Parametric Optimization of Isotropic and Composite Axially Symmetric Shells Subjected to External Pressure and Twisting

Abstract

The present paper is devoted to the problem of the optimal design of thin-walled composite axially symmetric shells with respect to buckling resistance. The optimization problem is formulated with the following constraints: namely, all analyzed shells have identical capacity and volume of material. The optimization procedure consists of four steps. In the first step, the initial calculations are made for cylindrical shells with non-optimal orientation of layers and these results are used as the reference for optimization. Next, the optimal orientations of layers for cylindrical shapes are determined. In the third step, the optimal geometrical shape of a middle surface with a constant thickness is determined for isotropic material. Finally, for the assumed shape of the middle surface, the optimal fiber orientation angle θ of the composite shell is appointed. Such studies were carried for three cases: pure external pressure, pure twisting, and combined external pressure with twisting. In the case of shells made of isotropic material the obtained results are compared with the optimal structure of uniform stability, where the analytical Shirshov’s local stability condition is utilized. In the case of structures made of composite materials, the computations are carried out for two different materials, where the ratio of E₁/E₂ is equal to 17.573 and 3.415. The obtained benefit from optimization, measured as the ratio of critical load multiplier computed for reference shell and optimal structure, is significant. Finally, the optimal geometrical shapes and orientations of the layers for the assumed loadings is proposed.

1. Introduction

Axially-symmetric, thin-walled structures have received considerable attention mainly due to their practical applications such as underwater pressure hulls, space vehicles, or pressure tanks [1,2,3,4]. Material and structural optimization becomes a central concept in their design because of the adaptability of composite materials to given design situations/requirements. To achieve an optimized structure with improved characteristics (buckling resistance), not only design parameters such as layer thicknesses and ply angles (stacking sequence configuration) can be employed but also shape/topology of thin-walled shells should be taken into account.

In general, the shape/topology optimization methods for improving buckling capacity are based on the variations of the shell meridian (with a positive Gaussian curvature) that can be represented in the form of ellipsoids, barrel-shaped, egg-shaped, or Cassini curves [2,3,4,5,6,7,8,9,10,11,12,13]. Unfortunately, the thin-walled structures are highly sensitive to the buckling phenomena. Moreover, the geometrical shape of the middle surface, variable or constant wall thickness, material properties as well as different forms of imperfections have a significant impact on this problem [9,10,11,12,13,14,15].

Among different axially-symmetric shapes of the shells, the so-called barrel-shaped structures are of special interest. Parametric optimization due to loss of stability of barrel-shaped shells subjected to axial compression and a combination of axial compression and pressure was presented by Błachut [16,17]. The experimental and numerical results of buckling of barrel-shaped shells under external pressure were reported by Błachut [18]. A good agreement between them was observed. The numerical simulations were carried out with the use of Abaqus. In the case of barrel-shaped shells, the critical pressure could be significantly higher in comparison with cylindrical shells with an equivalent volume of material. Finally, the optimal shape of the middle surface was determined by Błachut [19] with the use of the simulated annealing algorithm. The shape of the middle surface was described by the super-ellipse, where the parameters of this curve were considered as design variables.

The buckling phenomenon of the barrel-shaped shells was also studied by Jasion and Magnucki [20]. They analyzed the impact of a decreasing radius of meridional curvature on the value of critical external pressure. Next, Jasion [21] investigated the axially-symmetric shells with positive and negative Gaussian curvature under external pressure. Two kinds of analysis with the use of ABAQUS software were conducted, namely linear eigenvalue buckling prediction and fully non-linear post-buckling analysis. A significant difference in the behavior of the shells with negative Gaussian curvature and barrel-shaped shells was observed. In the post-buckling range, the latter shells reveal the unstable load-deflection path. Jasion and Magnucki [22] also studied the buckling phenomenon of barrel-shaped tanks filled with liquid. Finally, Magnucki and Jasion [23] investigated the buckling of the barrel-shaped shell subjected to radial pressure. The problem was described analytically and solved approximately with the use of Bubnov’s-Galerkin’s method. Next, the analytical results were verified with the use of the finite element method. A good agreement of the analytical and numerical results (membrane stresses, buckling load, and buckling shape) was reported.

Here it is worth noting that the external load can be also defined as the displacement of the particular part of a thin-walled structure. The example of this kind of buckling problem was presented and discussed on the example of the thin-walled column by Stawiarski and Krużelecki [24].

Very interesting works, which concern the buckling phenomena of the egg-shaped shells subjected to external pressure, were presented by Zhang and his colleagues [25,26,27]. In the first work, the influence of the geometrical dimensions on the critical buckling load was investigated. The shape of the middle surface mimics that of the goose egg. The analysis was carried out numerically in ABAQUS and compared with the analytical solution. Next, the non-linear buckling analysis was performed for the variable and constant thickness shells. The obtained results were also verified experimentally. Finally, the impact of the geometrical imperfection on the buckling and post-buckling behavior was also investigated.

The barrel-shaped, pseudo-barrel, and cylindrical shells of a constant mass with the corrugated middle surface were analyzed by Sowiński [28]. The problem of elastic stability of these structures was solved with the use of the finite element method. The explicit finite element solvers (LS-DYNA) were also used for the dynamic buckling behavior of spherical shell structures colliding with an obstacle block under the sea [4]. There were also proposed finite elements for nonlinear buckling and post-buckling analyses of plates and shells subjected to large deflection and rotation [29,30].

