Document Type : Research Paper
Authors
^{1} Department of Computer Science, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731753153, I. R. Iran.
^{2} Composite and Nanocomposite Research Laboratory, Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan,873175315, Iran
Abstract
Keywords
Main Subjects
Minimal Mass and Maximal Buckling Load of Composite HexagonalTriangle Grid Structure using FSDT under External Hydrostatic Pressure
^{a} Department of Computer Science, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731753153, I. R. Iran
^{b} Composite and Nanocomposite Research Laboratory, Department of Solid Mechanics, Faculty of Mechanical Engineering,
University of Kashan, Kashan, 873175315, Iran
KEYWORDS 

ABSTRACT 
Grid stiffeners; Buckling load; Composite shell; Optimization; Genetic algorithm. 
Gridstiffened composite shells are one of the most important structures in many industries. These structures based on their fabrication method, provide both high strength and light structural weight. In this study, buckling analysis under external hydrostatic pressure is performed to obtain critical buckling pressure and the optimum values of parameters for stiffeners. Firstorder shear deformation theory (FSDT) based on the Ritz method is used to calculate the critical buckling load of these structures. The effects of shell thickness, angle of helical stiffeners, rib section area, and the stiffeners number into the buckling load are determined. Comparing the calculated buckling load for stiffened and nonstiffened structures shows that stiffeners significantly optimize structural performance. Furthermore, optimization of stiffener parameters is done by Genetic Algorithm. The results show that the introduced structure has the minimum mass. So, the stiffener parameters would be better. According to the results, the optimum dimensions for stiffener buckling load for the optimal stiffener have been increased by about 80% compared to nonstiffened. 
A composite grid structure is a structure of composite onedirectional tapes that are joined together to form a continuous set as twodimensional (planar) or threedimensional (spatial). Composite grid structures are more capable than metal structures due to strength, low weight, flexibility in design, easy construction, and the ability to withstand various environmental conditions. Shells that have been stiffened with grid structures are an appropriate alternative for composite, sandwich, or filled metal panels. The main objective of using grid structures is the optimization of longitudinal properties of composite materials in structures. Although grid structures are being used as a new technology in many industries, especially the aerospace industry. However, in the past, several valuables of research have been conducted on this kind of structure.
Genetic algorithm (GA) as an evolutionary approach is suitable for tackling optimization problems. Many researchers reported that GA has a good performance to find the nearoptimal solutions of discrete optimization problems such as composite structures [14].
Kim [5] studied the construction of composite grid cylinders, in which building and buckling strength analysis of a stiffened cylinder with the same grid were studied. He investigated the effects of a vertical compressive force on the buckling and rib failure and stability of the entire structure have been studied. In [5], he focused on grid composite panels instead of grid cylinders and examined buckling modes, rib failure, shell, and the entire structure failure. Authors in [6] studied the optimization of a rotating structure with variable curvature. This structure was composed of a shell and a composite grid structure. The purpose of that optimization problem was to minimize the weight of this structure, so that with the change in the size and rib spacing, this structure had the lowest weight, and also required strength under local buckling.
After that, Zhang et al. [7] published an article in which they introduced two new grid structures and calculated their mechanical properties. These two structures were a combination of known structures. They also validated the obtained properties for these structures with the finite element method. Yazdani et al. [8] performed an experimental study on the composite grid shells buckling under axial load. They concluded that increasing the number of helical ribs has more effective than adding circumferential rings or changing the grid type. On the other hand, shells with diamondshaped grids had a more favorable performance in axial loading [8].
Jingxuan et al. [9] placed an advanced grid stiffened (AGS) composite under axial load. They determined the structural strength and failure threshold of the structure. The results were compared with the finite element method by ANSYS commercial software. Other researchers [10] explained the grid composite structures, their construction process, and the latest achievements and their applications in the space industry. In this study, ribs in the grid structure strengthen the structure and reduce its weight.
