# Optimal design of a functionally graded corrugated cylindrical shell subjected to axisymmetric loading

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## Abstract

In this work we consider an optimal design of corrugated cylindrical shells. The functionally graded (FG) corrugation with amplitude following a given shape is defined. The problem is solved as a linear one assuming the axially symmetric loading. The method of estimation of the stress–strain state and, consequently, the shell optimal design based on homogenization approach is proposed. The illustrative examples of the optimal FG corrugation shell subjected to hydrostatic load have demonstrated high efficiency of the employed method.

## Keywords

Cylindrical shell Corrugation Functionally graded structure Homogenization approach Optimal design## 1 Introduction

The methods of computation and optimization of corrugated shells have been considered mainly for the case of a regular corrugation. As an example, we can mention metallic corrugated core sandwich panels. In particular, an influence of different cores geometry has been taken into account, including square, triangular and trapezoidal cores [1, 2]. An optimal shape of the corrugated profile under longitudinal deformation has been proposed in work [3], whereas the influence of the length and amplitude of corrugations has been studied in references [4, 5]. On the other hand, it is possible to design more effective functionally graded (FG) corrugation, where both the amplitude and the step are changed by means of following a given rule [6, 7]. The latter option of the optimal design has been investigated to a lesser extent. The reason is mainly motivated by occurrence of additional modeling and computational difficulties in comparison with the known problems exhibited by regularly graded corrugated structures. The usual equivalent plate model of corrugated structures [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] is not suitable for the FG corrugation. Although commercial codes allow one to analyze corrugated structures by approximation of corrugations using shell or solid elements [18, 19, 20, 21, 22, 23, 24, 25], such approach is not practical due to a large number of corrugations.

Therefore, the key and most important problem is formulation and then effective solution of the problem of optimal design of the FG corrugated shells. Employment of numerical approaches for optimal design of FG corrugations implies occurrence of numerous difficulties in the sense of the time-consuming computations [26, 27, 28, 29, 30, 31, 32, 33]. The homogenization approach seems to be promising in this case [34, 35, 36, 37]. This asymptotic approach exploit the smallness of a single corrugation size with respect to the characteristic size of the corrugated structure. Namely, we use here one of the variants of the homogenization approach, modified to the needs of computation and optimal design of the FG structures [38, 39, 40, 41].

Although the employed methodology of the paper is relatively simple, its technical realization is not easy. A corrugated shell is substituted by a certain orthotropic shell with the averaged characteristics. We use the homogenization approach, which is one of the most feasible methods for analysis of such transition. However, the application of the homogenization method to the corrugated shells requires much more attention and study than its application to, for instance ribbed or perforated shells or even composite materials. This problem does not belong to simple ones because the governing equations of the initial corrugated shell and its counterpart orthotropic model are derived with regard to different reference surfaces. In order to keep reliable transition into orthotropic shell, the initial governing relations should be projected onto the axes of the cylindrical surface.

In our case, it can be done only by splitting the initial/input relation into bending and tangential components which yields complicated analytical formulas. In the next step, the multiple homogenization approach is used to study the projectional equations. The problem of optimization is solved with the variational approach.

The presented case study validated our theoretical investigations.

The paper is organized as follows. First, we employ the fundamental relations with regard to axially symmetric deformation (Sect. 2). In Sect. 3, the homogenization of the fundamental relations of the axially symmetric deformations is carried out. Section 4 deals with an optimal design of a FG corrugation under action of the distributed loading. Section 5 presents an example of the optimization procedure when the shell is subjected to hydrostatic pressure. Finally, Sect. 6 presents concluding remarks.

## 2 Basic relations for axially symmetric deformation

*n*stands for a number of corrugation waves.

*u*,

*v*,

*w*) shown in Fig. 7. Normal and tangential displacements with respect to the middle surface \((u^{\prime }, v^{\prime }, w^{\prime })\) can be expressed via the introduced projections by the following formulas

*u*,

*w*into two components: for the tangential deformations \((u^{t}, w^{t})\) and for the bending ones \((u^{b}, w^{b})\)

## 3 Homogenization of relations of axially symmetric deformations

*x*, hence

*C*is defined by the boundary conditions.

