On free vibration of laminated skew sandwich plates: A nite element analysis

The present work emphasizes the determination of the fundamental frequency of skew sandwich plates with orthotropic core and laminated facings using different design parameters. Finite elements CQUAD4 and CQUAD8 of MSC/NASTRAN are used for obtaining fundamental frequencies, which are validated against available literature results. The in uence of the skew angle, the ratio of the length-to total thickness (a/h) of the sandwich plate, and the ratio of the thickness of the core to face sheet (tc/th) on the fundamental frequency of skew sandwich plates are studied. Also, the in uence of parameters such as the number of layers in the face sheet, laminate sequence, and ber orientation angle on the fundamental frequency of laminated skew sandwich plates have been studied. It is found that the CQUAD8 element yields better results than the CQUAD4 element in the present study. The fundamental frequencies are found to increase with the increasing skew angle. The variation in fundamental frequency is negligible when the number of layers is large in the face sheet.


Introduction
Skew sandwich plates are now a day frequently used in numerous areas like aeronautical, automobile, civil engineering, and in most structural applications. In skew sandwich plates, the e ect of shear deformation is considerably more as compared to laminated composite skew plates, which was the reason behind the widespread applications of such plates. Also skew sandwich plate exhibits less weight, more sti ness, more structural e ciency, and more durability. Much research was made on sandwich plates on the free vibration behavior for more than two decades.
A linear analysis for bending and vibration of sandwich plates was employed for analytical and experimental investigations [1]. Also, re ned plate theory was proposed on sandwich plates [7]. The free vibration analysis using higher-order shear deformation theory of sandwich plates [18], laminated composite and sandwich plates [6], skew sandwich plate with laminated composite faces were presented [8]. Free vibrations and buckling of the sandwich panel with a exible core was investigated using a new improved high-order sandwich panel theory [11]. Free vibration analysis of laminated composite and sandwich plates using trigonometric shear deformation theory was performed [15]. Quasi-3D shear deformation theory was employed for thermo-mechanical bending analysis of functionally graded material (FGM) sandwich plates [35] and buckling and the post-buckling response was recorded from functionally graded carbon nanotube (FG-CNT) -magnesium (Mg) nanocomposite plate with interphase e ect [31]. The modi ed sti ness method was applied to the dynamic analysis of sandwich plates [2]. An experimental modal study was conducted on a cantilever exible plate underwater due to the hydrodynamic e ect [32]. A study dealing with the comparison of free vibration responses obtained from four theories on composite truss core sandwich plates were presented. The natural frequencies of the sandwich plate are calculated by using the classic laminated plate theory, the rst-order shear deformation theory, Reddy's third-order shear deformation theory, and a Zig-Zag theory [12]. Various shear deformation theories [13] were considered for the comparison based on the displacement elds [14].
Finite element analysis of composite sandwich plates was carried out based on Mindlin's plate theory [3]. The bending behavior [16] and free vibration response [17] using a four nodded rectangular nite element formulation based on a layer-wise theory, Static analysis [9], and the free vibration response [10] using an improved dis-crete Kirchho quadrilateral element based on third-order zigzag theory were presented. The p-Ritz method [5] on Skew sandwich plates and a numerical study were made on a sandwich plate to improve the dynamic e ects of geometric design variables and material alteration [4]. The vibration parameters of sandwich plates were predicted by a spline nite strip method [19], harmonic quadrature element method [26].
Free vibration analysis of plates and sandwich plates was discussed using C o iso-parametric nite element model [20], Two new C assumed strain nite element [21], C nite element model [22]. Fundamental exural frequencies of isotropic and laminated composite skew plates [23], skew sandwich composite plates [27,28] have been obtained using nite elements. Also, the experimental and nite element studies were carried out on free vibration of isotropic and laminated composite skew plates [24,25]. The nonlinear static, buckling, and vibration analysis of viscoelastic micro-composite beam reinforced by various distributions of boron nitride nanotube (BNNT) with initial geometrical imperfection by modi ed strain gradient theory (MSGT) using nite element method (FEM) was presented [33]. A critical review of available literature for the prediction of the behavior of laminated composites and sandwich structures under hygrothermal conditions was carried out [34].
The present research focuses on the free vibration studies on laminated sandwich skew plates with simply supported and clamped boundary conditions. The face sheet consists of a laminated composite reinforced with graphite-epoxy and a heavy core (orthotropic). The key objective is to investigate the in uence of the number of layers in the face panel, the ratio [a/h], the ratio [tc/t f ], the e ect of ber orientation, the e ect of the laminate sequence, the e ect of boundary conditions, the e ect of the skew angle on the sandwich plate's free vibration response. The paper is organized as follows: Firstly, for the free vibration analysis of the sandwich plate, convergence of the results gathered by both CQUAD4 and CQUAD8 elements is evaluated. The validation of the result by the present approach is compared to those available in the literature using converged element density. By implementing the mechanical properties as implemented in [20] for both the orthotropic face sheet (GFRPC) and the orthotropic core (Heavy), computational analysis is nally carried out to describe the e ect of various geometric parameters, boundary conditions, and skew angle.

