Document Type : Research Paper
Authors
Department of Mechanical Engineering, Tezpur University, Napaam, Sonitpur, 784028, Assam, India
Abstract
Keywords
Main Subjects
Modeling the Buckling Characteristics of Pineapple Leaf Fibre Reinforced Laminated Epoxy Composites
Department of Mechanical Engineering, Tezpur University, Napaam, Sonitpur, 784028, Assam, India
KEYWORDS 

ABSTRACT 
Pineapple leaf fibre; Eglass fibre; Buckling analysis; Finite element modeling. 
Pineapple leaf is a natural fibre possessing superior mechanical strength which can be used as a reinforcing component in natural fibrebased composites. In general, composites can endure a wide variety of loads while in service. This work reports the buckling analysis of pineapple fibre reinforced epoxy composites and compared it to an isotropic composite reinforced with a synthetic fibre such as Eglass. The effects of changing the fibre volume fraction and plate aspect ratio, on physical buckling behaviour have been reported. The elastic parameters were calculated analytically, whereas the buckling studies were carried out using the finite element method. Buckling was shown to be significantly influenced by the changes in volume fractions, plate aspect ratio, and buckling mode. Additionally, the influence of design parameters such as optimum ply angle for composite stackingsequence was also investigated under no shear conditions. It was observed that pineapple leaf fibre composites yielded better buckling characteristics than contemporary synthetic Eglass fibre composites. 
Natural fibre composites are being used to replace most structural applications in today's world, whether in the automation, aviation, or shipping industries. Because synthetic fibres are nonbiodegradable and only reusable up to a certain point, their widespread use has had a negative and damaging influence on sentient beings. Natural fibres are crucial as they are biocompatible and biodegradable, have a high strengthtoweight ratio, have less density, and are nontoxic. Additionally, they are also abundantly available as most of the parts of fruits and plants are wasted due to a lack of knowledge regarding their economic value [1,2]. Natural fibres have recently been used in industrial and automation applications, although they still lack some of the properties that synthetic fibres can supply. Senthilkumar et. al. [1] analysed pineapple leaf fibre composite for free vibration and damping properties. Earlier, Saha et. al. investigated the mechanical, thermal, and biodegradation behaviour of pineapple leaf particulates and found satisfactory results [2]. Peças et. al. [3] reviewed several papers based on the usage of fibres in various industries and on their processing and utilisation in automotive industries. Mohammed et. al. [4] presented various surface treatment methods and the impact of chemical treatments on fibre matrix adhesion, along with the physical and mechanical characteristics of several fibres. Jawid et. al. [5] presented various properties and qualities of pineapple leaf fibres. They also stated that pineapple leaf fibres are extensively used in textile industries and automation industries. Sema et. al. [6] provided insight into pineapple leaf fibres cultivation and their availability in portions of northeast India such as Karbi Anglong, North Cachar hills, West and East Garo hills, and Barak valley which produces almost 40% of the total pineapple of India. Mishra et. al. [7] investigated the fibre loading condition and its impact resistance for pineapple leaf fibre composites for various weight fractions. It was found that these composites have an impact strength of around 80.29 J/m for a 30% weight fraction. Borah et. al. [8] studied the structural responses of various fibre composites and presented a comparative study for varying damping ratios with plate aspect ratio and volume fractions. Jalili et. al. [9] investigated the effects of flax fibre on the multiobjective optimum design of hybrid laminated composite for maximum buckling with different material configurations keeping the cost as a priority. Hosseinzadeh et. al. [10] conducted the multiobjective optimization for flax fibres composite by minimizing the cost and maximizing the frequency gaps in order to assess the capabilities of fibrereinforced composites. Le et. al. [11] investigated the viscoelastic beahviour of a laminated composite under axial loading conditions using Abaqus. Le et. al. [12] studied the buckling behaviour of transversely isotropic multilayered beams with thin and soft interfaces for a better understanding of wrinkle formation under compressive loading. Chai et. al. [13] presented the stability behaviour of typical laminated composite plates with all sides simply supported and bound to inplane stress conditions, utilising a total potential energy method in conjunction with the RayleighRitz technique. Darvizeh et. al. [14] investigated the buckling of composite plates, wherein the mathematical modeling for typically laminated plates was constructed using the generalized differential quadrature rule (GDQR) and the R–R approach. Adali et. al. [15] stated that if the plies are symmetric about the middle, a balanced laminated is achieved where the coupling is absent. This implies that a balanced configuration has the advantage of reducing the and values for any angle ply laminates as these values can be neglected and are considered zero. Furthermore, they also mentioned that for laminates with several layers, and values might be rather modest in comparison to other values. Bert et. al. [16] found that among various stacking sequences, simply supported plates and symmetrical angle ply laminates had the largest buckling mode under uniaxial and biaxial loads. There has been a lot of research done on the mechanical and thermal properties of natural fibres, however, relatively little attention has been devoted to the structural deformations that may occur as a result of loading in composites.
The overall aim of this research is to investigate the buckling modes of failure of pineapple leaf fibre composite as a function of varying stacking sequences. Additionally, the buckling characteristics of the pineapple leaf fibre composites are compared with a synthetic fibre composite in order to understand the performance capabilities of natural fibre composites under buckling. The numerical analysis is performed using Kirchhoff’s plate theory and Navier’s solutions under noshear conditions. Furthermore, buckling characteristics were modelled by FEM analysis.
The rest of the paper is organized as follows: Section 2 formulates the numerical method and theory for buckling. Additionally, it describes the boundary conditions applied and the modeling of laminated composite incorporating the applied loads. In section 3, the detailed outcomes have been explained, and finally, concluding remarks regarding the outcome of the research have been stated in section 4.
The elastic modulus, shear modulus, and Poisson's ratio of a composite lamina were calculated using the rule of mixture, as given by Jones [17]. The requisite fibre qualities were derived from several studies.

