Elastic solution of axisymmetric thick truncated conical shells based on first-order shear deformation theory/Tampriu storu asimetriniu nupjautos kugines formos kevaliniu strukturu analize taikant pirmos eiles slyties deformacijos teorija.

Ghannad, M. ; Nejad, Zamani M. ; Rahimi, G.H. 等

1. Introduction

Generally, shells are curved structures which exhibit significant stiffness against forces and moments. Scientists have paid an enormous amount of attention to shells, resulting in numerous theories about their behavior. Of different kinds of shells, due to their extensive use in rocket and shuttle cones, conical shells have been of especially importance. Given the limitations of the classic theories of thick wall shells, very little attention has been paid to the analytical solution of these shells.

Naghdi and Cooper [1], assuming the cross shear effect, formulated the theory of shear deformation. Mirsky and Hermann [2], derived the solution of thick cylindrical shells of homogenous and isotropic materials, using the first shear deformation theory. 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 semilong cylindrical shells. Using the shear deformation theory and Frobenius series, Suzuki et. al. [5], obtained the solution of free vibration of cylindrical shells with variable thickness, and Takashaki et. al. [6], obtained the same solution for conical shells. Applying a threedimensional (3D) method of analysis, the free vibration frequencies and mode shapes of spherical shell segments with variable thickness are determined [7]. A paper was also published by Kang and Leissa [8] where equations of motion and energy functionals were derived for 3D coordinate system. The field equations are utilized to express them in terms of displacement components . Assuming that the material has a graded modulus of elasticity, while the Poisson's ratio is a constant, Tutuncu and Ozturk [9] investigated the stress distribution in the axisymmetric structures. They obtained the closed-form solutions for stresses and displacements in functionally graded cylindrical and spherical vessels under internal pressure. However, it needs to be pointed out that in deriving and, thus, plotting circumferential stress, Tutuncu & Ozturk made a mistake, which was pointed out by Shi et. al. [10] and Ghannad et. al. [11]. Assuming that a heterogeneous system is composed of the elements with different properties, in the paper [12] the reactions of pipeline systems to shock impact load and the possibilities of the simulation and evaluation of dynamic processes are investigated. Another general analysis of one-dimensional steady-state thermal stresses in a hollow thick cylinder made of functionally graded material (FGM) was obtained [13]. Eipakchi et. al. [14], obtained the solution of the homogenous and isotropic thick-walled cylindrical shells with variable thickness, using the first-order shear deformation theory (FSDT) and the perturbation theory. The stress state of two-layer hollow bars in which they are exposed to axial load is analyzed [15]. 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 [16], obtained closed form solutions for one-dimensional steadystate 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 [17]. Zamani Nejad and Rahimi [18], obtained stresses in isotropic rotating thick-walled cylindrical pressure vessels made of functionally graded material as a function of radial direction by using the theory of elasticity.

In the present study, the general solution of the thick truncated conical shells will be presented, making use of the FSDT. The governing equations, which are a system of ordinary differential equations with variable coefficients, have been solved analytically using the matched asymptotic method (MAM) of the perturbation theory.

2. Analysis

In the classical theory of shells, the assumption is that the sections that are straight and perpendicular to the mid-plane remain in the same position even after deformation. In the first-order shear deformation theory, the sections that are straight and perpendicular to the mid-plane remain straight but not necessarily perpendicular after deformation and loading. In this case, shear strain and shear stress are taken into consideration.

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 (1)

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. (1), x and z must be within the following ranges

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

R( x) and inner and outer radii ([r.sub.i], [r.sub.o]) of the cone are as follows (Fig. 1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

The general axisymmetric displacement field (Ux, Uz), in the first-order Mirsky-Hermann's theory could

be expressed on the basis of axial displacement and radial displacement, as follows

[U.sub.x] = u (x) + [phi] b(x)z ] (4)

[U.sub.z] = w (x) + [psi] (x) z)]

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.] (5)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [sigma.sub.i] and [epsilon.sub.i] are the stresses and strains in the axial

(x), circumferential ([theta]), and radial (z) directions. u and E are Poisson's ratio and modulus of elasticity, respectively.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

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 (11)

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.] (12)

and the external work is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

where [P.sub.x] and [P.sub.2] are components of internal pressure P along axial and radial directions, respectively. The variation of the strain energy is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

The resulting Eq. (14) will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

and the variation of the external work is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

The resulting Eq. (16) will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Substituting Eqs. (5) and (6) into Eqs. (15) and (17), and drawing upon calculus of variation and the virtual work principle, we will have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

And the boundary conditions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

Eq. (19) states the boundary conditions which must exist at the two ends of the cone.

In order to solve the set of differential Eqs. (18), forces and moments need to be expressed in terms of the components of displacement field, using Eqs. (5) to (10).

In Eqs. (18), it is apparent that u does not exist, but du/dx does. In the set of Eqs. (5), du/dx is needed to calculate displacements. Taking du/dx as v , and integrating the first equations in the set of Eqs. (18)

[RN.sub.X] =-[zeta] [P.sub.x] (R - h/2) dx + [C.sub.0] (20)

Thus, set of differential Eqs. (18) could be derived as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

The set of Eqs. (21) is a set of linear nonhomogenous equations with variable coefficients. The general method for solving these equations is the Frobenius method, which requires approximating displacements in terms of power series which are functions of x , substituting in the respective equations and applying boundary conditions in order to calculate the constants.

