Semi-analytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination

Composites: Part B 45 (2013) 587–600

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Semi-analytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination Mohammad H. Kargarnovin ⇑, Mohammad T. Ahmadian, Ramazan-Ali Jafari-Talookolaei, Maryam Abedi School of Mechanical Engineering, Sharif University of Technology, 14588-89694 Tehran, Iran

a r t i c l e

i n f o

Article history: Received 25 February 2012 Received in revised form 7 April 2012 Accepted 5 May 2012 Available online 14 May 2012 Keywords: A. Layered structures B. Delamination B. Vibration C. Analytical modeling

a b s t r a c t A rather new semi-analytical method towards investigating the free vibration analysis of generally laminated composite beam (LCB) with a delamination is presented. For the first time the combined effects of material couplings (bending–tension, bending–twist, and tension–twist couplings) with the effects of shear deformation, rotary inertia and Poisson’s effect are taken into account. The semi-analytical solution for the natural frequencies and mode shapes are presented by incorporating the constraint conditions using the method of Lagrange multipliers. To verify the validity and the accuracy of the obtained results, they were compared with the results from other available references. Very good agreements were observed. Furthermore, the effects of some parameters such as slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration, and delamination parameters on the dynamic response of the delaminated beam are examined. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Usage of advanced composite materials having high strengthstiffness, lightweight, fatigue resistance properties, etc., are widely practiced nowadays in various structural designs like aircraft, helicopters, automobiles, marine and submarine vehicles. However, it should be noted that the composite materials are very sensitive to the damage induced during their fabrication or service life. One of the commonly encountered types of defects or damages in the multi-layered composite structures is delamination. Such delamination damage is known to cause a degradation of overall stiffness and strength. Hence, existence of any delamination will change vibrational characteristics such as mode shapes and natural frequencies of such damaged structure. More importantly, the size and location of the delamination will play crucial role in these changes. Thus, any changes in the measured values for the natural frequencies and mode shapes in multi-layered composite will signify the presence of some invisible delaminations. As a result, considerable analytical, numerical, and experimental efforts have been expended to study the vibrational characteristics of the delaminated beam. The composite material for some specific applications may usually require the utilization of angle ply, anti-symmetric and unsymmetric laminates. Thus in such structural composite layups, appearance of bending–tension, bending–twist, and tension– twist couplings and Poisson’s effect will make the analysis more ⇑ Corresponding author. Tel.: +98 21 6616 5510; fax: +98 21 6600 0021. E-mail address: [email protected] (M.H. Kargarnovin). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.05.007

complicated. However, the classical laminated plate theory (CLPT), which neglects the transverse normal and shear stresses, has been used to study thin laminates but it has been pointed out by numerous researchers that, since the composites have a very low transverse shear modulus compared to their extensional modulus, the CLT is not adequate for the analysis of dynamic response even for beam with high slenderness ratios. Therefore, shear deformation is a better alternative in the analysis of such multi-layered composite structures. The first-order shear deformation theory (FSDT) assumes linear variation of in-plane displacements through the thickness. Since FSDT violates equilibrium conditions at the top and bottom faces of the plate, the shear correction factors are required to correct the unrealistic variation of the shear strain/ stress through the thickness. In order to avoid the usage of the shear correction factors, the higher-order shear deformation theories (HSDTs) based on power series expansion of displacements have been developed. A review of various shear deformation theories for the analysis of laminated composite plates is available in Refs. [1–3] and are not duplicated here. During the past decades, the free vibrations of the delaminated composite beams based on the classical theory have received considerable attentions by many researchers [4–15], but only few publications were devoted to include partially the Poisson’s effects and the influences of the couplings, shear deformation and rotary inertia for beams [16–23]. Wang et al. [4] have investigated the free vibrations of an isotropic beam with a through-width delamination by dividing the beam into four Euler–Bernoulli sub-beams that are joined together. By applying appropriate boundary and continuity conditions, the

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response of the beam has been obtained as a whole. However, the vibration modes were physically inadmissible for off-midplane delaminations. That is the delaminated layers have been assumed to deform ‘freely’ without touching each other and thus have different transverse deformations. Later on, Mujumdar and Suryanarayan [5] proposed a model based on the assumption that the delaminated layers are ‘constrained’ to have identical transverse deformations. This was referred to as the ‘constrained mode’ in contrast to the ‘free mode’ proposed by Wang et al. [4]. The predicted frequencies based on the ‘constrained mode’ were closer to the experimental values presented in [5]. Similar approach has been used by Tracy and Pardoen [6] for the LCB with hinged–hinged boundary conditions. Free vibrations of delaminated beam based on the CLT with respect to postbuckled states are studied by Yin and Jane [7], Jane and Chen [8] and Chang and Liang [9]. Della and Shu have presented the analytical solutions to the free vibration of composite beams with two non-overlapping [10], two overlapping [11,12], double [13,14] delaminations and bimaterial beams with single delamination [15]. Both the ‘free mode’ and ‘constrained mode’ models have been used in these papers. The free vibration analysis of composite beams with delamination using the finite element method has been presented by Ju et al. [16]. The presented model includes bending-extension coupling and transverse shear deformation. Two computational models have been used. In the first model, that is ‘‘free mode’’, all the nodal degrees of freedom in the delaminated region are independent except those at the connecting nodes at the ends of delaminations. In the second model, that is ‘‘constrained mode’’, the transverse nodal deflections of the elements of the upper sub-beam are assumed to be equal to the corresponding transverse nodal deflections of the elements of the lower sub-beam. Valoor and Chandrashekhara [17] have investigated the vibration analysis of the symmetric LCB with a delamination by considering the Poisson’s effect, transverse shear deformation and rotary inertia. In this work, it has been assumed that the sub-beams located at the delaminated region have the identical transverse displacement and rotation (the constrained mode). The ‘constrained mode’, however, failed to predict the opening in the mode shapes found in the experiments by Shen and Grady [18]. To capture the opening in the mode shapes found in the experiments [18], Luo and Hanagud [19] proposed an analytical model based on the Timoshenko beam theory, which uses piecewise-linear springs to simulate the ‘open’ and ‘closed’ behavior between the delaminated surfaces. Later on, shear deformation theory has been used to investigate the dynamic instability associated with composite beams with delamination that are subjected to the dynamic compressive loads using the finite element method [20]. Both transverse shear and rotary inertia effects are taken into account and the delaminated region was modeled using the ‘constrained mode’ model. The dynamic response of multi-layered composite beam-type structures with delaminations has been studied by Brandinelli and Massabo [21]. The delaminated region has been modeled by the linear elastic springs place uniformly between two surfaces in the delaminated region. Zhu et al. [22] have been presented a new finite element formulation, referred to as the Reference Surface Element (RSE) model, for numerical prediction of dynamic behavior of delaminated composite beams and plates. The ‘free mode’ and ‘constrained mode’ for dynamic analysis of delaminated composite beams have been unified in the RSE formulation. Liu and Shu [23] have developed analytical solutions to investigate the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration have been adopted. The main objective of this paper is to present a novel approach for the free vibrations of a thick LCB with a single delamination, in