Rahimi et al. [11] implemented and evaluated a grid composite cylinder with ANSYS commercial software. They examined the effect of the rib profile on the cylinder resistance under axial pressure load by changing the cylinder diameter and rib profile. Weber and Middendorf [12] integrated the interaction between adjacent skin fields into the calculation of the local skin buckling load by applying periodic boundary conditions at opposite panel edges. They considered the selfstiffening effect of the gridstiffened structures due to interaction with adjacent skin fields, significantly increasing the buckling resistance of such structures.
Liu and Paavola [13] evaluated a general analytical sensitivity analysis method for the composite laminated panels and shells. This method is applied to both classical laminate plate theory (CLPT) and firstorder shear deformation theory (FSDT) based on the finite element methods. Deveci [14] optimized the buckling of the composite laminates using a hybrid algorithm under the Puck failure criterion constraint. They proposed an optimization method to find the optimum stacking sequence designs of laminated composite plates in different fiber angle domains for maximum buckling resistance.
Civalek [15] worked on buckling analysis of the composite shells with different material properties by the discrete singular convolution (DSC) method. Ghasemi et al. [16] presented a multiobjective optimization of a composite cylindrical shell under external hydrostatic via an improved version of the evolutionary algorithm of NSGAII. The parameters of mass, cost, and buckling pressure as fitness functions and failure criteria as optimization criteria were considered. Also, the kind of material, the number of layers, and fiber orientations have been considered as design variables.
Hajmohammad et al. [17] developed a practical analytical approach to reach the optimal fiber orientation in the design of fiberreinforced polymer pressure vessels (FRPPVs) subjected to hydrostatic pressure. The genetic algorithm (GA) is applied to achieve the optimal orientation pattern with minimum weight and maximum buckling load.
Ghasemi et al. [18] presented a multistep optimization method to predict the optimal fiber orientation in glass fiberreinforced polymer (GFRP) composite shells. The proposed method contains a regenerated genetic algorithm (GA) coupled with an analytical approach to assessing the failure of the tubular structure.
Soltani et al. [19] investigated the lateral buckling analysis and layup optimization of the laminated composite of web and flanges tapered thinwalled Ibeams based on maximizing lateraltorsional stability strength and minimizing the mass of the structure. The critical factors of fitness function as lateral buckling strength and the mass of the structure with critical limitations such as ply angle, number of layers for the web and flanges, and the thickness of all section walls are considered to be optimized using the nondominated sorting genetic algorithm (NSGAII) and properly defined objective function.
The natural frequency analysis of vertical functionally graded (FG) microplates partially in contact with fluid was investigated by Khorshidi et al [20]. Thermal buckling, bending, and free vibration analyses of microscaled functionally graded GNPs reinforced porous nanocomposite annular plate were considered by Amir et al [21].
Fig. 1. Grid cylindrical shell
The study aims to gain the critical buckling pressure for grid composite structures under external hydrostatic pressure. In this study, a stiffened composite cylindrical shell using a grid structure with internal stiffeners hexagonal  triangle grid under external hydrostatic pressure is analyzed. With the development of smeared method and also using the linear FSDT, buckling critical pressure can be calculated. By calculating the critical buckling pressure for different modes, the effect of stiffener parameters on buckling pressure can be obtained. Moreover, given that in many industries, the use of lighter structures is in priority, grid structures were optimized to reduce weight, and identify the optimal dimensions of stiffeners in this certain problem. In the end, the optimization of composite grid shells was designed using a Genetic Algorithm (GA) and the optimum parameters of stiffeners are introduced. Two scenarios of GA based on their population size and iteration number were performed to obtain the optimized parameters.
Figure 1 shows a cylindrical shell stiffened by grid stiffeners. This shell has length L, thickness t, and diameter D. The following assumptions were considered in this structure:
Displacements are provided by the FSDT in cylindrical coordinates in the below equations, that u, v, and w are displacement components at any arbitrary point of the shell, , , and are the displacement components in the middle level, and are rotation from and axis, respectively [22].
(1) 

