Fiber Path Optimization in a Variable-Stifness Cylinder to Maximize Its Buckling Load Under External Hydrostatic Pressure

The optimum paths and orientations for curvilinear and straight fibers in composite cylindrical shells of variable and constant stiffness under external hydrostatic pressure are obtained and compared. The optimization was performed for several length-to-diameter ratios of the shells. Results showed that the buckling pressure of the variable-stiffness shell was by about 30℅ higher than that of the constant-stiffness one. This increase in the buckling pressure was obtained by locating fibers in the hoop direction in the middle of the cylinder and orienting the fibers toward the longitudinal direction on cylinder edges.

1. Introduction

Composite cylindrical shells are widely used in marine industries, such as submarines, AUVs, oil and gas pipelines, etc. In these applications, cylindrical shells are exposed to an external hydrostatic pressure. Laminated composites with 0, 90, and ±45° fiber orientations are commonly used in practical applications. Since fibers orientations are constant, these materials are called constant-stiffness composites. Recently, many efforts have been made to optimize the mechanical performance of composites. Designers have tried to find an arrangement of composite components that would satisfy a complex of design requirements, such as strength, stiffness, weight, and price. Therefore, composites with a variable stiffness (VS), in which fibers are placed along desired paths, exhibiting a better performance than straight-fiber ones were considered. Fibers can be stacked along curved paths by using an automated fiber placement (AFP) technique. In such a case, composite stiffness varies from one point to another, and its mechanical performance can be improved by varying the stiffness continuously.

Tatting [1] analyzed and optimized the buckling and failure loads of a cylindrical shell made of a curvilinear-fiber composite and subjected to different loading conditions, such as axial compression, internal and external hydrostatic pressure, bending moment, torsion, and transverse shear loads. The curvilinear-fiber composites, under bending moments and transverse shear loads, were found to show the enhancement of load. In the case of a cylindrical shell under an external hydrostatic pressure, a maximum increase in the buckling load equal to 9% was obtained, and it was concluded that, using curvilinear fibers, a considerable growth in the buckling pressure will not happen. Bloom et al. [2] optimized the buckling load of a variable stiffness cylindrical shell under a bending moment. In the optimum state, the buckling moment of the shell was by up to 17% higher than that of a constant stiffness (CS) composite cylinder. Bloom et al. [3] performed stress-strain tests on a variable-stiffness composite cylinder under a bending moment. Test data were compared with results of a finite-element analysis, and a good agreement was obtained between them. Rouhi et al. [4] optimized the buckling load of a variable-stiffness composite shell under a bending moment. A variation in its radius showed no significant effect on the buckling load, but a variation in the aspect ratio increased the buckling load significantly. In [5], a multiobjective optimization method was used to optimize the buckling load of a variable-stiffness composite cylinder subjected to a bending moment in either of two opposite directions. Ghayoor et al. [6] investigated the bending-induced buckling performance of variable-stiffness elliptical composite cylinders in comparison with that of their constant-stiffness counterparts. The axial stiffness and buckling and postbuckling loads of variable-stiffness composite cylinders, with and without a cutout, in axial compression were investigated by Wu et al. [7, 8]. The buckling and postbuckling of a variable-stiffness composite cylinder under axial compression were studied by White et al. [9]. In [10], the optimization of buckling load for a variable-stiffness composite shell, with and without a longitudinal reinforcement, under a bending moment was studied. It was concluded that the stress distribution due to the longitudinal reinforcement and the presence of curvilinear fibers led to a further increase in the buckling load. In [11], maximization of the fundamental frequency and minimization of the maximum displacement of a cylindrical shell subjected to torsion and bending loads were performed. The bending-induced buckling performance of variable-stiffness composite cylinders with different percentages of fiber steering plies was investigated in [12]. In [13], the maximum axial buckling capacity variable-stiffness elliptical composite cylinders was optimized, and the result was compared with that of the best constant-stiffness counterpart. In [14], a variable-stiffness composite cylinder under a bending loading was optimized. The buckling load of the cylinder was found to be by 40.22 and 31.25% higher than those of constant-stiffness and quasi-isotropic cylinders, respectively.