which the solution method is based on the employment of Legendre polynomials in conjunction with Lagrange multipliers. To the best of author’s knowledge this work is the first one to be reported on the analysis of a delaminated thick LCB that the Poisson’s effect, material couplings, rotary inertia and shear deformation have been considered. To do this, (a) by considering the Poisson’s effect, the coupling of the bending–tension, bending–twist, and tension– twist, rotary inertia and shear deformation, the kinetic and potential energies expression of a LCB are derived, (b) by choosing the Legendre polynomials and imposing its orthogonality properties, the problem’s constraints and then corresponding functional are made, (c) by extremizing the functional, a system of linear algebraic equations, which contain the Legendre polynomial coefficients and Lagrange multipliers, are obtained, (d) next, substituting these linear algebraic equation in the constraint relations results in a system of homogenous linear algebraic equations with the Lagrange multipliers as unknowns, and (e) finally employing the eigenvalue technique, the natural frequencies and the corresponding mode shapes of the LCB with delamination are obtained. 2. Governing differential equations of the delaminated LCB 2.1. Geometrical modeling Fig. 1 shows a laminated beam having a single through width delamination. The composite beam has a length of L with a rectangular cross-section of b  h. The delamination dimension is L2  b and it is located at L1 with respect to the left end of the LCB as shown in Fig. 1. It should be noted that in this study only a single delamination is considered. However, this study can easily be extended to include the multiple delaminations. In order to model geometrically this single delamination, consider Fig. 2 in which the LCB can be viewed as a combination of four intact beams (i.e. four sub-beams of 1 to 4) connected at the delamination boundaries x = L1 and x = L1 + L2 and common surface between sub-beams 2 and 3 (see Fig. 3). In this way, we will have four sub-beams of 1 to 4 with lengths and thicknesses of Li  hi(i = 1–4) where L2 = L3, L4 = L  L1  L2, h1 = h4 = h and h2 and h3 are the thicknesses of sub-beams 2 and 3, respectively. 2.2. Derivation of kinetic and potential energies for the delaminated LCB One of the efficient ways of deriving dynamic characteristics of a system is to use the variational principle. In implementing this method, one has to derive the kinetic and potential energies of the whole system to form the functional. Furthermore, the boundary conditions or any other constraints at whole system have to be incorporated using Lagrange multipliers.

Fig. 1. A schematic of generally LCB with single delamination.

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Fig. 2. The delaminated beam is modeled by four interconnected sub-beams.

respectively. Also Q ij are the transformed material constants, ks is the shear correction factor and Aij(i, j = 4, 5) are the transverse shear stiffnesses. For a laminated beam, the membrane forces N y^ and N^xy^ and the bending moment My^ are zero [21] and Eq. (1) can be rewritten as:

fN^x ; M^x ; M^xy^ gT ¼ ½afe^x ; j^x ; j^xy^ gT þ ½bfey^ ; e^xy^ ; jy^ gT

ð3Þ

in which

2

A11 6 ½a ¼ 4 B11 B16 Fig. 3. Typical interface locations of the delamination for a [0/90]2s graphite/epoxy composite beam.

Based on this methodology, for the problem under consideration, i.e. laminated composite beam with delamination, primarily the kinetic and potential energies for this beam are derived, then the functional are established in which the boundary, continuity and compatibility conditions are introduced into the functional using Lagrange multipliers. By extremizing the functional, a system of linear algebraic equations including the Lagrange multipliers is obtained. Through lengthy mathematical manipulations, the natural frequencies and the corresponding mode shapes of the LCB with delamination are obtained where the detailed procedures are coming in the following sections. 2.2.1. Kinetic and potential energies for each sub-beam In this study we consider an intact LCB comprising of an anisotropic material. The laminate is made of many unidirectional plies stacked up in different orientation with respect to a reference axis. The length, width, thickness and number of layers of the intact LCB are represented by L, b, h, and n, respectively. The laminated plate constitutive equations based on the firstorder shear deformation theory can be expressed as [24]:

8 9 2 N^x > A11 > > > > > 6 > > > > N A12 ^ > y > > > 6 > < N ^^ > = 6 6 A16 xy ¼6 > M^x > > 6 > 6 B11 > > > > 6 > > My^ > > 4 B12 > > > > : ; M^xy^ B16

A12

A16

B11

B12

A22

A26

B12

B22

A26

A66

B16

B26

B12 B22

B16 B26

D11 D12

D12 D22

B26

B66

D16

D26

9 38 B16 > e^x > > > > > ey^ > > > B26 7 > > 7> > > > 7> < B66 7 e^xy^ = 7 > j^x > D16 7 > 7> > > > 7> > jy^ > > D26 5> > > > > : ; j^xy^ D66

Q y^^z



 ¼

Q ^x^z

A44

A45

A45

A55



c^y^z c^x^z

Aij ¼

Q kij ð^zk  ^zk1 Þ;

k¼1 n X   1 Q kij ^z3k  ^z3k1 Dij ¼ 3 k¼1 n X ks Q kij ð^zk  ^zk1 Þ Aij ¼ k¼1

Bij ¼ 12

A12 6 ½b ¼ 4 B12 B26

D11 D16

A16 B16 B66

B12

3

7 D12 5 D26

38 0 9 B16 > < e^x > = 7 D16 5 j^x > > : j^xy^ ; D66

B11 D11 D16

ð4Þ

ð5Þ

where (A11 , B11 , etc.) are the coefficients of the ([a]  [b][c]1[b]T) and the matrix [c] is given by:

