Composite Structures 94 (2012) 2424–2432

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Non-linear vibrations of variable stiffness composite laminated plates

Pedro Ribeiro ⇑, Hamed AkhavanDEMec/IDMEC, Faculdade de Engenharia, Universidade do Porto, R. Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Available online 30 March 2012

Keywords:VibrationsNon-linearLaminated compositesVariable stiffnessPlatesFirst-order shear

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.03.025

⇑ Corresponding author. Tel.: +351 22 508 1721; faE-mail addresses: pmleal@fe.up.pt (P. Ribeiro), ham

a b s t r a c t

It is the objective of this work to analyze vibrations of variable stiffness composite laminated plates(VSCL), and investigate the differences between the oscillations of these plates and traditional laminates.The analysis is based on numerical experiments and a new p-version finite element with hierarchic basisfunctions, which follows first order shear deformation theory and considers geometrical non-linearity, isderived. Considering first linear oscillations, the natural frequencies and mode shapes of different VSCLare computed and compared with the ones of constant stiffness laminates. The linear natural frequenciesof the present model are also compared with the ones computed using a recently developed higher-ordermodel for VSCL. After, numerical tests are carried out in the time domain and, for the first time in VSCL,taking geometric non-linearity into account, to investigate the response to external forces. The non-linearordinary differential equations of motion are solved by Newmark’s method. It is verified that the varia-tion of the fibre orientation can lead to significant differences in the amplitudes of the non-linearresponse.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The vast majority of investigations on composite layered platesassume that the fibres are straight. This is not necessarily true,since tow-placement technology is capable of controlling fibretows individually and placing them onto the surface of a laminatewith curvilinear fibre paths [1,2]. Traditional, constant-stiffness,laminated composite panels have a number of well known advan-tages over panels in other materials, including the fact that theyare lightweight. However, the possibility to vary the fibre orienta-tion offers new design options and can be used to further optimisethe laminate in comparison to the conventional straight-fibre lam-inates, improving the structural response and achieving additionalweight reduction. On the other hand, machines that efficiently dis-pose fibres with a variable orientation are a somewhat recent real-ity and there are many unexplored issues regarding the mechanicalbehaviour of this type of variable stiffness panels.

One of the first publications on variable stiffness compositelaminates (VSCL) is Ref. [3], by Hyer and Lee, who obtainedimprovements in buckling resistance of plates with a hole by usingcurvilinear fibre paths. Another early study was published by Gür-dal and Olmedo [4], where the in-plane elastic response of panelswhere the fibre angle varies as a function of one spatial coordinatewas modelled. Tatting, Gürdal and co-workers [1,5,6] used the ex-tra freedom that is provided by curvilinear fibre paths to optimiseflat panels for maximum buckling load, taking into account

ll rights reserved.

x: +351 22 508 1445.ed@fe.up.pt (H. Akhavan).

‘manufacturing limits of tow-placement machines and exploringother properties, as for example the overall in-plane stiffness.Increases of the buckling load up to twice the value of conventionalstraight-fibre panels were achieved. A study on the effect ofthermal stresses on the buckling performance of VSCL again indi-cates that these panels can perform better than constant stiffnessones [7].

The number of publications that address vibrations of lami-nated fibre panels with curvilinear fibre paths is particularly small.In [8] the maximisation of the fundamental frequency of variablestiffness composite plates was treated; numerical results showedthat a significant increase in the optimal fundamental frequencymay be achieved using VSCL. In [9] a study is conducted with thesame goal – maximising the fundamental frequency – but nowon VSCL conical shells. Also in this case a higher fundamental fre-quency is found in VSCL than in constant stiffness shells. More re-cently two journal papers have been published. It is shown in [10]that plates with curvilinear fibres result in higher fundamental fre-quencies than plates with straight fibres and that there are specificoptimum fibre orientations along the plate boundaries. In [11] athird-order shear deformation theory (TSDT) model for linear VSCLwas presented. Maps of natural frequencies as functions of tow-orientation angles were determined in demonstrative examplesand it the variations in the vibration frequencies of VSCL and CSCLwere discussed. The literature review hence reveals that VSCL offerthe possibility to achieve better performances, but, also, that thereare many unexplored issues concerning vibrations of VSCL. In par-ticular, the authors are not aware of any paper on non-linear vibra-tions of VSCL.

P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432 2425

The finite element method may be used to analyse VSCL. In thecommon version – the so called h-version – of the finite elementmethod, finer meshes may be needed in VSCL then in constantstiffness panels, due to the curved tow paths. This mesh require-ment leads to models with a large number of degrees of freedom[8], which require long computational times, particularly in non-linear analysis.

In this paper vibrations of variable stiffness laminates are inves-tigated. The linear modes of vibration are defined and the re-sponses of different plates to harmonic and to step forces areanalysed, in order to verify to what extent variable stiffness lami-nates respond differently from constant stiffness laminates. Forthis purpose, a p-version finite element with hierarchic basis func-tions is developed. The element follows first-order-shear deforma-tion theory [12] and considers geometrical non-linearity [12–18],which becomes important when the vibration amplitudes are notsmall in comparison with the thickness, as is known to occur inaeronautics and aerospace structures. In the p-version of the FEMthe accuracy of the approximation is improved by increasing thenumber of shape functions over the elements. The p-version FEMhas been applied to investigate free and forced geometricallynon-linear vibrations in [15–17] and other studies, always requir-ing a relatively reduced number of degrees of freedom. Newmark’smethod [19] is used to solve the differential equations of motion inthe time domain.

