Issue Int. J. Simul. Multidisci. Des. Optim. Volume 12, 2021 Computation Challenges for engineering problems 18 8 https://doi.org/10.1051/smdo/2021017 26 August 2021

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1 Introduction

Sayyad et al. [1012] has done the detailed analytical study of buckling and vibration on anti-symmetric laminated plates by using novel trigonometric theory which includes shear deformation effect, to understand the effects of slender ratio, modular ratio and fibre angle of the composite plate. Ferreira et al. [13] numerically analysed the vibration and instability of composite plates under irregular geometries by considering the radial basis function on FSDT. Gunasekaran et al. [14] investigated analytically the variations in vibration frequencies of FG-graphene reinforced plates under arbitrary edge loads. Arunkumar et al. [15] numerically studied the bending and dynamic characteristics of the foam-filled sandwich panel by using FEM. Kumar et al. [16] have benchmarked a numerical solution for the vibration response of laminated plates, with the help of 2D FE model incorporated with a higher-order zigzag theory which is considered interlaminar shear stress. Alam and Asnani [17] used the variational principle to predict the dynamic and damping factor of fibre-reinforced plastic plates subjected to simply supported boundary conditions. Chandra et al. [18] given a brief literature review about a damping study in fiber-reinforced composite materials by using micromechanical, micromechanical, and viscoelastic methods. Sudhakar et al. [19] done a detailed investigation of rotating tapered laminates on frequencies and its damping factors, with the help of FEM based on the FSDT. Arunkumar et al. [20] done a detailed numerically investigated the acoustic radiation behaviour of sandwich panel by considering the inherent material damping. Jeyaraj et al. [21] presented the numerical study of acoustic radiation characteristics of the FRP structure under the thermal load by considering inherent material damping by following a combined FEM/BEM approach. Gunasekaran et al. [22] theoretically investigation the influence of arbitrary loads on the sound radiation behaviour of simply supported isotropic plate, with the help of strain energy and TSDT approach. Xiang et al. [23] used nth order shear deformation theory to analyse the free vibration behaviour of laminated composite plates.

In most of the practical cases the FRP structures are subjected to NUE stresses loads. Examples are local instability of stiffener, walls, floor and skin panels of wings of an aircraft are widely made up of FRP structures. During the on-flight conditions, these structures experience normal compression with uniform pressure. Due to this the panel edges are subjected to combined tensile and compressive stresses. So, it is important to investigate effect of NUE loadings on buckling and free vibration characteristics of a FRP structure.

Form the above detailed literature evaluation, it is concluded that the investigation of inherent material damping and free vibration characteristics of the fibre reinforced polymer laminated plate under different NUE loads is still an unexplored region of research. Present paper, mandates analytical investigation on the effects of non-uniform in-plane edge loads on the inherent material damping and natural frequencies of boron fibre reinforced polymer laminate.

2 Methodology

All side simply supported boron-FRP plate, under the non-uniform uniaxial in-plane load considered for the present study is shown in Figure 1. With the help of Strain energy approach, the buckling strength of boron-FRP plate is predicted analytically. By using Reddy's third order shear deformation theory (TSDT) approach, dynamic response of boron-FRP plate is obtained, to understand the variation in natural frequencies and inherent material damping. Equation (1) represents the common load intensity formulation of different NUE loads. To obtain the buckling load value (Pcr) of boron-FRP under NUE loads, the main characteristic equation (Eq. (2)) is solved by using second order approximation for Alpha (α) is 0, 1 and 1.5 and third order approximation for Alpha (α) is 2 case. For the free vibration analysis and material inherent damping of the FRP plate is obtained by solving equations (3) and (4) by considering Reddy's TSDT approach, the reader is referred to Reddy [2] and Jeyaraj et al. [21] for more details of the derivation.(1) (2) (3) (4)where D11, D12, D66 and D22 are boron-FRP plates bending stiffness components.m is buckling mode number of x axis direction, whereas n is in y axis direction of boron-FRP plate. is circular natural frequency and φk is respective mode shape, ηk is modal loss factor, ω is exciting frequency, [M] mass matrix, [K] is stiffness matrix where suffix R and I denotes the real and imaginary.

 Fig. 1FRP plate subjected to non-uniform edge loads.

3 Validation studies

All side simply supported metal plate, under the NUE loads studied by Kang and Lessia [3] is chosen for the buckling characteristic validation. Kang and Lessia [3] used minimum buckling load to characterize the buckling strength of the plate under different NUE loads. Buckling load value calculated by current method shows a upright agreement with Kang and Lessia [3] which is shown in Table 1. The minimum buckling load is given as(5)

Table 1

Comparison buckling load with Kang and Leissa [3].

3.2 Free vibration and damping

Since there are no clear study has been found in inherent damping characteristics of simply supported FRP plate, based on finite element approach of commercial software ANSYS is used for the validation of the present analytical method. A single layer boron-FRP with the dimension (1 × 1 × 0.001 m3) is used for validation of free vibration and inherent damping of boron-FRP plate. It is found that calculated complex natural frequency and inherent material damping of present analytical method is closely matching with ANSYS results which is shown in Table 2.

Table 2

Comparison of complex natural frequency and damping with ANSYS.