2

A22

6 ½c ¼ 4 A26 B22

A26 A66 B26

B22

matrix

3

7 B26 5 D22

Using Eq. (2), the transverse shear force–strain relation for the LCB can be also expressed as [25]:

Q ^x^z ¼ A55 e^x^z ¼

A55 

A245 A44

!

e^x^z

ð6Þ

The strain–displacement relationship can be written as [24]:

@u

@w

@w

@w

e^0x ¼ ^ ; j^x ¼ ^^x ; j^xy^ ¼ ^y^ ; e^x^z ¼ w^x þ ^ @x @x @x @x

ð7Þ

^ where u and w are the LCB mid-plane displacements in the ^x and y directions, and w^x and wy^ are the mid-plane bending slopes. Now, the potential energy,U, for the beam can be calculated using the following relationship [25]:

U¼

1 2

Z

L



0

 N^x e^0x þ M^x j^x þ M ^xy^ j^xy^ þ Q ^x^z e^x^z bd^x

ð8Þ

In the next step, we can express the potential energy in terms of displacement components using Eqs. (5)–(8) as follows:

 ð2Þ

n X

2

7 D16 5; D66

8 9 2 > A11 < N^x > = 6 M^x ¼ 4 B11 > > : ; M^xy^ B16

  Q kij ^z2k  ^z2k1 ;

k¼1

"

ð9Þ

Next, we turn to the kinetic energy of the LCB using the following relation [25]:

T¼

In the above equations, N^x , N y^ , and N^xy^ are the in-plane forces, M^x and M y^ are the bending and M^xy^ is the twisting moments, Q y^^z and Q ^x^z are the resultant shear forces, ðe^x ; ey^ ; e^xy^ Þ are the mid-plane strains, j^x and jy^ are the bending and j^xy^ is the twisting curvatures, cy^^z and c^x^z are the shear strains, Aij, Bij and Dij(i, j = 1, 2, 6) are the extensional, bending–extension coupling, and bending stiffnesses,

L

0

ði; j ¼ 1; 2; 6Þ ði; j ¼ 4; 5Þ

Z

A11 2 u þ B11 u;^x w^x;^x þ B16 u;^x wy^;^x þ D16 w^x;^x wy^;^x 2 ;^x # D11 2 D66 2 A55  2 þ w^x;^x þ wy^;^x þ w^x þ w2;^x þ 2w^x w;^x bd^x 2 2 2

U¼

where n X

3

B16

Using Eq. (1), the ðey^ ; e^xy^ ; jy^ Þ components can be replaced by ðe^x ; j^x ; j^xy^ Þ and by substituting the results in Eq. (3) one obtains:

ð1Þ

and



B11

1 2

Z

L

0

h   i I1 u2;t þ w2;t þ I3 w2^x;t þ w2y^;t bd^x

ð10Þ

in which

ðI1 ; I3 Þ ¼

Z

h=2

qð1; ^z2 Þd^z

h=2

It should be mentioned that in all above relations, the symbol ‘‘,’’ used as a subscript stands for the differentiation with respect to any variable followed after it.

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2.2.2. Kinetic and potential energies for the whole delaminated LCB Referred to Fig. 2, the interaction between the delaminated subbeams 2 and 3 can be modeled as a distributed soft spring with stiffness of k [19]. In this way the total potential energy for the delaminated LCB by inclusion of this soft spring can be expressed in terms of displacements using Eq. (9) as follows:

" A11;i 2 U¼ u þ B11;i ui;^x w^xi;^x þ B16;i ui;^x wy^i;^x þ D16;i w^xi;^x wy^i;^x 2 ^x 0 i¼1 # D11;i 2 D66;i 2 A55;i  2 2 þ w þ w þ w^xi þ wi;^x þ 2w^xi wi;^x bd^x 2 ^xi;^x 2 y^i;^x 2 4 Z X

Clamped–Hinged beam (C–H):

U 1 ð1Þ ¼ 0;

Wx1 ð1Þ ¼ 0;

Wy1 ð1Þ ¼ 0;

W 1 ð1Þ ¼ 0

½Immovable ðI : C—HÞ : U 4 ð1Þ ¼ 0; Movable ðM : C—HÞ : Nx4 ð1Þ ¼ 0;

Wy4 ð1Þ ¼ 0;

M x4 ð1Þ ¼ 0;

W 4 ð1Þ ¼ 0

Li

1 þ kðw2  w3 Þ2 2

ð18bÞ Hinged–Hinged beam (H–H):

U 1 ð1Þ ¼ 0;

Mx1 ð1Þ ¼ 0;

Wy1 ð1Þ ¼ 0;

W 1 ð1Þ ¼ 0

½Immovable ðI : H—HÞ : U 4 ð1Þ ¼ 0; Movable ðM : H—HÞ : Nx4 ð1Þ ¼ 0;

ð11Þ

Wy4 ð1Þ ¼ 0;

M x4 ð1Þ ¼ 0;

W 4 ð1Þ ¼ 0

Moreover, the total kinetic energy of the delaminated LCB is:

Z 4  i X 1 Li h  2 T¼ I1;i ui;t þ w2i;t þ I3;i w2^xi;t þ w2y^i;t bd^x 2 0 i¼1

ð12Þ

ð18cÞ Clamped–Free beam (C–F):

½Immovable ðI : C—FÞ : U 1 ð1Þ ¼ 0;

The above relations for the kinetic and potential energies will be used in the section (3) to form the functional.