Życzkowski and Krużelecki [31] determined the optimal shape of a cylindrical shell loaded by overall bending moment with the use of the stability constraint in the local form. Moreover, they proposed a concept of the shell of uniform stability. This concept was next utilized by Krużelecki and Trzeciak [32] in the case of optimal design of axially-symmetric shells under hydrostatic pressure and by Życzkowski et al. [33] for buckling of axially-symmetric shells under thermal load. In order to obtain the solution of the optimization problem the variational approach was applied. Barski and Krużelecki [34,35,36] and Barski [37] used the simulated annealing algorithm and Bezier’s curves in order to obtain the solution of the optimization problem of shells subjected to combined loadings. Such studies were carried for the following loading conditions: bending moment with shearing force taken into account, bending moment, axial force and twisting, and finally bending moment and external hydrostatic pressure. The concept of the shell of uniform stability was also utilized. The relationship, which describes the distribution of the wall thickness is not available in the analytical closed form. Therefore, the special, iterative algorithm for wall thickness estimation was also developed. In the latter work, the shape of the transversal cross-section of the middle surface was also optimized.

The experimental and theoretical studies of buckling and post-buckling behavior of composite shell structures were recently discussed and presented in the papers [38,39,40,41,42,43,44]. Such analyses were related to the structures with cut-outs and reinforcements [38], local damages [39], multilayered composite shells [40], honeycomb cylindrical shells [41], conical shells [42], and local buckling of multilayered shells and plates with cut-out under tensile loadings [43,44]. Concluding the above-mentioned studies, it can be said that an advantage of using fiber-reinforced composites over conventional metallic materials is that the former can be tailored to specific requirements of certain applications. However, the large number of design variables (i.e., fiber orientations angles and thicknesses of a particular layer) and the complex mechanical behavior associated with such materials make the structural design much more complex than those involving conventional materials. Also, for many 2D design problems (beams, plates, and shells), there are multiple designs with similar performance. These designs may have very different stacking sequences, but very similar or almost identical values of the extensional [A], coupling [B], and bending [D] stiffness matrices. Moreover, the number of design variables is strongly affected by the type of kinematical hypothesis used. Broad reviews of the existing approaches to the definition of design variables were presented by Verchery [45] and Muc and Chwał [46]. In such cases, it is important to produce all or most design alternatives. It is necessary to point out that the effectiveness of the optimum design, especially for composite structures, strongly depends on the proper choice of two elements: (i) the type of design variables and (ii) an appropriate optimization algorithm.

Generally, the discussed above papers mainly concern the barrel-shaped shells made of isotropic material. However, in the case of composite structures, the shape of the middle surface and the optimal configuration of the composite material (orientation of the layers) should be optimized. Therefore, such an optimization problem seems to be very difficult to solve. The main novelty of the currently presented approach is that the optimization problem is split into two steps. In the first step, the optimal shape of the middle surface is looked for. Next, for the optimal shape of the middle surface, the optimal configuration of the material is estimated. Both steps are realized by relatively simple procedures. However, the proposed approach seems to be very fast, robust, effective, and leads to a reliable solution. According to the author’s knowledge, the works where these two optimization problems (shape of the middle surface and configuration of the composite material) are studied at the same time are very rare. The current work is devoted to the problem of optimal design of the composite thin-walled and barrel-shaped structures. The shape of the middle surface, as well as the configuration of composite material are optimized, for which the external load reaches the maximum concerning elastic static buckling.

The paper consists of five sections and two appendices (Appendix A and Appendix B). The introduction and literature review are presented in Section 1. The formulation of the investigated problem as well as a description of the design variables is given in Section 2. The description of the numerical model and the method of optimization is discussed in Section 3. The results of the performed studies, including optimization of the axially symmetric shells to three different loading conditions (pure external pressure, pure twisting, complex twisting with external pressure) for three different isotropic and composite materials are presented and discussed in Section 4. The most important results and conclusions are summarized in Section 5. Moreover, some necessary formulations are given in Appendix A. The analytical model applied in the optimization technique and its verification with the use of the finite element analyses is also presented in detail in Appendix B.

2. Formulation of Optimization Problem

The subject of the optimization is the axially symmetric barrel-shaped shell presented in Figure 1. The total length of the analyzed structure is designated as L₀ and constant uniform thickness as H. It is assumed that the structure is subjected to three different loading conditions: pure external hydrostatic pressure p₀, pure twisting moment M_t, and a combined load, which consists of external hydrostatic pressure p₀ and twisting moment M_t.

Each layer of a laminate can be identified by its location in the laminate (the pair z_i, z_i-1, i = 1,…,N—the total number of plies), its fiber orientation θ_i (Figure 1), and material properties. Let us assume that each ply in the laminate is made from the same composite material. The value of the external pressure p₀ is defined as the load multiplier. The twisting moment M_t, which acts at both ends of the structure is expressed as follows:

$$M_t=m_t\hspace{0.17em}\pi \hspace{0.33em}{p}_0\hspace{0.17em}{R}_0^3$$

(1)

where: m_t is the dimensionless parameter introduced to have a possibility for free choice of the magnitude of the twisting moment (m_t = 0 means that the twisting moment does not act), and R₀ denotes a constant radius of cylindrical reference shell of a constant length L₀ and a constant thickness H₀. Due to the fact that the studied structure is loaded by uniformly distributed hydrostatic pressure p₀, which acts on the end caps as well, the axial compression has to be taken into account (see Appendix B, Equation (A16)). To eliminate a ‘circumferential bending’ of the wall of a shell under hydrostatic pressure and take advantage of membrane stresses under this load, our considerations are limited to shells with the axially symmetric shape of a middle surface. In other words, a circular profile of the cross-section constitutes a necessary condition of a membrane state for shells under uniform pressure (Vlassov [47]).

