Improved high order free vibration analysis of thick double curved sandwich panels with transversely flexible cores

Improved high order free­ vibration analysis of thick double curved sandwich panels with transversely flexible cores

K. Malekzadeh Fard^I; M. Livani^I; Faramarz Ashenai Ghasemi^II

^IDepartment of Structural Analysis and Simulation, Space Research Institute, Malek Ashtar University of Technology, Tehran-Karaj Highway, Post box: 13445-768, Tehran, Iran E-mail: [e-mail], [e-mail] ^IIDepartment of Mechanical Engineering, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Postal Code 16788-15811,Tehran, Iran E-mail: [e-mail]

ABSTRACT

This paper dealt with free vibration analysis of thick double curved composite sandwich panels with simply supported or fully clamped boundary conditions based on a new improved higher order sandwich panel theory. The formulation used the first order shear deformation theory for composite face sheets and polynomial description for the displacement field in the core layer which was based on the displacement field of Frostig's second model. The fully dynamic effects of the core layer and face sheets were also considered in this study. Using the Hamilton's principle, the governing equations were derived. Moreover, effects of some important parameters like that of boundary conditions, thickness ratio of the core to panel, radii curvatures and composite lay-up sequences were investigated on free vibration response of the panel. The results were validated by those published in the literature and with the FE results obtained by ABAQUS software. It was shown that thicker panels with a thicker core provided greater resistance to resonant vibrations. Also, effect of increasing the core thickness in general was significant decreased fundamental natural frequency values.

Keywords: Free vibration, Double curved sandwich panel, Boundaryconditions, Improved higher order sandwich panel theory.

1 INTRODUCTION

Structural efficiency is an important attribute for aircraft structures. A higher order theory approach, used by Kant and Patil (1991), replaced sandwich structure with an equivalent higher order shear deformable structure, which lacked the ability to determine local buckling modes and imperfection effects on the overall behavior. Using the three-dimensional elasticity equations, Bhimaraddi (1993) studied the static response of orthotropic doubly curved shallow shells. He assumed that the ratio of the shell thickness to its middle surface radius is negligible as compared to unity. The higher order sandwich panel theory was developed by Frostig et al. (1994, 2004), who considered two types of computational models in order to express governing equations of the core layer. The second model assumed a polynomial description of the displacement fields in the core that was based on displacement fields of the first model. Their theory did not impose any restrictions on distribution of the deformation through thickness of the core. Singh (1999) studied free vibration of the open deep sandwich shells made of thin layers and a moderately thick core. Rayleigh-Ritz method was also used to obtain natural frequencies. The improved higher order sandwich plate theory (IHSAPT), applying the first-order shear deformation theory for the face sheets, was introduced by Malekzadeh et al. (2005, 2006).

NOTATIONS

GREEK LETTERS

Zenkour (2005a, b) presented a comprehensive bending, buckling and free vibration analysis of simply supported functionally graded (FG) ceramic-metal sandwich plates. The sandwich plate face sheets were assumed to be isotropic material. Two-constituent material distribution through thickness was assumed to vary according to a power law distribution. Garg et al. (2006) investigated free vibration analysis of simply supported composite and sandwich doubly curved shells. Their formulation included Sander's theory based on equivalent single layer approach. Free vibration of FG material sandwich rectangular plates with simply supported and clamped edges was studied by Li et al. (2008). The governing equations based on the three-dimensional linear theory of elasticity were derived and also they considered two common types of FG sandwich plates, namely sandwich plate with FG face sheets and homogeneous core and sandwich plate with homogeneous face sheets and FG core.

Experimental and analytical investigations of bending and free vibration response of layered FG beams were carried out by Kapuria et al. (2008), who demonstrated capability of the zigzag theory in modeling mechanics of such beams. Rahmani et al. (2009) studied free vibration analysis of open single curved composite sandwich panel with a flexible core using a higher order sandwich panel theory. Their formulation used classical shell theory for the face sheets and an elasticity theory for the core layer. Cetkovic and Vuksanovic (2009) investigated global and local responses of laminated and sandwich structures using Reddy's theory, finite element solution and based on equivalent single layer approach. Biglari and Jafari (2010a) presented a simple three layer theory in order to study vibration and static analysis of open single curved sandwich structures. In their model, they used Donell's theory for the face sheets and considered inconsistent linear stress variation in the core layer. They (2010b) also studied the free vibrations of doubly curved sandwich shell with flexible core based on a refined general-purpose sandwich panel theory. In their theory, the in-plane stresses of the core were neglected.