(1) 

(2) 
Major Poisson’s ratio ( ) and Minor Poisson’s ratio ( ) can be obtained as;

(3) 

(4) 
Inplane shear modulus ( ) of the composite can be obtained as;

(5) 
where are the shear modulus of fibre and matrix respectively and can be calculated as;

(6) 
In recent years, the usage of composites in structural applications has expanded dramatically. These composites are in the form of thin laminates that may be formed according to the pattern that is required. When thin laminates with no outofplane loads are examined, the lamina is regarded to be in a plane stress state, i.e., , and . This yields the values which are known as reduced compliance coefficients. The compliance matrix [S], inverting the compliance matrix [S]and the reduced stiffness matrix [Q]is obtained as shown by Jones [17];

(7) 


(8) 


(9) 


(10) 


(11) 

where, are called reduced stiffness coefficients, and are related to elastic constants as explained in [17]. A2D coordinate system is used to represent an angle lamina as shown in Figure 1. The axes displaying coordinate systems 12 are referred to as local axes, whereas the axes xy are referred to as global axes. The angle formed by two axes is denoted by the symbol . The local and global stresses are related through transformation matrix T as described by Jones [17];

Fig. 1. Local and global axes of an angle lamina 
The expression of global stress and global strain can be written as given by Jones [17];

(12) 
where, elements are called transformed reduced stiffness matrix. The elements can be found as given by Jones[17].
The location of laminas in the composite laminate is shown in figure 2.

Fig. 2. Locations of laminas in a laminate 
For the orientation of fibres in a stacking sequence of laminas in a different direction, the flexural stiffness and bending stiffness matrix [D] are calculated as follows [18];

(13) 


(14) 

Equation of motion in terms of displacements in a constitutive relationship for a laminated plate in full matrix form can be written as shown by Chai and Hoon[13]:

(15) 
where, and are called the membrane strains and surface curvatures respectively [13].
For a given plate, the total loss in potential energy of the deformed plate is taken from Chai and Hoon [13]:

(16) 
Using the strain energy displacement resultants and consecutive relations for the laminated plate by using the above expressions and solving the form given in Eq. 15 and substituting in Eq. 16 the strain energy as given by [13]:

(17) 
where, are reduced flexural stiffness values and .
Assuming that the plate is subjected to stress resultants the potential energy acquired as a consequence of external loading is stated in [13]:

(18) 
According to Hamilton's variational principle, real displacements among permissible routes accompanied by a dynamical method as stated by Darvijeh [14] is given by,

(19) 

(20) 
where is the initial variation and are specified times, upon applying Hamilton’s principle for a midplane symmetric plate, with coupling terms i.e. , and neglecting the shear effects of loading . Under these conditions, the equation governing deflection is given by Reddy[19].