The geometry of the investigated shell is uniquely defined when the shape of a middle surface, as well as wall thickness, is determined. As is mentioned above, our considerations are restricted to the axially symmetric shape of a middle surface, where the profile of each cross-section of a shell is a circle. Thus, the radius of longitudinal R₁ and circumferential R₂ curvature are given as follows:

$$R_1=\frac{{\left(1+{R^{\prime }}^2\right)}^{\frac{3}{2}}}{-R^{″}},\hspace{1em}{R}_2=R{\left(1+{R^{\prime }}^2\right)}^{\frac{1}{2}}$$

(2)

where (‘) = d/dx and R is a distance from the axis of a structure to a point of a middle surface. Moreover, it is assumed that the wall thickness H of a studied structure is constant (it is not a design variable) and its value is to be determined from the optimization constraint, which presumes that the optimal structure and reference shell have the same volume of material. Due to the symmetry of the structure, only the right-hand part of a shell, namely 0 ≤ x ≤ L₀/2, is to be only considered, and the condition R′(0) = 0 of symmetry of the structure is assumed.

Finally, it is assumed that the investigated structure is made of an isotropic material (where the mechanical properties are described by Young’s modulus E₀ and Poisson’s ratio ν₀) as well as transversely isotropic composite material (where the mechanical properties are described at last by E₁, E₂, G₁₂, ν₁₂) of symmetric, angle-ply configuration. Such a configuration is uniquely defined by a single quantity, namely the value of fiber orientation angle ±θ°. The applied symmetric composite material consists of eight layers of equal thickness, where H_i= H/8, i = 1,2,…,8 with layers orientations defined as [−θ, +θ, −θ, +θ]_S.

Thus, the optimization problem can be stated as follows. We look for such a structure, for which the value of the critical loading parameter p₀ = p_cr reaches the maximum, namely:

$$p_{cr}\Rightarrow \max$$

(3)

Design variables are considered with the following quantities: (a) shape of the middle surface, which is described by function R = R(x) and (b) the fiber orientation angle ±θº in the composite material of angle-ply configuration. It is assumed that the shape of the middle surface is described by a simple parametric formula, namely:

$$R\left(x\right)=R^c\left(1-m{\left(\frac{2\cdot x}{L_0}\right)}^2\right)$$

(4)

The parameter R^c denotes the radius of the middle surface measured in the geometrical center of the shell, R^c = R(0), the value of which will be determined later. In the performed study, the parameter m is assumed as the design variable, which defines the geometrical shape of the middle surface of the shell. In other words, parameters m and θ are considered as the design variables. The optimization problem is defined as stated above under the following constraints:

• The volume of the material of the optimal structure and reference shell is identical.
• The capacity of the optimal structure and reference shell is equal.
• The minimal radius at both ends of the shell is constrained by a lower bound, R(L₀) > R_adm.
• The slope of the meridian is limited by upper bound, |R′| ≤ R_adm.
• The current study is limited to the convex shell (the positive Gaussian curvature), where R″ ≤ 0.

The lower bound values are assumed as follows R_adm ≥ 0 and R′_adm ≥ 0. The value of parameter R^c in definition (4) for the assumed value of m is determined with the use of equality constraint (2) (constraint (A2) in Appendix A). Further, it should be noted here that the value of the wall thickness of the optimal structure can be readily computed with the use of the equality constraint (1) (constraint (A1) in Appendix A) from the list presented above. Moreover, there is no defined restriction imposed on the wall thickness, for example, lower or upper bound. Restrictions can lead to non-smooth structures, which are not allowed in the present study. All necessary formulas, as well as dimensionless definitions of the particular quantities, are discussed in Appendix A.

3. Methods of Finding Solution

3.1. Optimization Procedure

The method proposed here of description of the shape of the middle surface by Equation (4) seems to be very simple. However, according to the author’s experience [34,35,36,37], the differences between the results obtained from proposed here with a parametric optimization procedure and the results obtained with the use of analytical methods (variational approach) or other advanced numerical algorithms, where the shape of the middle surface is described by, for example, Bézier’s curves, are not significant and does not exceed 7%. On the other hand, the simple parametric procedure is very effective and always provides a good estimation of the global optimum. Moreover, in the case of advanced optimization algorithms, like evolutionary algorithms or simulated annealing algorithms, the number of calls of objective function significantly increases in comparison with the parametric procedure.

Although the current paper concerns mainly the shell of the uniform thickness, the obtained results of optimization can be readily compared with the results, which one can get when the shell of uniform stability is assumed. The condition “the shell of uniform stability” means that local stability is satisfied in the form of equality not only at a dangerous point but at any point of a shell. In the latter case, the solution of the optimization problem can be found easily with the use of simple analytical formulas, which are detailed discussed in Appendix B. Moreover, this comparison can be done in two possible cases. In the first case, having previously determined the optimal shape of the middle surface, the distribution of the wall thickness is computed with the use of the Shirshov’s local buckling condition (Appendix B) [48]. In the second case, the optimal shape of the middle surface is looked for together with the optimal distribution of the wall thickness. Thus, the optimal shape of the middle surface can be slightly different in comparison with the shape, which is obtained when the shell of uniform thickness is assumed. It is rather obvious that the highest profit from optimization should be expected in the last case. It is worth noting here that Shirshov’s local stability condition has an approximate character. Therefore, a special procedure, which is based on the finite element method, designed to verify the accuracy and effectiveness of the Shirshov’s local stability condition and other necessary analytical formulas describing mainly the membrane resultant stresses, is also described in Appendix B.