To the best of knowledge of present authors, the effect of external hydrostatic pressure on variable-stiffness composite shells has rarely been investigated in available research works. As already mentioned, Tatting [1] concluded that curvilinear fibers are not useful for increasing the buckling pressure of cylinders under external hydrostatic pressure, but he assumed that the boundary conditions of cylinder edges are identical. In the current work, the buckling load of variable-stiffness composite shells under external hydrostatic pressure is optimized in the case of different boundary conditions at cylinder edges. For this purpose, the range of geometric parameters is extended, and more cylinders with different aspect (length to diameter) ratios are investigated. The genetic algorithm (GA) optimization used in our work is a simple and robust method to find the global optimum, which is commonly employed for variable-stiffness designs [1, 4,5,6, 15]. For the optimization, an in-home genetic algorithm code in the Python language was developed and linked to the Abaqus finite-element software.

2. Modeling method

In order to compute the buckling load of a composite cylindrical shell under an external hydrostatic pressure, shell dimensions, properties of the composite material, the number of layers, the orientation of fibers at each point of each layer and boundary conditions have to be given as inputs of the problem. The orientation of curvilinear fibers at any point of the shell depends on the coordinate of that point; therefore, a relation between fiber orientations and coordinates of shell points have to be derived.

The axial variation of stiffness can increase the buckling pressure of variable-stiffness composite cylindrical shells under an external hydrostatic pressure [1]. To this end, fibers have to be located in the hoop direction in the middle of the cylinder and in the longitudinal direction on cylinder edges. For modeling a composite cylinder made with curvilinear fibers, the path equation for the reference fiber have first to be determined. The direction angle θ of fibers (see Fig. 1) was is determined by the equation

$$ \theta (x)={\theta}_1+2\left({\theta}_0-{\theta}_1\right)\mid x-L/2\mid /L $$

Fig. 1.

Path of curvilinear fibers on a composite cylindrical shell.

according to which, the fiber orientation angle is θ₀ at edges and θ₁ at the middle of the cylinder. In the present study, several laminated composite cylinders with different length to diameter ratios and stacking sequence [±θ]_12s were considered.

For boundary conditions, all three displacements of one end of the cylindrical shell were taken to be restrained, the other end was assumed in the axial direction (restrained in the cross section plane), and rotations of both its ends were considered free. The plies were made of a 0.146-mm-thick AS4/3501-6 carbon/epoxy material, whose mechanical properties were as follows [16]: E₁ = 147 GPa, E₂ = 9 GPa, G₁₂ = 5 GPa, ν₁₂ = 0.3, X_T = 2004 MPa, X_C = 1197 MPa, Y_T = 53 MPa, Y_C = 204 MPa, and S₁₂ = 137 MPa, where E₁ and E₂ are the longitudinal and transverse moduli, G₁₂ is the shear modulus, ν₁₂ is the Poisson ratio, and X_T, X_C, Y_T, Y_C, and S₁₂ are the longitudinal tensile, longitudinal compressive, transverse tensile, transverse compressive, and in-plane shear strengths, respectively.

In order to obtain the mode shapes and buckling pressure, the linear buckling analysis of Abaqus software was used. The S4R element and Orient user subroutine were used for meshing and modeling the fiber orientation at each point of the cylindrical shell surface, respectively.

3. Optimization

In the present study, the design variables are θ₀ and θ₁, which can vary between 0 and 90°, and the objective of optimization is to maximize the buckling pressure of a composite cylinder under an external pressure. To prevent the wrinkling of tows in the variable-stiffness composite cylinders, the constraint on the minimum permitted turning radius of fiber, equal to 0.635 m [17], has to be considered. The minimum permitted turning radius of fiber is 0.635 m [17]. Mathematically, the optimization problem is defined as:

$$ \operatorname{Maximize}\ {P}_{\mathbf{cr}}\ \mathrm{at}\ {0}^{{}^{\circ}}\le {\theta}_0,\kern0.5em {\theta}_1\le {90}^{{}^{\circ}},\mathrm{and}\ R\le 0.635 $$

where P_cr is the buckling pressure and R is the in-plane radius of the steered fiber path.

As mentioned in the Introduction, the genetic algorithm was used in the optimization process. For this purpose, an in-home genetic algorithm (GA) optimization program was written in the Python language and linked to the Abaqus finite-element software. At the first stage, the initial population (initial sample points), including a number of chromosomes (consists of 20 random θ₀ and θ₁ angles between 0 and 90°) was randomly produced by the GA. At the next stage, the buckling load of each individual was evaluated at the fitness evaluation stage by the Abaqus software. Then, the selection, crossover, and mutation stages were performed by the GA. The above-mentioned stages were continued until the stopping criteria were satisfied. When the difference between the best and mean values of the buckling load for at least 10 generations was smaller than 3%, the stopping criteria was reached. For a better presentation of the optimization process, its flowchart is illustrated in Fig. 2.