Finite element formulation
For thick plates the following equation (1) holds good: (1) Using ve components u, v, w, θx, θy, the displacement of the plate are fully described where u, v, and w are displacements along Cartesian x, y and z-directions also θx (w, x, and φx) and θy(w, y, and φy) are total (bending and shear) rotations about y-and x-axes, respectively, whereas, u , v , and w are the mid-plane translations along x, y and z directions, respectively. Nodal displacements are used to describe the displacement δ j at any point within the element by the following equation.
Where N j are isoparametric shape functions [30]. The stiness matrix of the plate element assumes the form. Where, {ϵ} being the strain vector, and {δ} the nodal displacement vector.
[B] is the strain-displacement matrix, and [D] is the sti ness matrix given below. Where, Here, Q ij is the element of o -axis stress-strain relations. Q ij k relates stresses and strains in a k th layer by the relation Here σ , σ , and σ denote σx, σy and τxy respectively and ϵ , ϵ , ϵ denote ϵx, ϵy, γxy respectively. Whereas σ l k =Q lm k ϵm k where l,m= 4,5 and κ is the shear correction factor taken as 0.8334. The mass matrix of the plate element is given by [ρ] being the density matrix functions. The integration in every case is carried out over the area of the plate element. Generally, a 3-point Gauss quadrature is adopted to compute the bending sti ness of the elements, whereas 2-point integration is applied to calculate the shear sti ness, mass matrix, and element force vector. The governing equations, without damping being accounted for free vibration is

Convergence and validation . Convergence
The geometrical representation of the sandwich plate is as shown in Figure 1. The skewed sandwich plate with global and local coordinate systems is as shown in Figure 2. The displacement boundary conditions cannot be applied directly, due to the inclination of displacements to the skew edges. To overcome this, a local coordinate system (x , y ) normal and tangential to the skew edges is preferred. A total number of elements in the plate model is optimized to get exact and consistent values. Consequently, it is essential to analyze the convergence of the values. The convergence was made on simply supported and clamped skew sandwich plates using CQUAD4 (four-node plate element) and CQUAD8 (eight-node isoparametric curved shell element) elements of MSC/NASTRAN. Skew sandwich plates with varying aspect ratio, length to thickness ratio, and the ratio of a thickness of core to facing for skew angles 0 • , 15 • , 30 • , and 45 • using both the elements are evaluated. The converged detailed results are conveyed in Table 1. The material properties used are, for face sheets

. Validation
Validation of the results from the elements used in the present study is made by matching up the values for the natural frequency found in the present study to the available literature values. The comparison is shown in Table 2 and 3, for clamped and simply supported boundary conditions respectively of a skew sandwich plate in Hz. The material constants employed are similar to those used in [19]. The values found in the study are in good harmony with the literature results. Also for simply supported sandwich skew plates, the material constants are referred to as in [8].
Non-dimensional frequency parameter (K f ) of simply supported ve-layered symmetric laminated composite skew sandwich plates with orthotropic core was determined by using the formula idation results for simply supported boundary conditions are shown in Table 6. The material properties employed for the study were as mentioned in [22]. From Table 1 to 4 it is observed that the CQUAD8 element gives accurate and converged results as then the CQUAD4 element. From now CQUAD8 is adopted in further work.

Results and discussion
The present numerical study considers a variety of parameters, such as aspect ratio, a ratio of length to thickness of sandwich plates, ration thickness of face sheet to thickness of the core, skew angle, and boundary conditions of the sandwich skew plates.

. Study on the e ect of number of layers
The e ect of the number of layers on the fundamental frequency is assessed and results are graphically presented in Figure 3 and 4 in non-dimensional form K f as well as the mode shapes in Table 5. The aspect ratio kept constant to 1, skew angle, and the number of layers in the face sheet is varied for all sides simply supported and clamped edge condition. The following observations were made from the results, • An initial increase in the layers increases the sti ness of the plate, later the added layers do not contribute to the sandwich plate's vibration response. Adding the number of layers in the face sheet allows the sandwich skew plate to accumulate in its weight. The largest impact is the core thickness that takes the majority of  shear stress. The K f initially increases up to 4 layers, as the number of layers of the face sheet increased and after this, the shift is constant or insigni cant. • The clamped condition has no degree of freedom free to rotate or oscillate in the plate element. This makes the plate sti er compared to the simply supported one. Because of this, the value of K f is higher for all sides' clamped condition than all sides simply supported. • With the skew angle of the sandwich skew plates is increased, the value of K f is found increasing in all cases of the parametric study.