In the second step of the optimization procedure, for the previously found optimal shape of the middle surface, the optimal fiber orientation angle θ is looked for. It is assumed that now the studied structure is made of the composite material which is symmetric to the middle surface, angle-ply configuration. As is mentioned above, the applied composite material consists of 8, equally thick H_i = H/8, i = 1, 2,…, 8 layers. It is assumed that the feasible fiber orientation angles θ belong to the range [0°, 90°]. The solution of the optimization problem is estimated by repeatedly computing the critical buckling load multiplier q_cr for the starting value of the angle θ equal to 0° and increment dθ = 5°. From among the 19 calculated values of the critical load multiplier q_cr, as a solution of the optimization problem, a value of the angle θ is to be chosen, for which the value of critical load multiplier q_cr is the highest. It is worth noting here that for smaller angle increment dθ, the obtained solutions do not vary significantly. The fiber orientation angle is assumed to be θ = 0° when the fiber is parallel to the circumferential direction.

3.2. Numerical Model and Material Properties

In order to compute the critical buckling load multiplier q_cr the finite element method is applied. The whole calculations are carried out with the use of commercially available software ANSYS 13.0. The investigated structure is modeled as a shell with the use of higher-order SHELL281 elements. These elements have in each node three transitional and three rotational degrees of freedom. As is shown in Figure 2, the regular mapped mesh is applied. Assuming a cylindrical global coordinate system, where the axis of symmetry is Z-axis (Figure 1), the imposed boundary conditions are as follows: for Z = −L₀/2, and Z = L₀/2, UR = 0 and for Z = 0, UZ = 0 (symmetry condition), where UR and UZ is the radial and axial component of displacement, respectively. Moreover, to eliminate the possibility of rigid rotation about axis Z of the whole structure, for Z = 0, Uθ = 0, which is prescribed in any single point, where Uθ is the circumferential component of displacement. The external hydrostatic pressure is modeled as a normal surface load applied to all elements. The axial compression, caused by external pressure, as well as twisting, is applied to both ends of the shell as a nodal force FZ and Fθ, the values of which are:

$$FZ=\pm \frac{\pi \hspace{0.17em}R{\left(\pm 0.5L_0\right)}^2\hspace{0.17em}{p}_0}{N},\hspace{1em}F\theta =\pm \frac{m_{t\hspace{0.17em}}\pi \hspace{0.17em}{R}_0^2\hspace{0.17em}{p}_0}{N}$$

(5)

where N is a total number of nodes, which are located at the end edges of the structure. The ‘±’ sign denotes that the returns of the Fθ at the ends of the shell have to be the opposite. The obtained values of critical buckling load multiplier in the case of external pressure and twisting for simple cylindrical shells made of isotropic material agrees with those (theoretical and empirical) which are presented by Bushnell [49].

The calculations are performed for the shell of constant total length L₀ = 1000 mm, and of the following radius R₀ = 500, 375, 250, 187.5, 125 mm. Taking into consideration the dimensionless quantities, which are presented in Appendix A (Equation (A3)), it corresponds to μ = R₀/L₀ = 0.5, 0.375, 0.25, 0.188, 0.125. Moreover, the computations are carried out for two values of a parameter γ = H₀/R₀ = 0.004, 0.008. Therefore, in the first case H = 2, 1.5, 1, 0.75, 0.5 mm and in the second case H = 4, 3, 2, 1.5, 1 mm. Finally, after performing the appropriate convergent tests of numerical solutions, it is assumed that the size of the finite element is equal to l_e = 15 mm in all cases.

In the first step of the optimization procedure, the applied isotropic material is assumed to be steel. In the second step of the optimization problem, there are considered to be two different composite materials, namely carbon fibers/epoxy resin and glass fibers/polyester resin. The mechanical properties of the applied materials are collected in

Table 1. Mechanical properties of applied materials [50,51].

Material	E1 [GPa]	E2 [GPa]	G12 [GPa]	ν12	E1/E2
steel	210.0	210.0	80.77	0.30	1.000
carbon fiber/epoxy resin	181.0	10.3	7.17	0.28	17.573
glass fiber/polyester resin	28.0	8.2	2.80	0.29	3.415

.

4. Results

The computations are carried out separately for the shells subjected to pure hydrostatic pressure (Section 4.1), pure twisting (Section 4.2), and finally for different combinations of mentioned previously components of loading (Section 4.3). The optimization procedure is applied to the above three loading cases, in which different materials and geometries were investigated. According to the optimization procedure, which is described in Section 3, the results are presented in two parts.

The first part for each particular loading condition is related to the optimization of the geometry of the isotropic barrel shell. This optimal barrel shape shell curvature is described by the determined radii r(0) and r(1) or parameter m. For such purpose, the dimensionless values of the critical buckling load multiplier computed for the reference q₀, for the optimal shell q^max, and for the shell of uniform stability q^ustb are calculated using the Formula (A3). It is assumed that associated shells of uniform stability have the same shape of the middle surface as the structures, which are the solution of the optimization problem. The values q^ustb are computed according to the analytical formulas discussed in Appendix B. Finally, n and n₀ refer to the number of the circumferential half-waves measured in the circumferential direction for the optimal and reference shell, respectively.