Movable ðM : C—FÞ : Nx1 ð1Þ ¼ 0;

Wy1 ð1Þ ¼ 0; Nx4 ð1Þ ¼ 0;

3. Solution method

Wx1 ð1Þ ¼ 0;

W 1 ð1Þ ¼ 0 M x4 ð1Þ ¼ 0;

Mxy4 ð1Þ ¼ 0;

Q xz4 ð1Þ ¼ 0 ð18dÞ

As a general way of dealing with the free vibration analysis in a continuous system, at the first step the technique of separation of variables has to be used. The choice of the assumed displacement function not necessarily satisfying the boundary conditions is the main advantage of the Lagrange multiplier technique which provides more freedom in selecting the deflection function [26]. In the present work, for the spatial function, the simple Legendre polynomials are chosen as displacement functions which as we can see later the orthogonality condition of these polynomials simplifies the calculation of energy term in delaminated LCB. Knowing that the time part of the beam response will be represented by harmonic functions, following general forms for the ui ð^ x; tÞ; wxi ð^ x; tÞ; wyi ð^ x; tÞ and wi ð^ x; tÞ are assumed:

ui ð^x; tÞ ¼ U i ð^xÞejxt ;

wxi ð^x; tÞ ¼ Wxi ð^xÞejxt ;

wi ð^x; tÞ ¼ W i ð^xÞejxt

ði ¼ 1; 2; 3; 4Þ

wyi ð^x; tÞ ¼ Wyi ð^xÞejxt ; ð16Þ

in which U i ð^xÞ, Wxi ð^xÞ, Wyi ð^xÞ and W i ð^xÞ are the spatial functions and x is the circular frequency. As mentioned above, the spatial function can be expressed in terms of the simple Legendre polynomials given by:

U i ðxÞ ¼

Wyi ðxÞ ¼

nt X U im Pm ðxÞ; m¼0 nt X

Wyim Pm ðxÞ;

Wxi ðxÞ ¼

nt X

Wxim Pm ðxÞ;

m¼0 nt X

W i ðxÞ ¼

m¼0

W im Pm ðxÞ;

ð17Þ

m¼0

Here, Pm is the simple Legendre polynomial of degree m and the ^ x has been transformed to x using ^ x ¼ ðx þ 1ÞLi =2j1
Furthermore, we have twenty-four continuity and compatibility conditions ðC:C:sÞ at delamination boundaries at the left and right ends of the sub-beams 2 and 3 as follows:C:C:s at the left end of sub-beams 2 and 3:

W 1 ð1Þ ¼ W 2 ð1Þ;

Wx1 ð1Þ ¼ Wx3 ð1Þ;

2 0 2 W ð1Þ ¼ U 2 ð1Þ; U 1 ð1Þ þ h3 W 01 ð1Þ ¼ U 3 ð1Þ L1 1 L1 Q xz1 ð1Þ ¼ Q xz2 ð1Þ þ Q xz3 ð1Þ; Nx1 ð1Þ ¼ N x2 ð1Þ þ Nx3 ð1Þ Mxy1 ð1Þ ¼ M xy2 ð1Þ þ M xy3 ð1Þ

Mx1 ð1Þ ¼ Mx2 ð1Þ þ Mx3 ð1Þ  h2 Nx2 ð1Þ þ h3 Nx3 ð1Þ

½Immovable ðI : C—CÞ : U 4 ð1Þ ¼ 0; Movable ðM : C—CÞ : Nx4 ð1Þ ¼ 0; Wx4 ð1Þ ¼ 0;

Wy4 ð1Þ ¼ 0; W 4 ð1Þ ¼ 0

ð19Þ

C:C:s at the right end of sub-beams 2 and 3:

W 4 ð1Þ ¼ W 2 ð1Þ;

Wx4 ð1Þ ¼ Wx3 ð1Þ;

W 4 ð1Þ ¼ W 3 ð1Þ;

Wx4 ð1Þ ¼ Wx2 ð1Þ; Wy4 ð1Þ ¼ Wy2 ð1Þ; Wy4 ð1Þ ¼ Wy3 ð1Þ

2 0 2 W ð1Þ ¼ U 2 ð1Þ; U 4 ð1Þ þ h3 W 04 ð1Þ ¼ U 3 ð1Þ L1 4 L1 Q xz4 ð1Þ ¼ Q xz2 ð1Þ þ Q xz3 ð1Þ; Nx4 ð1Þ ¼ N x2 ð1Þ þ Nx3 ð1Þ

U 4 ð1Þ  h2

Mxy4 ð1Þ ¼ Mxy2 ð1Þ þ M xy3 ð1Þ Mx4 ð1Þ ¼ M x2 ð1Þ þ Mx3 ð1Þ  h2 Nx2 ð1Þ þ h3 Nx3 ð1Þ

ð20Þ

A variational principle is formulated based on the kinetic and potential energies by a procedure similar to one followed by Washizu [26]. This variational principle along with the constraint conditions is used to solve any kind of vibration problems. The functional F to be extremized is given by the following expression:

F ¼UþT  

U 1 ð1Þ ¼ 0; Wx1 ð1Þ ¼ 0; Wy1 ð1Þ ¼ 0; W 1 ð1Þ ¼ 0

Wx1 ð1Þ ¼ Wx2 ð1Þ; Wy1 ð1Þ ¼ Wy2 ð1Þ; Wy1 ð1Þ ¼ Wy3 ð1Þ

U 1 ð1Þ  h2

8 X i¼1

Clamped–Clamped beam (C–C):

W 1 ð1Þ ¼ W 3 ð1Þ;

32 X

ki ðBC 0 sÞ 

20 X ki ðC:C:s at x ¼ L1 Þ i¼9

ki ðC:C:s at x ¼ L1 þ L2 Þ

ð21Þ

i¼21

ð18aÞ

where ki (i = 1, 2, . . . , 32) are the Lagrange multipliers. Substituting the assumed series for Ui(x), Wxi(x), Wyi(x) and Wi(x) (i = 1, 2, 3, 4) in Eq. (21) and making some simplifications yields:

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

2 m1 Z nt bX 4 2 c X A11;i b 1 X 4 F¼ U im ð2m  4k1  1ÞPm2k1 1 ðxÞ Li 1 m¼1 i¼1 k ¼0

Z

(

1

Pn ðxÞPm ðxÞdx ¼

1

1 2nþ1

n¼m

0

n–m

1

bX nt 2 c X 2B11;i b U in ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ Li n¼1 k ¼0

It should be mentioned that the relations related to BC’s and C:C:s in terms of Legendre polynomials are given in Appendix A. The necessary extremizing conditions are given by [26]:

Z

@F @F @F @F ¼ ¼ ¼ ¼ 0; @U im @ Wxim @ Wyim @W im

n1



2



nt X

1

Um

1 m¼1

m1 bX nt 2 c X ð2m  4k1  1ÞPm2k1 1 ðxÞ Wxn

ðm ¼ 0; 1; 2; . . .Þ and ði ¼ 1; 2; 3; 4Þ

n¼1

k1 ¼0

bXc 2B16;i b  ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ Li k ¼0 n1 2

Using Eq. (23) in conjunction with Eq. (22) results in a system of linear algebraic equations which, in matrix form can be written as:

2



Z

nt 1X

U im

1 m¼1

m1 bX nt 2 c X ð2m  4k1  1ÞPm2k1 1 ðxÞ Wyin

½Afq1 ; q2 ; q3 ; q4 gT ¼ fBg

n¼1

k1 ¼0

qi ¼ fU i0 ; U i1 ; . . . ; U im ;

2

Z

nt X

1

Wxm

m1 bX 2 c

1 m¼1

ð2m  4k1  1ÞPm2k1 1 ðxÞ

Wxi0 ; Wxi1 ; . . . ; Wxim ;

nt X

Wxn

n1 bX 2 c 2D16;i b  ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ Li k ¼0



ng 1X

Wxim

m1 bX 2 c

1 m¼1





n1 bX 2 c

n¼1

k2 ¼0

Wyin

Z

nt X

1

Wyim

1 m¼1



þ

ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ m1 bX 2 c

D66;i b Li

ð2m  4k1  1ÞPm2k1 1 ðxÞ

k1 ¼0

nt X

n1 bX 2 c

n¼1

k2 ¼0

Wyin

4. Numerical results

ð2n  4k2  1ÞPn2k2 1 ðxÞdx

In order to check on the accuracy of the solution technique in this paper, primarily the obtained results using our method are compared with those results for which we could obtain out of existing literatures. In the next step a delaminated Timoshenko beam under different boundary conditions is studied for which the exact solutions do not exist. On the following sub-sections detailed analyses are presented.

nt A55;i bLi X 2 A b W2 þ 55;i Li 4 m¼0 2m þ 1 xim



Z

1

nt X

W im

1 m¼1



ð2m  4k1  1ÞPm2k1 1 ðxÞ

k1 ¼0

4.1. Natural frequencies

n1 bX nt 2 c X W in ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ A55;i b

n¼1



m1 bX 2 c

Z

k2 ¼0 n1 bX nt 2 c X Wxim Pm ðxÞ W in ð2n  4k1  1ÞPn2k1 1 ðxÞdx

nt 1X

1 m¼0

n¼1

k1 ¼0

# nt x Li b X 2   I1;i U 2im þ I1;i W 2im þ I3;i W2xim þ I3;i W2yim 4 m¼0 2m þ 1 2

þ

nt 8 X kL2 b X 2  2 W 2m þ W 23m  2W 2m W 3m  ki ðBC’sÞ 4 m¼0 2m þ 1 i¼1



20 32 X X ki ðC:C:s at x ¼ L1 Þ  ki ðC:C:s at x ¼ L1 þ Ld Þ i¼9

ð22Þ

i¼21

In the calculation of the functional, we have used the following properties of Legendre polynomial [27]:

P0n ðxÞ ¼

n1 bX 2 c ð2n  4k  1ÞPn2k1 ðxÞðn P 1Þ

k¼0

ð25Þ

The natural frequencies and corresponding mode shapes of beams can be calculated using Eqs. (24) and (25). In calculating the natural frequency, the determinant of the coefficient matrix in Eq. (25) is computed for various values of frequency starting from a near zero value. Zero crossing of the determinant is identified and the corresponding value of frequency is the natural frequency of the beam in question.

ð2m  4k1  1ÞPm2k1 1 ðxÞ

k1 ¼0

nt X

Wyi0 ; Wyi1 ; . . . ; Wyim g ði ¼ 1; 2; 3; 4Þ

½Cfk1 ; k2 ; . . . ; k32 gT ¼ 0

2

Z

W i0 ; W i1 ; . . . ; W im ;

Solving Eq. (24) for Wim, Wxim, Wyim and Uim (i = 1, 2, 3, 4) and substituting the results into the constraint Eqs. (A.1)–(A.6) results in a system of homogenous linear algebraic equations in terms of the Lagrange multipliers:

n¼1

k1 ¼0

ð24Þ

in which the right hand side of Eq. (24) consists of Lagrange multipliers and:

n1 bX 2 c D11 b  ð2n  4k2  1ÞPn2k2 1 ðxÞdx þ L k ¼0



ð23Þ

Example 1. In order to show the accuracy of the presented method for a delaminated LCB, in addition to the convergence evaluation, checking on the validity of the results are also carried out. The beam is 266.7 mm long, 25.4 mm wide and 1.778 mm thick in which the delamination is located at the midplane and has a length of 101.6 mm with L1 = 117.5 mm. The eight-ply laminated beam with lay-up of [0/90/90/0]s glass/epoxy is considered having material properties taken from [17]. The obtained results for the natural frequencies of above beam under I: C–F boundary conditions (clamped at x = 0) are given in Table 1. As one can see in this table the obtained results for the first four natural frequencies are calculated using up to 125 terms in the series (i.e. Eq. (17)) and compared with some experimental and analytical results reported in the literatures. Moreover, it becomes clear that in our solution no significant variation can be seen in the results if more than 100 terms are used. Therefore, in all following case studies, presented results are obtained using only first 100 terms in the series. In addition, the comparison of our results indicates very good agreement with other references.

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

Table 1 Convergence study for a delaminated LCB. Mode number

1st 2nd 3rd 4th

Experiment [28]

n

Impulse

Sine sweep

[17]

[23]

10

25

50

75

100

125

16 98 223 441

17 99 223 440

15.73 96.86 224.77 458.32

15.96 94.95 256.74 454.26

15.85 95.85 226.35 456.74

15.50 95.60 226.12 456.33

15.50 95.56 225.96 456.04

15.50 95.56 225.92 455.92

15.50 95.56 225.92 455.87

15.50 95.56 225.92 455.87

Table 2 Fundamental frequencies (Hz) of the LCB with central delamination located at interface 1. Delamination length (mm)

Intact 25.4 50.8 76.2 101.6 a

Present a

Free

Cons.

81.87 80.18 75.07 66.78 55.79

81.87 80.18 75.07 66.78 55.79

Della and Shu [11]

Luo and Hanagud [19]

Hu et al. [29]

Shen and Grady [18]

Free

Cons.

