Analytical solutions of coupled functionally graded conical shells of revolution

Nomenclature

1 Introduction

The invention, conception, and manufacturing of functionally graded materials (FGMs) introduce a new class of composite materials that should be considered in the investigations of structural behaviour and their response to loading and boundary conditions (Figure 1).

Figure 1

FGM categories. (a) Continuous gradients and (b) discontinuous gradients.

For discontinuous gradients three types of gradients are introduced: 1) gradient composition, 2) gradient microstructure, and 3) gradient porosity.

For FGMs the review of possible compositions of materials is discussed in previous studies [1–4].

It is necessary to underline that the analysis of structures made of FGMs can be divided into the following areas:

1. Bending and stiffness of beam, plate, shells, or 3-D structures – see refs [5–18]; the formulation of the governing equations can be expressed by the Love–Kirchhoff theory, the first-order shear deformation theory, the zigzag theory, or higher order 2-D shell theories.

2. Vibration and flutter analysis – various approaches and reviews of the existing works in the open literature are presented in previous studies [19–25].

3. Buckling analysis – the number of research works is continuously increasing – see refs [26–33].

4. Thermal stresses – the above-mentioned problems are directly connected with the thermal response of constructions; therefore, the fundamental papers are associated with different mechanical problems [34,35,36,37,38,39,40].

5. Optimization problems – the examples of such an analysis are discussed in previous studies [41–43].

The derivations of the structural response understood in the sense of deformations, buckling loads, or free vibrations can be carried out in different ways, i.e. analytical, the Rayleigh–Ritz method, and the Bubnov–Gallerkin method or finite difference method called as the generalized differential quadrature method. However, due to the material behaviour plotted in Figure 1, as far as the author is concerned, there are no finite element (FE) commercial packages that allow the analysis of the structural configuration plotted in Figure 1. In general, it is necessary to introduce hierarchical formulation for 3-D structures or to divide the construction to 15–20 or more FEs in the thickness direction to characterize variable material configuration in 2-D approach.

In general, two different forms of conical shells can be considered (Figure 2). In addition, shallow conical shells that are conical segments can also be analysed.

Figure 2

Forms of conical shells. (a) Complete shell and (b) truncated shell.

The present work is devoted to the formulations and analytical solutions for axi-symmetric conical shells to illustrate the differences and similarities between functionally graded and isotropic conical structures. It should be mentioned that it is possible to also investigate conical shells made of laminates or nanostructures in the identical way as shown in the study of Muc and Muc-Wierzgoń [44]. These problems have not been addressed in this article and will be the objectives of future works.

The analytical description of axisymmetric isotropic shells of revolution is demonstrated in monographs [45,46,47]. Galletly and Muc [48,49] and Muc [33] studied buckling and deformations of laminated shells of revolution considering symmetric configurations only. Due to coupling effects between membrane and bending stress resultants, the analysis of arbitrarily laminated shells was usually carried out using numerical methods, e.g. Czebyshev expansions [50] or power series expansions [51,52].

Although a great number of works had been carried out for static analysis of axisymmetric shells using isotropic and laminated materials a general lack of information for axisymmetric structures made of FGMs is observed. As it is pointed out by Moita et al. [40], few works are found in the literature, and most of them are related to cylinders, circular, and annular plates solved using numerical methods.

The aim of this article is to demonstrate analytical solutions for axi-symmetric conical shells made of FGM and subjected to the normal uniform pressure. The solutions are obtained using Bessel functions and compared with the available published results for isotropic shells. The influence of coupling effects on shell deformations is studied in detail.

2 Formulation of the problem

The shells of revolution can be defined in the curvilinear orthogonal coordinates φ and θ of a point on the shell mid-surface. It is convenient to take the spherical coordinates, where the angle φ defines the location of a point along the meridian and θ describes the location of a point along the parallel circle – the circumferential coordinate (Figure 1). R ₁ = R ₁(φ) is the principal radius of the meridian and R ₂ = R ₂(φ) is the principal radius of the parallel circle. Let us note that r = R ₂sin(φ) and the Lame parameters are defined as follows: ALam = R ₁, BLam = r.

When an arbitrary shell of revolution is subjected to rotationally symmetrical loads, its deformations and stress resultants and couples do not depend upon the circumferential variable θ. The axisymmetric deformations of shells of revolution are described by two displacement components of the shell mid-surface u and w in the meridional and normal directions to the shell cross-section, respectively (Figure 3).

Figure 3

Cross-section of an axisymmetric shell of revolution.

For shells of revolution under axi-symmetrical loads the strain displacement equations take the following form:

where ε denotes the membrane strain, κ represents the change in curvature, and 1 and 2 correspond to the meridional φ and circumferential directions θ, respectively. The constitutive equations are written as follows:

where

The terms N _r and M _r represent the stress resultants and moment resultants, respectively. The variation in elastic modulus E characterizes the distribution of porosity along the thickness direction z and is defined as follows:

where the symbols t and b refer to the material properties on the top and bottom surfaces, n is power index, and ν is the Poisson’s ratio.