In the second part of each optimization procedure, there are performed calculations for transversely anisotropic cylindrical and barrel shells. First of all, the calculations are made for cylindrical geometry with orientations of fibers θ = 90°. This allows for the determination of the value of the critical load multiplier for reference structure designated as q^com₀. In the next step, the optimal fiber orientation angle for cylindrical shell θ_cyl is determined. It should be noted, that the layers orientations are defined as [−θ_cyl, +θ_cyl, −θ_cyl, +θ_cyl]_S, as is presented in Figure 1. For the optimal configuration of the layers in the cylindrical shell, the maximum load multiplier for cylindrical shell q^com_cyl is calculated. The results are presented in the form of the ratio q^com_cyl and q^com₀, which gives information concerning the increase of the critical buckling load in the optimal cylindrical shell. In the final step, the optimal configuration of the fibers orientation angle θ^max is determined for the optimal barrel shape of the middle surface. The same as for cylindrical shape, it means that the layers orientations for barrel shape shell are defined as [−θ^max, +θ^max, −θ^max, +θ^max]_S, as presented in Figure 1.

4.1. Composite Shell under Hydrostatic Pressure

The results of optimization in the case of the shell under hydrostatic pressure are collected in

Table 2. Results of optimization for shell under hydrostatic pressure.

μ	q0·10−6	qmax/q0	qustb/q0	h	n/n0	r(0)	r(1)
γ = 0.004
0.500	2.500	3.461	3.767	0.937	36/16	1.115	0.753
0.375	2.162	3.476	3.693	0.938	32/14	1.159	0.647
0.250	1.773	3.570	3.547	0.961	28/10	1.216	0.500
0.188	1.517	3.220	3.233	0.987	30/10	1.216	0.500
0.125	1.223	2.670	2.750	1.007	20/8	1.216	0.500
γ = 0.008
0.500	11.811	3.012	3.189	0.937	26/12	1.115	0.753
0.375	10.364	2.980	3.082	0.938	24/12	1.159	0.647
0.250	8.457	3.046	2.975	0.961	22/10	1.216	0.500
0.188	7.115	2.761	2.757	0.987	22/8	1.216	0.500
0.125	5.903	2.238	2.279	1.007	14/6	1.216	0.500

,

Table 3. Results of optimization for composite shell under external pressure, E₁/E₂ = 17.573.

μ	γ	Cylindrical Pipe			Optimal (Barrel) Shape
		qcom0·10−6	qcomcyl/qcom0	θcyl [°] 1	qmax/qcom0	θmax [°] 1	n/n0
0.500	0.004	0.781	2.091	10	5.495	15	22/20
0.375		0.683	2.084	0	5.457	10	18/18
0.250		0.569	2.028	0	5.460	0	16/16
0.188		0.495	2.099	5	4.785	0	16/14
0.125		0.411	1.996	0	3.903	0	10/12
0.500	0.008	3.659	2.126	5	4.851	25	18/16
0.375		3.217	2.045	0	4.768	15	14/14
0.250		2.655	2.065	5	4.723	0	12/12
0.188		2.353	1.998	0	4.068	0	10/12
0.125		1.938	2.189	10	3.386	0	8/10

¹ layer orientations are defined as [−θ, +θ, −θ, +θ]_S.

and

Table 4. Results of optimization for composite shell under external pressure, E₁/E₂ = 3.415.

μ	γ	Cylindrical Pipe			Optimal (Barrel) Shape
		qcom0·10−6	qcomcyl/qcom0	θcyl [°] 1	qmax/qcom0	θmax [°] 1	n/n0
0.500	0.004	1.489	1.368	0	4.080	10	28/18
0.375		1.300	1.357	0	4.140	0	24/16
0.250		1.053	1.377	0	4.354	0	22/12
0.188		0.913	1.357	0	3.880	0	22/10
0.125		0.741	1.423	0	3.210	0	16/10
0.500	0.008	6.949	1.372	0	3.579	15	22/14
0.375		6.079	1.364	0	3.609	5	18/12
0.250		5.001	1.348	0	3.719	0	16/10
0.188		4.391	1.410	0	3.238	0	16/10
0.125		3.552	1.345	0	2.727	0	12/8

¹ layer orientations are defined as [−θ, +θ, −θ, +θ]_S.

and Figure 3 and Figure 4. The determined optimal barrel shape geometries for the isotropic shell are summarized in Table 2. Generally, the optimal shape of the middle surface is far from the cylinder. It is worth noting here that starting from μ = 0.25 the inequality constraint (3), which limits the length of the radius of the shell at both ends is active, r(1) ≥ 0.5. The highest profit from optimization is observed for the shells for which the parameter μ is relatively large. Together with decreasing the value of ratio μ, the profit from optimization also decreases. However, in the case of μ = 0.25, the profit is slightly higher in comparison with the shell, where μ = 0.375. It can be explained by the fact, that for that structure (μ = 0.25) the character of the observed buckling pattern is changed, which is shown in

Table 5. Buckling patterns of reference and optimal shells under external pressure, γ = 0.008.

Ref. Cylindrical Metal Shell	Optimal Metal Barrel Shell	Ref. Composite Cylindrical Shell θ = 90°	Optimal Composite Barrel Shell
μ = 0.500
μ = 0.375
μ = 0.250
μ = 0.188
μ = 0.125

.

In Table 5, the corresponding buckling patterns of the cylindrical references are also presented. For shorter shells, the half-waves are observed near both ends. For the longer structures, the buckling pattern is shifted to the center of the structure. Generally, the total number of half-waves for the optimal structure is over two times higher in comparison with the reference one. The obtained profit is also slightly higher for the thinner shells. It is worth stressing here that in the case of the shells of uniform thickness the observed values of h are not very far from the unity. Thus, the thicknesses of the real optimal structures do not vary significantly. Additionally, it should be stressed here that the profit from optimization q^ustb, computed under the assumption of the shell of uniform stability, is similar to that which is obtained for the shell of uniform thickness. This is an important conclusion from a practical point of view because the structure of uniform thickness is easier to manufacture.