(21) 
To determine a nonzero deflection the inplane forces are [19];
, , 
(22) 
where k is also known as the plate buckling constant.
Navier's solutions are particularly useful for orthotropic laminates. The Navier solution makes use of two sinusoidal functions that meets the boundary condition in exact form. The number of waves in the two directions is a function of the solution, the dimensions of the plate, and the properties of the material as expressed in [19].

(23) 
Substituting Eq. 23 into Eq. 21 we obtain (for any m and n)

(24) 
For uniaxial compressions of a rectangular laminate, and for biaxial . Where,

(25) 


(26) 


(27) 

The smallest value of , for any m, occurs for , [19]. Hence critical buckling load is a function of . For Navier’s solution, the nondimensionalized buckling load is found using this formula as expressed in Reddy [19].

(28) 
The FEM modeling of fibre composite was carried out in ANSYS composite prepost for the design and stacking sequence of laminated composite. SHELL181 type element was used in the present FE analysis as the SHELL181 is pertinent for the analysis of thin to substantially thick shell structures. The SHELL181 element contains four nodes, each with six degrees of freedom.
The boundary condition of the square plate for the eigenvalue buckling analysis was considered as simply supported (SSSS) on all 4 edges. The plate edges were transformed to the nodal named selection, making it easier to define the boundary conditions at the extreme boundary limits of edges. The nodal displacement along the zaxis was zero for all four edges of the plate, rotation along the yaxis was fixed for the top and bottom nodal edges, and rotation along the xaxis was fixed for the left and right nodal edges of the plate. Two additional nodalnamed selections were created on the plate, one in middle (considering plate coordinates) where displacement along x,y was zero at the center and the other nodalnamed selection at the side edge along the xaxis where displacement along y was set to zero. Figure 3 shows the stacking sequence of the composite.

Fig. 3. Angle ply stacking sequence of 
For the eigenvalue buckling analysis, the dimensions of the composite plate were kept at 120 mm 120 mm 1.6 mm based on ASTM standards as provided by Komur et. al. [20], where the ply thickness is 0.0004 meters or 0.4mm each, for 4 plies. For eigenvalue buckling in ANSYS, after applying the boundary condition, a load of 1000 N/mm (line pressure) was applied on the two opposite sides of the composite plate for uniaxial loading and on all four sides of the composite plate for biaxial loading conditions of the simply supported plate (SSSS) (refer figures 4 and 5).

Fig. 4.Uniaxial loading of a square composite plate 

Fig. 5. Biaxial loading of a square composite plate 
The eigenvalue buckling in ANSYS gradually starts applying the load until it reaches the bifurcation point as shown in Figure 6 (A point where deformation occurs or starts).

Fig. 6.Diagram for bifurcation point in buckling 
At this point, the composite buckles, and the rest is examined in postbuckling behaviour, also known as nonlinear buckling. The loading of 1000 N/m is applied as a Load Multiplier or Load Factor, which is given as;
The mesh convergence study was carried out and the optimal results were found for an element size of 12 mm mesh for 120 mm 120 mm 1.6 mm plate based on ASTM standards [20]. It had better mesh metric quality with an aspect ratio of 1 (best). There was no mesh refinement required as the plate considered was square or rectangular (for higher plate aspect ratios) and had no curvature to it, where the outcomes were identical with no or very minimal error. The plate wrapping factor details the wrap (mathematical deviation of a mesh element) of mesh elements. A greater wrapping factor means a lower or poorer quality mesh structure, and the ideal wrapping factor is 0. The mesh wrapping factor was 0 for the generated mesh in this study. By default, the element order was set to program controlled and physics preference was set to mechanical, thus for which ANSYS creates quad 4 elements, and the number of elements and nodes were found to be 100 and 121 respectively.
It is one of the most critical aspects of failure analysis for a material that is undergoing deformation. Buckling is often determined in the situation of lengthy columns; although it can occur in the majority of practical instances, including thin flat plates. Buckling is a failure mode in which a structure deflects or deforms as a result of abrupt compressive force. This study uses Navier's model to investigate the impact of uniaxial and biaxial loading under a given load on all four sides of an orthotropic composite plate. Navier's model is used for orthotropic plates with midplane symmetry and the effects of shear have been neglected. The variation of buckling load is studied with plate aspect ratio and the number of half waves in the xdirection. For the validation of the analytical result, the same values are considered as given in Reddy [19].
In Table 1, the influence of plate aspect ratio and modulus on nondimensionalized buckling loads of rectangular laminates for uniaxial loading conditions (k=0) and biaxial loading conditions (k=1) are shown, where, varied, , , . All layers are considered to be of equal thickness [13].
Table 1. Comparison of Navier's nondimensional buckling load [19] with the analytical result
k 


