Analysis of Laminated Composite Plates Vibration Behavior Using a New Simple Finite Element Based on Reddy’s Third Order Theory
Khmissi Belkaid, Nadir FERGANI
Research Center in Industrial Technologies CRTI, B.O. Box 64 CHERAGA 16014 Algiers, ALGERIA Corresponding author, Ph.D, E-mail: [email protected], Phone :+213659006816
Abstract.In this paper a 2D quadrilateral finite element has been developed based on Reddy’s third order shear deformation theory for the natural vibration behavior analysis of composites laminated plates. The developed element is a C0 four-nodded isoparametric with seven degrees of freedom (7DOF) at each node. Each node has only three translation components, two rotations and two higher order rotational degrees. In particular, the selective numerical integration technique is introduced in the present FE formulation in order to achieve good results and to alleviate the locking phenomenon. The performance and reliability of the proposed formulation are demonstrated by comparing the author’s results with those obtained using the three-dimensional elasticity theory, analytical solutions and other advanced finite element models. The results indicate that the proposed formulation is promising in terms of the accuracy and the convergence speed for both thin and thick plates.
Keywords:Third Order Shear Deformation Theory; Laminated Composite Plates; Finite Element;Vibration Behavior
INTRODUCTION
Nowadays, the multilayer composite materials are found increasingly wide applications essentially in all industrial sectors, such as aerospace, automotive, civil engineering and shipbuilding. This considerable use is probably due to the notable benefits of this materials type namely; an excellent ratio rigidity weight, good corrosion resistance, fatigue resistance and mainly more advantages which their properties are adaptable for different situations. A review has been published by Mouritz et al. [1] for the recent composite structures applications and their developments on the ships and submarines. On the other hand, the multilayer composite structures analysis is still questionable and soliciting accurate theories to describe their complicated mechanical behavior. Therefore, modeling thick multilayer structures requires refined theories taking into account good expression of the transverse shear deformation effect through their thicknesses and in particular the interlaminar. Several theories take into account the transverse shear effect have been proposed for the analysis of multilayer composite structures from the equivalent single layer approach [2]. The classical laminated plate theory (CLPT) [3,4] is one of the oldest and simplest theory to describe the plates kinematic, however it is stilljust adequate for the thin structures analysis due of the negligence of the transverse shear effect. While, the first order shear deformation theory [FSDT] [5,6] is among the simplest theory which takes into account the transverse shear deformation. Unfortunately, these theories require to use the correction factors [6-9]. Consequently, to remedy this problem many higher order shear deformation theories (HSDT) have been proposed in the literature [10,11], in accurately assessing the deformation and stress of transverse shear multilayer plates without to need factors correction. Moreover, Reddy’s theory (TSDT) is the most frequent higher order theory for multilayer plates analysis when it is able to assess the stresses and transverse shear deformations with a small number independentunknown on the layer number [12,13]. However, the third order theory of Reddy [TSDT] encounters a problem when finites elements are applied by requiring the second order derivative C1 [14], it is identical in the finite elements development of thin plate based on the classical theory [15]. Recently, Belkaid et al. [16,17] developed a four node finite element C1 with five degrees of freedom in each node based on Reddy’s third-order shear deformation theory for the bending behavior analysis of isotropic and laminated composites plates.
Therefore, many finite element models (2D) based on Reddy’s third order theory have been proposed in the literature for various geometries and nodes, and also different number degrees of freedom for the analysis behavior of laminated composite plates. Moreover,a review has been published by Zhang and Yang (2009) [18] contains the recent development finites elements for the vibration behavior of laminated composite plates.The main objective of this paper is todevelop a newly simple finite element less expensive in terms of accuracy and stability on the basis of Reddy’s third order theory (TSDT) by adopting the approach of equivalent single layer, and able to analyze the
natural vibration behavior of isotropic and laminated composite plates by ensuring the right compromise between the cost and precision.
KINEMATIC
The displacement field according the Reddy’s third order shear deformation theory (TSDT) [1] can be expressed as follows:
= + −4
3ℎ +
= + −4
3ℎ +
=
(1)
Where , , are displacements to the median plane of the plate and , are rotations about the axes y and x respectively, and h is the thickness of the plate.
