Vibration and Buckling Analysis of Skew Sandwich Plate using Radial Basis Collocation Method

Vibration and Buckling Analysis of Skew Sandwich Plate using Radial Basis Collocation Method

J. Singh , R. B. Prasad^*

Mechanical Engineering Department, Madan Mohan Malviya University of Technology, Gorakhpur, India

1. Introduction

Because of their advantageous features like high strength and stiffness-to-weight ratios, thermal properties, and a variety of other multi-physical aspects, laminated composites and sandwich configurations are used in many lightweight structures in aviation, structural, maritime, and civil engineering applications. By giving designers of mechanical parts the flexibility to customize the distribution of materials of various qualities in accordance with the loading routes, these designs also enable custom optimization. Laminated composites are constructed by layering piles of composites with specific fiber orientations in each layer to create the structures. Shi et al. [1] applied a semi-analytical approach to the buckling response of sandwich plates. Karakoti et al. [2] investigated skew-edge sandwich plates via the finite element method. Katariya et al. [3] studied statics and natural frequency of skew sandwich composite plates using the HSDT model.

Radial basis function-based mesh-free methods have been used for the analysis of plates by Singh et al.[4] for free vibration of laminated composite plates, and Solanki et al. [5] for the flexure behavior of laminated plates. Singh et al. [6] for buckling of square sandwich plates, Shukla and Singh [7] for flexure analysis of angle-ply rectangular plates, Solanki et al. [8] for nonlinear vibrations of square laminated composite and sandwich plates, Singh et al. [9] for flexure of laminated composite and sandwich plates. RBFs have been also used to analyze the buckling behavior of functionally graded materials rectangular plates by Kumar et al. [10, 11]. Civalek [12] obtained the frequencies and buckling loads of skew laminated composite plates using the discrete singular convolution method. Ashour [13] studied the free vibration response of symmetric laminated angle-ply thin skew plates using the finite strip transition matrix method. Malekzadeh and Alibeygi Beni [14] investigated the frequency response of skew FGM plates via DQM.

The buckling response of a CFRP plate has been studied by Yidris et al. [15] Civalek and  Jalaei [16] analyzed the shear buckling of an FG skew plate.  Vibrational analysis of FG skew sandwich plates of geometric distortions has been carried out by Khanke and Tande [17]. Kiani and Żur.[18] studied the effect of vibrations on graphene platelet-reinforced composite skew plates resting on point supports. Investigating Responses due to the nonlinear moving load of FG-GPLRC skew plates has been carried out by Noroozi and Malekzadeh [19].

Sayyad and Ghugal [20] used sinusoidal beam theory for analysis of functionally graded sandwich curved beams. Flexure response of laminated plates [21] stress analysis of orthotropic laminated doubly-curved shells on rectangular planform under concentrated Force have been analysed by Sayyad and Ghugal [22].

Bending, buckling, vibration 0f rectangular laminated composite and sandwich plates has been studied by Sayyad and Ghugal [23–27].

Ghugal and Sayyad [28] has studied stress analysis of thick laminated plates using trigonometric shear deformation theory. Laminated composite, sandwich and FGM beams have been analyzed by Shinde and Sayyad [29].

Thermoelastic analysis of laminated plates has been done by Sayyad et al. [30] using four variable plate theory.

The literature review indicates that there has been limited research on the buckling and vibration analysis of skew-laminated sandwich plates. Therefore, this paper aims to address these gaps by using theoretical approaches.

The paper derives the GDEs for buckling and vibration characteristics and reveals the influence of skew angles. The GDEs are derived via Hamilton’s principle and discretized via the RBF approach.

Overall, this study provides valuable insights into the behavior of skew laminated plates, which can be applied in various engineering applications, such as aviation and civil engineering.

2. Mathematical Formulation

Figure 1 shows the geometry of a skew plate with a skew angle ( ) where thickness ‘h’ along the z-axis has been considered.

Figure.1.  Geometry of skew plate

The displacement variables are expressed as Srivastava and Singh [31], [32]:

For the present analysis, the transverse shear stress has been considered as proposed by Touratier [33].

The GDEs of the skew plate along with boundary conditions are derived using Hamilton’s principle, which is expressed as [34]:

where, L is Lagrangian and defined as

where,

KE = Kinetic energy,  UE = Strain energy,

VE = Potential energy due to external loads

The expressions for kinetic energy and strain energy of the plate can be written as[35]:

The potential energy due to in-plane mechanical loading can be expressed as [36],[37];

In Equation (6)  and  are the applied in-plane compressive loadings in x and y directions and shear loading, respectively. The governing differential equations of the plate are obtained by collecting the coefficients of , ,  ,_x and _y can be expressed as:

where,

The simply supported boundary conditions are taken as:

where,

3. Solution Methodology

A skew domain with NB boundary nodes and NI interior nodes is obtained and shown in
Figure 2.

Fig. 2. Geometry of skew plate with nodes

The field variables  and  of Eq (1) are assumed in terms of radial basis function as [38], [39]:

where,  are unknown coefficients,  is RBF, is the radial distance between the nodes, and c is the shape parameter. For the present analysis, RBF taken is a thin plate spline  with c=3.