2. Formulation

First-order shear deformation theory (FSDT) is adopted here,because it allows considering the effects of shear deformationand rotary inertia with a reasonably small computational cost[12,15–17]. Therefore, the displacement components in the x, yand z directions, respectively represented by u, v and w are givenby

uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ z/0yðx; y; tÞ vðx; y; z; tÞ

¼ v0ðx; y; tÞ � z/0x ðx; y; tÞ wðx; y; z; tÞ ¼ w0ðx; y; tÞ ð1Þ

where u0, v0 and w0 are the values of u, v and w at the reference sur-face, and /0

x and /0y are the independent rotations about the x and y

axis of lines normal to the middle surface. The origin of the Carte-sian coordinate system is located in the centre of the undeformedplate; axis x and y define the plate’s middle reference plane.

In each p-version finite element for VSCL the middle plane dis-placements are given by

u0ðn;g; tÞv0ðn;g; tÞw0ðn;g; tÞ/0

x ðn;g; tÞ/0

yðn;g; tÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

Nuðn;gÞT 0 0 0 0

0 Nuðn;gÞT 0 0 0

0 0 Nwðn;gÞT 0 0

0 0 0 N/x ðn;gÞT 0

0 0 0 0 N/y ðn;gÞT

266666664

377777775

quðtÞqv ðtÞqwðtÞq/xðtÞ

q/yðtÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð2Þ

Symbols n and g represent local coordinates; vectors Ni(n,g)contain the shape functions and vectors qi(t) the generalised dis-placements (i = u, v, w, /x and /y). In the p-version FEM, the meshis not changed and the discretization error is reduced by increasingthe number of shape functions. It was proved by analyzing the er-ror and verified in several numerical examples that the p-version ofthe FEM generally converges faster than the h-version [15–17,21].

The basis functions here used to form vectors Ni(n,g) are the onespreviously employed in Refs. [12,15], which can be consulted fordetails on these functions. The set of basis functions is hierarchic,i.e. the finite element space Sp�1, spanned by polynomial basisfunctions with degree up to p � 1, is contained by the space Sp,spanned by basis functions up to degree p. Nu(n,g) appears twicein Eq. (2) because we are using the same functions for the twomembrane displacement components.

In our case one element suffices, and the local and global coor-dinates are simply related by

x ¼ an=2 y ¼ bg=2 ð3Þ

with a representing the plate length and b the plate width.The displacement relations of FSDT considering solely the Von

Kármán non-linear terms [18,23] (only the non-linear terms re-lated with the transverse displacement), are:

ex

ey

cxy

8><>:

9>=>; ¼

1 0 0 �z 0 00 1 0 0 �z 00 0 1 0 0 �z

264

375 em

0 þ enl

eb0

( ) !ð4Þ

em0 ¼

u0;x

v0;y

u0;y þ v0

;x

8><>:

9>=>; eb

o ¼�/0

y;x

/0x;y

�/0y;y þ /0

x;x

8>><>>:

9>>=>>; enl ¼

ðw0;xÞ

2=2

ðw0;yÞ

2=2

w0;xw0

;y

8>><>>:

9>>=>>; ð5Þ

where ex, ey represent the membrane strain components in the x andy directions and cxy the membrane shear strain. Partial derivation isrepresented by a comma. In Eqs. (4) and (5), function arguments arenot written in order to simplify the notation (e.g. u0

;x actually isu0;xðx; y; tÞ); simplifications of this kind are often adopted in this text.

The following linear strain displacement relation is employedfor the transverse shear strains:

czx

cyz

( )¼

w0;x þ /0

y

w0;y � /0

x

( )ð6Þ

Although the fibre orientation varies in the VSCL under analysis,stresses and strains are still related by

r1

r2

s23

s13

s12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ðkÞ

¼

Q11 Q 12 0 0 0Q21 Q 22 0 0 0

0 0 Q 44 0 00 0 0 Q 55 00 0 0 0 Q 66

26666664

37777775

ðkÞ e1

e2

c23

c13

c12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ðkÞ

ð7Þ

with numbers 1, 2 and 3 indicating the principal material axes x1, x2

and x3. Eq. (7) is common to traditional orthotropic lamina [20,24],the difference is that in the VSCL under analysis the orientation ofthe principal material axis is not constant in each lamina.

Refs. [20,24] give the plane-stress reduced stiffnesses Qij asfunctions of the longitudinal and transverse moduli of elasticity(ek

1; ek2, respectively), Poisson ratios ðmk

12; mk21Þ and shear moduli

ðgk12; g

k13; g

k23Þ. The same references also give transformed stress–

strain relations in a form similar to

rxðx; y; z; tÞryðx; y; z; tÞsxyðx; y; z; tÞ

8><>:

9>=>;ðkÞ

¼Q11ðx; yÞ Q 12ðx; yÞ Q 16ðx; yÞQ12ðx; yÞ Q 22ðx; yÞ Q 26ðx; yÞQ16ðx; yÞ Q 26ðx; yÞ Q 66ðx; yÞ

264

375ðkÞ

�exðx; y; z; tÞeyðx; y; z; tÞcxyðx; y; z; tÞ

8><>:

9>=>;ðkÞ

syzðx; y; tÞszxðx; y; tÞ

� �ðkÞ¼ Q44ðx; yÞ Q 45ðx; yÞ

Q45ðx; yÞ Q 55ðx; yÞ

" #ðkÞcyzðx; y; tÞczxðx; y; tÞ

� �ðkÞð8Þ

2426 P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432

But we note that in Eq. (8) the transformed reduced stiffnessesQk

ijðx; yÞ are written as functions of coordinates x and y because, un-like it is traditional, they are not constant over the laminate. Spe-cifically, in the remainder of this work the reduced stiffnesses arefunctions of h(x) and can be written as in Eqs. (9) and (10):