Free vibration analysis of thick orthotropic plates was performed by Ghugal et al. (2011) using a trigonometric shear deformation theory. In their theory, the zero shear stress conditions on the top and bottom surfaces of the plates were satisfied. Rahmani et al. (2012) studied the free vibration and buckling analyses of circular cylindrical composite sandwich shells subjected to external loads based on the Love-Kirchhoff assumptions for the face sheets. In their theory, the in-plane stresses of the core and the out of stresses of the face sheets were neglected. Mochida et al. (2012) studied free vibration response of doubly curved shallow shells using approximate Galerkin method. Classical theory of elasticity and von-Karman's non-linear deformation theory were used by Rafieipour et al. (2013) to investigate free vibration analyses of laminated composite plates.

Using developed four-node quadrilateral element and the zigzag theory, Yasin and Kapuria (2013) studied the static and free vibration analysis of singly- and doubly-curved composite and sandwich shallow shells. In their theory, the transverse normal stresses were neglected. Ghavanloo and Fazelzadeh (2013) examined free vibration analysis of simply supported doubly curved shallow shells. Their formulation was based on Novozhilov's linear shallow shell theory. Using Donell's nonlinear shallow shell theory and Kirchhoff's hypothesis, Awrejcewicz et al. (2013) studied free vibration analysis of doubly curved orthotropic shallow shells. Viola et al. (2013) used a 2D higher order shear deformation theory with nine parameters in order to analyze free vibration analysis of the thick laminated doubly curved shells and panels. Their main assumptions were based on small deflections and negligible normal stress and strain. A high-order model for the analysis of circular cylindrical composite sandwich shells subjected to low-velocity impact loads was presented by Khalili et al. (2014). In their theory, the impact behavior of the cylindrical composite sandwich shells was described by a high-order sandwich shell theory. Malekzadeh et al. (2014) applied the first-order shear deformation theory to study effects of some geometrical, physical and material parameters on response of the composite plates embedded with shape memory alloy (SMA) wires.

This study investigated free vibration analysis of double curved thick composite sandwich panels using a new improved higher order double curved sandwich panel theory and the second computational model of Frostig (2004). In this work, analytical solution of the displacement field of the core was presented in terms of the polynomials with unknown coefficients according to the second computational model of Frostig (2004). Furthermore, the formulation included accurate stress-resultant equations for composite sandwich structures, in which the terms

were imported in equations and exactly integrated. These coefficients could be very important in the structural analysis of thick double curved composite sandwich structures. Simply supported and fully clamped boundary conditions were considered. In order to assure accuracy of the present formulations, convergence of the results was examined in details.

2 THEORETICAL FORMULATION

2.1 Basic Assumptions

Consider a double curved thick composite sandwich panel which is composed of two composite laminated face sheets and a flexible core layer. Thickness of the top face sheet, bottom face sheet and core is h_t, h_b and h_c, respectively. The sandwich panel is supposed to have length a, width b and total thickness h, as shown in Figure 1.

Orthogonal curvilinear coordinates are also shown in Figure 1 where t and b refer to the top and bottom face sheets of the panel, respectively. Curvature radii of the top face sheet, bottom face sheet and core in x-z and y-z planes are respectively. The assumptions used in the present analysis were based on small deformations of linearly elastic materials.

2.2 Mathematical Formulation

Mathematical formulation consists of deriving the governing field equations of motion along appropriate boundary conditions of the face sheets and core. They are derived using the Hamilton's principle (Frostig 2004).

Using the first shear deformation theory, displacements u, v and w of the face sheets along the x, y (longitudinal) and z (thickness) axes are expressed through the following relations (Reddy 2003):

where is vertical coordinate of the each face sheet , measured upward from the mid-plane of each face sheet. Kinematic equations of the face sheets are as follows:

where:

For the thick core layer, displacement fields are based on the second Frostig's model (2004) as follows:

Based on small deformations, kinematic relations of the core layer are as follows:

Assuming perfect bonding between the top and bottom face sheet-core interfaces, compatibility conditions are as shown below:

Using Equations (4) and (6) and some simplifications, compatibility conditions can be written as follows:

It can be seen in Equation (7) that the number of unknowns in the core layer is reduced to five. These unknowns

. Therefore, generally, all unknowns for a double curved composite sandwich panel are fifteen as follows:

The governing equations are derived using the Hamilton's principle which requires that:

where denote variation of kinetic energy and strain energy, respectively. Also, t is time duration between times denotes variation operator.

