Applied Mathematical Sciences, Vol. 9, 2015, no. 109, 5401 - 5415

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.56427

Geometrically Non-Linear Vibrations of

Fully Clamped Asymmetrically

Laminated Composite Skew Plates

Moulay Abdelali Hanane

Université Mohammed V Agdal-

Ecole Mohammadia D’ingénieurs

Avenue Ibn Sina, B.P.765

Agdal, Rabat, Morocco

Benamar Rhali

Université Mohammed V Agdal-

Ecole Mohammadia D’ingénieurs

Avenue Ibn Sina, B.P.765

Agdal, Rabat, Morocco

Copyright © 2015 Moulay Abdelali Hanane and Benamar Rhali. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, a theoretical model based on Hamilton’s principle and spectral

analysis is used to obtain a solution for geometrically non-linear vibration of

symmetrically and asymmetrically cross ply laminated skew plates. The

Analysis was considered regarding the influence of different parameters such as

fiber orientation, ply properties, plate skew angle and aspect ratio. Comparison

between the numerical and analytical results and those available in previous

studies showed a good agreement. The associated non-linear bending stresses

were calculated and compared showing a higher increase with increasing the skew

angle for asymmetrically laminated composite skew plates.

Keywords: Non-linear vibrations, Composite laminated skew plate, asymmetric,

bending stress

5402 Moulay Abdelali Hanane and Benamar Rhali

1 Introduction

In fields like Aerospace, mechanical and civil engineering, thin laminated

composite skew plates are commonly used. Very often, such structures are

working in severe environmental conditions and subjected to high vibration

amplitudes. This induces non-linear effects in the lamina of the composite

material. Analytical methods are important for understanding the influence of

different parameters on the response of the structure, and complete and validate

the numerical methods. Numerous studies have been devoted to analytical and

numerical methods [1-2] for investigating plate large vibration amplitudes.

The lack of symmetry about the mid-plane of asymmetric laminated composite

plates leads to bending- stretching coupling inducing errors if linear theories are

applied [3]. Whitney and Leissa [4] considered the effect of this coupling in a

non-linear dynamic plate theory. The unbalanced residual stresses created after

curing an asymmetric plate transforms the original plane design of the plate [5].

However, use of asymmetric laminated plates may be interesting if the laminate is

subjected to a non-symmetrical thermal field. Asymmetric composite laminates

have been the subject of different technique and design approaches. Few years

ago, asymmetric laminates have been studied using the classical lamination theory

[6, 7]. Asymmetric composite beams subjected to large vibration amplitudes were

analyzed using the finite element method by Singh et al. [8]. They presented the

bending-extension coupling by two second order, ordinary, differential non-linear

equations. The non-linear free vibration response of simply supported angle-ply

plates has been studied by Bannet [9]. Chandra and Raju examined the large

vibration amplitudes of cross-ply and angle-ply plates using a two-term

perturbation solution for non-symmetric laminates [l0-12]. Ugo Icardi [13-14]

analysed a non-linear asymmetric cross-ply laminate using a higher order layer

theory and applying a third -order zig-zag approach. He concluded that the use of

the equilibrium equations of elasticity provides a good accuracy for the value of

the transverse shear stress. Hegaze [15] has been interested by the dynamic

analysis of unsymmetric stiffened and unstiffened composite laminated plates

with a higher-order finite element. Kharazi and Ovesy [16] used the first-order

Shear Deformation Theory (FSDT) and the Rayleigh–Ritz method to study the

large deflections of unsymmetric delaminated composite plates. Yazdani and

Ribeiro [17] presented a geometrically non-linear static analysis of unsymmetric

composite plates with curvilinear fibers using the p-version layer approach. They

studied the effect of unsymmetric stacking sequences and fiber orientations.

