Document Type : Research Article
Authors
Mechanical Engineering Department, Madan Mohan Malviya University of Technology, Gorakhpur, India
Abstract
Keywords
Main Subjects
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
KEYWORDS 

ABSTRACT 
Skew plate; HSDT; Sandwich; Meshfree; Vibration; Buckling. 

This paper presents the free vibration and buckling responses of a skew sandwich plate using higherorder shear deformation theory (HSDT). The governing differential equations (GDEs) for the skew sandwich plate are obtained using Hamilton's principle, which states that the actual motion of a system minimizes the total potential energy of the system. The GDEs obtained are discretized using radial basis function (RBF), which is a meshfree based numerical method. The vibration and buckling results for skew sandwich plates using meshfree methods and the effect of node distribution are not available in the open literature to the best of the author's knowledge. Numerous results are presented showing the nondimensional frequency and buckling parameters of the skew sandwich plates for different values of the plate geometry, material properties, and boundary conditions. These results provide insights into the vibration and buckling behavior of skew sandwich plates and can be used to optimize the design and performance of these plates for various applications, such as aerospace structures, marine structures, and civil engineering structures. Convergence studies of present results are checked, and the results obtained are also validated with the results available in the open literature. The effect of spantothickness ratio, coretoface thickness ratio, aspect ratio, boundary conditions, boundary node distribution, and skew angle is examined. The results presented in this paper can be useful for engineers and researchers working in the field of structural mechanics and can contribute to the development of safer and more efficient structures. 
Because of their advantageous features like high strength and stiffnesstoweight ratios, thermal properties, and a variety of other multiphysical 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 semianalytical approach to the buckling response of sandwich plates. Karakoti et al. [2] investigated skewedge 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 functionbased meshfree 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 angleply 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 angleply 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 plateletreinforced composite skew plates resting on point supports. Investigating Responses due to the nonlinear moving load of FGGPLRC 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 doublycurved 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 skewlaminated 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.
Figure 1 shows the geometry of a skew plate with a skew angle ( ) where thickness ‘h’ along the zaxis has been considered.
Figure.1. Geometry of skew plate
The displacement variables are expressed as Srivastava and Singh [31], [32]:

(1) 
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]:
(2) 
where, L is Lagrangian and defined as
(3) 
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]:
(4) 

(5) 
The potential energy due to inplane mechanical loading can be expressed as [36],[37];
(6) 
In Equation (6) and are the applied inplane 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:
(7) 

(8) 

(9) 

(10) 

(11) 
where,
(12) 

(13) 
The simply supported boundary conditions are taken as:
(14) 
where,
(15) 
A skew domain with NB boundary nodes and NI interior nodes is obtained and shown in
Figure 2.
CASE1 
CASE2 
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]:
(16) 
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.
(17) 
The GDEs along with boundary conditions are discretized and expressed in compact matrix form as:
(18) 
For Buckling analysis =0 and =1
For Vibration analysis =1 and =0
here, [K] is the stiffness matrix obtained using equations [711] for interior nodes and using equation [14] for boundary nodes. [M] is the mass matrix obtained using equation [714]. The final expressions for vibration analysis can be expressed as [40]
(19) 
The eigenvectors (V) and eigenvalues (D) are calculated as:
(20) 
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:
(21) 
Finally, the buckling load is calculated as
(λ) = (D).
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].
E22/E11=0.543, 
E1=1, 
=0.3, 
G12/E11=0.2629, 
G13/E11=0.1599, 

G23/E11=0.2668, 
=1. 

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. CASE1 represents the position when boundary nodes are at the corner while CASE2 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]
No. of Nodes 
CASE=1 
CASE2 

=0 
=45 
=0 
=45 

5x5 
16.2389 
11.5657 
18.0937 
10.6235 
7x7 
11.7651 
20.0860 
5.3638 
18.1613 
9x9 
10.3759 
15.3905 
10.3735 
16.1812 
11x11 
10.0651 
14.4978 
10.0258 
15.2517 
13x13 
9.9713 
14.0399 
9.9311 
14.6074 
15x15 
9.9339 
13.7013 
9.8979 
14.1129 
Srinivas and Rao [43] 
9.8104 
 
9.8104 
 
Table 2. Buckling parameter for different skew angles ( ) [Rf=10]
No. of Nodes 
CASE=1 
CASE2 