Free

Cons.

Free

Cons.

Average test

Cons.

81.88 80.47 75.36 66.13 55.67

81.88 80.47 75.36 66.14 55.67

81.86 81.84 76.81 67.64 56.95

81.86 81.84 76.81 67.64 56.95

81.87

81.87

76.52

76.52

56.56

56.56

79.83 78.17 75.37 67.96 57.54

82.04 80.13 75.29 66.94 57.24

Cons. stand for constrained mode.

Table 3 Fundamental frequencies (Hz) of the LCB with central delamination located at interface 2. Delamination length (mm)

Intact 25.4 50.8 76.2 101.6

Present

Della and Shu [11]

Luo and Hanagud [19]

Hu et al. [29]

Shen and Grady [18]

Free

Cons.

Free

Cons.

Free

Cons.

Free

Cons.

Average test

Cons.

81.87 79.30 75.45 66.57 56.88

81.87 79.30 75.45 66.57 56.88

81.88 80.58 75.81 67.05 56.86

81.88 80.58 75.81 67.05 56.86

81.86 80.86 76.62 68.80 59.34

81.86 80.86 76.62 68.80 59.34

81.87

81.87

76.89

76.89

57.69

57.69

79.83 77.79 75.13 66.96 48.33

82.04 81.39 78.10 71.16 62.12

Table 4 Fundamental frequencies (Hz) of the LCB with central delamination located at interface 3. Delamination length (mm)

Intact 25.4 50.8 76.2 101.6

Della and Shu [11]

Luo and Hanagud [19]

Hu et al. [29]

Shen and Grady [18]

Free

Present Cons.

Free

Cons.

Free

Cons.

Free

Cons.

Average test

Cons.

81.87 80.38 79.09 76.05 70.54

81.87 80.38 79.13 76.32 71.87

81.88 81.53 80.09 76.75 70.92

81.88 81.53 80.13 77.03 72.28

81.86 82.01 80.74 77.52 71.73

81.86 82.02 80.79 77.82 73.15

81.87

81.87

80.45

80.50

71.21

72.61

79.83 80.12 79.75 76.96 72.46

82.04 81.46 79.93 76.71 71.66

To check further on the accuracy of the presented method, other cases are considered which are analyzed in the following example. Example 2. Consider a cantilever eight-ply LCBs with lay-up of [0/ 90]2s and dimensions of 127  12.7  1.016 mm3. The beam is made of T300/934 graphite/epoxy with material properties adopted from [19]. In our following analysis four different delamination lengths namely: 25.4 mm, 50.8 mm, 76.2 mm, and 101.6 mm are considered one at the time. The fundamental frequencies of centrally located delamination in a LCB with delamination positioned at the interfaces 1–4 one at the time are compared in Tables 2–5, respectively both for the ‘free mode’ and ‘constrained mode’. Good agreement is seen between the calculated by the present method, the experimental and analytical results of [18,19,10] and finite element results [29]. Tables 6 and 7 show the second and third bending frequencies of the delaminated beam, with various delamination lengths located centrally. Again, good agreement is observed between the present results, the results obtained using finite element models based on the

higher-order theory (HOT), 3D NASTRAN software [20] and analytical results by [10]. These comparisons further validate the capability of the present method for reliable frequency prediction of the LCB. Example 3. In this example, the natural frequencies of the carbon/ cyanate cantilever beam with dimensions 30 cm length and 5 cm width, and 0.218 cm thickness are compared with those available in the literatures. The angle-ply stacking sequence with lay-ups of [30/30]4s and [45/45]4s are considered having material properties taken from [30]. The delamination is located at the second interface measured from the mid-plane. As can be seen from Table 8, good agreement is observed. Example 4. In this example and examples 4–6 the LCB material is taken as AS4/3501 Graphite–Epoxy, having the following mechanical properties [25]:

E11 ¼ 144:8 GPa; E22 ¼ 9:65 GPa; G12 ¼ 4:14 GPa; G13 ¼ 4:14 GPa; G23 ¼ 3:45 GPa;

q ¼ 1389:23 kg=m3

t12 ¼ 0:33; ð26Þ

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600 Table 5 Fundamental frequencies (Hz) of the LCB with central delamination located at interface 4. Delamination length (mm)

Intact 25.4 50.8 76.2 101.6

Present

Luo and Hanagud [19]

Shen and Grady [18]

Free

Cons.

Free

Cons.

Average Test

Cons.

81.87 80.53 79.39 76.75 72.13

81.87 80.53 79.41 76.89 72.82

81.86 82.03 80.87 77.60 69.43

81.86 79.96 68.92 62.50 55.63

79.87 79.96 68.92 62.50 55.63

82.04 81.60 80.38 77.70 73.15

L1 ¼

Table 6 Second frequencies (Hz) for the LCB with central delamination (at interface 1). Delamination length (mm)

Intact 25.4 50.8 76.2 101.6

Present

Della and Shu [11]

Radu and Chattopadhyay [20]

Free and cons.

Free and cons.

HOT

NASTRAN 3D

513.17 449.11 443.16 394.19 328.62

513.20 495.09 465.96 399.07 315.50

513.30 509.24 469.02 369.08 325.79

510.70 504.18 478.66 399.36 305.75

Intact 25.4 50.8 76.2 101.6

Present

Radu and Chattopadhyay [20]

Free and cons.

HOT

NASTRAN 3D

1427.006 1197.652 790.745 745.831 721.045

1432.435 1217.572 826.281 751.384 735.589

1423.881 1172.821 772.516 739.660 717.908

Table 8 First five natural frequencies of the delaminated beams for different stacking sequence. [30/30]4s

L1 ; L

L2 ¼

L2 ; L

 ¼ h2 h 2 h

The first five normalized frequencies of an unsymmetric delaminated LCB [90/0] with slenderness ratio (L/h = 15) under various boundary conditions referred to the constrained mode are presented in Table 9. In this calculations a central delamination with L2 ¼ 0:2 is considered. A close inspection of the results in this table reveals that the longitudinal vibration does not play a significant role on the vibration analysis in this case until the fourth and fifth modes (see Table 9).