Dhurvey and Mittal [18] presented a review on various studies on composite

laminates with ply drop-off. Pirbodaghi, Fesanghary and Ahmadian [19] analyzed

the non-linear vibrations of laminated composite plates resting on non-linear

elastic foundations using the homotopy analysis method (HAM). This study

showed that only a first-order approximation of the HAM leads to highly accurate

solutions for this type of non-linear problems.

Geometrically non-linear vibrations 5403

The aim of this paper was the extension of the theoretical model developed by

(Benamar Bennouna and White 1993) [23] using a technique of homogenization

to analyze the geometrical non-linear free dynamic response of asymmetrically

laminated skew plates in order to investigate the effect of non-linearity on the

non-linear resonance frequencies and the non-linear fundamental mode shape at

large vibration amplitudes. The general formulation of the model for non-linear

vibration of laminated plates at large vibration amplitudes is presented. Fully

clamped boundaries have been considered here and periodic displacements were

assumed. The results obtained have been compared to the previously published

ones. The first non-linear mode shape and the associated bending stresses have

been investigated. The amplitude dependence of the non-linear resonance

frequency ratio nl/l on the vibration amplitude associated to the first non-linear

mode shape, for various values of the plate skew angle, the number of layers, the

aspect ratio and the fiber orientation are discussed.

The results showed a hardening type of nonlinearity with a higher increase in the

nonlinear frequency ratio with increasing the skew angle and the amplitude of

vibration for the case of symmetrically cross ply laminated composite skew plates,

compared to the asymmetric case. The bending stress also increases with

increasing the skew angle and reaches its maximum value for the skew angle =45°

in the case of an asymmetrically laminated skew plate. The maximum is obtained

near to the plate centre.

2 Geometrically Non-Linear Vibrations of Fully Clamped

Laminated Composite Skew Plates

2.1 Constitutive Equation of a Laminated Skew Plate at Large Deflections

Consider the skew plate with a skew angle shown in Fig.1. For the large

vibration amplitudes formulation developed here, it is assumed that the material of

the plate is elastic, isotropic and homogeneous. The thickness of the plate is

considered to be sufficiently small so as to avoid the effects of shear deformation.

The skew plate has the following characteristics: a, b, S: length, width and area

of the plate; x-y: plate co-ordinates in the length and the width directions; -η, H:

Skew plate co-ordinates and plate thickness; E, : Young’s modulus and

Poisson’s ratio; D, : plate bending stiffness and mass per unit volume.

5404 Moulay Abdelali Hanane and Benamar Rhali

Fig.1 Composite Skew plate in x-y and -η co-ordinate system

For the classical plate laminated theory, the strain-displacement relationship for

large deflections are given by:

{𝜀} = {𝜀0} + 𝑧 {𝑘} + {0} (1)

in which {𝜀0}, {𝑘} and {0} are given by:

{ε0} =

[

∂U

∂x∂V

∂y

∂U

∂y+

∂V

∂x]

; {k} = [

kx

ky

kxy

] =

[

−∂2W

∂x2

−∂2W

∂Y2

−2 ∂2W

∂xy ]

; {0} = [

x0

y0

xy0

] =

[ 1

2(∂W

∂x)2

1

2(∂W

∂y)2

∂W

∂x ∂W

∂y ]

(2)

U, V and W are displacements of the plate mid-plane, in the x, y and z directions

respectively. For the laminated plate having n layers, the stress in the Kth layer can

be expressed in terms of the laminated middle surface strains and curvatures as:

{σk} = [Q̅]k{ε} (3)

in which {𝜎}𝑘𝑇 = [𝜎𝑥𝜎𝑦𝜎𝑥𝑦 ] and terms of the matrix [�̅�] can be obtained by the

relationships given in reference (Timoshenko 1975) [20]. The in-plane forces and

bending moments in a plate are given by:

Geometrically non-linear vibrations 5405

[NM

] = [ A B B D

] [{ε0} + {0}

{k}] (4)

A, B and D are the symmetric matrices given by the following Equation 5. [𝐵] = 0

for symmetrically laminated plates (Mallick 1993) [21].

