Document Type : Research Paper
Authors
Department of Mechanical Engineering, Semnan University, P.O. Box 35131191, Semnan, Iran
Abstract
Keywords
The Effect of Different Weighting Ratios, Length, and Thickness on Weighted Sum of the First Natural Frequency and Critical Buckling Load for a Laminated Composite Circular Cylindrical Shell
R. Ashrafian^{*}, A. Ghoddosian
Department of Mechanical Engineering, Semnan University, P.O. Box 3513119111, Semnan, Iran
KEYWORDS 

ABSTRACT 
Weighted Sum Natural Frequency Buckling Load Laminated Composite Circular Cylindrical Shell 
In this study, a weighted sum, consisting two nondimensionalized quantities critical buckling force and natural frequency, is employed to maximize the objective function for a laminated composite circular cylindrical shell. The function is considered to find the optimum solutions as the goal. Orientation angels of fibers are mentioned in a wellknown configuration as candidate design, and critical buckling force and natural frequency values are derived with the first order shear deformation theory. The composite shell is considered with 8 layers, also the boundary conditions are assumed to be fully simply support and to satisfy boundary conditions displacement and slope components are defined in form of double Fourier series. After combination of differential operators and Fourier series, eventually the matrix L is found and Galerkin method gains function values. For this purpose, a program based on MATLAB is employed for the process. Validations of numerical results show that the used method is moderately satisfactory and acceptable in predicting the critical buckling force and the natural frequency of the shell in comparison with other works. As the conclusion, the effect of different weighting ratios, shell lengthtoradius ratios, and shell thicknesstoradius ratios on the optimal designs are investigated and the results are compared. 
Composite physical properties are achieved by combining different materials to meet specific requirement. Enormous benefits that these materials posses attract researchers to explore more into it to script its behavior in a welldefined form to the users. On the other hand, the structures are quite often are subjected to inplane or external loads which may cause buckling. In addition, the vibration can be problematic when the excitation frequency coincides with the shell’s resonance frequency. Such loadings may occur at different times under inservice conditions, necessitating a design approach that is capable of taking in to account these various loading conditions.
In the recent years, numerical approaches are focused on the problems of the composite structures. For example, Liu et al. [1] investigated nonlinear breathing vibrations of an eccentric rotating composite laminated circular cylindrical shell, which is subjected to the lateral and temperature excitations. It was carried out based on Donnell thin shear deformation theory, von Kármántype nonlinear relation and Hamilton’s principle. Pitton et al. [2] by a methodology involving the design of an Artificial Neural Network (ANN) predicted the approximation of the buckling load and of the prebuckling stiffness of a composites cylindrical shell. Zhang et al. [3] focused on the resonant responses and chaotic dynamics of a composite laminated circular cylindrical shell with radially prestretched membranes at both ends and clamped along a generatrix. Matsunaga [4] by employing the method of power series expansion of displacement components, a set of fundamental dynamic equations of a twodimensional higherorder theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials studied vibration and buckling through Hamilton’s principle. Ungbhakorn, and Singhatanadgid [5] employed the similitude invariant and the scaling laws of the symmetric crossply laminated circular cylindrical shells for buckling and free vibration problems by applying the similitude transformation to the governing differential equations directly. Lam and Loy [6] studied the influence of boundary conditions for a thin laminated rotating cylindrical shell. The analysis was carried out using Lovetype shell theory and solved using Galerkin's method. Lee [7] defined free vibration and dynamic response for the CFRP and GFRP crossply laminated circular cylindrical shells under impulse loads and investigated by using the firstorder shear deformation shell theory. The modal analysis technique was used to develop the analytical solutions of the simply supported cylindrical shells. Walker and Smith [8] presented a methodology for using genetic algorithms with the finite element method to minimise a weighted sum of the mass and deflection of fibre reinforced structures with several design variables. Free vibration of laminated composite shells with cutouts are presented by a nine noded curved C0 finite element (FE) formulation developed by Kumar et al. [9] based on higher order shear deformation theory (HSDT) using Sander's approximations. A research by Sepiani et al. [10] investigated the free vibration and buckling of a twolayered cylindrical shell made of inner functionally graded (FG) and outer isotropic elastic layer, subjected to combined static and periodic axial forces. Wagner et al. [11] believed that the worst geometric imperfection is a mathematical concept which should deliver in theory a
lower bound for the buckling load of unstiffened cylindrical shells, and the corresponding knockdown factors could be used as base for improved shell design guidelines in order to reduce weight and cost of unstiffened shells. The problem of local buckling of a thin composite laminated cylindrical shell under external pressure is studied by Mikhasev et al [12]. Geier et al. [13] by a research argued that the buckling loads of laminated cylinders can strongly depend on the position of the differently oriented layers within the shell. Labans et al. [14] presented a research that two laminated composite shells, one with a conventional straight fiber laminate denoted the classical laminated shell and the second one with a variable angle tow reinforced composite, had been excited and their natural frequencies and mode shapes had been measured and monitored as a function of the axial compression load. Ng and Lam [15] studied the vibration and critical speed of thin isotropic cylindrical shells under constant axial loads. In the analysis, Donnell's theory for a thinwalled cylindrical shell is used. Liu and chu [16] investigated Nonlinear vibrations of thin circular cylindrical shells. Based on Love thin shell theory, the governing partial differential equations of motion for the rotating circular cylindrical shell are formulated using Hamilton principle. Kassegne and chun [17] argued that Fiber reinforced composite materials continue to experience increased adoption in different employment. Walker et al. [18] obtained the multiobjective design of a symmetrically laminated shell with the objectives defined as the maximization of the axial and torsional buckling loads. The ply angle is taken as the optimizing variable.
Employing a weighted sum, consisting two nondimensionalized quantities critical buckling force and natural frequency, is considered to maximize objective function for a laminated composite circular cylindrical shell. Orientation angels of fibers are considered as design variable. Critical buckling force and natural frequency values are derived with the first order shear deformation theory. Eventually, the effect of different weighting ratios, shell aspect ratio, and shell thicknesstoradius ratios on the optimal designs are investigated and the results are compared.
A circular cylindrical shell, the schematic of the klayer of the shell and coordinates are shown in Fig. 1. Based on firstorder shear deformation theory, the equilibrium equations for a shell under axial loads are as follows and the deformations are assumed to be small [19]:
Fig. 1. Klayer laminated composite circular cylindrical shell and coordinate
(1) 
, and are defined by the following relation [19]:
(2) 
All of the equivalent material properties for each layer is obtained with regard to ‘rule of mixture’. Equation constitution of composite shell based on classical laminate theory are defined by the following relations [7]:

(3) 
The A, B, D, and H matrices are defined as follows, where [19]:
(4) 

(5) 
equals in the last portion of Eq. (4). Then [19]:

(6) 

The boundary conditions for the cylindrical shell with fully simply support are considered as [10]:
(7) 
The external excitations are taken to be zero in order to solve the buckling and free vibration problems. After substituting Eq. (7) into the equations of motion, the results are simplified in the following form:
(8) 
where:
(9) 
All components of matrix L are expanded in the appendix segment. To satisfy the boundary conditions, u, v, w, and are defined by the following double Fourier series [7]:
(10) 
T is the function of time in the above equations. Also A, B, C, D and E are the constant coefficients of the natural mode shapes. Galerkin method is employed to solve the differential equations [6].
(11) 
In the buckling analysis, the material and the geometry of the shell are assumed to be perfect and no imperfection exists. To calculate the buckling load, the static solution is performed (i.e. T= 0). After Solving Eq. (8) by Galerkin method and its simplification, the following equation was obtained [5]:
(12) 
Determinant of the coefficients is set to be zero; thus, the buckling loads equation is derived as [5]:
(13) 
In Eq. (13), are the constant coefficients and is the axial critical buckling load [5].
To solve the free vibration problem, the function of time is assumed as follows [5]:
(14) 
A method similar to the buckling analysis method is employed to be derived the following set of equations [5]:

(15) 
The determinant of the coefficients is set to be zero, thus the characteristic frequency equation is derived as [5]:
(16) 
where are the constant coefficients, after solving the Eq. (16), natural frequencies are calculated, and substitution of these frequencies in Eq. (15) infer the constant coefficients of the mode shapes [5].
The following nondimensionalized quantities are introduced to the all computations [8]:
(17) 
where and correspond to lamination angles for eight layers. Maximization of the fundamental natural frequency and critical buckling load of the shell with the laminate configurations given by a combination of , , ply angles is considered as a optimization problem to find the best orientation angles of fibres in the laminated cylindrical shells. The objective function, OF, can be describes as follows [8]:

(18) 
where and are the weighting factors which have following conditions , . As the first case, the single objective designs can be obtained as special cases by setting α=0 or α =1. As the second case, to equalize the effect of each component of the objective function formula (OF), α should be set to 0.5. Furthermore, α=0.25 or α=0.75 provide a space between the first case and the second case. In this study, five laminate configurations are considered as candidate designs [8].
The material properties related to Fig. 2 are shown in Table 2. Fig. 2 indicates the comparison between the frequency results for a fully simply supported (SSSS) boundary condition of the shell for the present research and Ref. [6]. To ensure the accuracy of the model for buckling of the cylindrical shell, the buckling loads obtained by Ref. [5] was compared with the same results obtained by the present work. Table 4 shows the discrepancy between the results. Table 3, shows the ply properties of the composites used in Table 4 and Ref. [5]. As it is shown in Figs. 2 and Table 4, the results are in good agreement.
The numbers (m, n) in the parenthesis represent the buckling modes, i.e., number of axial half waves (m) and number of circumferential waves (n).
Numerical results are given for graphite/epoxy material, and the laminated cylindrical shell is constructed of equal thickness layers. H/R = 0.2 and L/R = 1 are considered.
Table 1. Laminate Configurations
Prompt 
Laminate Configuration 
C1 

C2 

C3 

C4 

C5 
Fig. 2 Comparison of variation of frequency with circumferential wave number for the SSSS composite cylindrical shell with Ref. [6], , , stacking sequence: [90/0/90]
Table 2. Relative material properties with Fig. 2
19 
7.6 
4.1 
4.1 
4.1 
0.26 
1643 
Table 3. Ply properties of the composites used in Ref. [5]

Ply Thickness (mm) 

Graphite/Epoxy 
0.127 
0.24 
5.65 
10.8 
132 
Kevlar/Epoxy 
0.127 
0.34 
2.07 
5.50 
76.8 
Eglass/Epoxy 
0.127 
0.26 
4.14 
8.27 
38.6 
Table 4. Comparison between the results for buckling loads of a simply supported anisotropic crossply cylindrical shell with the lay up ,
Configuration 

Buckling Loads (KN/m) 

Material 
Ratio (L/R) 

Present Study 
Ref. [5] 
Discripancy (%) 
Graphite/Epoxy 
1 

85.44 (11,3) 
84.23 (11,3) 
1.41 

3 

85.44 (11,9) 
84.23 (11,9) 
1.41 
Kevlar/Epoxy 
1 

41.09 (10,3) 
39.38 (10,3) 
4.16 

2 

40.62 (10,5) 
38.96 (10,5) 
4.08 
Eglass/Epoxy 
1 

43.72 (12,3) 
42.97 (12,3) 
1.71 

3 

43.72 (12,9) 
42.97 (12,9) 
1.71 
The effect of five different weighting ratios on the optimal results is given for different five laminate configurations in Figs. 3 and 4 and Table 5. As seen from Fig. 3 and 4, the best laminate configuration is C1 for all the quantities of . The (OF)_{MAX} is almost the same for C2, C3, C4, and C5 laminate configurations, whereas C1 is completely different, and obtain the best quantity of OF 1.97 with Theta( ) 39.43. As seen from Fig. 3 and 4, as the weighting ratio increases, the (OF)_{MAX} decreases for all the states. Overall C1 displays the best result and maximized OF amongst other states. The best fiber angle in Fig. 3 varies around . As mentioned above, the largest amount assigned to the objective function was 1.97. This value belonged to α=0. This truely indicate the greater importance and influence of natural frequency than critical buckling force. Therefore, the values and orientation angles of fiber that lead to the maximum values of the natural frequency will most definitely have the maximum values of the weight function.
The effect of shell lengths on the optimal results is given in Fig. 5 for five laminate configurations. As seen from the figure, as the shell length increases, the OF decreases. Also, the best fiber angle decreases for all state’s despite of the rise of shell lengths, with the exception of C1. The optimum fiber angles and (OF)_{max} are given for all laminate configurations for shell lengths in Table 6.
The effect of shell thickness on the optimal results is given in Fig. 7 for five different laminate configurations. As seen from Fig. 7, as the shell thickness increases, the OF increases. Each case of the Fig. 7 clearly demonstrate an intersection between 70 deg and 80 deg. The best fiber orientation angles for all cases alters in 40 deg approximately. The optimum fiber angles and (OF)_{max} are given for all laminate configurations for shell thickness in Table 7.
In this study, maximization of a weighted sum of the frequency and buckling load under external load for laminated composite circular cylindrical shell is investigated. Five shell configurations with eight layers are considered as candidate designs. The best design enjoys the highest quantity for OF, which equals to a weighted sum of the objectives nondimensionalized quantities of the critical buckling force and the first natural frequency.
Present results can lead designer to apply optimal laminate configuration which can be functional in an acceptable manner. The effect of different weighting ratios, length, and thickness on the optimal results are investigated. Graphs, demonstrating the relation of the fiber angle with OF, illustrate that the maximum OF occurs at a specific value of the fiber angle and this value can be several times higher than the OF at other fiber angles. This fact emphasizes the importance of optimization in modern design to obtain the best performance of laminated composite shells. Eventually, it can be said that the fundamental frequency generally has a more significant effect than the buckling load on the maximum OF, and the weighting ratio generally has not a marked effect on the fiber angles.
Nomenclature
Radius, (mm) 