(2) 

(3) 

Smeared method for evaluating interactions between the shell and the stiffener was presented by Jaunky et al. [23]. Kidane et al. [24] have provided a method to study the axial buckling of the grid composite cylinder by using the same theory. To make the equivalent of stiffeners, strains and forces in the stiffeners are calculated. Then, by considering the stiffness matrix obtained for stiffeners, a shell with appropriate thickness is determined which has stiffness exactly equal to stiffeners. At this step, determining shell is replaced instead of the total stiffeners, and finally, the shell is added to the primary shell, and all of them are considered as a structural unit.

(4) 

(5) 
By loading the entire structure, reaction forces in stiffeners are made as an axial force in the stiffeners section which is shown with . However, during this loading, stiffeners can bear shear loads and planar shear loads, which are expressed with and , respectively. Inside axial and shear forces applied to the stiffeners are shown in Fig.2.
Fig. 2. Forces applied to the stiffeners
By solving the strains in the direction of stiffeners and the perpendicular on stiffeners, stiffeners forces are obtained concerning strains.
(6) 

(7) 

(8) 

Moments applied to the stiffeners at the interface between the shell and stiffener, are caused due to shear forces. Moments applied to the unit cell are shown in Fig.3.
Fig. 3. Moments applied to the stiffeners
According to the presented method in the previous section, the applied moment on the sides of the unit cell can be obtained as equations (9), (10), and (11).
(9) 

(10) 

(11) 

By taking a plane with normal vector Z, components of shear strains are:

(12) 
The shear forces caused by shear strains are obtained as follows:
(13) 

(14) 

(15) 

By analyzing the forces, moments, and shear forces exerted on the unit cell transforming from these equations as matrix multiplication, stiffness matrix can be achieved as shown in equation (16). Since the shell is a laminated composite, the shell stiffness matrix can be written as equation (17) [2526].
It is noteworthy that the axial stiffness , coupled stiffness (bendingaxial) , bending stiffness and shear stiffness are obtained via the equations (18), (19), (20), and (21) where is the shear correction factor with the amount of [22].
Finally, the equivalent stiffness matrix for a composite grid shell is the sum of the shell stiffness matrix and the stiffeners stiffness matrix as shown in the equation (22).

(16) 


(17) 



(18) 


(19) 


(20) 


(21) 


(22) 
In the Ritz method, the total energy function ( ) is achieved from the sum of the values of strain energy and work done by external forces. On the other side, to create balance, the total energy function of the structure must be minimized. In other words, to minimize the total energy, total potential energy should be differentiated concerning displacement field coefficients , and , and by putting them equal to zero, the coefficient matrix is obtained [2728]:

(23) 

(24) 

(25) 
To have nontrivial solutions in the above equation, the determinant of the coefficient matrix must be zero ( ). After the extension of this equation, the buckling characteristic equation is obtained, and by solving this equation, the buckling load for the different and is achieved. It should be noted that the minimum amount of , is the buckling critical load.
We have verified the results caused by the buckling of composite cylindrical shells with those of reference [29]. We have considered a cylindrical shell with a diameter and lengths. The mechanical properties of shell material (without stiffeners) are provided in Table 1. Also, the layers' angle is [90/60/45]sym.
Table 1. Mechanical properties of the shell material
Properties 







Figure 4 shows the buckling load for a shell with and without stiffeners based on the thickness. The obtained results demonstrate that using grid stiffeners for all shells (with different thicknesses) increases buckling load.
Figure 5 shows the buckling load and specific buckling load concerning the helical rib angle for three different thicknesses. As can be seen, by increasing the helical rib angle for each thickness, the buckling load first is increased, and then after a certain angle (about 75 to 85 degrees) is reduced.
Fig. 4. Buckling load for a grid composite shell with and without stiffeners
Fig. 5. The effect of stiffeners angle on the buckling load
Figure 6 shows the buckling load for different rib heights. It is evident that for all shell thicknesses, increasing the height of ribs increases the buckling load.
Fig. 6. The effect of rib height on the buckling load
Figure 7 shows the effect of the rib section area on the buckling load. The results show that by increasing the stiffener section area, the buckling load increases, and the larger the ribs section area, the higher the buckling load.
Fig. 7. The effect of the stiffeners section area
on the buckling load
Figure 8 shows the effect of the helical stiffener number on the buckling load. The results indicate that by increasing the number of helical stiffeners, the buckling load will be increased.
Fig. 8. The effect of a couple of stiffeners numbers
on the buckling load
Genetic algorithm, inspired by genetic science and Darwin's evolutionary theory, is based on natural selection. Genetic algorithms are commonly used to generate highquality solutions to optimization problems. The Genetic Algorithm (GA) finds nearoptimal solutions to problems by relying on operators such as mutation, crossover, and selection. Figure 9 shows the flowchart of GA with all steps involved from the beginning until the end.
To reduce the weight of our proposed composite grid shell by Genetic Algorithm, we should first determine the objective function based on the weight of the structure. Equation (26) presents the weight function, which is the sum of the stiffeners' weight and shell weight.