The deformations associated with displacement field (1), given as follows:
= = + ( + )
= = + ( + )
= = 0
= + + = +
= + + = +
= + + = + ( + )
(2)
THE CONSTITUTIVE EQUATIONS
The stress-strain relationships laminated to the k-ièmelayer after the transformation to the global coordinate (x, y, z) [2] are given using the transformation matrix and the stress-strain relationships become as follows:
=
̅̅̅ ̅̅̅ ̅̅̅
; = ̅̅ ̅̅ (4)
Fig 1.The geometry and coordinate system of the composite laminate plate
VIRTUAL WORK PRINCIPLE
The static equations of the theory can be derived from the principle of virtual work [1] by expressing the variation of strain energy as follows:
( + + + + ) +
= ( ̈ + ̈ + ̈ )
(5)
FINITE ELEMENT FORMULATION
The present finite elementis a C0four nodes isoparametric finite element having seven DOF for each node (Fig.2).
Three displacements ( , , ), two rotations ( , ), and two higher order rotations , where = + , = + . In addition, the analytical integration can be converted to Gauss points numerical integration techniques [3]. For more performance of the model in terms of accuracy and stability and to avoid shear locking phenomenon, the present formulationhas been used with the selective numerical integration technique of Gauss points (2×2) points for membrane and flexional contribution and (1×1) point for the transverse shear contribution. (See Fig. 3.).
Fig 2. Description of the normalized isoparametric element Finally leads to the followingassembled equations for free vibration analysis.
M K 0
Where M, K are respectively the global matrices of mass and rigidity of the plate, {∆}, ∆̈ are respectively the vector of the nodal variables and the vector of nodal acceleration of the system defined at time t.
After evaluating the stiffness matrix and the mass matrix for all elements,the equations of motion, for the analysis of the free vibration, can be established under theproblem with generalized eigenvalues as follows:
K 2
M 0RESULTS AND DISCUSSION
In the first example, the convergence of the developed quadrilateral new Reddy type element is studied for the natural frequencies behavior of a simply supported (SS1) thick isotropic square platewith a length-to- thickness ratioof a/h=10. An10×10 finite element mesh for the whole plate is used in this example.
The comparison was made in Table 1. with thosefrom 3D elasticity (exact) solution given by Srinivas S et al. [4], and higher-order theoryand classical plate theory (CPT) [5]and the finite element solution given by A.K. Nayaka et al [6].It is seen that the resultsobtained from the present element are very close to those ofthe 3D elasticity solution.
The CPT overestimates thefrequencies and the effect of transverse shear deformationincreases with increasing mode numbers.The present finite element results presented in Table 1appear to be better than thefinite element results of A.K. Nayaka et al.Moreover, in this study a demonstration has been considered on the fourvibration modes of an isotropic plate(Fig. 3.).
Table. 1 Comparison of non-dimensional frequencies = ⁄ of a square simply supported plate with v= 0.30 ; a/h =10
Modes Exact HSDT Nayak4 CPT Present
(1,1) 0.0932 0.0931 0.0934 0.0955 0.093229 (1,2) 0.2226 0.2222 0.2253 0.2360 0.22479 (2,2) 0.3421 0.3411 0.3463 0.3732 0.34476 (1,3) 0.4171 0.4158 0.4299 0.4629 0.42806 (2,3) 0.5239 0.5221 0.5368 0.5951 0.5331
(1,4) – 0.6545 0.6940 0.7668 0.6892
(3,3) 0.6889 0.6862 0.7081 0.8090 0.70037 (2,4) 0.7511 0.7481 0.7864 0.8926 0.77733
Fig. 3.First 4naturals vibration modes of simply supported isotropic square plate
Exemple2
In this example, we consider different simply supported cross-ply laminated plates with ratio a/h = 10, in order to determine the first modenaturals vibration for different orthotropic ratio E1/E2. Satisfactory results are obtained from the present element (Table 2) with 10*10 mesh comparing to those obtained analyticallywith the 3D elasticity solution[7] and with numerical models by Nayak[6].
