out through the FEM show good agreement.

Received June 02, 2010 Accepted September 27, 2010

References

[1.] Naghdi, P.M., Cooper, R.M. Propagation of elastic waves in cylindrical shells, including the effects of transverse shear and rotary inertia. -Journal of the Acoustical Society of America, 1956, v.29, No1, p.56-63.

[2.] Mirsky, I., Hermann, G. Axially motions of thick cylindrical shells. -Journal of Applied Mechanics-Transactions of the ASME, 1958, v.25, p.97-102.

[3.] Greenspon, J.E. Vibration of a thick-walled cylindrical shell, comparison of the exact theory with approximate theories. -Journal of the Acoustical Society of America, 1960, v.32, No.5, p.571-578.

[4.] Ziv, M., Perl, M. Impulsive deformation of Mirsky Hermann's thick cylindrical shells by a numerical method. -Journal of Applied Mechanics-Transactions of the ASME, 1973, v.40, No.4, p.1009-1016.

[5.] Suzuki, K., Konnon, M., Takahashi, S. Axisymmetric vibration of a cylindrical shell with variable thickness. -Bulletin of the JSME-Japan Society of Mechanical Engineers, 1981, v.24, No.198, p.2122-2132.

[6.] Takahashi, S., Suzuki, K., Kosawada, T. Vibrations of conical shells with variable thickness. -Bulletin of the JSME-Japan Society of Mechanical Engineers, 1986, v.29, No.285, p.4306-4311.

[7.] Kang, J.H., Leissa, A.W. Three-dimensional field equations of motion and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness. -Journal of Applied Mechanics-Transactions of the ASME, 2001, v.68, No.6, p.953-954.

[8.] Partaukas, N., Bareisis, J. The stress state in two-layer hollow cylindrical bars. -Mechanika. -Kaunas: Technologija, 2009, Nr.1(75), p.5-12.

[9.] Nejad, M.Z., Rahimi, G.H. Deformations and stresses in rotating FGM pressurized thick hollow cylinder under thermal load. -Scientific Research and Essays, 2009, v.4, No3, p.131-140.

[10.] Nejad, M.Z., Rahimi, G.H., Ghannad, M. Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system. -Mechanika. -Kaunas: Technologija, 2009, Nr.3(77), p.18-26.

[11.] Brazenas, A., Vaiciulis, D. Determination of stresses and strains in two-layer mechanically inhomogeneous pipe subjected to internal pressure at elastic plastic loading. -Mechanika. -Kaunas: Technologija, 2009, Nr.6(80), p.12-17.

[12.] Vlachoutsis, S. Shear correction factors for plates and shells. -International Journal for Numerical Method in Engineering, 1992, v.33, No7, p.1537-1552.

[13.] Ghannad, M., Rahimi, G.H., Khadem, S.E. General plane elasticity solution of axisymmetric functionally graded cylindrical shells. -Journal of Modares Technology and Engineering, 2010, v.41. Article in Press.

M. Ghannad *, M. Zamani Nejad **

* Mechanical Engineering Faculty, Shahrood University of Technology, Shahrood, Iran, E-mail: ghannad.mehdi@gmail.com

** Mechanical Engineering Department, Yasouj University, Yasouj P. O. Box: 75914-353 Iran, E-mail: m.zamani.n@gmail.com, m_zamani@mail.yu.ac.ir

Table
Numerical results of the different solutions

       [[sigma].sub.r], MPa   [[sigma].sub.[theta]], MPa

FSDT          -27.56                    155.62
FEM           -28.16                    156.11
PET           -28.16                    156.16

       [[sigma].sub.x], MPa   [[sigma].sub.r], mm

FSDT          36.24                 0.03826
FEM           36.22                 0.03843
PET           38.40                 0.03827