10 
20 
25 
40 

Non dimensional 
Numerical 
Non dimensional 
Numerical 
Non dimensional 
Numerical 
Non dimensional 
Numerical 
Non dimensional 
Numerical 

0 
0.5 
13.9 
13.85 
18.12 
18.09 
21.87 
21.85 
22.87 
22.85 
24.59 
24.57 

1 
5.65 
5.608 
6.347 
6.317 
6.961 
6.942 
7.124 
7.108 
7.404 
7.393 

1.5 
5.233 
5.191 
5.277 
5.247 
5.31 
5.291 
5.318 
5.32 
5.332 
5.321 

1 
0.5 
11.12 
11.08 
12.69 
12.63 
13.92 
13.88 
14.24 
14.21 
14.76 
14.73 

1 
2.852 
2.804 
3.174 
3.158 
3.481 
3.471 
3.562 
3.554 
3.702 
3.696 

1.5 
1.61 
1.597 
1.624 
1.614 
1.634 
1.628 
1.636 
1.631 
1.641 
1.637 

In all cases the critical buckling mode is
(m n) = (1 1), except for and (k=1); for which the modes are (1 1), (1 2), (1 2),
(1 2), and (1 3) for modulus ratios 5, 10, 20, 25, and 40, respectively [19]. It was observed that larger aspect ratios resulted in increased modes of buckling, as seen in Figure 7.

Fig. 7.Buckling load (nondimensionalized), vs plate aspect ratio [19] 
The nondimensionalized buckling load
versus plate aspect ratio for laminates with material parameters are , are plotted in Figure 8.

Fig. 8. Buckling load (nondimensionalized), vs plate aspect ratio (numerical) 
When the plate aspect ratio is less than 2.5, it collapses into a single halfwave in the xdirection (refer to Figure 9) [19].

Fig. 9. Buckling load (nondimensionalized) vs number of half wavelengths ‘m’ [19] 
The plate bends into larger and more halfwaves in the xdirection as the aspect ratio increases [19]. Notice that intersections of two successive modes correspond to specific aspect ratios (refer to Figure 10). As a result, there are two buckled mode configurations for each of these plate aspect ratios.

Fig. 10. Buckling load (nondimensionalized) vs number of half wavelengths m [19] (numerical) 
Herein, PaLF composites are compared with a synthetic fibre composite (Eglass), as PaLF fibres have better mechanical properties compared to most of the other NF's [5]. For synthetic fibre, Eglass was considered as it had almost comparable mechanical properties with PaLF [5,21]. The fibre and matrix properties are shown in Table 2.
Table 2. Materials Properties
Fibres 
Density ( ) 
Young's Modulus (GPa) 
Poisson's Ratio 
References 
Pineapple 
1500 
82 
0.3 
[35] 
EGlass 
2500 
73 
0.22 
[21] 
Epoxy 
1200 
3.78 
0.35 
[8, 22] 
Table 3 shows nondimensionalized buckling load for uniaxial loading.
Table 3. values for unidirectional loading 