Fig. 2.

Flowchart of the optimization process.

4. Results and discussion

To validate the results obtained, the numerical and experimental data given in [7] were utilized. In [7], a [±45/±θ]_s lay-up was used for a composite cylinder with a diameter of 0.413 m and length of 0.889 m. If the points at 0 and 180° of cylinder cross section are called the crown and keel, respectively, the fiber orientation angle θ varies from ±10° on the crown and keel to ±45° on shell sides (90 and 270° of cross section). The shells were made of an IM7/8552 carbon/epoxy material. The experimental and calculated buckling loads found in [7] were 76,560 and 67,046 N, respectively, differing by 13%, but our analysis gave 70,716 N with only an 8% error.

The Nasrtan and Abaqus finite-element softwares were used for the analysis in [7] and in the present study, respectively.

To study the convergence of the genetic algorithm code, an optimization problem for a cylinder with a length of l m, diameter of 0.35 m, and stacking sequence of [±θ]_12s, made of the AS4/3501-6 carbon/epoxy composite, was solved. In this case, 100 generations with a population size (the number of individuals) equal to 20 per generation and 3591 elements for the cylindrical shell were considered. The best and mean values of the buckling pressure converged to each other after about 80 generations.

The maximum buckling pressure and optimum lay-up for cylinders of diameter 0.5 m and lengths 0.25 m to 3.5 m were obtained. In all the cases, the failure pressure was higher than the buckling pressure. In Table 1, the maximum buckling load under an external hydrostatic pressure for eleven variable-stiffness and eleven constant-stiffness composite cylinders are presented.

Table 1. Maximum Buckling Pressure and Optimum Lay-Up for Cylinders of Diameter 0.5 m and Lengths 0.25 to 3.5 m

The resulting fiber paths were circumferentially shifted along the cylinder surface. To visualize the steered fiber paths, the optimum fiber path of a steered ply is shown in Fig. 3 for a cylinder with a length of 1.75 m, diameter 0.5 m, and the optimum stacking sequence ±[17/90]_12s.

Fig. 3.

Optimum fiber path of a steered ply in a VS cylinder.

To illustrate the effect of curvilinear fibers on the enhancement of buckling pressure, this enhancement vs. the length-to-diameter ratio L/D shows in Fig 4.

Fig. 4.

Enhancement Δ of the buckling pressure of VS cylinders of CS with D = 1 (a), 1.5 (b), 2.0 (c), and 2.5 m (d) vs L/D compared with that of a CS cylinder.

As is seen, the greatest increase in the buckling pressure was 28% at L/D = 3.5. At L/D > 5, no enhancement was observed. In this case, on increasing the length of cylinder, the lateral pressure had a greater effect than the axial load on buckling of the cylinder. Therefore, for this case, [90]_48T will be the optimum lay-up. It should be noted that, with further increase in length of the cylinder, the effect of axial load on buckling will increase, and the cylindrical shell will buckles like a column under an axial load. In this case, the optimum lay-up is not [90]_48T. According to Table 1, for shells of length exceeding 0.5 m, the fiber angle should be 90° in the cylinder midsection. Thus, using 90° fibers (in the circumferential direction) in the cylinder midsection and inclining the fibers to some degree in the longitudinal direction, toward cylinder edges, the buckling hydrostatic pressure will increase. In the first-mode buckling, the maximum deformation occurs in the cylinder midsection. Therefore, to increase the buckling pressure, fibers in the circumferential direction have to be located at the midsection. In a short cylinder, the constraint of the minimum allowable radius of curvature does not allow one to locate fibers at 90° in the cylinder midsection. This is shown in Table 1 for cylinders of length 0.25 and 0.375 m.

The strengthening of midsection ensures the existence of only one half-wave in the longitudinal direction. In fact, fibers in the circumferential direction play the role of light stiffeners for the cylinder. The optimum lay-ups and maximum buckling loads for cylinders with lengths of 1, 1.5, 2, and 2.5 m and several diameters are presented in Tables 2 to 5. These results were obtained without considering the minimum permitted turning radius of curvilinear fibers.