Q k11ðh

kðxÞÞ ¼ U1 þ U2 cosð2hkðxÞÞ þ U3 cosð4hkðxÞÞ

Q k12ðh

kðxÞÞ ¼ U4 � U3 cosð4hkðxÞÞ

Q k22ðh

kðxÞÞ ¼ U1 � U2 cosð2hkðxÞÞ þ U3 cosð4hkðxÞÞ

Q k16ðh

kðxÞÞ ¼ 12

U2 sinð2hkðxÞÞ þ U3 sinð4hkðxÞÞ

Q k26ðh

kðxÞÞ ¼ 12

U2 sinð2hkðxÞÞ � U3 sinð4hkðxÞÞ

Q k66ðh

kðxÞÞ ¼ U5 � U3 cosð4hkðxÞÞ

ð9Þ

Expressions similar to the former, but for lamina with straightfibres, can be found for example in [25]. We could not find in theliterature corresponding expressions for Q k

44ðhkðxÞÞ, Q k

45ðhkðxÞÞ and

Qk55ðh

kðxÞÞ, and established the following:

Q k44ðh

kðxÞÞ ¼ U6 þ U7 cosð2hkðxÞÞ

Q k45ðh

kðxÞÞ ¼ �U7 sinð2hkðxÞÞ

Q k55ðh

kðxÞÞ ¼ U6 � U7 cosð2hkðxÞÞ

ð10Þ

Ui are defined as:

U1 ¼18ð3Q 11 þ 3Q 22 þ 2Q12 þ 4Q 66Þ

U2 ¼12ðQ 11 � Q22Þ

U3 ¼18ðQ 11 þ Q22 � 2Q 12 � 4Q66Þ

U4 ¼18ðQ 11 þ Q22 þ 6Q 12 � 4Q66Þ

U5 ¼18ðQ 11 þ Q22 � 2Q 12 þ 4Q66Þ

U6 ¼12ðQ 44 þ Q55Þ

U7 ¼12ðQ 44 � Q55Þ

ð11Þ

Ui, i = 1–5, can again be read in [25] and two additional parametersneeded for transverse shear, U6 and U7, were here introduced. Eqs.(9) and (10) have the advantage that parts of the expressions forthe transformed reduced stiffness are invariant, which may becomeuseful when examining their variation. In our particular case, it hasanother important advantage and this is computational: the expres-sions of the reduced stiffness become simpler, in the sense that theyinvolve less trigonometric terms, and therefore lead to simplerintegrals.

The membrane forces V, the moment resultants M and thetransverse forces T (all per unit length) are given by the constitu-tive relations

Vðx; y; tÞMðx; y; tÞ

� �¼

AðxÞ BðxÞBðxÞ DðxÞ

� �em

0 ðx; y; tÞ þ enlðx; y; tÞeb

0ðx; y; tÞ

( ) !ð12Þ

Tðx; y; tÞ ¼ CðxÞcðx; y; tÞ ð13Þ

where matrices A(x), B(x), C(x) and D(x) are functions of x. Theextensional stiffness matrix is

AðxÞ ¼Pnk¼1

hk

U1ðhkðxÞÞ U4ðhkðxÞÞ 0

U4ðhkðxÞÞ U1ðhkðxÞÞ 0

0 0 U5ðhkðxÞÞ

2664

3775

0BB@

þ U2ðhkðxÞÞ1 0 0

0 �1 0

0 0 0

264

375 cosð2hkðxÞÞ

þ U3ðhkðxÞÞ1 �1 0

�1 1 0

0 0 �1

264

375 cosð4hkðxÞÞ

þ 12

U2ðhkðxÞÞ0 0 1

0 0 1

1 1 0

264

375 sinð2hkðxÞÞ

þ U3ðhkðxÞÞ0 0 1

0 0 �1

1 �1 0

264

375 sinð4hkðxÞÞ

1CA ð14Þ

Only laminates symmetric about their middle plane will be ana-lysed and, therefore, the bending-extensional coupling matrix B(x)is null: there is no linear coupling between the membrane defor-mations and bending. The bending stiffness matrix is given by

DðxÞ ¼Pnk¼1

ðz3k � z3

k�1Þ3

U1ðhkðxÞÞ U4ðhkðxÞÞ 0

U4ðhkðxÞÞ U1ðhkðxÞÞ 0

0 0 U5ðhkðxÞÞ

264

375

0B@

þ U2ðhkðxÞÞ1 0 00 �1 00 0 0

264

375 cosð2hkðxÞÞþ

þ U3ðhkðxÞÞ1 �1 0�1 1 00 0 �1

264

375 cosð4hkðxÞÞ

þ U2ðhkðxÞÞ0 0 10 0 11 1 0

264

375 sinð2hkðxÞÞ

þ U3ðhkðxÞÞ0 0 10 0 �11 �1 0

264

375 sinð4hkðxÞÞ

1CA ð15Þ

Matrix C(x), which relates transverse forces per unit length withthe transverse shear, is given by

CðxÞ ¼ kPnk¼1

hk U6ðhkðxÞÞ1 00 1

� �þ U7ðhkðxÞÞ

1 00 �1

� �cosð2hkðxÞÞþ

�

þ U7ðhkðxÞÞ0 �1�1 0

� �sinð2hkðxÞÞ

�ð16Þ

In the present case, the virtual work of the internal forces can bewritten as follows (strain arguments not represented; X indicatesintegration over x and y):

dWin ¼ �Z

Xdem

0 TAðxÞem0 þ deb

0TDðxÞeb0 þ dem

nlAðxÞem0 þ dem

0 AðxÞemnl

�þ dem

nlAðxÞemnl þ dem

nlAðxÞemnl þ dczyCðxÞczy þ dcxzCðxÞcxz

idX

ð17ÞIt is accepted, as in laminates with straight fibres, that the mass

distribution of the plane is homogeneous, therefore the virtualwork of inertia forces is the common one, already employed in firstorder shear deformation p-version FEM formulations in Ref. [12].The virtual work of the external forces is also common to tradi-tional laminates, and can be found, for example, in [16].