Table 2 Effect of degree of orthotropy layers on the fundamental frequency = 10ℎ / of simply supported symmetric square laminates with a/h = 5
No of
layer Reference E1/E2
3
3 10 20 30 40
Exact 2.6474 3.2841 3.8241 4.1089 4.3006
Present 2.632 3.278 3.712 3.954 4.118
Nayak Q9 2.6238 3.2645 3.6978 3.9413 4.1053 5
Exact 2.6587 3.4089 3.9792 4.3140 4.5374
Present 2.646 3.39 3.944 4.272 4.492
Nayak Q9 2.6367 3.3727 3.9290 4.2576 4.4789 9
Exact 2.6640 3.4432 4.0547 4.4210 4.667
Present 2.65 3.432 4.04 4.402 4.648
Nayak Q9 2.6367 3.3727 3.9290 4.2576 4.4789
CONCLUSIONS
In this paper, a new four nodes isoparametric finite element model has been proposedon the basis to Reddy’s third order shear deformation theory by adopting the equivalent single-layer approach. The present element has seven degrees of freedom for each node, three displacements, two rotations and two higher order rotations , , , , , , where = + , = + when these degrees of freedom are approximated from the bilinear interpolation functions of Lagrange. The formulation is able to take into account the transverse shear effect in order to analyze the natural vibration behavior of isotropic and laminated composite plates thin and thick without to need correction factors. Also for more performance of the present finite element, the selective numerical integration technique is conducted in order to get accurate results without including numerical locking phenomena where the plate is thin.
REFERENCES
1.
A. Mouritz et al., "Review of advanced composite structures for naval ships and submarines",Composite Structures, 53, No.1, 21-42 (2001).
2.
J. N. Reddy, "An evaluation of equivalent-single-layer and layerwise theories of composite laminates",Composite Structures, 25, No.1, 21-35 (1993).
3. G. Kirchhoff, "Über die Schwingungen einer kreisförmigen elastischen Scheibe",Annalen der Physik,
157, No.10, 258-264 (1850).
4. A. E. H. Love, A treatise on the mathematical theory of elasticity. Cambridge University Press. (2013).
5.
E. Reissner, "The effect of transverse shear deformations on the bending of elastic plates",Journal of Applied Mechanics, 12, A69-A77 (1945).
6.
J. M. Whitney, "The Effect of Transverse Shear Deformation on the Bending of Laminated Plates",Journal of Composite Materials, 3, No.3, 534-547 (1969).
7.
N. Pagano, "Exact solutions for composite laminates in cylindrical bending",Journal of Composite Materials, 3, No.3, 398-411 (1969).
8.
J. Whitney, "Shear correction factors for orthotropic laminates under static load",Journal of Applied Mechanics, 40, No.1, 302-304 (1973).
9.
Y. Stavsky and R. Loewy, "On vibrations of heterogeneous orthotropic cylindrical shells",Journal of Sound and Vibration, 15, No.2, 235-256 (1971).
10. J. N. Reddy and D. Robbins, "Theories and computational models for composite laminates",Applied
Mechanics Reviews, 47, No.6, 147-169 (1994).
11.
Y. Ghugal and R. Shimpi, "A review of refined shear deformation theories of isotropic and anisotropic laminated plates",Journal of Reinforced Plastics and Composites, 21, No.9, 775-813 (2002).
12.
J. N. Reddy, "A simple higher-order theory for laminated composite plates",Journal of Applied Mechanics, 51, No.4, 745-752 (1984(a)).
13.
J. N. Reddy, "A refined nonlinear theory of plates with transverse shear deformation",International Journal of Solids and Structures, 20, No.9, 881-896 (1984(c)).
14.
A. H. Sheikh and A. Chakrabarti, "A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates",Finite Elements in Analysis and Design, 39, No.9, 883-903 (2003).
15.
K Belkaid and A TATI, Analysis of laminated composite plates bending using a new simple finite element based on Reddy’s third order theory, Revue Des Composites Et Des Materiaux Avances. 25 (1) 89-106.(2015)
16. K Belkaid and A Tati and R Boumaraf, A simple finite element with five degrees of freedom based on
Reddy’s third-order shear deformation theory, Mechanics of Composite Materials. 52 (2) 257-270(2016).
17.
J. L. Batoz and M. B. Tahar, "Evaluation of a new quadrilateral thin plate bending element",International Journal for Numerical Methods in Engineering, 18, No.11, 1655-1677 (1982).
18.