Table 2. Maximum Buckling Pressures and Optimum Lay-Ups for Cylinders with a Length of 1 m and Diameters of 0.2 to 3.5 m

Table 3. Maximum Buckling Pressures and Optimum Lay-Ups for Cylinders with a Length of 1.5 m and Diameters of 0.3 to 3 m

Table 4. Maximum Buckling Pressures and Optimum Lay-Ups for Cylinders with a Length of 2 m and Diameters of 0.4 to 3.5 m

Table 5. Maximum Buckling Pressures and Optimum Lay-Ups for Cylinders with a Length of 2.5 m and Diameters of 0.5 to 4 m

As is seen in Tables 2 to 5, in order to achieve a maximum buckling pressure for cylindrical shells made with curvilinear fibers, the fibers have to be located in the circumferential direction at the midsection and inclined to some degree to the longitudinal direction toward shell edges. The trend of enhancement of the buckling pressure seen in Tables 2 to 5 is similar to that described in Table 1. For a more clear illustration, the percentage of buckling pressure enhancement of the variable-stiffness composite cylinder is shown in Fig. 4b-d in comparison with those of constant-stiffness composite cylinders of four lengths (see Fig. 4a).

Figure (4b-e) indicates that there is no increase in the buckling pressure of VS (variable-stiffness) cylinders in comparison with that of CS (constant-stiffness) ones with four different lengths and aspect ratio of 5. This is due to the fact that, for a cylinder of smaller diameter, the lateral pressure has a greater effect on the buckling load, and the optimum lay-up will be [90]_48T. Therefore, the aspect ratios of 5 and higher are not appropriate for this kind of structures. The greatest increase in buckling pressure was 30%, which occurred for a cylinder with an aspect ratio of about 3 and length of 1 m.

Furthermore, investigating the buckling mode shapes of the cylinders, it was observed that curvilinear fibers could change the circumferential wavenumbers of mode shapes compared with those of their CS counterparts. In the cases where the enhancement of buckling pressure was considerable, the displacements due to buckling arouse in the shell half-length close to its right end. For example, as is evident in Fig. 6, at D = 0.6 m and L = 2.5 m (see Table 5), the wavenumber of the circumferential mode shape for the optimum constant-stiffness cylinder with a [90]_48T lay-up was n = 2, while for the optimum variable-stiffness cylinder with a ±<10/90>_12s lay-up, n = 3. The displacements due to buckling here also occurred in the cylinder half-length close to its right end.

Fig. 6.

Buckling mode shape of composite cylinders with L = 2.5 m and D = 0.6 m: CS cylinder with a [90]_48T lay-up (a) and VS cylinder with a ±<10/90>_12s lay-up (b).

As can be seen in Fig. 6, the buckling mode shape is not symmetric with respect to the shell midspan, which is explained by the asymmetric boundary conditions — the right end of the shells is free, but the left one is fixed in the axial direction.

Conclusions

Owing to the lack of adequate studies on the buckling optimization of variable-stiffness composite cylindrical shells subjected to an external hydrostatic pressure, the current research was devoted exactly to this problem. In past studies, it was concluded that curvilinear are not useful for increasing the buckling pressure of cylinders operating under an external hydrostatic pressure if the boundary conditions at both their ends are identical. In the present study, considering different boundary conditions at cylinder ends, the buckling pressure of several composite cylindrical shells with different length to diameter ratios was optimized. The results obtained show that, under an external hydrostatic pressure, the buckling pressure of a variable-stiffness composite cylinder was by up to 30% higher than that of a constant-stiffness composite cylinder.

Locating fibers at 90° (in the circumferential direction) in the cylinder midsection and orientating fibers in the longitudinal direction on cylinder edges, the external hydrostatic pressure could be increased. In fact, since the maximum buckling deformation occurs in the cylinder midsection, the buckling pressure can be increased by strengthening this region locating fibers in the circumferential direction. Fibers in the circumferential direction play the role of stiffeners of the shell. Furthermore, when the enhancement of buckling pressure is considerable, the curvilinear fibers change the wavenumber of circumferential buckling mode shapes in comparison with that in the case of straight fibers, and buckling displacements take place in the shell half-length close to its free end.

In the case of a cylindrical composite shell, the enhancement of buckling pressure using curvilinear fibers depends on its length to diameter ratio. The optimum length to diameter ratio of composite cylindrical shells with curvilinear fibers under an external hydrostatic pressure is about 3 to 4.