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Using disk form multilayers, a semi-analytical solution has been derived for determination of displacements and stresses in a rotating cylindrical shell with variable thickness under uniform pressure. The thick cylinder is divided into disk form layers form with their thickness corresponding to the thickness of the cylinder. Due to the existence of shear stress in the thick cylindrical shell with variable thickness, the equations governing disk layers are obtained based on first-order shear deformation theory (FSDT). These equations are in the form of a set of general differential equations. Given that the cylinder is divided into

Thick cylindrical shells with variable thickness have widely been applied in many fields such as space fight, rocket, aviation, and submarine technology. Given the limitations of the classic theories of thick wall shells, very little attention has been paid to the analytical and semi-analytical solutions of these shells. Assuming the transverse shear effect, Naghdi and Cooper [

In this paper, elastic analysis has been presented for rotating thick cylindrical shells under internal pressure with variable thickness using disk form multilayers.

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.

Geometry of a thick cylindrical shell with variable thickness

Thick cylindrical shell with variable thickness.

The location of a typical point

The general axisymmetric displacement field

The kinematic equations (strain-displacement relations) in the cylindrical coordinates system are

The stress-strain relations (constitutive equations) for homogeneous and isotropic materials are as follows:

The normal forces

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

With substituting strain energy and work of external forces, we have [

Equation (

In order to solve the set of differential equations (

The coefficients matrices

The set of differential equations (

In this method, the thick cylinder with variable thickness is divided into disk layers with constant height

Dividing of thick cylinder with variable thickness to disk form multilayers.

Therefore, the governing equations convert to nonhomogeneous set of differential equations with constant coefficients.

The length of middle of an arbitrary disk (Figure

Geometry of an arbitrary disk layer.

The radius of middle point of each disk is as follows:

The coefficients matrices

Defining the differential operator

The above differential equation has the total solution including general solution for homogeneous case

For the general solution for homogeneous case,

We have

In general, the problem for each disk consists of 8 unknown values of

In this problem, the boundary conditions of cylinder are clamped-clamped ends; then we have

A cylindrical shell with

The effect of the number of disk layers on the radial displacement is shown in Figure

Effect of the number of disk layers on the radial displacement.

In Figures

Radial displacement distribution in middle layer.

Radial stress distribution in middle layer.

Circumferential stress distribution in middle layer.

Radial displacement distribution in different layers.

The distribution of radial displacement at different layers is plotted in Figure

Distribution of circumferential stress in different layers is shown in Figure

Circumferential stress distribution in different layers.

Figure

Shear stress distribution in different layers.

The effects of angular velocity

Radial displacement distribution in middle layers.

Circumferential stress distribution in middle layers.

Shear stress distribution in middle layers.

Figures

According to Figure

In the present study, we have the following.

Based on FSDT and elasticity theory, the governing equations of thick-walled disks are derived.

A thick cylindrical shell with variable thickness is divided into disks with constant height.

With considering continuity between layers and applying boundary conditions, the governing set of differential equations with constant coefficients is solved.

The results obtained for stresses and displacements are compared with the solutions carried out through the FEM. Good agreement was found among the results.

Adventures of the semi-analytical using disk form multilayers are as follows.

First shear deformation theory and perturbation theory result in the analytical solution of the problem with higher accuracy and within a shorter period of time.

The solutions are complicated and time consuming.

The shells with different geometries, and different loadings, and different boundary conditions, with even variable pressure, could be more easily solved.

The method is very suitable for the purpose of calculation of radial stress, circumferential stress, shear stress, and radial displacement.

The authors declare that there is no conflict of interests regarding the publication of this paper.