*R*, the elastic characteristics of which present the averaged characteristics of the studied corrugated cylindrical shell. If we pass to a regular corrugation with the constant amplitude and take \(A_2 =1\) in Eq. (32), it coincides with the obtained in [5, 6]. In practice, since for the majority of the corrugated cylinders the following inequality holds: \(\max z\ll 1\); hence, we further assume \(A_2=1\). One of the important advantages of our approach in comparison with the structurally orthotropic theory [3] consists in the possibility of improvement of the obtained averaged solution. For instance, relation (31) allows one to express \(w_1^b \) by \(w_0^b \)

## 4 Optimal design of the FG corrugated shell under distributed loading

One of the key design specifications of the considered corrugated structures (Fig. 2) is their stiffness. It is important to ensure maximum rigidity of the designed structures with minimal weight. Weight of corrugated cylinders (Fig. 3), for a given thickness of the shell wall, is determined by the length of the generatrix playing the role of a constraint. In a number of works (see [42] and the references cited therein), external loading has been employed as a measure of stiffness properties of structures. One may proceed in a way similar to the mentioned one, i.e., to minimize weight for a given stiffness of a studied structure.

*q*.

*k*as follows

*q*direction to satisfy a positive value under the root term. The sign in front of the root defines two possible optimal designs differing in the direction of convexity of the corrugation first half-wave. Therefore, in the case of the sinusoidal shape of corrugation, the variable amplitude of its FG profile is governed by the following equation

## 5 Example of optimization

*n*,

*S*. In order to define sizes of the smooth (not corrugated) border parts of the shell \(( {0,x_1 }), ( {1-x_2 ,1} )\), the deflections \(w_b^{( i)} \), being solutions to Eq. (51), are assumed in the following way

*S*. The optimal profile of the sinusoidal corrugation (57), (60) for \(n=6; S= 3\pi \) is shown in Fig. 8.

The maximum deflection (54) of the optimally FG corrugated sheath (Fig. 8) was compared with the maximum deflection of a regularly corrugated shell having the same length of the curvilinear axis. The maximum deflection of the regular shell is defined with the help of the homogenized equation (39) keeping constant corrugation amplitude of \(H=0.0436\). The carried-out analysis has shown that the relative decrease in the maximum deflection achieves about 50%.

In the case of the optimal profile (Fig. 8), we take minus sign in front of the root in Eq. (60). If we take the flexibility (37) as the optimality criterion and we do not take into account the derivatives (stresses), then the sign choice does not play any role. If in the constraints or the optimality criteria the derivatives participate, then the optimality problem must be solved with allowance for corrections to the averaged solution. In the latter case, the sign in relations (60) plays an important role, since it defines the direction of convexity of the first corrugation half-wave, which consequently influences the local distribution of stresses.

## 6 Concluding remarks

Overall, the carried-out optimization of the FG corrugation with variable amplitude has been realized by using two approaches: isolating the smooth sections near the edges of the shell and changing the height of the corrugation according to the law determined by the change in the load.

The first approach only works for uniform loads. Using this approach could, for example, increase the sensitivity of the FG round corrugated diaphragms [5]. The second approach is aimed at changing the height of the corrugation, and it is especially effective for non-uniform external load like hydrostatic loading, wind, etc. It can be expected that the use of the FG corrugation with variable amplitude would be effective also in the problems of stability of the corrugated structures. However, it should be mentioned that the employment of FG corrugation increases the cost of producing such structures.

It should be also emphasized that optimal FG corrugation, for any type of smoothly distributed external load and any boundary conditions, the deflection FG corrugated part will be uniform (55). The influence of the type of loading and the boundary conditions is expressed in the law of variation of the corrugation amplitude (60), in the sizes of the smooth border parts of the shell, and in the uniform deflection of the corrugated part.

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