(Aij, Bij, Dij) = ∫ Qij(k)(1, z, z2)

H2⁄

−H2⁄

dz (5)

Here the 𝑄𝑖𝑗(𝑘)

are the reduced stiffness coefficients of the kth layer in the plate

co-ordinates. The transverse displacement function W may be written as in

reference (Ribeiro and Petyt 1999) [22] in the form of a double series:

W = {Ak}T{W} sinkωt (6)

Where {Ak}T = {a1

k, a2k, … , an

k} is the matrix of coefficients corresponding to the

kth harmonic, {W}T = {w1, w2, … , wn} is the basic spatial functions matrix, k is

the number of harmonics taken in to account, and the usual summation convention

on the repeated index k is used. As in reference (Benamar, Bennouna and White

1993) [23], only the term corresponding to k=1 has been taken into account, which

has led to the displacement function series reduced, to only one harmonic: i.e.

W = ai wi(x, y)sinωt (7)

Here the usual summation convention for the repeated indexes i is used. i is

summed over the range 1 to n, with n being the number of basic functions

considered. The expression for the bending strain energy Vb, axial strain energy Va

and kinetic energy T are given in reference (Harras 2001) [24] in the rectangular

co-ordinate (x,y). The skew co-ordinates are related to the rectangular co-ordinate

(,) by: =x-y tan ; =y/cos. So, the strain energy due to bending Vb, axial

strain energy Va and kinetic energy T are given in the -η co-ordinate system. In

the above expressions, the assumption of neglecting the in-plane displacements U

and V in the energy expressions has been made as for the fully clamped rectangular

isotropic plates analysis considered in reference (Benamar, Bennouna and White

1993) [23]. Discretization of the strain and kinetic energy expressions can be

carried out leading to:

Vb =1

2sin2(ωt)aiajkij ; Va =

1

2sin4(ωt)aiajakalbijkl ; T =

1

2ω2cos2(ωt)aiajmij (8)

in which mij, kij and bijkl are the mass tensor, the rigidity tensor and the geometrical

non-linearity tensor respectively. Non-dimensional formulation of the non-linear

vibration problem has been carried out as follows.

5406 Moulay Abdelali Hanane and Benamar Rhali

𝐰𝐢(, 𝛈) = 𝐇𝐰𝐢∗ (

𝐚,𝛈

𝐛) = 𝐇𝐰𝐢

∗(∗, 𝛈∗) (9)

where ∗and 𝛈∗are non-dimensional co-ordinates ∗ =

𝐚 𝐚𝐧𝐝 𝛈∗ =

𝛈

𝐛 one then

obtains:

kij =aH5E

b3 kij∗ ; bijkl =

aH5E

b3 bijkl∗ ; mij = ρH3abmij

∗ (10)

where the non-dimensional tensors m*ij, k*ij and b*ijkl are given in terms of

integrals of the non-dimensional basic function wi*, non-dimensional extensional

and bending stiffness coefficient A*ij and D*

ij , skew angle and aspect ratio α.

Upon neglecting energy dissipation, the equation of motion derived from

Hamilton’s principle is:

δ∫ (V − T)2π

0= 0 (11)

where V= Vb +Vm+Vc. The expression for the bending strain energy Vb, the

membrane strain energy Vm, the coupling strain energy Vc and the kinetic energy T.

The skew co-ordinates are related to the rectangular co-ordinate (,) by: =x-y

tan ; =y/cos. So, the strain energy due to bending Vb, the membrane strain

energy Vm, the coupling strain energy Vc and the kinetic energy T are given below.

In order to simplify the expressions and reduce the problem to that of a

homogeneous skew plate, the change of variable z1=z-c is made, with c =Bij

Aij.

Consequently, the coupling term Vc vanishes and the bending strain and the

membrane strain energy expressions are written in the -η co-ordinate system.

