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 35131-19111, Semnan, Iran

1.        Introduction

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 well-defined form to the users. On the other hand, the structures are quite often are subjected to in-plane 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 in-service 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 Kármán-type 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 pre-buckling 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 pre-stretched 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 two-dimensional 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 cross-ply 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 Love-type shell theory and solved using Galerkin's method. Lee [7] defined free vibration and dynamic response for the CFRP and GFRP cross-ply laminated circular cylindrical shells under impulse loads and investigated by using the first-order 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 two-layered 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 knock-down 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 thin-walled 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 non-dimensionalized 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 thickness-to-radius ratios on the optimal designs are investigated and the results are compared.

2.      Governing Equations

A circular cylindrical shell, the schematic of the k-layer of the shell and coordinates are shown in Fig. 1. Based on first-order 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. K-layer laminated composite circular cylindrical shell and coordinate

,  and  are defined by the following relation [19]:

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]:

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

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

3.      Boundary Conditions

The boundary conditions for the cylindrical shell with fully simply support are considered as [10]:

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:

where:

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]:

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].

4.      Buckling Analysis

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]:

Determinant of the coefficients  is set to be zero; thus, the buckling loads equation is derived as [5]:

In Eq. (13),  are the constant coefficients and  is the axial critical buckling load [5].

5.      Free Vibration Analysis

To solve the free vibration problem, the function of time is assumed as follows [5]:

A method similar to the buckling analysis method is employed to be derived the following set of equations [5]:

The determinant of the coefficients is set to be zero, thus the characteristic frequency equation is derived as [5]:

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].

6.      Problem Formulation

The following non-dimensionalized quantities are introduced to the all computations [8]:

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]:

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].

7.      Verification and Numerical Result

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

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

Table 3. Ply properties of the composites used in Ref. [5]

Table 4. Comparison between the results for buckling loads of a simply supported anisotropic cross-ply cylindrical shell with the lay up    ,

7.1.        Effect of weighting ratios on the results

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.

7.2. Effect of shell lengths on the results

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.

7.3. Effect of shell thickness on the results

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.

8.      Conclusions

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 non-dimensionalized 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

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

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

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

Fig. 6 Effect of shell thickness on the optimal results for different five laminate configurations (L/R = 1, α = 0.5).

Appendix

References

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