The first variation of kinetic energy (assuming homogeneous conditions for displacement and velocity with respect to time coordinate) can be written as follows:

where

and

denotes the second derivative in time. The first variation of internal potential energy of the sandwich panel is as follows:

where:

Using the Hamilton's principle (Equations (8)-(10)) and the kinematic relations (Equations (1)-(7)), equations of motion are obtained as follows:

In Equations (11)-(25),

are moments of inertia for the core layer, as indicated below:

Stress resultants per unit length for the core layer are defined as demonstrated below:

Due to the radii of curvature are much larger than the thickness of the face sheets, in this paper 1+z/R for the face sheets can be approximated to 1. Therefore, stress resultants per unit length for the face sheets can be defined as follows:

Where K_s is the shear correction factor.

Constitutive equations for in-plane stress resultants based on the first order shear deformation laminate theory are defined as (Reddy, 2003):

where

are extensional, bending-extensional coupling and bending stiffnesses, respectively.

3 ANALYTICAL SOLUTION

Displacement fields based on the double Fourier series for a double curved composite sandwich panel with simply supported boundary condition at the top and bottom face sheets are assumed to be in the following form

:

where

.

When all edges are clamped, functions in the above series expansions must be replaced with, respectively.

In Equation (30),

are the Fourier coefficients and m and n are half wave numbers along x and y directions, respectively. By substituting stress resultants (Equation (27)), compatibility conditions (Equation (7)) and displacement field (Equation (30)) in the governing equations (Equations (11)-(25)), applying the Galerkin method and collecting coefficients, the eigenvalue equation is obtained as follows:

where

where

and {c} is displacement vector for all the values of m and n. Also [K] and [M] are stiffness and mass matrices, respectively. The above eigenvalue equation can be solved for various eigenvalues and associated eigenvectors.

4 VALIDATION OF THE RESULTS

In order to validate the present method and demonstrate its capability, the results obtained from the present theory were compared with recent theoretical and numerical results found in the literature. Also, the results were compared with those obtained from finite element ABAQUS software.

Example 1: Free vibration analysis of a flat composite sandwich panel with SSSS B.C.

In this example, a flat sandwich panel with laminated face sheets, PVC foam core and simply supported boundary conditions (SSSS) was considered. The lay-up sequences for face sheets were [0/90/0] and the sandwich panel was symmetric about the mid-plane. Mechanical properties of the face sheets and foam core are given in Table 1.

In Table 2, the results obtained from the present theory (IHSAPT) were compared with those obtained from the first model of Frostig, the higher order equivalent single layer theory (HSDT-ESL) and FE modeling in ANSYS code.

The results were presented for the first four dimensionless natural frequencies of a square sandwich panel with and . The maximum difference between the present theory and the higher order equivalent single layer theory (FSDT-ESL) was 9.45 percent. Due to core flexibility in the current theory, the obtained natural frequencies from current theory were lower than the natural frequencies obtained from the FSDT-ESL. Also, the present results were in good agreement with those obtained from finite element ANSYS software and the first higher order Frostig's theory. In Table 3, dimensionless natural frequencies of sandwich panel are presented with the material properties given in Table 1 and lay-up sequences [45/-45/45] for the face sheets.

As can be seen in Table 3, the results of the present theory were in good agreement with those of IHSAPT (Malekzadeh et al. 2005). Considering of different displacement fields for the core layers, there was the discrepancy between the results. In the present theory, displacement field for the core was assumed to be a polynomial with unknown coefficients. But, Nayak et al. (2002) and Malekzadeh et al. (2005) used equivalent single layer Reddy's theory and 3D elasticity theory for the core, respectively.

Example 2: Free vibration analysis of an open single curved composite sandwich panel with SSSS B.C.