. E ect of ratio of t c /t f
Aspect ratio and a/h ratio kept constant as 1 and 10 respectively, only the ratio tc/t f is varied. The results are obtained for antisymmetric cross-ply, 5 layers simply supported and clamped boundary conditions for di erent skew angles. The K f values are graphically presented in Figure 5 and 6. From the graph, the following observations are drawn. Core Thickness, which takes the most of shear stress, is the key in uencer for the vibration response of the sandwich skew plate. With the ratio of tc/t f is increased, the  core thickness will also increase relative to the face sheet thickness. The higher the core thickness, the sandwich skew will become less sti , and the K f value for a given skew angle will be greatly decreased.

. E ect of ratio of a/h
Aspect ratio and tc/t f ratio kept constant as 1 and 10 respectively, only the ratio a/h is varied. The results are obtained for antisymmetric cross-ply, 5 layers simply supported and clamped boundary conditions for di erent skew angles. The K f values are graphically presented in Figure 7 and 8. From the graph, the following observations are drawn. A potential in uencer is the core thickness compared to face sheet thickness. It is inappropriate to add more layers to the face sheet rather than vary the core thickness. The length of the sandwich plate kept constant only variable is the total thickness of the sandwich skew plate. When the ratio of a/h is increased, the K f value decreases considerably for a given skew angle.

. E ect of laminate sequence
A symmetric angle ply laminated skew sandwich plate is considered. Aspect ratio 1, a/h=10, and tc/t f =10 kept constant, only skew angle and ber angle are varied for the study.

. Symmetric three layer angle ply skew sandwich plates
The results are obtained for the symmetric 3 layers simply supported and clamped boundary conditions. The K f values are graphically presented in Figure 9 and 10 also the mode shapes in Table 6. From the graph, the following observations are drawn. For the 0 • skew angle, the K f increases as an increase in the value of ber angle. As the ber angle is increased for skew angle15 • , 30 • , and 45 • , the value of K f initially decreases and then increases.

. Symmetric ve layer angle ply skew sandwich plates
The results are obtained for the symmetric 5 layers simply supported and clamped boundary conditions. The K f values are graphically presented in Figure 11 and 12, and mode shapes in Table 7. From the graph, the following observations are drawn. As the ber orientation angle increases, the K f value increases and reaches a maximum value or symmetric about 52.5 • then decreases for simply supported and 50 • for clamped boundary conditions.

Conclusion
Sandwich skew plates exhibit excellent high sti ness to weight ratio as compared to other laminated structures.
The material properties at the interface of the face sheet and core components create complexities to accurately evaluate the mechanics of the sandwich skew plates by the analytical method. The nite element method (FEM) provides the exibility in designing the structure and recording the response of the skew sandwich plate e ortlessly. The present analysis uses CQUAD4 and CQAUD8 elements to evaluate the vibration response of the skew sandwich plate. A convergence study is performed by imposing simply supported and clamped boundary edge conditions. Results obtained by the present method are validated with those available in the literature. Aspect ratio, skew angle, the thickness of face sheet and core, number of layers in the face sheet, edge conditions, etc are considered in evaluating vibration response of skew sandwich plates. Concluding remarks are made after performing numerical analysis as: • Both CQUAD4 and CQUAD8 elements have good agreement with the available literature results. But CQUAD8 element yields more converged, accurate results since the element has 8 nodes while CQUAD4 has 4 nodes. • The number of layers in the face sheet, when increased, the K f initially increases up to 4 layers due to the initial increase in the sti ness of the face sheet, after that the change is constant or negligible. • When increasing the core thickness (increasing tc/t f and a/h ratios) an increase in total plate thickness, the sti ness of the plate decreases, K f value decreases considerably for a given skew angle. Higher core thickness does not contribute to sti ness and vibration response of the skew sandwich plate. • While the skew angle is increased, the side length shortens. This leads to an increase in sti ness of the skew sandwich plate. Because of which the increased value of K f is observed for a given ratio of tc/t f and a/h. • Considerable in uence is observed while studying ber orientation on the sandwich skew plate for vibration response. For 3 layers and 5 layers symmetrically laminated composite sandwich plate, the value of K f initially decreases then increases. A similar variation can be seen [5] for both simply supported and clamped boundary conditions. • The value of K f is higher for all side clamped condition than all sides simply supported. In the clamped edge condition, the plate becomes sti er than simply supported edge condition.