In the next step of the proposed optimization procedure, the reference structure is changed. Now the reference shell is a cylindrical shell with such an angle-ply configuration for which the critical buckling load multiplier q^com₀ reaches the lowest possible value. It occurred that the lowest value of q^com₀ is always obtained for fiber orientation angle θ = 90°.

It should be noted here that the curves presented in Figure 3 and Figure 4 are prepared under the assumption that the optimal fiber orientation angle θ is observed for the shell, where the shape of the middle surface is determined previously in the first step of the optimization procedure for isotropic material.

Generally, the profit from optimization increases together with increasing ratio E₁/E₂. For E₁/E₂ = 17.573, the highest values of q^max/q^com₀ ratio are observed for relatively short structures with parameters μ = 0.5, and μ = 0.375. However, in the case of E₁/E₂ = 3.415, the highest value of q^max/q^com₀ ratio is obtained for the shell with parameter μ = 0.25. For these shells, the optimal fiber orientation angles θ^max vary from 5° to 25°. In the case of longer structures μ < 0.375 the optimal angle θ^max = 0°. In both cases of E₁/E₂ ratio, the profit form optimization is higher for the shells with parameter γ = 0.004. In comparison with the cylindrical shells (Table 3 and Table 4), the profit from optimization is over two times higher. Thus, one can say that the change of the shape of the middle surface has a decisive impact on the final results. The obtained buckling patterns of optimal structures are very similar to those which are presented in Table 4. Finally, it is worth stressing here that the obtained results, namely the values of q^max/q^com₀ ratio, are significantly higher in comparison with the results which are presented for isotropic shells for all studied cases.

4.2. Composite Shell under Twisting

The results of optimization of the shell subjected to twisting are presented in Figure 5 and Figure 6 and

Table 6. Results of optimization for shell under twisting.

μ	q0·10−6	qmax/q0	qustb/q0	h	n/n0	r(0)	r(1)
γ = 0.004
0.500	7.823	1.688	1.611	0.957	34/20	1.093	0.803
0.375	7.368	1.671	1.587	0.965	32/18	1.113	0.753
0.250	6.749	1.672	1.575	0.973	26/16	1.170	0.619
0.188	6.329	1.702	1.569	0.987	22/14	1.216	0.500
0.125	5.770	1.712	1.474	1.007	18/12	1.216	0.500
γ = 0.008
0.500	33.552	1.638	1.502	0.957	24/16	1.093	0.803
0.375	31.668	1.619	1.487	0.959	22/14	1.126	0.728
0.250	29.129	1.623	1.468	0.970	18/12	1.182	0.590
0.188	27.334	1.651	1.453	0.987	16/12	1.216	0.500
0.125	24.928	1.651	1.365	1.007	14/10	1.216	0.500

,

Table 7. Results of optimization for composite shell under twisting, E₁/E₂ = 17.573.

μ	γ	Cylindrical Pipe			Optimal (Barrel) Shape
		qcom0·10−6	qcomcyl/qcom0	θcyl [°] 1	qmax/qcom0	θmax [°] 1	n/n0
0.500	0.004	2.535	1.695	20	2.951	55	38/24
0.375		2.428	1.666	15	2.802	55	36/22
0.250		2.297	1.617	0	2.596	55	30/20
0.188		2.209	1.589	0	2.531	55	24/18
0.125		2.086	1.549	0	2.386	15	14/16
0.500	0.008	10.083	1.715	25	2.864	50	28/18
0.375		10.245	1.695	20	2.777	55	26/18
0.250		9.627	1.651	15	2.636	55	22/16
0.188		9.262	1.612	10	2.555	50	16/14
0.125		8.775	1.581	0	2.366	30	10/12

¹ layer orientations are defined as [−θ, +θ, −θ, +θ]_S.

,

Table 8. Results of optimization for composite shell under twisting, E₁/E₂ = 3.415.

μ	γ	Cylindrical Pipe			Optimal (Barrel) Shape
		qcom0·10−6	qcomcyl/qcom0	θcyl [°] 1	qmax/qcom0	θmax [°] 1	n/n0
0.500	0.004	4.552	1.238	0	2.052	50	36/22
0.375		4.362	1.233	0	1.972	55	34/20
0.250		4.100	1.221	0	1.894	55	28/18
0.188		3.911	1.214	0	1.885	20	20/16
0.125		3.602	1.214	0	1.933	0	16/14
0.500	0.008	19.259	1.244	20	2.016	50	26/16
0.375		18.368	1.240	0	1.952	50	24/16
0.250		17.280	1.232	0	1.881	55	20/14
0.188		16.562	1.219	0	1.859	25	16/12
0.125		15.539	1.207	0	1.832	15	12/12

¹ layer orientations are defined as [−θ, +θ, −θ, +θ]_S.

and

Table 9. Buckling patterns of reference and optimal shells under twisting, γ = 0.008.

Ref. Cylindrical Metal Shell	Optimal Metal Barrel Shell	Ref. Composite Cylindrical Shell θ = 90°	Optimal Composite Barrel Shell
μ = 0.500
μ = 0.375
μ = 0.250
μ = 0.188
μ = 0.125

. In the case of the shell of uniform thickness, the profit from optimization does not vary significantly with respect to parameter μ as well as γ and it does not exceed q^max/q₀ = 1.712 (Table 6). In comparison with a shell under external pressure, the profit from optimization is much smaller. Moreover, to the contrary of the shell of uniform stability, together with decreasing parameter μ the profit from optimization slightly increases. A clear discrepancy between the shell of uniform thickness and the shell of uniform stability is observed. The results obtained for the shell of uniform stability are generally worse in comparison with the shell of uniform thickness. As is discussed in Appendix B, these discrepancies can be caused by a very rough estimation of membrane resultant stress caused by twisting. The wall thickness h, like previously in the case of the structure under external pressure, is also very close to unity. This fact is also very important from a practical point of view. For the shells, where μ ratio is equal to μ = 0.125 the wall thickness for the optimal structure is even slightly greater in comparison with the cylindrical reference shell.