To complete the set of fundamental equations we add the equilibrium equations derived using principle of virtual work [45,46,47]:

where T denotes the transverse shear force, and q is the normal pressure on the shell mid-surface.

Using the classical Meisner approach to the analysis of axisymmetric shell deformations [45,46,47], the fundamental equations can be reduced to the system of two differential equations for two variables β and U = R ₂ T being the functions of the meridional coordinate φ:

where prime over the symbols denotes the differentiation with respect to φ variable and

where φ _e is a coordinate of the shell edge and the constant C characterize an axial coordinate of the external load applied at the top edge. Inserting that the couple term B = 0, equations (7) and (8) are reduced to the classical equations for isotropic axisymmetric shells.

Equations (7) and (8) can be rewritten in a more compact form as follows:

where A = A ₁₁, B = B ₁₁, D = D ₁₁.

If the radius of the shell curvature R ₁, the shell thickness t, the Young’s modulus E, and the Poisson ratio ν are constant, equations (7), (8), (10), and (11) can be reduced to the following complementary differential equations (the symbol c denotes that F = 0 – equation (9)):

[ L ( U c ) + i μ 2 U c ] [ L ( U c ) − i μ 2 U c ] = 0 , μ 2 = ( R 1 A − B ) 2 A D − B 2 − ν 2 ,

Each of the equations has complex solutions as AD > B ². If the coupling term B tends to 0, the above relation describes the deformations of isotropic axisymmetric shells.

3 Solution of governing relation for axi-symmetric conical shells

For conical shells the generators of the mid-surface are straight and so it is appropriate to change the variables introducing ds = R ₁dφ (compare Figure 3 with Figure 4). Replacing the variable φ with the variable s in equations (7), (8), (10), and (11), we introduce the following relationships:

where φ is the constant along the shell meridian and R ₁ does not vary with the s variable. The generators are inclined at an angle α with the axis of symmetry – φ = π/2 – α. Let us note that the curvature 1/R ₁ tends to 0. The equations correspond to those for cylindrical shells as α = 0.

Figure 4

Geometry of a conical shell.

In addition, let us examine deformations of conical shells having the constant thickness, i.e. t(s) = const. Setting R ₁ → ∞ and using relation (13), the system of equations (7), (8), (10), and (11) is reduced to the following form (F(φ) = 0):

Let us note that inserting B = 0, the equations (14) and (15) are reduced to the classical Meisner equations for isotropic shells.

Elimination of each variable (T or β) in equations (14) and (15) leads to fourth-order differential equations, i.e.:

Each of the equations (7), (8), (10), and (11) can be solved for one of the dependent variables. The solutions can be represented in the following way:

where the superscripts c and p denote the complementary (F = 0) and particular (F ≠ 0) solutions, respectively. The complementary solutions satisfy the following relation:

Making the transformation of variables y = 2 μ x , the differential equations (17), (18) or (20) are reduced to the classical form of the Bessel equation. Each of the equations has complex solutions (AD > B ²). There are conjugated, i.e.:

where S _i are the real functions. The solutions W _i are represented by the Kelvin functions:

where J ₂ is the second-order Bessel function of the first kind, and Y ₂ is the second-order modified Bessel function of the second kind. They can be found easily by the single Mathematica command DSolve. Finally, the complementary solution of equation (20) can be expressed as:

For FGM conical shells the deformation functions are controlled by the value of the parameter μ ² in equation (20). For isotropic shells this value is the function of the ratio 1/t

but for FGM shells it is also the function of the coupling effects expressed by the term B – equation (20). Figure 5 illustrates the influence of the coupling effects on the variations of the parameter μ ². It is represented by two values: the ratio E _t/E _b and the porosity index n – equation (5).

Figure 5

Variations of the dimensionless controlling parameter μ 2 / μ isotr 2 .

The computations of the complementary solutions for the kinematic parameter β ^c can also be found in the analytical way using equation (7):

Then, from the equilibrium equation (6) one can obtain the analytical form of the stress resultants N ϕ c , N θ c .

Finally, the kinematic complementary displacements u ^c and w ^c can be derived in the closed analytical way using the relation (26), the fundamental definitions in equations (1) and (2), and the constitutive relation (3):

The computations can be made easily applying the symbolic packages, e.g. Mathematica.

It is necessary to distinguish two classes of problems (Figure 2) that should be introduced independently, referred directly to the convenience and simplicity of analytical solutions. For shells closed at the apex (Figure 2a). singularity occurs at φ = 0. Shells opened at the apex always have the convergent solutions (Figure 2b). It is necessary to point out that analytical solutions (19) are always expressed by real functions.

3.1 Shells closed at the apex

The solutions for equation (17) are presented in the handbook [53]. The change of variables,

(28) x = sin 2 φ , β c = z sin φ ,

leads to the following hypergeometric equation [3]:

(29) x ( x − 1 ) d 2 z d x 2 + [ c − ( a + b + 1 ) x ] d z d x − a b z = 0 .