Insertion of Equations 8 into Equation 11, and derivation with respect to the

unknown constants ai, leads to the following set of non-linear algebraic equations:

𝟐𝐚𝐢𝐤𝐢𝐫∗ + 𝟑𝐚𝐢𝐚𝐣𝐚𝐤𝐛𝐢𝐣𝐤𝐫

∗ − 𝟐𝛚∗𝐚𝐢𝐦𝐢𝐫∗ = 𝟎, r=1…n. (12)

These have to be solved numerically. To complete the formulation, the procedure

developed in (Harras 2001) [24] is adopted to obtain the first non-linear mode. As

no dissipation is considered here, a supplementary equation can be obtained by

applying the principle of conservation of energy, which leads to the equation:

𝛚∗𝟐 =𝐚𝐢𝐚𝐣𝐤𝐢𝐣

∗ +(𝟑/𝟐)𝐚𝐢𝐚𝐣𝐚𝐤𝐚𝐥𝐛𝐢𝐣𝐤𝐥∗

𝐚𝐢𝐚𝐣𝐦𝐢𝐣∗ (13)

This expression for ω*2 is substituted into Equation 12 to obtain a system of n

non-linear algebraic equations leading to the contribution coefficients ai, i=1 to n. ω

and ω* are the non-linear frequency and non-dimensional non-linear frequency

parameters related by:

Geometrically non-linear vibrations 5407

ω2 =D

b4cos4θω∗2

(14)

To obtain the first non-linear mode shape of the skew plate considered, the

contribution of the first basic function is first fixed and the other basic functions

contributions are calculated via the numerical solutions of the remaining (n-1)

non-linear algebraic equations.

2.1. Bending stress analysis

Many analytical and numerical methods used to model the geometrically non-linear

vibration of plates and beams considered transverse displacement only [25]. It is

well known that the estimation of the total non-linear stress must include the

in-plane displacement U and V. But, the scope of the present work permitted only

consideration of bending stress, which allows only a qualitative understanding of

the non-linearity effects. The bending stress associated to the fundamental

non-linear mode shape was examined, which allows a quantitative understanding

of the non-linearity effects. The maximum bending strains b and b obtained for

z=H/2 are given by:

εb =H

2(∂2W

∂2). (15)

εb =H

2(∂2W

∂2

1

cosθ−

2tanθ

cosθ

∂2W

∂∂+

∂2W

∂2tan2θ). (16)

For z=H/2 which corresponds to a layer with =90°, one has Q16=Q26=0. For the

skew plate, the associated stresses b and b can be obtained by applying the

plane stress Hooke’s law as:

b =HQ11

2(∂2W

∂2) +

HQ12

2(∂2W

∂2

1

cosθ−

2tanθ

cosθ

∂2W

∂∂+

∂2W

∂2tan2θ). (17)

b =HQ12

2(∂2W

∂2) +

HQ22

2(

∂2W

∂2

1

cosθ−

2tanθ

cosθ

∂2W

∂∂+

∂2W

∂2tan2θ). (18)

In terms of the non-dimensional parameters defined in the previous section, the

non- dimensional stresses *b and *b are then given by:

∗b = (2 ∂2W∗

∂∗2 + (∂2W∗

∂∗2

1

cosθ−

2tanθ

cosθ

∂2W∗

∂∗ ∂∗ + 2 ∂2W∗

∂∗2 tan2θ)). (19)

∗b = (γ

∂2W∗

∂∗2

1

cosθ− γ

2tanθ

cosθ

∂2W∗

∂∗ ∂∗ + γ2 ∂2W∗

∂∗2 tan2θ + 2 ∂2W∗

∂∗2 ). (20)

where =b/a, =Q11/Q12 and =Q22/Q12.