The number of half-waves is much higher in the case of optimal structures in comparison with reference ones and the number of half-waves decreases together with the decreasing value of parameter μ. Similarly, as in the case of shells under external pressure, the number of half-waves observed for the optimal structure decreases together with an increasing value of γ. On the contrary of the structure under external pressure, all half-waves are localized at both ends of the shell, what is depicted in Table 9. The center of the structure seems to be unaffected by the buckling phenomenon. Finally, starting from μ = 0.188, the inequality constraint (3), where r(1) ≥ 0.5, is active for both values of parameter γ.

As before, in the case of the second step of the optimization procedure, where the optimal fiber orientation angle is looked for, it occurred that the lowest value of the critical load multiplier is obtained for cylindrical reference where the fiber orientation angle θ = 90° in all analyzed cases.

To the contrary of shells under hydrostatic pressure, in the case of shells under twisting the profit from optimization is the highest for the structures where the μ = 0.5 for all investigated cases. Generally, a clear maximum can be observed in Figure 5 and Figure 6, which correspond to the fiber orientation angle θ^max = 50° or θ^max = 55°. For the shells, where μ = 0.188, 125 the value of θ^max is changing. The profit from optimization depends mainly on the ratio E₁/E₂. If the E₁/E₂ ratio is greater, the observed results are also higher. The impact of parameter γ on the results are not significant as in the case of shells subjected to hydrostatic pressure. In comparison with the cylindrical shells (Table 7 and Table 8), the profit from optimization is significantly higher.

Therefore, similarly to the case of shells under hydrostatic pressure, the change of the shape of the middle surface has a significant impact on the final results from optimization. Moreover, the obtained maximal values of q^max/q^com₀ are higher in comparison with results, which are presented in the case of corresponding isotropic structures. Finally, as mentioned in the previous section, the buckling patterns of the optimal structures, which are made of composite material with optimal fiber orientation angle θ^max, are very similar to those which are presented in Table 9. However, the number of half-waves varies significantly.

4.3. Shell under Combined Loadings

In the case of the shell structures subjected to combined loadings the main aim of carried out computations is to determine the impact of the gradually increasing twisting (the value of parameter m_t) on the solution of the optimization problem. Taking into consideration the discrepancy between the analytical and numerical results observed for the shells of uniform stability (Appendix B) under pure twisting, the appropriate comparisons are not presented now. The calculations are made for isotropic metal shells and composite shells with mechanical properties described by the ratio E₁/E₂ = 17.573. The results obtained for the isotropic shells for parameter γ = 0.004 and γ = 0.008 and for different values of ratio μ are summarized in Figure 7. As can be observed (Figure 7), for the isotropic shells for which the ratio μ = 0.5, even a small contribution of twisting in combined loadings causes a visible reduction of the profit from optimization. In the case of the other investigated shells of uniform thickness, the obtained results are similar, which are depicted in Figure 7.

In the case of composite shells, the optimization is performed for two geometries defined by the parameters γ and μ. In the first case, in which the shell has shortened length, the parameters are equal to γ = 0.004 and μ = 0.5. For such geometry, there have been two optimal geometries of the barrel-shaped shell found (see Table 2 for pure hydrostatic pressure, and Table 6 for pure twisting). In the second case, in which the shell has a shape of a long pipe, the parameters are equal to γ = 0.008 and μ = 0.125. In this case, only one shape of the optimal shape is studied. As mentioned above, m_t = 0 means that the studied shell is subjected to external pressure only. On the other hand, if the parameter m_t → ∞, the profit from optimization, measured as q^max/q₀, is expected to be identical, as in the case of pure twisting.

As can be observed in

Table 10. Results of optimization for composite shell under combined loadings, γ = 0.004 and μ = 0.5 and E₁/E₂ = 17.573.

mt	Cylindrical Pipe			Optimal (Barrel) Shape: rC = r(ξ = 0) = 1.115, r(ξ = 1) = 0.753, m = 0.325		Optimal (Barrel) Shape: rC = r(ξ = 0) = 1.093, r(ξ = 1) = 0.803, m = 0.265
	qcom0·10−6	qcomcyl/qcom0	θcyl [°]	qmax/qcom0	θmax [°] 1	qmax/qcom0	θmax [°] 1
0	0.781	2.090	10	5.495	15	5.359	10
0.5	0.780	2.088	10	5.221	20	5.109	15
1	0.778	2.085	10	4.988	55	4.865	60
2	0.770	2.069	5	4.616	55	4.542	60
5	0.722	1.960	10	3.872	55	3.853	55
9	0.646	1.892	15	3.476	55	3.485	55
15	0.555	1.820	15	3.249	55	3.274	55
25	0.460	1.764	15	3.101	55	3.135	55
∞-Pure twisting	2.535	1.695	20	2.916	50	2.951	55

¹ layer orientations is defined as [−θ, +θ, −θ, +θ]_S.

and

Table 11. Results of optimization for composite shell under combined loadings, γ = 0.008 and μ = 0.125, E₁/E₂ = 17.573.

mt	Cylindrical Pipe			Optimal (Barrel) Shape: rC = r(ξ = 0) = 1.216, r(ξ = 1) = 0.500, m = 0.589
	qcom0·10−6	qcomcyl/qcom0	θcyl[°] 1	qmax/qcom0	θmax [°] 1
0	1.938	2.189	10	3.386	0
0.5	1.937	2.189	10	3.385	0
1	1.936	2.186	10	3.383	0
2	1.929	2.180	5	3.372	0
5	1.891	2.100	0	3.302	0
9	1.808	1.956	0	3.146	0
15	1.669	1.825	0	2.913	0
25	1.466	1.724	0	2.678	15
∞-Pure twisting	8.775	1.581	0	2.366	30

¹ layer orientations are defined as [−θ, +θ, −θ, +θ]_S.