In this example, free vibration analysis of an open single curved composite sandwich panel with foam core was investigated. Lay-up sequences for face sheets were [0/90]. Mechanical properties of the face sheets and core are given in Table 4.

In the Table 5 dimensionless first natural frequency for thin sandwich panels with three different ratios of radius to width were presented. In this table, results of the present theory (IHSAPT) were compared with those obtained from the first Frostig's model, first order shear deformation theory, higher order equivalent single layer theory and FE modeling in ANSYS code. As can be seen in Table 5, the current results were in good agreement with FE modeling in ANSYS code. Table 5 also shows that results of different theories for the thin sandwich panel were in better agreement with each other than those for the thick sandwich panel.

Example 3: Free vibration analysis of a flat composite sandwich panel with CCCC B.C.

Now, free vibration analysis of a sandwich panel with laminated face sheets and fully clamp boundary conditions (CCCC B.C.) is studied. There has been few research on free vibration of sandwich structures with CCCC B.C.. Therefore, in order to validate the current results, the sandwich structures were modeled in ABAQUS. This example used mechanical properties, given in Table 1.

At first, convergence of the first five dimensionless natural frequencies was investigated (Table 6).

It can be seen from this table that the natural frequencies converged after about 13×13 expressions (m=n=13). Table 6 also shows that the lowest convergence rate of the natural frequencies occurred in the first mode shape.

In Table 7, the current results were compared with the presented FE model by ABAQUS code and the higher order equivalent single layer theory (HSDT-ESL) presented by Nayak et al. (2002).

As can be seen in Table 7, the results of the present method were in very good agreement with those of FE model in ABAQUS; but, there was little difference between the current results and those of the HSDT-ESL. It is because the HSDT-ESL model did not consider flexibility of the core layer.

Example 4: Free vibration analysis of a composite sandwich cylindrical panel with SSSS B.C.

In this example, free vibration analysis of a composite sandwich cylindrical panel with SSSS B.C. is investigated. Lay-up sequences for face sheets are [0/90/0]. Mechanical properties of the face sheets and core are given in Table 8.

In Table 9 the dimensionless first natural frequency of sandwich panels with two different ratios of radius to width are presented. In this table, results of the present theory (IHSAPT) are compared with the three layer theory presented by Rahmani et al. (2012) and with the three dimensional elasticity solutions given in Yasin and Kapuria (2012). As can be seen in

Table 9, the present and Rahmani et al. (2012) results show better agreement than Yasin and Kapuria (2012) ones. Because the present paper and Rahmani et al. (2012) used the high order sandwich panel theory, while Yasin and Kapuria (2012) used the zig-zag theory.

5 RESULTS

According to the above examples, all formulations of free vibration analysis were validated. Now, some examples are considered and the obtained results are presented and discussed.

Example 1: Free vibration analysis of a double curved composite sandwich panel with SSSS and CCCC B.Cs.

In this example, free vibration analysis of a double curved composite sandwich panel with SSSS and CCCC B.Cs. was investigated. Mechanical and geometrical properties of the sandwich structure are given in Table 10. Lay-up sequences of the top and bottom face sheets were [0/90/0] and the sandwich panel was symmetric about mid-plane.

In Table 11, the dimensionless natural frequencies of the double curved composite sandwich panel for the first four mode shapes with both boundary conditions are presented.

As expected, the natural frequencies of the double curved sandwich structures with CCCC B.C. were higher than those with SSSS B.C.. As shown in Table 11, the first dimensionless natural frequency for both boundary conditions occurred in mode shape (m,n)=(1,1). In Figure 2 mode shapes of the face sheets are presented at the first natural frequency for the double curved composite sandwich panel with both boundary conditions.

Example 2: Effects of geometrical parameters and types of boundary conditions on the free vibration analysis of a double curved composite sandwich panel

In this example, effects of various parameters on free vibration response of a double curved composite sandwich panel with SSSS and CCCC B.Cs. were investigated. Properties of the sandwich structure are given in Table 10. Lay-up sequences of the face sheets were [0/90/0] and the sandwich panel was symmetric about the mid-plane.

First, effects of h_c/h (core to panel thickness ratio) and R₁/R₂ (ratio of radii of curvatures) ratios on the first dimensionless natural frequency were studied. It is clear that core to panel thickness ratio and ratio of radii of curvatures had a significant effect on free vibration and dynamic analysis of the sandwich panels. In Figure 3(a)-(b), effects of h_c/h and R₁/R₂ ratios on the first natural frequency (for both boundary conditions) are shown, respectively. In these figures, length (a) and width (b) of sandwich panel were constant and did not change by thickness of the panel.