5408 Moulay Abdelali Hanane and Benamar Rhali

The relationships between the dimensional and non-dimensional stresses are:

=Q12H2

2b2 ∗. (21)

3. Results and Discussion

The aim of this section is to apply the theoretical model presented above to

analyze the geometrical non-linear free dynamic response of skew fully clamped

symmetrically and asymmetrically laminated plates in order to investigate the

effect of non-linearity on the non-linear resonance frequencies and non-linear

fundamental mode shape at large vibration amplitudes. Convergence studies are

carried out, and the results are compared with those available from the literature

through a few examples of laminated composite skew thin clamped plates with

different fiber orientation and aspect ratio. The material properties, used in the

present analysis are: (1) Plate 1, 3 and 6-layers symmetrically cross-ply

APC2/AS4 Peek/carbon laminate square plate [0°/90°/0°] and [90°/0°/90°]sym ET

= 131 GPa; EL = 5 GPa; GLT=2.855 Gpa LT = 0.25. (2) Plate 2, square

antisymmetrically cross-ply [0°/90°] fiberglass reinforced plastic clamped plate;

EL=206.84 Gpa; ET=5.171 Gpa; GLT=2.855 Gpa; LT = 0.25. (3) Plate 3, 4-layers

cross-ply asymmetrically laminated plate [0°/90°/0°/90°]; with the same

proprieties as plate 2. Plate 4, 4-layers cross-ply symmetrically laminated plate

[0°/90°/90°/0°] with the same proprieties as plate 2, where E, G and are Young’s

modulus, shear modulus and Poisson’s ratio. Subscripts L and T represent the

longitudinal and transverse directions respectively with respect to the fibers. All

the layers are of equal thickness. Calculation was made by using 18 functions

corresponding to three symmetric beam functions in the direction and three

symmetric beam functions in the η direction, and three anti-symmetric beam

functions in the direction and three anti-symmetric beam functions in the η

direction. Table 1 shows nonlinear to linear frequency ratios (nl/l) for square

symmetrically cross-ply APC2/AS4 laminated plate 1 obtained using only 18

well-chosen plate functions. It can be seen that there is a good convergence with

results presented in reference (Harras 2001) [24]. The variation of

non-dimensional nonlinear frequency ratio nl/l with respect to non-dimensional

maximum amplitude wmax/h is evaluated for antisymmetrically cross-ply [0°/90°]

fiberglass reinforced plastic clamped plate 2 for an aspect ratio =1 and is shown

in Tables 2, along with published results. It is observed that present results are in

close agreement with those of assumed direct numerical integration (five-point

Gauss rule) presented in reference [26]. Next, the nonlinear free vibration

behaviors of fully clamped cross-ply asymmetrically and symmetrically thin

square and skew plates 3 and 4 are examined in Table 3. It is observed that the

results obtained here closely match with increasing skew angle for the case of

asymmetrically cross ply laminated composite skew plates 3, however the

nonlinear frequency ratio increases with increasing skew angle and amplitude of

vibration for the case of symmetrically cross ply laminated composite skew plates

Geometrically non-linear vibrations 5409

4. It is indicating hardening type of nonlinear behavior. Furthermore, for the

chosen amplitude, it can be noted that, with the increase in skew angle, the degree

of nonlinearity is high.

Table 1 Nonlinear to linear frequency ratios (nl/l) for square (a/b=1) symmetrically

cross-ply APC2/AS4 laminated plate 1.

W/h [0°/90°/0°] [90°/0°/90°/90°/0°/90°]

nll

Present

results

Ref

[24]

Error

%

Present

results

Ref

[24]

Error

%

0,2 1,0074 - - 1,008 1,0074 0,06

0,4 1,0288 - - 1,0315 1,0295 0,19

0,5 1,0464 1,0455 0,09 1,0507 - -

0,6 1,0665 - - 1,0684 1,0645 0,36

0,8 1,1125 - - 1,1163 1,1105 0,52

1 1,1693 1,1671 0,18 1,1751 1,1657 0,8

Table 2 Nonlinear to linear frequency ratios (nl/l) for square antisymmetrically

cross-ply [0°/90°] fiberglass reinforced plastic clamped plate (a/h=150).