Stacking sequence 


PaLF 
Min. buckling mode PaLF 
EGlass 
Min. buckling mode Eglass 
% change 

Numerical 
FEM 
Numerical 
FEM 

[ ]s 
1 
20 
8.29 
8.18 

7.81 
7.71 

5.79 
25 
9.43 
9.31 

8.83 
8.71 

6.36 

30 
10.61 
10.47 

9.88 
9.75 

6.88 

2 
20 
8.18 
8.05 

7.81 
7.71 

4.52 

25 
8.82 
8.68 

8.62 
8.49 

2.27 

30 
9.53 
9.38 

9.28 
9.13 

2.62 

3 
20 
7.08 
6.99 

6.84 
6.76 

3.39 

25 
7.78 
7.69 

7.48 
7.39 

3.86 

30 
8.53 
8.43 

8.17 
8.07 

4.22 

[ ]s 
1 
20 
8.28 
8.09 

7.81 
7.62 

5.68 
25 
9.42 
9.21 

8.82 
8.62 

6.37 

30 
10.60 
10.36 

9.87 
9.65 

6.89 

2 
20 
8.28 
8.09 

7.81 
7.62 

5.68 

25 
9.42 
9.21 

8.82 
8.62 

6.37 

30 
10.60 
10.36 

9.87 
9.65 

6.89 

3 
20 
7.90 
7.70 

7.56 
7.37 

4.30 

25 
8.80 
8.58 

8.38 
8.17 

4.77 

30 
9.74 
9.51 

9.23 
9.01 

5.24 

[ ]s 
1 
20 
8.68 
8.59 

8.16 
8.06 

5.99 
25 
9.91 
9.82 

9.26 
9.16 

6.56 

30 
11.17 
11.08 

10.38 
10.29 

7.07 

2 
20 
8.66 
8.59 

8.14 
8.06 

6.00 

25 
9.88 
9.82 

9.23 
9.16 

6.58 

30 
10.96 
10.86 

10.36 
10.29 

5.47 

3 
20 
7.78 
7.72 

7.47 
7.40 

3.98 

25 
8.65 
8.60 

8.25 
8.19 

4.62 

30 
9.56 
9.52 

9.08 
9.03 

5.02 

[ ]s 
1 
20 
8.57 
8.45 

8.06 
7.94 

5.95 
25 
9.78 
9.65 

9.14 
9.01 

6.54 

30 
11.02 
10.89 

10.25 
10.11 

6.99 

2 
20 
8.55 
8.45 

8.04 
7.94 

5.96 

25 
9.75 
9.65 

9.11 
9.01 

6.56 

30 
10.98 
10.89 

10.21 
10.11 

7.01 

3 
20 
7.89 
7.80 

7.56 
7.47 

4.18 

25 
8.78 
8.71 

8.37 
8.28 

4.67 

30 
9.71 
9.65 

9.21 
9.14 

5.15 
From Table 3, the buckling load values for uniaxial loading for PaLF composite and Eglass composites can be obtained. The minimum buckling load for various stackingsequence of , , and has been provided for increasing volume fraction. Variations for increasing plate aspect ratios and buckling load are also shown. According to Table 3, the buckling load values for higher plate aspect ratios occur at higher mode shapes. This is because, as previously stated in Eq. 27, the minimum critical buckling load is a function of 'm', and thus the minimum value for critical buckling may occur at higher mode shapes as the aspect ratio increases for certain values of modes.
It is certain that the PaLF composite outperforms the Eglass composite in terms of outcomes. For plate aspect ratio ( ) and
( ) at [ ]s stacking sequence, the PaLF composite showed the best possible results. Similar findings were obtained for stacking sequence under uniaxial compression by [15,23].
Figure 11 shows the variation of nondimensionalized buckling load to plate aspect ratio for increasing mode shapes. The figure also shows the variation for different stackingsequence of the composite plate. It can be noted that the nondimensionalized buckling load is minimum for [ ]s stacking sequence, whereas the [ ]s and [ ]s stacking sequence results are better. Note that Figure 11 shows the nondimensionalized form for a continuously varying plate aspect ratio.