It can been seen in Figure 3(a) that, with increasing core to panel thickness ratio for both boundary conditions, the first dimensionless natural frequency decreased. With increasing h_c/h ratio, the first natural frequency decreased, too. Also, the difference between natural frequencies for SSSS and CCCC B.Cs. at h_c/h=0.1 was higher than that at h_c/h=0.9. In general, with increasing thickness of the core, natural frequencies increased because increasing in the thickness of the core increased stiffness of the sandwich panel. Figure 3(b) shows that, with increasing R₁/R₂ ratio for both boundary conditions until approximately R₁/R₂ =3, the first natural frequency decreased and converged to a constant value. This behavior occurred due to with increasing R₁/R₂ ratio, geometry of double curve panel converged in the cylindrical single curve panel. In addition, selection of SSSS B.C. for a double curved composite sandwich panel decreased the first dimensionless natural frequency of the panel, as can be seen in Figure 3(b).

Now, effects of fiber angle, i.e. lay-up sequence and type of boundary condition, on the first natural frequency are investigated. In Figure 4 (a)-(b), variations of the first natural frequency with fiber angle for a sandwich panel with SSSS and CCCC B.Cs. are presented, respectively.

As can be seen in Figure 4(a), maximum first natural frequency for SSSS B.C. occurred in fiber angle 45ºwhile Figure 4(b) shows that maximum first natural frequency for CCCC B.C. approximately occurred in fiber angle 55º. In addition, natural frequency of a double curved composite panel with CCCC B.C. for all fiber angles was more than that with SSSS B.C..

The normalized modal displacements corresponding to the first and second natural frequencies for a sandwich panel with SSSS and CCCC B.Cs. are shown in Figure 5, where amplitudes of the vibration in this figure are normalized. As is obvious in Figure 5(a), normalized displacements of the top face sheet was higher than those of the bottom face sheet for both boundary conditions. In this figure, the dot lines belong to CCCC B.C.. In the first mode shape, the top and bottom face sheets moved vertically in the same direction for SSSS and CCCC B.Cs. while Figure 5(b) shows that the top and bottom face sheets in the second mode shape with SSSS B.C. moved in the opposite direction those with CCCC B.C..

6 CONCLUSION

In this work, free vibration analysis of double curved composite sandwich panels with simply support and fully clamped boundary conditions was studied. The analysis was very general and valid for any type of core, any type of face sheets as well as the cases in which the conditions at the upper face sheet were different from those at the lower one along the same edge. Thickness of the upper face sheet might be different from that of the lower face sheet. Transverse shear and rotary inertia effects of face sheets were taken into consideration. Therefore, the upper and lower face sheets could be thick or thin, independent from each other.

The numerical study revealed that soft-core sandwich panels exhibited a complex behavior and vibration patterns of the sandwich panels were more complex than those of the homogeneous panels. The thicker panels with a thicker core provided greater resistance to resonant vibrations. The effect of types of boundary conditions, core to panel thickness ratio, ratio of radii curvature and fiber angle on dynamic response of double curved composite sandwich panels was also studied.

The results revealed that:

1. The present theory for free vibration analysis of double curved thick sandwich panels was more accurate than other theories.

2. By increasing the core to panel thickness ratio, the first natural frequency of double curved composite sandwich panels for SS and CC B.Cs. linearly decreased.

3. By increasing the radii of curvatures ratio until approximately R₁/R₂ =3, the first natural frequency of double curved composite sandwich panels for SS and CC B.Cs. decreased and after that converged to a constant value..

4. Maximum first natural frequency for SSSS B.C. occurred in fiber angle 45ºwhile maximum first natural frequency for CCCC B.C. approximately occurred in fiber angle 55º.

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Received 04.03.2014

In revised form 06.06.2014

Accepted 11.06.2014

Available online 26.09.2014

Publication Dates

• Publication in this collection
  08 Dec 2014
• Date of issue
  2014

History

• Received
  04 Mar 2014
• Accepted
  11 June 2014
• Reviewed
  06 June 2014