P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432 2427

Applying the principle of virtual work, equations of motion ofthe following form are obtained:

M11u 0 0 0 0

0 M22v 0 0 0

0 0 M33w 0 0

0 0 0 M44Rx 0

0 0 0 0 M55Ry

266666664

377777775

€quðtÞ€qvðtÞ€qwðtÞ€q/xðtÞ

€q/yðtÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ

K11Luu K12

Luv 0 0 0

K21Lvu K22

Lvv 0 0 0

0 0 K33Lc K34

Lc K35Lc

0 0 K43Lc K44

Lb þ K44Lc K45

Lb þ K45Lc

0 0 K53Lc K54

Lb þ K54Lc K55

Lb þ K55Lc

2666666664

3777777775

quðtÞqvðtÞqwðtÞq/xðtÞq/yðtÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ

0 0 KNLðqwðtÞÞ132 0 0

0 0 KNLðqwðtÞÞ232 0 0

KNLðqwðtÞÞ313 KNLðqwðtÞÞ

323 KNLðqwðtÞÞ

334 0 0

0 0 0 0 00 0 0 0 0

266666664

377777775

quðtÞqvðtÞqwðtÞq/xðtÞ

q/yðtÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

PuðtÞPvðtÞPwðtÞ

m/xðtÞ

m/yðtÞ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð18Þ

The superscripts 1–5 indicate the position of each sub-matrix.The mass matrix is constituted by sub-matrices Mii

k , with k = u, v,w, Rx or Ry, standing, respectively, for membrane in the directionof x, membrane in the direction of y, transverse, rotational aboutaxis x and rotational about y. The constant stiffness matrix is con-stituted by sub-matrices of the type Kij

LK, where k = uu, uv, vu, vv,c, b; with c standing for shear and b for bending. Matrices of typeKNLðqwðtÞÞ

ij2 and KNLðqwðtÞÞ

ij3 depend linearly on the transverse gen-

eralised coordinates qw(t) and matrix KNLðqwðtÞÞ334 depends qua-

dratically on these coordinates. The equations of motion havequadratic and cubic non-linear terms and the system is non-auton-omous, since the vector of generalised external forces, on the righthand side, is an explicit function of time.

Since FSDT is adopted, a shear correction factor should be used.In the present work the relation between the thickness and thesmallest plate length is at the most equal to, and generally lowerthan, 0.1 and the shear correction factor k = 5/6 is accepted asappropriate. Although several correction factors have been pro-posed by different authors and this choice may be contended, itis to some extent justified by a literature review. In fact, it is shownin Ref. [26] that k = 5/6 provides results that are rather accurate inmany situations and even for thick panels. Additionally, it is shownin Chapter 2 of Ref. [22] that with FSDT and k = 5/6 fairly accuratetransverse shear stresses of moderately thick (h/a 6 0.1) laminated

x

y

T0

T1

a/2

θ (x)

Fig. 1. Representation of fibre with variable orientation.

plates can be computed. Finally, the natural frequencies of thepresent model will be compared with the ones computed usingthe third-order shear deformation theory of Ref. [11], which dis-penses with the use of a shear correction factor. We will see thatthe present model performs rather well.

The fibre angles can be varied in several ways [4,7]. In this studya linear fibre variation function, which, in spite of its simplicity, al-lows one to generate panels with a reasonably large range of prop-erties, will be selected. Hence, in lamina k, the orientation of fibresalong the x axis is defined by

hkðxÞ ¼ 2ðTk1 � Tk

0Þa

jxj þ Tk0 ð19Þ

Therefore, the fibre angle is Tk0 at the panel midlength x = 0, and

Tk1 at the panel ends x = �a/2 and x = a/2. This leads to a fibre vary-

ing as represented in Fig. 1.

3. Numerical tests and discussion

To obtain an insight on the dynamic characteristics ofvariable stiffness laminates, numerical results from an eightlayer ½�2hTk

0jTk1i�S laminate are presented. By ½�2hTk

0jTk1i�S we

mean ½�hTk0jT

k1i; hT

k0jT

k1i;�hT

k0jT

k1i; hT

k0jT

k1i; hT

k0jT

k1i;�hT

k0jT

k1i; hT

k0jT

k1i;

�hTk0jT

k1i�. Although according to Eq. (19) each variable stiffness

layer is in this work characterised by only two parameters, Tk0

and Tk1, several laminate configurations can still be constructed.

We will make Tk0 either equal to +45� or �45�; more specifically:

in the layers immediately below and above the middle planeTk

0 ¼ þ45�, and Tk0 alternates between �45� and +45� in the

remaining layers. The panels differ in Tk1, which, when Tk

0 is +45�(�45�), ranges from �45� to +90� (+45� to �90�). Extreme varia-tions in the fibre curvature may lead to kinking [2]; hence, only areduced number of results from numerical tests on laminateswhere the fibre curvature is larger are here shown. Laminate[±h45|45i]S is a traditional constant stiffness laminate, that pro-vides a reference for comparison.

The material properties were taken from Ref. [27] and areproperties that may occur in a carbon fibre reinforced epoxy:E1 = 131.7� 109 N/m2, E2 = 9.86 � 109 N/m2, G12 = G13 = G23 =4.21 � 109 N/m2, t12 = 0.28, q = 1600 kg m�3. The length of all panelsis a = 300 mm. In the thinner panel, the total thickness is h = 1.2 mm (a/h = 250). Since FSDT is adopted and due to the relative lack of studieson vibrations of VSCL plates, it is interesting to investigate how thethickness interferes with the vibrations of those plates. For that pur-pose, we will also analyse thicker panels. Moreover, in the linear re-gime, results from the present FSDT model will be compared withdata computed using the model based on third-order-shear-deforma-tion theory, presented recently [11]. Thisallows us not only to validatethe present model, but also to ascertain how it performs in thickerlaminates. The boundaries are fully clamped.