, the obtained results strictly depend on the magnitude of the applied twisting. In both cases of the set of geometrical parameters (γ and μ) together with the increasing value of m_t (higher m_t means greater influence of twisting) the value of the critical load for the reference structures also decreases. The profit from optimization in all analyzed structures also decreases. In the case of cylindrical shells, the profit obtained from optimization is almost identical and for the relatively small values of twisting the profit is over q^com_cyl/q^com₀ = 2 and together with increasing the twisting is below q^com_cyl/q^com₀ = 1.5 for μ = 0.125. However, a significantly different tendency is observed in the case of the optimal value of the fiber orientation angle θ. For the relatively small magnitude of twisting, the optimal fiber orientation angles are identical for studied cylindrical shells θ^cyl = 10° but for the more participation of the twisting, the tendency is different. In the case of shorter structures, the value of θ^cyl increases to 15° and in the case of the longer ones the θ^cyl decreases to 0°.

The change of the shape of the middle surface of the structure causes a further increase in the profit from optimization. It is worth noting that in the case of the shorter shells the obtained values of the ratio q^max/q^com₀ are significantly larger in comparison with the longer ones. It should be stressed here that the optimal fiber orientation significantly differs for the structures where μ = 0.5 and μ = 0.125. In the first case, generally, the optimal fiber orientation angle is greater in comparison with those which are obtained in the latter case. For the shorter shells, the optimal value of θ^max is about 55° whereas for the longer ones the value of θ^max is almost constant and equal to 0°.

Finally, the shapes of the buckling patterns of the studied structures are presented in

Table 12. Buckling patterns of reference and optimal shells under twisting, γ = 0.004.

Ref. Composite Cylindrical Shell θ = 90°	Optimal. Composite Cylindrical Shell	Optimal Composite Barrel Shell
mt = 0.5
mt = 1
mt = 2
mt = 5
mt = 15

. Generally, it can be observed that in the case of cylindrical structures the buckling patterns obtained for all structures are very similar. The number of half-waves in the case of optimal cylindrical reference shells is generally greater in comparison with the optimal ones. However, in the case of shells of barrel shape, the buckling half-waves are generally localized in the neighborhood of the end caps of the structure. Therefore, the introduction of the variable thickness together with the change of the shape of the middle surface seems to be quite justified.

5. Conclusions

The presented work is devoted to the parametric optimization of axially-symmetric shells subjected to different loading conditions including pure external hydrostatic pressure, pure twisting, and a combination of external hydrostatic pressure and twisting. The proposed optimization procedure consists of two steps. In the first step, the optimal shape of the middle surface, as well as the constant uniform wall thickness, is looked for. In the next step, for the previously determined optimal shape of the middle surface, the optimal fiber orientation angle θ is determined. The obtained results can be summarized as follows:

• The proposed optimization technique, based on splitting the procedure into two steps (the shape of the middle surface and the layer configuration are optimized separately), gives considerable benefits in the case of anisotropic structures subjected to combined loadings, and allows for the determination of more optimal geometries for all investigated loading conditions.
• In the case of the isotropic structure under hydrostatic pressure, profit from optimization is the highest for structure, where μ = 0.5 and γ = 0.004. The obtained q^max/q⁰ ratio varies from 3.461 to 2.238.
• In the case of isotropic shells subjected to pure twisting the profit from optimization varies insignificantly (from 1.712 to 1.619) and is smaller in comparison with shells under hydrostatic pressure. A very slight impact of parameters μ and γ on final results is observed.
• Generally, in the case of composite shells, the profit from optimization is significantly higher as in the case of isotropic shells. For shells under hydrostatic pressure, the q^max/q^com0 ratio varies from 5.495 to 3.386 for E₁/E₂ = 17.573, and from 4.080 to 2.727 for ratio E₁/E₂ = 3.415. The value of parameters μ and γ as well as the E₁/E₂ ratio have a significant influence on the final results.
• For the composite shells subjected to twisting the impact of the parameters μ and γ are not significant except the E₁/E₂ ratio. The obtained results vary from 2.951 to 2.366 for E₁/E₂ = 17.573, and from 2.052 to 1.832 for E₁/E₂ = 3.415. A clear maximum in the relationship between q^max/q^com0 ratio and the fiber orientation angle θ is observed.
• In the case of structures subjected to combined load (external pressure and twisting), a significant profit of the application of the barrel-shaped shell is observed. A larger increase of this profit (q^max/q^com0 is from 5.359 to 2.951) is observed for shorter shells (γ = 0.004 and μ = 0.5). In the case of the longer shells, this profit is much smaller (from 3.386 to 2.366). Optimal fiber orientation angle θ^max is significantly different in such cases.
• In the case of the structures, where the participation of twisting is significant, it seems to be a justified extension of the optimization procedure by the introduction of the variable thickness as the additional design variable.