Table 7 Third frequencies (Hz) for the LCB with central delamination (at interface 1). Delamination length (mm)

Fig. 4. Interface location of the delamination for a[h/  h/h/  h]as graphite/epoxy composite laminate.

[45/45]4s

[30]

Present

[30]

Present

31.059 193.04 429.46T 534.80 644.24

29.870 192.27 429.02T 532.65 642.05

15.799 101.05 293.39 351.60 455.28T

15.408 100.74 292.97 351.02 454.86T

For all the problems, the width of the beam is taken as unity and the thickness of each layer in the LCB is equal. Also, the calculated natural frequencies in these examples are presented in a dimenrffiffiffiffiffiffiffiffiffi  2 sionless form X ¼ x= Eq11Lh4 with the shear correction factor of ks = 5/6 [25] and the following are other non-dimensional parameters used in our analysis:

Example 5. Consider a cantilever (I:C–F) angle-ply LCB with a central delamination located at the midplane, i.e. interface 1 and symmetric stacking sequence of [h/h/h/h]s (see Fig. 4). Fig. 5 illustrates the variation of non-dimensional frequency (xwo  xw) /xo vs. L2 on the ‘free mode’ (Fig. 5a) and the ‘constrained mode’ (Fig. 5b) as the layout angle changes. In this figure, x0, xwo, and xw are the fundamental frequency of an intact beam, of the LCB without Poisson’s effect and of the LCB with Poisson’s effect included, respectively. For the ‘free and constrained modes’, the inclusion of the Poisson’s effect causes more pronounced increase on the fundamental frequency in the LCB as the delamination length L2 decreases. Moreover, from these figures as it is expected, the Poisson’s effect produces no significant changes on the fundamental frequency for the unidirectional (h = 0°) or cross-ply (h = 90°) LCB. However, the fundamental frequency for an angle-ply beam where Poisson’s effect is not considered deviates significantly from the exact value (i.e. considering Poisson’s effect), especially for the layout angle between 30° and 60°. Note that the maximum difference occurs at (h = 45°) and for both cases of free and constrained modes this difference becomes 61.5%.

Example 6. Consider a delaminated beam as shown in Fig. 4 with h = 10°. The first five fundamental frequencies of such beam based on the free mode under various boundary conditions are calculated and presented in Table 10. The beam has the slenderness ratio of L/ h = 15 and the central delamination is located at interface 1 with L2 ¼ 0:2. A close inspection of the numbers given in this table reveals that retaining the longitudinal and torsional deformations in some cases will play a major role on the beam natural frequen-

Table 9 Comparison of non-dimensional first five natural frequencies of a beam with unsymmetric lay-up. Mode no.

1 2 3 4 5 a

C–C

C–H

H–H

C–F

Movable

Immovable

Movable

Immovable

Movable

Immovable

Movable

Immovable

2.6481 6.8694 11.2047 13.6017a 17.8525

2.7247 6.8694 12.1864 17.8079 24.7011

1.8803 5.8610 10.4575 13.2551a 17.0118

1.8833 5.9972 11.3872 16.7729 23.5622

1.1926 4.8650 9.0452 13.1385a 16.2337

1.3000 4.8688 10.7039 15.6184 22.6911

0.4660 2.5805 7.1052 12.3288 18.5324

0.4659 2.5801 7.1053 11.2716 14.1973a

Indicating the longitudinal vibrations mode.

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

Fig. 6. A LCB with [45/0/45/0] lay-up and a central delamination.

(a) Free Mode

Fig. 7. Effect of length-to-thickness ratio on dimensionless fundamental natural frequencies of the delaminated LCB with various delamination length.

(b) Constrained Mode Fig. 5. The influence of Poisson’s effect on the fundamental natural frequencies of the delaminated LCB (L/h = 15).

Table 10 Non-dimensionalized free mode frequencies of anti-symmetric angle-ply [10°/10°/ 10°/10°]as delaminated beam (L/h = 15).

a b

Mode no.

M:C–C

M:C–H

M:H–H

M:C–F

1 2 3 4 5 6 7 8

4.0746 10.0070 11.4243a 12.9340b 15.8929 17.3677b 23.1155 31.2636

2.9365 9.0788 11.3997a 12.8446b 15.2422 17.3504b 22.5811 30.9555

1.7465 8.0505 11.3903a 12.8115b 14.4463 17.3411b 22.3717 30.6541

0.8075 3.7746 4.5070T 10.7286 11.4672a 17.7072 15.2746b 24.9776

Indicating the longitudinal vibrations. Indicating the torsional vibrations.

Fig. 8. Effect of material anisotropy on the first three frequencies of the delaminated LCB with M:H–H boundary condition.

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

Fig. 9. First mode shape of a LCB with delamination located at interface 1 related to four different lengths.

cies (the modes with predominance of longitudinal and torsional vibrations). Another important aspect is that the order of natural frequency modes changes when the boundary conditions change. For example, the first longitudinal mode shifts from the third natural frequency for M:C–C, M:C–H, and M: H–H boundary conditions to the fifth natural frequency for M:C–F boundary conditions. This is due to the change in the system stiffness caused by the change in the boundary conditions. Example 7. The influence of the length and thicknesswise location of a central delamination on the normalized fundamental frequency of the I:C–C beam comprising of four lamina with unsymmetric lay-up shown in Fig. 6 is presented in Table 11. The dimensionless lengths of the delamination are 0.2, 0.4, 0.6, and 0.8. Lower and upper bounds of the natural frequencies are obtained out of solutions for the ‘free’ and ‘constrained’ modes of the delaminated layers, respectively as pointed out in Section 3.

Table 11 Dimensionless fundamental frequencies of the delaminated beams (L/h = 15). L2

0.2 0.4 0.6 0.8

Interface 1

Interface 2

Interface 3

Free

Cons.

Free

Cons.

Free

Cons.