Comparing the above relation to a standard form of the Gaussian hypergeometric equation one can find that:

(30) T c = A 1 S 1 + A 2 S 2 ,

where S 1 = sin ϕ Re F ( a , b , c , sin 2 ϕ ) , S 2 = sin ϕ Im F ( a , b , c , sin 2 ϕ ) ,

(31) a = [ 0.75 − 0.5 ( 1.25 + i μ 2 ) 1 / 2 ] , b = [ 0.75 + 0.5 ( 1.25 + i μ 2 ) 1 / 2 ] , c = 2 ,

and F(a,b,c,x) is a hypergeometric function. Two additional solutions S ₃ and S ₄ are singular at the shell apex and they would have to be suppressed [46].

Comparing the complementary solutions for isotropic and FGM shells (Figures 6 and 7) one can observe the effects of the μ 2 / μ isotr 2 ratio. Both the changes of the index parameter n and of the E _b/E _t ratio affect the solutions expressed by the functions S ₁ and S ₂. In general, the results illustrate the localized effects at the boundary φ = π/2. Such an analysis can be carried out easily due to the analytical form of the solutions in equations (30) and (31).

Figure 6

Solution function S ₁ for FGM conical shells.

Figure 7

Solution function S ₂ for FGM conical shells.

Further analysis of particular problems is reduced to finding the solutions for shells having the particular loading and boundary conditions and in this way to search for particular solutions T ^p and β ^p (equation (16)).

4 Conical shells under uniform external pressure

Let us consider the case of the conical shell loaded by the uniform external pressure q and clamped at the edge φ = π/2:

With this value the particular solutions:

since equations (14) and (15) are homogeneous. However, the effects of the distributed load q is brought by the equilibrium equation (6), i.e.:

The stress resultants can be derived from the equilibrium conditions (26) and the equations (30) and (32)–(34):

Using the above equations and the definitions (2) and (3) and the relation (7), the stress couples can be written in the following way:

Inserting the coupling stiffness B = 0, one can find the equations (35)–(37) for isotropic shells [45,46]. The symbols P ₁ and P ₂ are integration constants and for the boundary conditions in equation (32) are equal to:

Figure 8 demonstrates the distributions of the dimensionless bending moments M _φ /(qt ²) along the shell meridian. The results demonstrate the localized effects at the clamped edge that increase with the decrease of the thickness ratio. The coupling effects expressed by the controlling parameter μ ² in equation (17) have a significant influence on the values of the bending moments at the clamped edge – the growth of this value results in the increase of bending effects.

Similar results to those plotted in Figure 8 can be obtained using equations (35)–(38). However, the distributions of the stress resultants N _φ show that these values are almost equal to 0.5qR (the membrane state).

Figure 8

Distributions of the dimensionless bending moments M _φ .

5 Conical shells subjected to edge loadings

Let us consider the deformations of conical shells subjected to the edge loads in the form of bending moments M and edge forces H as illustrated in Figure 9.

Figure 9

Edge loadings.

Using the Legendre polynomial solutions described in Section 3.3 one can evaluate the distributions of meridional dimensionless bending moments (Figure 10). The curves represent the stress concentrations near the shell bottom edge. As it may be seen the unsymmetry of the shell wall construction (FGMs) results in the increase of the maximal bending moments. The influence of FGMs is nonlinear. As it is noticed by Kraus [46] the extension of the peaks in each curve is approximately equal to 2.8π.

Figure 10

Distributions of bending moments – φ = 60°, s ₁ = 100 [mm], s ₂ = 300 [mm], and R _θ = s ₂ ctg(φ) = 1,730 [mm].

6 The Geckeler approximation

Geckeler [54] proposed to reduce the equations (14) and (15) to the following form:

Differentiating twice each of equations (40) or (41) the problem is reduced to fourth-order differentiating equation.

Equation (41) demonstrates explicitly the effects of the unsymmetry of the wall construction due to the form of FGMs – equation (16).

Biderman [55] proved the convenience of the Geckeler solutions to the edge-effect problems discussed in the previous section. The difference between accurate and simplified solutions does not exceed 1%.

Cui et al. [56] formulated the approximations of the solutions for conical shells loaded by pressure.

7 Final remarks

The presented analytical solutions herein demonstrate evidently the significant influence of the coupling bending-membrane terms of the stiffness matrix illustrating the unsymmetry in the construction of shell walls of porous FGMs. Those effects are characterized by the controlling parameter μ. It is worth to mention that the similar controlling parameter is proposed and introduced in the description of flutter problems for structures made of FGMs or plates reinforced by nanostructures (nanoplates or nanotubes) – see the study of Muc and Muc-Wierzgoń [44].

Let us note that the distributions of the stress resultants and bending moments are sensitive to the variations of the thickness ratio t/R. In the present work, the classical Love–Kirchhoff hypothesis is used, but for thicker structures (the thickness-to-radius ratio >0.1) the higher order 2-D shell theories should be used, e.g. in the form proposed in the Appendix of the work in the study of Muc and Muc-Wierzgoń [44].

The present results are derived for conical shells; however, the identical analysis can be easily extended to the analysis of shells or pressure vessels having various forms – paraboloidal, hyperbolical, elliptical, and torispherical discussed from the numerical and optimization point of view, e.g. in the study of Moita et al. [40].