3.1. Linear modes – free vibrations

Natural frequencies and mode shapes of a few VSCL vibrating inthe linear regime are shown in this section. A convergence study iscarried out to start with.

In the first six natural frequencies, disposed in increasing order,of a square plate where [±2h45|0i]S and with thickness equal to a/250 are given. The frequencies are computed with diverse numbersof shape functions, but, to simplify the convergence study, the fol-lowing relation holds: po = pi = ph = p. The same relation holdsthroughout this work; it is nevertheless left here the note that anoptimum model for large amplitude oscillations generally hasmore membrane than transverse shape functions [15–17]. The fre-quencies computed with p = 17, a very large number of shape func-tions, offer reference values, which are used to compute the

Table 1Linear natural frequencies of [±h45|0i]S plate computed with different numbers of shape functions (po = pi = ph = p).

Mode p = 5 p = 6 p = 7 p = 8 p = 9 p = 11 p = 13 p = 17

1 863.16 (.48 � 10�2) 860.81 (.21� 10�2) 860.08 (.12� 10�2) 859.57 (.65� 10�3) 859.47 (.53� 10�3) 859.24 (.26� 10�3) 859.12 (.12� 10�3) 859.012 1393.0 (.63� 10�2) 1390.2 (.43� 10�2) 1384.9 (.42� 10�3) 1384.53 (.17� 10�3) 1384.4 (.82� 10�4) 1384.3 (.36� 10�4) 1384.3 (.19� 10�4) 1384.33 2100.6 (.35� 10�1) 2052.1 (.11� 10�1) 2038.7 (.42� 10�2) 2034.9 (.24� 10�2) 2032.0 (.94� 10�3) 2030.7 (.32� 10�3) 2030.3 (.12� 10�3) 2030.14 2257.2 (.39� 10�1) 2224.7 (.25� 10�1) 2175.1 (.17� 10�2) 2173.2 (.83� 10�3) 2171.7 (.14� 10�3) 2171.6 (.60e-4) 2171.5 (.27� 10�4) 2171.45 2609.9 (.20� 10�1) 2576.4 (.68� 10�2) 2563.1 (.16� 10�2) 2560.7 (.60� 10�3) 2559.9 (.30� 10�3) 2559.4 (.12� 10�3) 2559.2 (.52� 10�4) 2559.16 3323.4 (.56� 10�1) 3254.3 (.34� 10�1) 3209.5 (.20� 10�1) 3156.9 (.35� 10�2) 3149.4 (.11� 10�2) 3146.2 (.10� 10�3) 3146.0 (.41� 10�4) 3145.9

Table 2Linear natural frequencies of [±h45|30i]S, h = a/20, plate computed with different numbers of shape functions (po = pi = ph = p).

Mode p = 5 p = 6 p = 7 p = 8 p = 9 p = 11 p = 13 p = 17

1 9193.9 (6.5 � 10�3) 9189.2 (6.0� 10�3) 9166.2 (3.5� 10�3) 9164.6 (3.3� 10�3) 9152.9 (2.0� 10�3) 9145.2 (1.2� 10�3) 9140.2 (6.4� 10�4) 9134.32 16147 (1.2� 10�2) 16076 (7.6� 10�3) 16049 (5.9� 10�3) 16022 (4.2� 10�3) 16009 (3.4� 10�3) 15986 (2.0� 10�3) 15972 (1.1� 10�3) 159553 18483 (1.4� 10�2) 18394 (9.5� 10�3) 18349 (7.0� 10�3) 18307 (4.7� 10�3) 18294 (4.0� 10�3) 18263 (2.3� 10�3) 18243 (1.2� 10�3) 182214 24740 (2.2� 10�2) 24458 (1.1� 10�2) 24411 (8.7� 10�3) 24337 (5.6� 10�3) 24320 (4.9� 10�3) 24269 (2.8� 10�3) 24238 (1.5� 10�3) 242015 26909 (3.6� 10�2) 26335 (1.3� 10�2) 26198 (8.2� 10�3) 26157 (6.6� 10�3) 26100 (4.4� 10�3) 26053 (2.6� 10�3) 26022 (1.4� 10�3) 259866 31416 (6.3� 10�2) 30135 (2.0� 10�2) 29929 (1.3� 10�2) 29801 (8.6� 10�3) 29711 (5.6� 10�3) 29637 (3.1� 10�3) 29594 (1.6� 10�3) 29545

Table 3Linear natural frequencies of plates with diverse angle variations, a = 250h, computed with po = pi = ph = p = 7.

Mode [±2h45|�45i] [±2h45|�30i] [±2h45|�15i] [±2h45|0i]S [±2h45|15i]S [±2h45|30i]S [±2h45|45i]S [±2h45|60i]S [±2h45|75i]S [±2h45|90i]S

1 714.8 782.8 835.3 860.1 856.2 830.5 793.9 759.2 738.1 731.62 1168 1224 1302 1385 1460 1518 1523 1395 1272 12013 1873 1922 2026 2039 1940 1795 1665 1667 1710 17624 1933 2034 2072 2175 2336 2441 2429 2290 2058 1913

1 This notation is classical in isotropic plates, where (m,n) indicates a mode with malf-waves in the x direction and n half waves in the y direction. In laminatedmposites, only exceptionally the half-waves develop in the direction of the axis,

ence the use of expression ‘‘close to’’ in this text.