It is usually the case that the convergence of these series involves numerous terms. For instance, in paper [5], 50 terms and in paper [6], 100 terms are considered. In the present paper, MAM of the perturbation theory has been used to solve these equations.

3. Analytical solution for homogeneous truncated conical shell

We assume that Young's modulus and the Poisson's ratio are constant.

In order to calculate the matrices [[A].sub.4x] in the set of Eq. (21), the stress resultants, obtained in terms of displacement field, are substituted in Eqs. (7) to (10), and integrated.

Thus, forces and moments are obtained as follows

where k is the shear correction factor that is embedded in the shear stress term. It is assumed that in the static state, for conical shells k = 5/6 [19]. The parameters ju and a are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

The coefficients matrices [[A.sub.i].sub.4X4],, and force vec tor {f} are obtained by substituting Eqs. (22) to (25) into Eqs. (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (31)

Given that the set of differential equations (21) do not have exact solutions, for the purpose of solving, MAM of the perturbation theory has been used, in which the convergence of solution is fast.

Solving the equations with variable coefficients gives rise to solving a system of algebraic equations and two systems of differential equations with constant coefficients. These systems of equations have the closed forms solutions. To accomplish this, making use of the characteristic scales, the governing equations are made dimensionless

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (33)

Substituting dimensionless parameters the set of Eqs. (21) is

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (36)

The coefficients matrices [[A.sub.i].sub.4x4], and force vec tor {[F.sup.*]} are obtained as follows

[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)

where the parameters are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (41)

The set of Eqs. (33) is singular. Therefore, its solution must be considered in the area of boundary layer problems. For the purpose of solving, MAM of the perturbation theory has been used. As boundary conditions are clamped-clamped, one lies in [x.sup.*] = 0 and the other in [x.sup.*] = 1 . So, the solution of the problem contains an outer solution away from the boundaries and two inner solutions near the two boundaries [x.sup.*] = 0 and [x.sup.*] = 1 [20].

4. Results and discussion

The analytical solution described in the preceding section for a homogeneous and isotropic truncated conical shell with a = 40 mm, b = 30 mm, h = 20 mm and L = 400 mm will be considered. The Young's Modulus and Poisson's ratio, respectively, have values of E = 200 GPa and u = 0.3. The applied internal pressure is 80 MPa. The truncated cone has clamped-clamped boundary conditions.

Fig. 2 shows the distribution of axial displacement at different layers. At points away from the bounda ries, 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 change in axial and radial displacements in the lower boundary is greater than that of the upper boundary and the greatest axial and radial displacement occurs in the internal surface (z = - hj2). 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 is negative, and at other layers positive. The greatest circumferential stress occurs in the internal surface (z = - hf 2).

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.

In Figs. 6-9, the effects of the changes in tapering angles with different values of a on axial displacement, radial displacement, circumferential stress, and shear stress will be considered.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The distribution of axial displacement in the inner surface of the cone is shown in Fig. 6. The greater the tapering angle, the greater the displacement, which is significant. In cones with small tapering angles, the greatest axial displacement occurs in the lower boundary. The larger the tapering angles, the greater the axial displacement in the lower boundary and in the middle surface of the cone. The changes in the middle surface are the greatest.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The distribution of radial displacement in the inner surface of the cone is shown in Fig. 7. The greater the tapering angle, the greater the radial displacement. The greatest radial displacement occurs in the lower boundary. In cones with small tapering angles, the greatest axial displacement occurs in the lower boundary. The larger the tapering angles, the greater the axial displacement in the lower boundary and in the middle surface of the cone. The changes in the middle surface are the greatest. In a like manner, the distribution of the circumferential stress in the inner surface is illustrated in Fig. 8. As this figure suggests, the greater the tapering angle, the greater the circumferential stress. The distribution of shear stress is shown in Fig. 9. According to this figure, the shear stress at points away from the boundaries is insignificant, and at boundary layers the changes in tapering angles do not have a significant bearing on the shear stress.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

5. Conclusions

In this study, the analytical solution of a thick homogenous and isotropic conical shell is presented, making use of the FSDT. In line with the energy principle and the FSDT, the equilibrium equations have been derived. Using the MAM of the perturbation theory, the system of differential equations which are ordinary and have variable coefficients has been solved analytically. The axial displacement in thick conical shells at points away from the boundaries depends more on the length rather than the radius, whereas at boundaries, this depends on both length and radius. The radial displacement at all points in a conical shell depends on the radius and the length. The circumferential stress at different layers depends on the radius and the length. These changes are relatively great. The greatest values of stress and displacement belong to the inner surface. The shear stress at points away from the boundaries is insignificant, and at boundary layers it is the opposite. The axial displacement, radial displacement, and circumferential stress are heavily dependent on tapering angles and any change in the tapering angle brings about a change in them. However, the shear stress does not undergo similar changes with changes in tapering angles.

Received September 01, 2009

Accepted October 12, 2009

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M. Ghannad, M. Zamani Nejad, G. H. Rahimi

M. Ghannad *, M. Zamani Nejad **, G. H. Rahimi ***

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

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

*** Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran, E-mail: gh.rahimi.s@gmail.com