4 Results and Discussions

In this study cross-ply (0/90/0/90/0/90)s and angle-ply (− 45/45/− 45/45/− 45/45)s laminates made of boron-epoxy are considered, assumed that all edges are simply supported. The plate considered has a layer thickness of 0.16 mm and investigated for three aspect ratios (a/b) of 0.5, 1.25 and 2 with the constant slenderness ratio (b/h) as 200 by maintaining constant value of b = 0.4 m. The FRP plate has the properties of boron fibre reinforced polymer (BFRP) which is given as complex value for predicting the inherent material damping, where E1 = 211(1+0.0014j) GPa, E2 = 21.1(1+0.0008j) GPa, G12 = G13 = 7(1+ 0.011j) GPa, ν = 0.3 and ρ = 1882 kg/m3. The present investigations, dimensionless natural frequency and buckling load as given in equations (6) and (7)).(6) (7)

Table 3

4.2 Dynamic response

In this section, the variation of dimensionless natural frequencies with a rise in different NUE loads is shown in Figure 2. When boron-FRP plates are exposed to the corresponding NUE buckling load, the dimensionless natural frequencies of the fundamental mode approaches zero. It is also noted that it is impossible to attain higher buckling mode, for the NUE loads exposed at the shorter length of the plate. Due to this the boron-FRP buckles with (1,1) mode for all load cases when aspect ratio is 0.5. However, load factor cases α = (0,1) and α = (1.5,2) of the boron-FRP plate buckles at (1,1) and (2,1) mode for aspect ratio is 1.25, due to its own material inherent orientation properties and larger length of the plate exposed to the NUE loads. Whereas the boron-FRP plate buckles (2,1) for load factor case α = (0, 1, 1.5) and (3,1) mode for α = 2 case, when aspect ratio is 2. Form Figure 2., it is clearly noted that when the aspect ratio rises above one by default the buckling mode number for α = 2 will be higher when associated to rest of the cases, due to tensile force it obtains the more stability to buckle at higher mode.

Table 4 shows the impact of different NUE loads on the representation of modal loss factors (i.e., inherent material damping) associated with its dimensionless natural frequencies. As the free vibration analysis is done by increasing load fraction of buckling load for the respective boron-FRP plates, it is evident that reduce of frequency mode towards zero. But the modal loss factor values increase to a very high value, which causes diminished of the frequency mode at the full fraction of buckling load acting on the boron-FRP plates. This is due to the reducing stiffness effect caused by the NUE loads. For better understanding, the graphical representation of the effect of different NUE loads on the dimensional natural frequencies and inherent material damping is shown in Figure 3, with the constant aspect ratio (a/b = 1.25) is chosen for cross-ply and angle-ply boron-FRP plates. Figure 4. shows the effect of NUE load α = 2 cases on the free vibration mode shapes, it is clearly shown that rise in buckling load fraction. The boron-FRP plate buckles at (2,1) mode instead of (1,1) mode and observed that shift of mode shape towards the lower region due to the tensile forces acting at the top region of the boron-FRP plate.

 Fig. 2Effect of NUE loads on dimensional buckling load. (a) Cross ply, (a/b) = 0.5. (b) Angle ply, (a/b) = 1.25. (c) Cross ply, (a/b) = 2. (d) Angle ply, (a/b) = 2.
Table 4

Effect of NUE loads on dimensional natural frequencies and inherent material damping (a/b = 1.25).

 Fig. 3Effect of NUE loads on dimensional natural frequencies and inherent material damping. (a) Cross ply, (a/b) = 1.25. (b) Angle ply, (a/b) = 1.25. (c) Cross ply, (a/b) = 1.25. (d) Angle ply, (a/b) = 1.25. (e) Cross ply, (a/b) = 1.25. (f) Angle ply, (a/b) = 1.25.
 Fig. 4Effect of NUE loads on free vibration modes.

5 Conclusions

Strain energy approach is employed to determine the dimensionless buckling load, free vibration response of BFRP plate is obtained by using Reddy [2] TDST approach. It's observed that non-uniform edge load has signification effect on the dimensionless natural frequency and inherent material damping of the boron-FRP plate. Fundamental buckling mode for load case alpha = 2 always obtain the higher buckling mode, when compared to the other load cases. Shifting of modal frequencies are observed in the load case alpha = 2 and when the boron-FRP plate has higher aspect ratio (a/b). A very high rise in the modal loss factor (i.e., inherent damping values for the respective modal indices) is observed in the fundamental natural frequency of the boron-FRP plate when the load fraction is 0.975 Pcr for the corresponding non-uniform load cases.

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Cite this article as: Vijay Gunasekaran, Pitchaimani Jeyaraj, Lenin Babu Mailan Chinnapandi, Free vibration and inherent material damping characteristics of boron-FRP plate: influence of non-uniform uniaxial edge loads, Int. J. Simul. Multidisci. Des. Optim. 12, 18 (2021)

All Tables

Table 1

Comparison buckling load with Kang and Leissa [3].

Table 2

Comparison of complex natural frequency and damping with ANSYS.

Table 3

Table 4

Effect of NUE loads on dimensional natural frequencies and inherent material damping (a/b = 1.25).

All Figures

 Fig. 1FRP plate subjected to non-uniform edge loads. In the text
 Fig. 2Effect of NUE loads on dimensional buckling load. (a) Cross ply, (a/b) = 0.5. (b) Angle ply, (a/b) = 1.25. (c) Cross ply, (a/b) = 2. (d) Angle ply, (a/b) = 2. In the text
 Fig. 3Effect of NUE loads on dimensional natural frequencies and inherent material damping. (a) Cross ply, (a/b) = 1.25. (b) Angle ply, (a/b) = 1.25. (c) Cross ply, (a/b) = 1.25. (d) Angle ply, (a/b) = 1.25. (e) Cross ply, (a/b) = 1.25. (f) Angle ply, (a/b) = 1.25. In the text
 Fig. 4Effect of NUE loads on free vibration modes. In the text

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