2428 P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432

relative errors given in brackets. One verifies that when p = 5 thefirst five frequencies are computed with errors below 0.039(3.9%), with p = 7 the first five frequencies are computed with er-rors below 0.42% and with p = 9, the first six natural frequenciesare computed with errors below 0.11% (see Table 1).

Another example, now on a thicker plate, of the convergencetests that were carried out is shown in Table 2. This gives the firstsix natural frequencies of a square plate where [±2h45|30i]S andwith thickness equal to a/20. Taking again the frequencies of thep = 17 model as references, one verifies that with p = 5, the first fivefrequencies are computed with errors below 3.6%, with p = 7 thefirst five frequencies are computed with errors below 0.87% andthat with p = 9, the first six natural frequencies are computed witherrors below 0.56%.

A study on the combined effect of varying the fibre orientationand the panel thickness resulted in Tables 3–6. These tables pres-ent the first four natural frequencies of square plates, with thick-ness a/250, a/50 h, a/20 and a/10. The models employed have245 DOF (p = 7). The fibre orientations in the lamina of each plateare defined by [±2h45|0i]S, [±2h45|15i]S, [±2h45|30i]S, [±2h45|45i]S,[±2h45|60i]S, [±2h45|75i]S and [±2h45|90i]S. In the sole case ofplates where h = a/250 fibre variations [±2h45|-45i]S, [±2h45|-30i]S, [±2h45|-15i]S are also considered.

Although Tables 4–6 include results computed using the third-order shear deformation model of [11], the FSDT results of the ac-tual model are discussed first and on their own. The comparisonwith the TSDT is carried out in a later paragraph. One verifies, asexpected, that the frequencies vary with the fibre orientation.However, it is also seen that the variation occurs in a slightly differ-ent way for different thicknesses. In the thinner plates – a = 250hand a = 50h – the first natural frequency decreases steadily from[±2h45|0i]S to [±2h45|90i]S, in the thicker plates – a = 20h anda = 10h – the fundamental frequency slightly increases from[±2h45|0i]S to [±2h45|15i]S and only after decreases with increasingTk

1 until the minimum value is attained at [±2h45|90i]S. The total

variation of the fundamental frequency from plate [±2h45|0i]S toplate [±2h45|90i]S, in percentage and using the average value asweight, also decreases as the thickness increases. In the thinnerplate the fundamental frequency varies from 8.1% to �8.1% aboutthe average value; in plate a = 50h it varies from 7.7% to �7.7%,in plate a = 20h it varies from 5.9% to �6.7% and in plate a = 10hit varies from 3.3% to �5.2%.

The frequencies of the thinner plates where [±2h45|�45i],[±2h45|�30i], [±2h45|�15i] do not have any remarkable feature,with the exception of the plate with the larger fibre curvature,[±2h45|�45i], which is of all plates the one with lower fundamen-tal frequency.

In what modes two and three are concerned, their natural fre-quencies approach each other (the second natural frequency in-creases and the third decreases) as the fibre varies less along theplate, that is they become closer from [±2h45|0i]S until[±2h45|45i]S and deviate again (the second natural frequency de-creases and the third increases) until [±2h45|90i]S. Actually, ifone inspects the mode shapes, one sees that mode number twois first close to a (1,2) mode and after Tk

1 ¼ 45� becomes closer toa (2,1) mode,1 swapping with mode three. Hence, if we follow themode shapes instead of the value of the natural frequencies, wewould say that the frequency of modes close to (1,2) steadily in-crease, whilst the natural frequency of modes close to (2,1) steadilydecreases.

The natural frequency of the fourth mode attains its larger valuefor [±2h45|30i]S in the thinner plates – a = 250h and a = 50h – andfor [±2h45|45i]S in the thicker plates.

We then conclude that the way and extent to which the varia-tion of natural frequencies occur, somewhat depends on the plate

hcoh

T0=45º, T1=0º T0=45º, T1=15º T0=45º, T1=30º T0=45º, T1=45º T0=45º, T1=60º T0=45º, T1=75º T0=45º, T1=90º

xy

Fig. 2. Mode shapes of plates a/h = 250.

Table 4Linear natural frequencies of plates with diverse angle variations, a = 50h, computed with po = pi = ph = p = 7. The third-order shear deformation theory frequencies werecomputed using a model presented in [11].

Mode [±2h45|0i]S [±2h45|15i]S [±2h45|30i]S [±2h45|45i]S [±2h45|60i]S [±2h45|75i]S [±2h45|90i]S

FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT

1 4192.5 4274.3 4177.3 4258.4 4058.7 4137.2 3886.5 3961.7 3722.3 3794.2 3623.7 3693.7 3591.7 3661.22 6708.2 6836.7 7064.2 7199.4 7348.5 7488.4 7383.8 7522.8 6784.6 6913.7 6197.2 6315.4 5850.9 5962.53 9693.6 9881.8 9267.2 9447.2 8627.8 8796.7 8041.5 8201.3 8056.7 8214.9 8255.8 8417.2 8481.2 8646.84 10465 10661 11201 11407 11671 11881 11622 11832 11025 11227 9970.6 10160 9308.0 9486.9

Table 5Linear natural frequencies of plates with diverse angle variations, a = 20h, computed with po = pi = ph = p = 7. TSDT model presented in [11].

Mode [±2h45|0i]S [±2h45|15i]S [±2h45|30i]S [±2h45|45i]S [±2h45|60i]S [±2h45|75i]S [±2h45|90i]S

FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT

1 9342.6 9544.7 9349.8 9547 9166.2 9352.7 8859.3 9036.7 8535.0 8705.8 8323.9 8491 8239.4 8405.42 14660 14960 15391 15697 16049 16350 16242 16520 15090 15369 13875 14140 13145 134013 19823 20269 19265 19686 18349 18745 17408 17804 17468 17842 17755 18128 18014 183934 22371 22791 23641 24040 24411 24783 24439 24804 23640 24000 21879 22279 20692 21096

Table 6Linear natural frequencies of plates with diverse angle variations, a = 10h, computed with po = pi = ph = p = 7. TSDT model presented in [11].