(26) 

Fig. 9. Flowchart of Genetic Algorithm
The criterion of the optimization is buckling pressure. In this case study, the buckling pressure must not be more than . The buckling pressure could be obtained from the analytical solution in the previous section.
Since the weight function is a function of the shell parameters, so we also need to specify the variables. In this case, the couple number of helical stiffeners, the unit cell number in shell height, and also ribs thickness and width have been considered as the variables. Notably, lower and upper bounds for each variable were defined. In Table 2, variables are presented along with their upper and lower bounds.
Table 2. Variables and their ranges
Variable 
Ranges 
: Couple helical stiffeners number 

: Unit cell number in height shell 

: Rib width 

: Rib thickness 

Another point that should be addressed at the beginning of applying GA is the population size. In this study, the population size is considered between 15 and 20. The selection operator here is the Elitism operator.
According to the crossover, Mutation, and also elitism selection operators in the Genetic Algorithm, the parameters, the crossover probability, the mutation probability, and the elitism probability should be determined. The crossover possibility for each chromosome pair is considered 0.8. Also, the mutation probability which is the probability of doing a jumping act on each chromosome is considered 0.2, and the elitism possibility which is defined as the probability of chromosome selection is considered 0.35.
In the first optimization scenario, the population size is intended 20 and also the algorithm is repeated 20 times. The results are shown in Fig.10 and Fig.11.
Figure 10 shows the optimization process of the structure mass in terms of the iteration numbers, and it consists of two data sets, which are the minimum mass and the average mass for each generation. According to this figure, it is clear that as the iteration progress, the structure mass decreases. Therefore, a nearoptimal structure can be achieved.
Fig. 10. Optimization process of the structure mass in terms of algorithm iterations (first scenario)
Figure 11 shows the minimum mass of the structure for each generation in terms of iteration in a separate form. In this figure, the optimization process is more evident.
Fig. 11. The structure mass in terms of iterations
(first scenario)
In the second optimization scenario, the population size is 15 and the iteration number is 30. The results are shown in Fig.12 and Fig.13. the aim of this scenario is studying about the effects of increasing the number of interactions on the optimal solution found by the applied GA.
Fig. 12. Optimization process of the structure mass in
terms of algorithm iterations (second scenario)
Figure 12 shows the optimization process of the structure mass in terms of the iteration number, with the difference that the iteration number has increased compared to the first scenario. As can be seen, the more iteration number, the less amount of the mass for each generation. So, by repeating the algorithm, a better solution can be achieved.
Fig. 13. The structure mass in terms of iterations
(second scenario)
Figure 13 shows the minimum mass of the structure for each generation, in terms of the iteration number. As can be seen, the applied GA converges and the minimum mass has not changed for generations 11 to 30 and is a fixed value.
Table 3. The stiffeners' optimum size






Scenario Number 
2 
4 
9 
7 
76.5 
8.827 
First scenario 
2 
3 
10 
8 
72.3 
8.7 
Second scenario 
By applying GA for the mentioned scenarios, the nearoptimal size for stiffeners in each scenario is obtained. Also, according to obtained and for each scenario, the appropriate ribs’ angle can be calculated. Given the stiffener’s size for each scenario, the total mass of the structure is also specified. All information is given in Table 3.
The introduced structure in the second scenario has less mass compared to the first one. So the stiffener size would be better in the second scenario, and the structure will have the minimum mass. According to the results, the optimum dimensions for stiffener buckling load for the optimal stiffener is equal to 1.3273 MPa which was increased by 80%.
In this study first buckling of the nonstiffened shell has been obtained and then the results were compared. The main theory of the problem is the linear firstorder shear deformation theory. So at first, the buckling of a grid shell under these two theories is compared and then the effect of grid stiffeners on the buckling has been investigated. Then the effects of stiffeners’ various parameters on the buckling of grid shells are discussed. In the end, the optimization of composite grid shells was designed using a Genetic Algorithm and the optimum parameters of stiffeners are introduced. According to the results, the buckling load for the obtained optimal stiffener has been increased by 80%.
Nomenclature
: Distance from the middle surface; : Middle surface normal strains; : Middle surface shear strains; : Surface curvatures; : Crosssection area of stiffeners; : Longitudinal modulus of stiffeners; : ; : : Force applied to the stiffeners; : Resultant stress; : Resultant momentums; : Stiffness coefficients; : The k^{th} layer; : The (k+1)^{th} layer; : Equivalent stiffness matrix for shell; : Equivalent stiffness matrix for stiffeners; : Strain energy; : Potential energy; : Moments applied to unit cell; : Shear modulus; : Length of unit cell; : Width of unit cell; : Extensional stiffness of stiffeners; : Coupling stiffness of stiffeners; : Coefficient matrix; : Shear stiffness of stiffeners; : Reduced stiffness coefficients; : Bending stiffness of stiffeners; 
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
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