Length, (mm) 

Thickness, (mm) 

Displacement components in the axial, tangential and radial directions 

External forces 


External moments 
Internal forces 

Internal moments 

Shear forces 

Moment of Inertia, (mm4) 

Extensional, coupling, bending and thickness shear stiffness matrices 

Shear correction factor 
Fig. 3 The dependence of the OF (Objective Function) on fibre angle for five different laminate configurations
Table. 5 The best configurations and (Objective Function) _{MAX} of the five different candidates for weighting ratios
Laminate 

(OF)_{MAX} 

Configuration 


C1 
39.43 
39.43 
39.43 
39.43 
39.43 

1.97 
1.83 
1.68 
1.54 
1.40 
C2 
40.44 
40.44 
39.43 
39.43 
39.43 

1.69 
1.59 
1.49 
1.40 
1.30 
C3 
47.52 
47.52 
47.52 
47.52 
47.52 

1.71 
1.61 
1.51 
1.41 
1.31 
C4 
39.43 
39.43 
38.42 
38.42 
38.42 

1.70 
1.60 
1.50 
1.40 
1.31 
C5 
38.42 
38.42 
38.42 
38.42 
37.41 

1.61 
1.53 
1.44 
1.35 
1.26 
Fig. 4 The dependence of the OF (Objective Function) on fibre angle for different weighting ratios
Table. 6 The best configurations and (Objective Function) _{MAX} of the five different candidates for L/R ratio
Laminate 

(OF)_{MAX} 

Configuration 


C1 
32.06 
34.25 
37.04 
40.08 

1.86 
1.84 
1.81 
1.77 
C2 
59.02 
46.2 
45.81 
40.04 

1.98 
1.86 
1.74 
1.63 
C3 
59.51 
48.14 
47.71 
47.54 

1.99 
1.89 
1.78 
1.64 
C4 
59.78 
47.31 
48.62 
45.05 

1.94 
1.83 
1.73 
1.61 
C5 
44.23 
44.54 
43.89 
40.14 

1.83 
1.73 
1.62 
1.54 
Fig. 5 Effect of shell length on the optimal results for different five laminate configurations (H/R = 0.2, α = 0.5).
Table. 7 The best configurations and (Objective Function) _{MAX} of the five different candidates for H/R ratio
Laminate 

(OF)_{MAX} 

Confiruration 



C1 
39.52 
38.17 
36.24 
32.26 

1.48 
1.39 
1.34 
1.30 
C2 
40 
41.48 
40.04 
38.2 

1.37 
1.30 
1.26 
1.23 
C3 
47.54 
44.48 
45.14 
41.58 

1.39 
1.32 
1.27 
1.25 
C4 
40.09 
42.29 
41.48 
41.28 

1.37 
1.31 
1.26 
1.23 
C5 
39.43 
35.61 
35.23 
34.16 

1.32 
1.26 
1.22 
1.20 
Transformed stiffness matrix 

Number of axial half waves 

Number of circumferential waves 

Stiffness Matrix 

Mass Matrix 

Axial critical buckling load, (KN/m) 

Fundamental frequency, (Hz) 


Slopes in planes xz and z 
Density for each layer, (kgm^{3}) 