Mode [±2h45|0i]S [±2h45|15i]S [±2h45|30i]S [±2h45|45i]S [±2h45|60i]S [±21h45|75i]S [±2h45|90i]S

FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT FSDT TSDT

1 14418 14874 14570 14991 14551 14921 14316 14652 13920 14244 13570 13895 13370 137002 22335 22957 23321 23906 24304 24815 24719 25123 23477 23929 21902 22372 20882 213623 27264 28321 27018 27942 26510 27307 25826 26612 25894 26603 25990 26699 25967 267134 32946 33737 33935 34670 34730 35359 34985 35535 34485 34996 33242 33792 32134 32777

P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432 2429

thickness. Larger variations are seen to occur, in percentage terms,in thinner plates.

Still in what the natural frequencies are concerned, we nowinvestigate how the results obtained with the present FSDT model

compare with the ones of the TSDT model introduced in [11].Tables 4–6 show that the two models give similar results,although, as occurs in traditional composite laminates and in iso-tropic plates, the difference between FSDT and TSDT increases with

T0=45º, T1=0º T0=45º, T1=15º T0=45º, T1=30º T0=45º, T1=45º T0=45º, T1=60º T0=45º, T1=75º T0=45º, T1=90º

xy

Fig. 3. Mode shapes of plates a/h = 20.

- 1.25

-0.75

-0.25

0.25

0.75

1.25

600 600.2 600.4 600.6 600.8 601

t /T e

w(0, 0, t)

h

Fig. 4. Transverse displacement of middle point of plates a/h = 250, excited byharmonic forces at the respective linear fundamental frequency. T0 = 45� and: —T1 = 0�, T1 = 15�, T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

0.6

0.7

0.8

0.9

1

1.1

1.2

2 2.02 2.04 2.06 2.08 2.1

t (s)

w(0, 0, t)

h

Fig. 7. Steady-state transverse displacement of middle point of plates a/h = 250,excited by step force, initial cycles. T0 = 45� and: — T1 = 0�, T1 = 15�,T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

-0.5-0.4-0.3-0.2-0.10

0.10.20.30.40.5

800 800.2 800.4 800.6 800.8 801t/Te

w(0, 0, t)

h

Fig. 5. Transverse displacement of middle point of plates a/h = 250, excited byharmonic forces at the fourth natural frequency. T0 = 45� and: — T1 = 0�,T1 = 15�, T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

-0.25

0.25

0.75

1.25

1.75

2.25

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

t (s)

w(0, 0, t)

h

Fig. 6. Transverse displacement of middle point of plates a/h = 250, excited by stepforce, initial cycles. T0 = 45� and: — T1 = 0�, T1 = 15�, T1 = 30�, T1 = 45�,— T1 = 60�, T1 = 90�.

2430 P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432

the plate thickness. Nevertheless, taking the TSDT data as refer-ence, the absolute value of the relative difference between the fun-damental frequencies of the thicker plate (a = 10h) is, in all cases,smaller than 3.01%. Whichever the thickness, the maximum differ-ence in the fundamental frequencies – i.e. in the first natural fre-quencies – takes place in the VSCL plate [±2h45|0i]S. Themaximum difference between the TSDT and FSDT also occurs with

this fibre distribution and on the thicker plate: it is about 3.7%, onthe third natural frequency. These comparisons indicate that thepresent FSDT model is rather accurate until a/h = 10, particularlyif the first vibration mode is of interest.

An idea on the variation of mode shapes with the fibre orienta-tion is given by Figs. 2 and 3. The change in the first mode shape isrelatively minor and not evident in a perspective plot. The varia-tion in modes 2 and 3 is more perceptive, in particular the swapthat occurs at T0 = 45�, T1 = 45�. Finally, the change in modenumber 4 as the fibre orientation varies is rather evident. Thechange in the mode shapes takes place in a rather similar way inthe thin (a/h = 250) and in the moderately thick plate (a/h = 20).

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

400 400.2 400.4 400.6 400.8 401

t/Te

w(0, 0, t)

h

Fig. 8. Transverse displacement of middle point when the a/h = 20 plates areexcited by harmonic forces at the respective linear fundamental frequency. T0 = 45�and: — T1 = 0�, T1 = 15�, T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

0.0240.0250.0260.0270.0280.0290.030.0310.032

0.2 0.201 0.202 0.203 0.204 0.205 0.206 0.207 0.208 0.209 0.21

w(0, 0, t)

h

t (s)

Fig. 11. Steady-state transverse displacement of middle point when the a/h = 20plates are excited by step forces, steady state. T0 = 45� and: — T1 = 0�, T1 = 15�,

T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

400.2 400.4 400.6 400.8 401 401.2t/Te

w(0, 0, t)

h

Fig. 9. Transverse displacement of middle point when the a/h = 20 plates areexcited by harmonic forces at the respective fourth natural frequency. T0 = 45� and:— T1 = 0�, T1 = 15�, T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

-0.01

0.01

0.03

0.05

0.07

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

t (s)

w(0, 0, t)

h

Fig. 10. Transverse displacement of middle point when the a/h = 20 plates areexcited by step forces, initial cycles. T0 = 45� and: — T1 = 0�, T1 = 15�,T1 = 30�, T1 = 45�, — T1 = 60�, T1 = 90�.