Fig. 11. Buckling load (Nondimensionalised) vs plate aspect ratio for uniaxial loading of various stackingsequence at 25% volume fraction of PaLF composite 
Nondimensionalized form ( ) is important in understanding the concepts of how this parameter ( ) is directly related to flexural stiffness values, for changing plate aspect ratios. It could not be possible to understand directly just by checking the critical buckling load values. As critical buckling values will show the possible load at which it will start to buckle and not the modes of buckling. The nondimensionalized form could be used to study how a plate will behave under certain loads and in several modes that it will buckle into. The nondimensionalized buckling load has an inverse relation to as previously stated in Eq. 28. Also as stated by Joshi et. al. [23], at , equals the value of at and at equals the value of at . At and are equal, hence the maximum buckling load occurs at orientation. But the inverse relationship between and nondimensionalized buckling load results in changing the maximum buckling load values for [ ]sand [ ]s stacking sequence. Furthermore, it can also be observed how increasing the plate aspect ratio changes the buckling mode for any plate. As the plate aspect ratio grows, the plate buckles into more and more half waves in the xdirection. It is worth noting that the intersection of two successive modes for various stacking sequences, happens at a particular mode and plate aspect ratio. For higher angles of orientation of fibres apart from [ ]s stacking sequence, figure 11, shows how the plate buckles into two half waves as the plate aspect ratio reaches two.
Table 4 shows the buckling load values for biaxial loading for the PaLF composite and Eglass composite.
For the biaxial loading of the composite plate, it was observed that the minimum critical buckling load was lowered by a significant amount as compared to uniaxial loading. PaLF results are still better for comparison with Eglass fibre. The minimum critical buckling load for all stackingsequence at different volume fractions and plate aspect ratios were found at . Buckling load for biaxial loading is unsustainable for a higher plate aspect ratio because the plate lengthens and the buckling load decreases drastically.
It was observed that PaLF composite would be much more resistant to deformation when compared to Eglass due to its superior mechanical properties. Thus PaLF can be a suitable alternative for E glass in different types of composites. When compared with synthetic fibre composites; PaLF composites showed better buckling characteristics, thus, showing that PaLF composites outperform contemporary Eglass composites. The improvements were observed to be more in the case of square laminates and at higher weight fraction of fibres (i.e. 30 vol%). Square PaLF composite plates with 30 vol% fibres showed a buckling load increment of approximately about 7% when compared with similar Eglass composites. Thus, PaLF composite materials have huge potential to be used in structural components under buckling conditions.
Table 4. values for bidirectional loading 

Stacking sequence 


PaLF 
Min. buckling mode PaLF 
EGlass 
Min. buckling mode Eglass 
% change 

Numerical 
FEM 
Numerical 
FEM 

[ ] 
1 
20 
4.15 
4.09 

3.91 
3.85 

5.78 
25 
4.72 
4.65 

4.42 
4.36 

6.36 

30 
5.30 
5.23 

4.94 
4.87 

6.79 

2 
20 
1.63 
1.61 

1.60 
1.58 

1.84 

25 
1.75 
1.74 

1.72 
1.70 

1.71 

30 
1.90 
1.88 

1.85 
1.83 

2.63 

3 
20 
1.32 
1.30 

1.31 
1.29 

0.76 

25 
1.40 
1.39 

1.39 
1.37 

0.71 

30 
1.50 
1.48 

1.48 
1.46 

1.33 

[ ] 
1 
20 
4.14 
4.04 

3.90 
3.81 

5.80 
25 
4.71 
4.60 

4.41 
4.31 

6.37 

30 
5.30 
5.18 

4.93 
4.82 

6.98 

2 
20 
1.97 
1.92 

1.89 
1.85 

4.06 

25 
2.18 
2.13 

2.08 
2.04 

4.59 

30 
2.40 
2.35 

2.29 
2.24 

4.58 

3 
20 
1.72 
1.69 

1.66 
1.63 

3.49 

25 
1.90 
1.87 

1.83 
1.79 

3.68 

30 
2.09 
2.05 

2.00 
1.96 

4.31 

[ ] 
1 
20 
4.34 
4.29 

4.08 
4.03 

5.99 
25 
4.95 
4.91 

4.63 
4.58 

6.46 

30 
5.59 
5.54 

5.19 
5.14 

7.16 

2 
20 
1.82 
1.81 

1.77 
1.75 

2.75 

25 
2.00 
1.98 

1.93 
1.91 

3.50 

30 
2.18 
2.17 

2.10 
2.09 

3.67 

3 
20 
1.48 
1.46 

1.45 
1.43 

2.03 

25 
1.60 
1.59 

1.56 
1.55 

2.50 

30 
1.73 
1.72 

1.69 
1.67 

2.31 

[ ] 
1 
20 
4.29 
4.22 

4.03 
3.97 

6.06 
25 
4.89 
4.83 

4.57 
4.51 

6.54 

30 
5.51 
5.44 

5.12 
5.06 

7.08 

2 
20 
1.90 
1.88 

1.83 
1.81 

3.68 

25 
2.09 
2.07 

2.01 
1.99 

3.83 

30 
2.29 
2.28 

2.19 
2.18 

4.37 

3 
20 
1.59 
1.57 

1.54 
1.52 

3.14 

25 
1.73 
1.72 

1.68 
1.66 

2.89 

30 
1.89 
1.88 

1.83 
1.81 

3.17 
Figure 12 shows the variation of nondimensionalized buckling load to plate aspect ratio for increasing mode values m for biaxial loading. It can be noted that buckling load decreases drastically for increasing plate aspect ratio . This can also be viewed as, considering biaxial loading the plate is exerted to compressive forces from all sides, thus making the plate easy to buckle, so at higher plate aspect ratios the buckling effect will be significantly more resulting in a lower value of buckling load.