The GDEs along with boundary conditions are discretized and expressed in compact matrix form as:

For Buckling analysis =0 and =1

For Vibration analysis =1 and =0

here, [K] is the stiffness matrix obtained using equations [7-11] for interior nodes and using equation [14] for boundary nodes. [M] is the mass matrix obtained using equation [7-14]. The final expressions for vibration analysis can be expressed as [40]

The eigenvectors (V) and eigenvalues (D) are calculated as:

Frequency (ω) =

For buckling analysis, the final equation can be expressed as [41]:

where [K]G is the geometric matric obtained from equation [9]. [K]L denotes the stiffness matrix at domain nodes and [K]B for boundary nodes.

The eigenvectors (V) and eigenvalues (D) are calculated as:

Finally, the buckling load is calculated as
(λ) = (D).

4. Result and Discussion

Present section deals with numerical experimentations and validation of obtained results. A square sandwich plate (a/h=10) consisting of two orthotropic face layers of 0.1 times thickness and a core layer of 0.8 times thickness is considered. The material properties taken for the core layer are as Pandit et al. [42].

The elastic modulus of face sheets has been varied with a factor Rf.

The frequency and buckling responses are normalized as  and  respectively.

Two cases of node generation have been considered here to see the effect of change in the position of boundary nodes while applying the boundary conditions. CASE-1 represents the position when boundary nodes are at the corner while CASE-2 represents when boundary nodes are not at the corner.

The convergence and validation study for the fundamental frequency parameter and buckling parameter of a simply supported square sandwich plate are obtained and presented in Table 1 and Table 2 and the same has been depicted in Fig. 3 and Fig. 4 respectively. Results obtained are compared with results due by Srinivas and Rao [43] for free vibration analysis. It can be seen that the present results converged for both cases within 1% and became closer to the results of Srinivas and Rao [43] for vibration analysis at 15X 15 nodes. From Fig. 3 and Fig. 4, it is observed that convergence is better when a plate is rectangular as compared to that of for skew plate with a skew angle of 45.

Table 1. Frequency parameter  for different skew angles ()[Rf=10]

Table 2. Buckling parameter  for different skew angles ( ) [Rf=10]

Figure 3. Convergence of frequency parameter

Figure 4. Convergence of Buckling parameter 

Table 3 presents the comparison of present results with Pandit et al.[42] and Srinivas and Rao [43].

Present results are in good agreement with published results. Further, the effect of the span-to-thickness ratio with the variation of Rf is studied.

The results obtained for the fundamental frequency parameter and buckling parameter are obtained and shown in Fig. 5 and Fig. 6 respectively. It is observed that with increase in span to thickness ratio fundamental frequency parameter decreases while fundamental buckling parameter increases, and its effect is negligible after b/h=40. It can be also seen that with increase in Rf fundamental frequency parameter increases while fundamental buckling parameter decreases.

Table 3. Frequency parameter of a square sandwich plate of orthotropic layers

Figure5. Effect of span-to-thickness ratio on frequency parameter  for different Rf [=30]

Figure6. Effect of span-to-thickness ratio on buckling parameter  for different Rf  [=30]

The effect of core to face thickness ratio with skew angles are also considered. Results obtained are presented in Table 4 and Table 5 for vibration and buckling analysis respectively. With increase in skew angle both the parameters increases while with increase in core to face thickness ratio it decreases.

Table 4. Frequency parameter  for different skew angles ()[ Rf =10, b/h==20] 

Table 5. Buckling parameter  for different skew angles ()[ Rf =10, b/h==20]

Further, the effect of boundary conditions with aspect ratio are also considered presented in Fig. 7. It is observed that with increase in aspect ratio, the buckling parameter decreases for both the cases.

The effect of skew angle for different modes are presented in Table 6 and Table 7 for vibration and buckling analysis respectively.

Table 8 shows the results for buckling parameter with different skew angles and in-plane loadings of all edges simply supported.

Fig. 8 and Fig. 9 shows the contours of different mode shapes. Present methodology shows the capability to obtain the contours for different mode shapes.

Figure7.  Effect of boundary conditions of buckling parameter

Table 6. Frequency parameter  for different skew angle ()[ Rf =10, b/h==20]

Table 7. Buckling parameter  for different skew angle ()[ Rf =10, b/h==20]

Table 8. Buckling parameter  for different square skew angle ()[ Rf =5, b/h==10, SSSS]

Figure 8. Different mode frequency of sandwich plate [Rf=5, hc/hf=8, =30]

Figure 9. Different modes  of the sandwich plate under in-plane loading conditions [Rf=5, hc/hf=8, =30]

5. Conclusions

In this study, the buckling and free vibration characteristics of laminated sandwich plates are investigated using the HSDT.

• The present solution methodology is easy to implement for acceptable results.
• With an increase of span to thickness ratio, the frequency parameter decreases and after a/h =40, its effect is negligible.
• With an increase in core-to-face thickness ratio, frequency and buckling parameter decreases.
• With an increase in skew angle, the frequency parameter as well as the buckling parameter increases.
• There is a considerable effect on results with node distribution on boundaries.
• New results for vibration and buckling analysis of skew sandwich plates have been produced, which can be used for validation by other research groups.

The results of this study can be useful for optimizing the design and performance of laminated sandwich plates for various applications, such as aerospace structures, marine structures, and civil engineering structures.

By understanding the buckling and free vibration behavior of these plates, engineers can design structures that can withstand expected loads and vibrations, thereby increasing the safety and reliability of the structures.

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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