Midsurface engineering strains 

Transverse shear strains 


Curvature and twist of the shell 
Natural frequency, (Hz) 
Fig. 6 Effect of shell thickness on the optimal results for different five laminate configurations (L/R = 1, α = 0.5).
Appendix
References
[1] Liu, T., Zhang, W., Mao, J.J. and Zheng, Y., 2019. Nonlinear breathing vibrations of eccentric rotating composite laminated circular cylindrical shell subjected to temperature, rotating speed and external excitations. Mechanical Systems and Signal Processing, 127, pp.463498.
[2] Pitton, S.F., Ricci, S. and Bisagni, C., 2019. Buckling optimization of variable stiffness cylindrical shells through artificial intelligence techniques. Composite Structures, 230, p.111513.
[3] Zhang, W., Liu, T., Xi, A. and Wang, Y.N., 2018. Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes. Journal of Sound and Vibration, 423, pp.6599.
[4] Matsunaga, H., 2007. Vibration and buckling of crossply laminated composite circular cylindrical shells according to a global higherorder theory. International journal of mechanical sciences, 49(9), pp.10601075.
[5] Ungbhakorn, V. and Singhatanadgid, P., 2003. Similitude and physical modeling for buckling and vibration of symmetric crossply laminated circular cylindrical shells. Journal of composite materials, 37(19), pp.16971712.
[6] Lam, K.Y. and Loy, C.T., 1998. Influence of boundary conditions for a thin laminated rotating cylindrical shell. Composite structures, 41(34), pp.215228.
[7] Lee, Y.S. and Lee, K.D., 1997. On the dynamic response of laminated circular cylindrical shells under impulse loads. Computers & structures, 63(1), pp.149157.
[8] Walker, M. and Smith, R.E., 2003. A technique for the multiobjective optimisation of laminated composite structures using genetic algorithms and finite element analysis. Composite structures, 62(1), pp.123128.
[9] Kumar, A., Chakrabarti, A. and Bhargava, P., 2013. Vibration of laminated composite cylindrical shells with cutouts using higher order theory. CONTRIBUTORY PAPERS, p.195.
[10]Sepiani, H.A., Rastgoo, A., Ebrahimi, F. and Arani, A.G., 2010. Vibration and buckling analysis of twolayered functionally graded cylindrical shell, considering the effects of transverse shear and rotary inertia. Materials & Design, 31(3), pp.10631069.
[11]Wagner, H.N.R., Hühne, C., Rohwer, K., Niemann, S. and Wiedemann, M., 2017. Stimulating the realistic worst case buckling scenario of axially compressed unstiffened cylindrical composite shells. Composite Structures, 160, pp.10951104.
[12]Mikhasev, G., Seeger, F. and Gabbert, U., 2001. Local Buckling of Composite Laminated Cylindrical Shells with Oblique Edges under External Pressure. Technische Mechanik. Scientific Journal for Fundamentals and Applications of Engineering Mechanics, 21(1), pp.112.
[13]Geier, B., MeyerPiening, H.R. and Zimmermann, R., 2002. On the influence of laminate stacking on buckling of composite cylindrical shells subjected to axial compression. Composite structures, 55(4), pp.467474.
[14]Labans, E., Abramovich, H. and Bisagni, C., 2019. An experimental vibrationbuckling investigation on classical and variable angle tow composite shells under axial compression. Journal of Sound and Vibration, 449, pp.315329.
[15]Ng, T.Y. and Lam, K.Y., 1999. Vibration and critical speed of a rotating cylindrical shell subjected to axial loading. Applied Acoustics, 56(4), pp.273282.
[16]Liu, Y. and Chu, F., 2012. Nonlinear vibrations of rotating thin circular cylindrical shell. Nonlinear Dynamics, 67(2), pp.14671479.
[17]Kassegne, S.K. and Chun, K.S., 2015. Buckling characteristic of multilaminated composite elliptical cylindrical shells. International Journal of Advanced Structural Engineering (IJASE), 7(1), pp.110.
[18]Walker, M., Reiss, T. and Adali, S., 1997. Multiobjective design of laminated cylindrical shells for maximum torsional and axial buckling loads. Computers & structures, 62(2), pp.237242.
[19]Reddy, J.N., 2004. Mechanics of laminated composite plates and shells: theory and analysis. CRC press.