W/h [0°/90°]

nll

Present results Ref [26] Relative error %

0,3 1,0507 1,048 0,3

0,6 1,1798 1,1827 0,2

0,9 1,3604 1,3762 1,2

1,2 1,5451 1,6069 4

1,5 1,8038 1,861 3,2

1,8 2,0104 2,1304 6

2,1 2,253 2,4099 7

5410 Moulay Abdelali Hanane and Benamar Rhali

Table 3 Nonlinear to linear frequency ratios (nl/l) for skew cross-ply asymmetrically

and symmetrically clamped plate.

W/h [0°/90°/0°/90°] [0°/90°/90°/0°]

nl/l

Skew angle ° Skew angle °

0° 30° 45° 0° 30° 45°

0,2 1,0086 1,0087 1,0087 1,0076 1,0097 1,0134

0,4 1,036 1,0354 1,0344 1,0309 1,039 1,0518

0,6 1,0797 1,0764 1,0736 1,0653 1,0818 1,1113

0,8 1,1393 1,1318 1,1247 1,1122 1,1393 1,187

1 1,2126 1,2053 1,1926 1,1716 1,211 1,2795

1,2 1,2756 1,2657 1,2589 1,2327 1,2838 1,3717

Figure 2 and Figure 3 shows respectively the non-linear stresses along the

section *=0.5 at W*max=2 for the rectangular case =0° and the skew cases

=30° and =45° for the asymmetrically and symmetrically composite skew

plates 3 and 4. It can be noticed that the stress was increasing with increasing

skew angle and had maximum for the skew angles =45° in the asymmetrically

laminated skew plate near to the centre of the plate.

Fig. 2 Comparison between the non-dimensional bending stress distribution associated

with the fundamental non-linear mode of a fully clamped asymmetrically composite plate

3 for =0(°), =30(°) and =45(°) with W* max =2 and =1 along the section *=0.5.

=30°

=0°

=45°

𝜎∗

∗

Geometrically non-linear vibrations 5411

Fig. 3 Comparison between the non-dimensional bending stress distribution associated

with the fundamental non-linear mode of a fully clamped symmetrically composite plate

4 for =0(°), =30(°) and =45(°) with W* max =2 and =1 along the section *=0.5.

Conclusion

The theoretical model established and applied to beams, plates and shells

(Benamar, Bennouna and White 1993) [23] has been successfully applied to

calculate the first non-linear mode shape of fully clamped skew symmetrically

and asymmetrically laminated plates. A technique of homogenization was used

for asymmetrically laminated plates for various values of the skew angle, the

number of layers, the plate thickness ratio and different fiber orientation. The

present formulation has been verified with the available results in the literature.

Limited numerical studies are conducted to examine the effect of the skew angle,

the fiber orientation, and the aspect ratio on the large-amplitude frequency of

composite skew plates. As has been shown in this work, the fundamental

non-linear mode shape of the fully clamped skew asymmetrically and

symmetrically laminated plate can be obtained with a sufficient accuracy by using

18 basic plate functions. This provides a more accurate description of the

non-linear mode shape, compared with previous results based on the use of nine

symmetric basic plate functions since it allows the non-symmetry induced by the

fiber orientation to be taken into account. The present study reveals a hardening type of nonlinearity for the asymmetrically and symmetrically laminated skew plate

=30°

=0°

=45°

𝜎∗

∗

5412 Moulay Abdelali Hanane and Benamar Rhali

and the nonlinear frequency ratio in general increases with the increase in the

skew angle. The asymmetrically laminated plate presented higher values of

bending stresses compared with the symmetrically laminated plate in the

direction.

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Received: June 24, 2015; Published: August 25, 2015