=0 
=45 
=0 
=45 

5x5 
1.5920 
1.3842 
3.1811 
3.4106 
7x7 
1.6854 
5.0083 
4.3924 
7.8329 
9x9 
2.5082 
6.7541 
2.7162 
8.4169 
11x11 
2.5336 
7.4413 
2.5470 
8.4519 
13x13 
2.5355 
7.3135 
2.4984 
8.0468 
15x15 
2.5723 
7.0777 
2.4791 
7.7763 
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 spantothickness 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
References 
Rf 

1 
2 
5 
10 
15 

Pandit et al.[42] 
4.7283 
5.6871 
7.6933 
9.7870 
11.1816 
Srinivas and Rao [43] 
4.7419 
5.7041 
7.7148 
9.8104 
11.2034 
Present 
4.7544 
5.7134 
7.7232 
9.8979 
11.2123 
Figure5. Effect of spantothickness ratio on frequency parameter for different Rf [=30]
Figure6. Effect of spantothickness 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]
Coretoface thickness ratio 

0 
15 
30 
45 
60 

2 
3.5614 
3.6574 
4.1697 
5.5115 
9.2500 
4 
3.2055 
3.2555 
3.6937 
4.8455 
8.1457 
6 
2.9469 
2.9916 
3.3934 
4.4501 
7.4817 
8 
2.7547 
2.8052 
3.1844 
4.1827 
7.0308 
10 
2.6046 
2.6615 
3.0253 
3.9822 
6.6909 
12 
2.4834 
2.5452 
2.8974 
3.8224 
6.4186 
Table 5. Buckling parameter for different skew angles ()[ Rf =10, b/h==20]
Coretoface thickness ratio 

0 
15 
30 
45 
60 

2 
3.7971 
4.0713 
5.3611 
8.4687 
15.1909 
4 
3.0755 
3.1932 
4.1918 
6.4315 
11.7088 
6 
2.5993 
2.6951 
3.5371 
5.4216 
9.8650 
8 
2.2714 
2.3775 
3.1181 
4.8068 
8.7795 
10 
2.0308 
2.1472 
2.8174 
4.3786 
7.9987 
12 
1.8462 
1.9685 
2.5867 
4.0555 
7.3618 
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 inplane 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]
Modes 
Skew angle 

0 
30 
60 
75 

1 
1.9115 
2.0409 
3.7674 
12.2436 
2 
2.7678 
3.0178 
5.4304 
14.3424 
3 
4.1911 
4.7656 
7.8702 
17.0028 
4 
6.1185 
5.5724 
9.4601 
19.8703 
5 
6.2410 
6.3289 
11.0003 
23.4864 
6 
7.0057 
6.5226 
12.0317 
25.7191 
7 
8.2896 
8.2046 
12.0317 
30.6092 
8 
8.5213 
8.7674 
13.8923 
30.6092 
Table 7. Buckling parameter for different skew angle ()[ Rf =10, b/h==20]
Modes 
Skew angle 

0 
30 
60 
75 

1 
1.0920 
1.4481 
4.8325 
14.5690 
2 
2.2901 
2.3145 
5.9576 
14.6059 
3 
2.9371 
2.8061 
6.7724 
15.8205 
4 
3.6965 
2.9614 
7.3035 
15.8245 
5 
4.9850 
3.9249 
8.1120 
17.1792 
6 
5.1586 
4.4415 
8.8965 
18.3184 
7 
5.2388 
5.4227 
9.5923 
20.1938 
8 
5.5972 
5.6897 
10.5197 
20.4136 
Table 8. Buckling parameter for different square skew angle ()[ Rf =5, b/h==10, SSSS]
CASE 
Skew angle 


0 
15 
30 
45 
75 

CASE1 
Uniaxial 
2.2580 
1.5514 
2.7289 
3.3884 
12.0979 
Biaxial 
1.1293 
0.8432 
1.0850 
1.2930 
4.6472 

Pure Shear 
3.8800 
3.8396 
2.5300 
2.2732 
11.8956 

CASE2 
Uniaxial 
2.2680 
2.3005 
3.0076 
3.3884 
12.2728 
Biaxial 
1.1341 
1.1118 
1.1405 
1.2439 
4.9167 

Pure Shear 
3.8412 
3.4927 
2.5288 
1.9828 
12.5058 
CASE1 
CASE2 



Mode1 
Mode1 
Mode4 
Mode4 
Figure 8. Different mode frequency of sandwich plate [Rf=5, hc/hf=8, =30]
CASE1 

CASE2 

Uniaxial loading 
Pure shear 
Figure 9. Different modes of the sandwich plate under inplane loading conditions [Rf=5, hc/hf=8, =30]
In this study, the buckling and free vibration characteristics of laminated sandwich plates are investigated using the HSDT.
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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