Fig. 12. Buckling load (Nondimensionalized) vs plate aspect ratio for biaxial loading of various stackingsequence at 25% volume fraction of PaLF composite 
It can be noted from figure 12 that, for higher values of plate aspect ratio the mode of buckling never shifts to higher modes, as the minimum buckling load for biaxial loading occurs at . Eq. 27 shows that nondimensionalized buckling is strongly associated to , and hence the dependency results in the minimal buckling mode.
Table 5 shows the variation of critical buckling load for uniaxial loading of a discrete array of stackingsequence with changing plate aspect ratio at 25 vol% vol fraction for PaLF composite.
According to Joshi et. al. [23] for a square plate under uniaxial compression (Eq. 24) adds significantly to the buckling load. The bending stiffness coefficient continues to fall as the orientation increases from to . As mentioned before, at , equals at and at equals at . At and are equal, hence the maximum buckling load occurs at orientation. Also, Joshi et. al. [23] observed that if the plate aspect ratio is less than one, then the contribution of decreases with decreasing aspect ratio and strives to push the optimal fiber orientation toward , while and attempts to hold it back. As a result, the optimal fiber orientation is between and . For an aspect ratio higher than one, contributes significantly and attempts to pull the fibre orientation towards , resulting in an effective fibre orientation between and . According to Joshi et. al. [23] changing the buckling mode to , affects the contribution of each attribute in Eq. 27. Because is multiplied by , the term no longer dominates at neartounity aspect ratios. This results in optimal fiber orientation angles ranging from and for aspect ratios in the range from one to two. As a consequence, nearly identical findings were achieved for this investigation, where the optimal fibre stacking sequence for increased plate aspect ratio was discovered to be for the stacking sequence.
Table 5. Variation of critical buckling load for uniaxial loading of a discrete array of stackingsequence with changing plate aspect ratio at 25 vol% vol fraction for PaLF composite 


[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
0.2 
142.440 

128.597 

133.008 

130.175 

0.4 
37.682 

34.228 

35.926 

35.070 

0.6 
18.494 

17.062 

18.184 

17.725 

0.8 
12.028 

11.420 

12.255 

11.970 

1 
9.309 

9.209 

9.817 

9.652 

2 
8.684 

9.209 

9.817 

9.652 

3 
7.689 

8.583 

8.601 

8.708 

4 
7.787 

8.356 

8.584 

8.598 

5 
7.845 

8.372 

8.699 

8.668 

6 
7.689 

8.434 

8.601 

8.708 

7 
7.712 

8.409 

8.556 

8.606 

8 
7.757 

8.356 

8.584 

8.598 

9 
7.689 

8.353 

8.601 

8.626 

10 
7.695 

8.372 

8.558 

8.625 

Figure 13 shows the fluctuation of buckling load for uniaxial loading to increasing plate aspect ratio for various ply stacking sequences. When previously noted, the best ply angle for a plate aspect ratio of & was determined to be approximately and as the plate aspect ratio increases the optimum ply angle shifted towards .

Fig. 13. Uniaxial buckling load vs plate aspect ratio for various stackingsequence of PaLF composite at 25% volume fraction from Table 5. 
This can be validated from figure 13 which shows that for larger plate aspect ratios, the optimal ply angle for uniaxial loads was about for the aforementioned simulation results using 4 plies stackingsequence. Joshi et. al. [23] found the best ply orientation angle for uniaxial loading at while considering 6 plies stackingsequence.
Now considering biaxial loading in the composite plate for various stackingsequence from Table 6, in conjunction with Figure 14, it can be seen that for biaxial loading the s stacking sequence performed best with a plate aspect ratio of one. When the plate aspect ratio exceeds one, the crossply stacking sequence of s shows the best possible results compared to other stacking sequences. This is also true that the crossply stacking sequence is mostly considered for biaxial loading in composite plates, as it can withstand a higher amount of loading. Hence it would be wise to go for s stacking sequence under biaxial loading in composites.
The uniaxial and biaxial loading of composites for critical buckling load study has been carried out considering the parameters that account for buckling as in Eq. 27. Figures 13 and 14 provide the information regarding the failure of a composite at the minimum value of buckling irrespective of mode, and depicts the change in plate aspect ratio to corresponding critical buckling.