P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432 2431

3.2. Forced oscillations

Numerical tests in the time domain are now carried out onplates where [±2h45|0i]S, [±2h45|30i]S, [±2h45|45i]S, [±2h45|60i]S

and [±2h45|90i]S, and two thicknesses are considered: a/h = 250and a/h = 20. The plates length and width are a = b = 0.3 m, theremaining properties were given above. Two types of variation ofthe excitation with time are considered: harmonic and a step in-put. The force amplitude does not change in space (it is a uniformpressure). The initial conditions are zero displacement and zerovelocity. Stiffness proportional damping is assumed and the pro-portionality damping parameter is defined as a = 0.01/x‘1, where0.01 is the value employed for a damping coefficient in laminatesin Ref. [28]. With the damping parameter defined in this way, ifeach plate is represented by a one degree of freedom system, thenondimensional damping ratio (damping coefficient over the crit-ical damping coefficient [29]) does not change from plate to plate.

Fig. 4 represents the transverse displacements of the middlepoints of a constant stiffness and several VSCL plates. Each plateis excited by a harmonic force at the respective fundamental linearfrequency and with amplitude 100 N/m2. Te is the excitation peri-od, employed to non-dimensionalise time in the horizontal axis.Plates where the fibre angle at the edges x = ±a/2 is higher showlower displacement amplitude, which is not surprising since thisfibre orientation leads to softening at the boundaries. Moreover,the first mode shapes of all the plates are rather similar, hencethe plates are affected in a similar way by the external force (seeFig. 2 for the mode shapes of the plates defined in linear vibrations.In the non-linear regime, the shape of the plate will change duringthe vibration, but, in most circumstances and for moderate vibra-tion amplitudes, it probably remains similar to the mode shapeof the linear system [12,30]). It is important to note that the VSCLwhere T1 = 0�, 15� and 30� experience smaller displacement ampli-tudes than the constant stiffness plate (T1 = 45�).

Fig. 5 represents the transverse displacements when the platesare excited by harmonic forces at the respective fourth linear fre-quency and with amplitude 500 N/m2. In this case the constantstiffness plate is the one that attains lower displacement ampli-tude. This is related with the particular shape that this plate as-sumes in its fourth vibration mode. Moreover, the fourth linearnatural frequency of this plate was the largest, demonstrating thatthe stiffness related to this mode is higher in this plate (Table 3).

Figs. 6 and 7 represent the transverse displacements of thesame plates, now excited by a step force with amplitude 103 N/m2. In this case, as occurred with the excitation at the fundamentalfrequency, the T1 = 0� VSCL plate experiences the lower displace-ment amplitudes and plates where the fibre orientation with re-spect to the x axis is larger at the x = ±a/2 edges experiencehigher amplitude deflections. Again, the constant stiffness plateis not the one with lower amplitude vibration displacement.

Now we pass to thicker plates, in order to investigate if the fibrevariation affects differently thick plates. Fig. 8 represents the trans-verse displacements of the middle points of plates where a/h = 20,excited by harmonic forces at the respective fundamental linearfrequency and with amplitude 105 N/m2. As occurred with the thinplate, as T1 increases, the displacement amplitude also increasesand in VSCL plates where the fibres run closer to parallel to theboundaries (T1 = 60� and T1 = 90�), a lack of stiffness result in con-siderably larger displacement amplitudes.

Fig. 9 show the steady-state displacements of the middle pointsin direction z when the plates are excited by harmonic forces at thefourth natural frequency. In this case all displacements are lowamplitude and the constant stiffness plate is the one where loweramplitudes are attained.

The last case study addresses the response of the a/h = 20 platesto a step load with amplitude equal to 5 � 105 N/m2. Although thisis quite a large excitation amplitude, the plates experience low

2432 P. Ribeiro, H. Akhavan / Composite Structures 94 (2012) 2424–2432

amplitude displacements in the transient phase (Fig. 10) and, par-ticularly, when steady state is achieved (Fig. 11). The VSCL plateswith T1 = 0� and T1 = 15� have nearly equal steady state displace-ment at the middle point. Three plates attain lower displacementamplitudes than the plate where T1 = 45� (the constant stiffnessplate). In the other VSCL here analyzed, larger displacement ampli-tudes are achieved, a behaviour we attribute to softening close toboundaries in these plates.

4. Conclusions

A computationally efficient model to study linear and non-lin-ear oscillations of VSCL was presented. The model follows a p-ver-sion FEM approach, employing a hierarchical set of displacementshape functions, and assumes constant shear deformation. Themode shapes and natural frequencies of diverse plates withclamped boundaries were computed, by considering the free, lin-ear, vibration problem. It was verified that both in thin and moder-ately thick plates, the frequencies and mode shapes change withthe diverse fibre orientation variations. The change in mode shapeswas found to be stronger in higher modes than in the first mode.The response to harmonic and impulsive forces was investigatedin the time domain. Also in this case, it was verified that plates thathave similar characteristics in everything except in the way the fi-bre varies within the lamina, experience different oscillationamplitudes. In case the mode mainly excited is the first, becausethis mode is rather similar in all plates, plates with fibres closerto perpendicular to the boundaries tend to vibrate with smalleramplitudes than plates with fibres running closer to parallel. If ahigher mode is excited, the oscillations amplitude depend stronglyon the respective mode shape, which, as said, can vary more signif-icantly from plate to plate with the fibre curvature. Hence, in linearand non-linear vibrations, the variation of the fibre orientationwithin a lamina allows one to obtain, with the same materialsand geometric properties, plates that have different dynamicbehaviour and, in a few cases, behave in a more rigid fashion.

Acknowledgements

This work was supported by the Portuguese Science and Tech-nology Foundation (FCT) through project PTDC/EME-PME/098967/2008. This support is gratefully acknowledged.

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