Fig. 14. Biaxial buckling load vs plate aspect ratio for stackingsequenceofPaLF composite at 25% vol. fraction 
Table 6. Variation of critical buckling load for biaxial loading of a discrete array of stackingsequence with changing plate aspect ratio at 25% volume fraction for PaLF composite 


[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
[ ] 
Min. buckling mode 
0.2 
136.9611 

123.6507 

127.8927 

125.1684 

0.4 
32.4844 

29.5073 

30.9704 

30.2325 

0.6 
13.5984 

12.5458 

13.3708 

13.0329 

0.8 
7.3341 

6.9632 

7.4723 

7.2986 

1 
4.6547 

4.6045 

4.9084 

4.8262 

2 
1.7367 

2.1296 

1.9848 

2.0701 

3 
1.3857 

1.8651 

1.5865 

1.716 

4 
1.2892 

1.7994 

1.467 

1.6139 

5 
1.25 

1.7747 

1.4158 

1.5711 

6 
1.2303 

1.7629 

1.3893 

1.5493 

7 
1.2191 

1.7564 

1.3737 

1.5365 

8 
1.212 

1.7524 

1.3637 

1.5284 

9 
1.2072 

1.7498 

1.357 

1.523 

10 
1.2038 

1.7479 

1.3522 

1.5192 

This work describes the possibility of using a PaLF composite in assisting researchers in identifying practical prospective applications in the field of acoustic and vibration isolation such as multidimensional earthquake isolation devices using viscoelastic dampers and vibration isolation pads used in industries [4], impact energy absorption widely incorporated in motorsports and automotive application [4] and anticrushing devices used in automotive applications, and superlight composite panels used in windmills and aerospace applications [4, 8]. The work could be further expanded to experimental findings. Additionally, this study can be broadly performed for different structures like honeycomb and stiffened structures considering the hybridization of composites.
The goal of the current study is to create an NF composite with enhanced buckling properties that can support axial loads without easily bending or deforming. Finite element analysis was used to perform buckling of thin laminated plates for the PaLF composite and compared it to an isotropic synthetic fibre like Eglass. Additionally, the model's correctness was determined by numerically comparing the findings. The generalized conclusions that can be drawn are as follows;
Based on this study, it is paramount to state that pineapple leaf fibrebased composite materials exhibit better buckling characteristics than contemporary synthetic fibre composites such as Eglass composites. Thus, they have huge potential to be used in structural components under buckling conditions.
Nomenclature

Young’s modulus in longitudinal direction (GPa) 

Young’s modulus in transverse direction (GPa) 

Major Poisson's ratio 

Minor Poisson's ratios 

Inplane shear modulus 

Reduced compliance coefficients 

Reduced stiffness coefficients 

Flexural stiffness 
[A] 
Extensional stiffness matrix 
[B] 
Coupling stiffness matrix 
[D] 
Bending stiffness matrix 

Stress (Pa or MPa) 

Shear stress (Pa or MPa) 

Strain 

Shear strain 
W 
Transverse displacement of a point on the plate 
Λ 
Natural frequency factor 

Natural frequency (Hz) 
t 
Time (s) 

Density of the composite material (kg/m3) 
f 
Frequency (Hz) 

Critical buckling load 

Nondimensional buckling load 
k 
Plate buckling constant 
ζ 
Damping ratio 
m 
Half wavelengths in the xdirection 
n 
Half wavelengths in the ydirection 
h 
Total thickness of the laminate (mm) 

Stress resultants 
a 
Length of composite plate along xaxis (mm) 
b 
Breadth of composite plate along y axis (mm) 
U 
Strain energy of plate 
V 
Potential energy in plate 
Π 
Total loss in potential energy of a deformed plate 

First variation according to Hamilton’s principle 
Acknowledgments
The authors want to thank the anonymous reviewers and all the persons concerned in helping with this paper.
Conflicts of Interest
The authors declare 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.
References
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