3.7005 2.6226 1.2342 0.7052

3.7622 3.6584 3.6509 3.5841

3.0314 3.0085 2.7862 2.1692

3.0314 3.0085 2.7862 2.1692

3.0401 3.0153 2.8031 2.1953

3.0401 3.0153 2.8031 2.1955

Based on the results given in this table, if this single delamination is located at the interfaces 2 or 3 the natural frequencies of the beam under free and constrained modes become the same that is no ‘‘opening mode’’ is seen. But when the delamination is located at the interface 1, the so-called ‘‘opening modes’’ may appear more easily even for short length delamination. It should be mentioned that though the thicknesswise distance of the interfaces 1 and 3 is the same from

596

M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

Fig. 10. First mode shape of a LCB with delamination located at interface 2 related to four different lengths.

the free surfaces, but the opening mode is only seen when the delamination is located at the interface 1. This could be due to the fact that the closeness of stiffnesses of sub-beams [0°] and [45°/0°/45°] when the delamination is placed at the interface 3. On the other hand, when the delamination is placed at the interface 1, the difference between the stiffnesses of the clustered sub-beams 2 with [0°/45°/0°] lay-ups and sub-beam 3 with [45°] lay-up is relatively high. Next we would like to see the effects of length-to-thickness ratio (L/h) and delamination length on the dimensionless fundamental frequency of a LCB with M:C–H boundary conditions. In all related calculations the central delamination is located at interface 1. Fig. 7 shows the variation of dimensionless fundamental frequency vs. length-to-thickness ratio (L/h) for various L2 . Referred to this figure, it is clear that for large values of L/h, the X1 approaches to a constant value and moreover, as L2 increases, the value of X1 decreases. In addition for the L/h < 15] the fundamental frequency increases with much faster rate than for L/h > 15. Now, consider a similar beam as presented in Fig. 6 with the M:H– H boundary condition. In this case again we take a central delamination with L2 ¼ 0:4 located at the interface 1 with beam

slenderness ratio of L/h = 15. Fig. 8 shows the effect of material anisotropy on the first two natural frequencies of the delaminated LCB. It is noted that the value of E11, is varied, while the other elastic constants are kept unchanged and similar to those of graphite/epoxy material coefficients. For both the free mode and constrained mode, increasing the material anisotropy (E11/E22) causes the decrease of natural frequencies of the beam. Moreover, when the material anisotropy is increased, the difference between the frequencies based on the free and constrained modes get more pronounced, therefore the delamination opening becomes more likely.

4.2. Mode shapes The first mode shape of a thick LCB (L/h = 10) with I:C–F boundary conditions where the delamination staying in different thicknesswise location under the free mode (k = 0) assumption is shown in Figs. 9–11 for four different delamination lengths. Note that the lay-up configuration and thicknesswise location of the delamination shown in Fig. 6 is also considered in this case. From

M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

597

Fig. 11. First mode shape of a LCB with delamination located at interface 3 related to four different lengths.

these figures, we can see that the first mode of such beam does not show any opening in the cases where the delamination is located at the interfaces 2 or 3 while in cases where the delamination is located at interface 1, except for the ðL2 ¼ 0:2Þ we can observe clearly the delamination opening mode. The reason for such behavior is the closeness or farness of sub-beams stiffnesses as was discussed in the results given in Table 11. Indeed, the orientation of clustered sub-beams will play a significant role on this issue. 5. Conclusion The free vibration analysis of the delaminated LCB has been investigated using an assumed series solution in conjunction with Lagrange multipliers. Lamination scheme of cross-/angle ply and symmetrical/unsymmetrical configurations have been considered. It is observed that the present method is a computationally efficient tool in predicting the natural frequencies, constrained and free mode shape of the beams. This method is particularly attrac-

tive because of the ease with which one can choose the generalized displacement functions. This fact is demonstrated by choosing Legendre polynomials that the orthogonal properties of which simplifies energy expressions considerably. The natural frequencies of the delaminated LCB obtained by this method are extremely close to those available exact solutions. The effects of length to depth ratios, lamina schemes and the material anisotropy on the vibrational characteristics are discussed. Since in this study the shear deformation, rotary inertia, material couplings (bend–stretch, shear–stretch and bend–twist couplings) and Poisson’s effect are considered, the results presented hereby are believed to be more accurate and can render a benchmark for the future research. Appendix A By substituting Eq. (17) in Eq. (18), the BC’s can be obtained as: Clamped–Clamped beam (C–C):

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M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600

nt X

ð1Þ U 1m ¼ 0;

m¼0 nt X

nt X I : C—F : ð1Þm U 1m ¼ 0 and M : C-F :

m¼0

ð1Þm Wy1m ¼ 0;

m¼0

"

"

nt X ð1Þm Wx1m ¼ 0;

m

m¼0

nt X

nt X

m¼0

m¼1

ð1Þm W 1m ¼ 0;

3 m1 bX 2 c ð2m  4k1  1Þð1Þm2k1 1 ¼ 05;

nt X I : C—C : U 4m ¼ 0 and M : C—C :

k1 ¼0

m¼0 nt  X

3

m1 2 c bX A11;1 U 4m þ B11;1 Wx4m þ B16;1 Wy4m ð2m  4k1  1Þ ¼ 05;

m¼1 nt X

k1 ¼0

Wx4m ¼ 0;

m¼0

ðA11;1 U 1m þ B11;1 Wx1m þ B16;1 Wy1m Þ

nt X

Wy4m ¼ 0;

m¼0

nt X

W 4m ¼ 0;

nt X ð1Þm Wx1m ¼ 0;

nt X ð1Þm Wy1m ¼ 0;

nt X ð1Þm W 1m ¼ 0;

m¼0

m¼0

m¼0

m1 bX nt 2 c X U 4m ð2m  4k1  1Þ ¼ 0;

m1 bX nt 2 c X Wx4m ð2m  4k1  1Þ ¼ 0;

m¼1

m¼1

k1 ¼0

m¼0

ðA:1Þ

nt X

Wy4m

m¼1

Clamped–Hinged beam (C–H): nt X ð1Þm U 1m ¼ 0;

nt X ð1Þm Wx1m ¼ 0;

m¼0

m¼0

nt X ð1Þm Wy1m ¼ 0;

nt X ð1Þm W 1m ¼ 0;

m¼0

m¼0

k1 ¼0 m1 bX 2 c ð2m  4k1  1Þ ¼ 0;

Wy4m ¼ 0;

m¼0

nt X W 4m ¼ 0;

Hinged–Hinged beam (H–H): nt X

m¼0

m¼1

m1 bX 2 c

ð2m  4k1  1Þð1Þm2k1 1 ¼ 0;

k1 ¼0

ðB11;1 U 4m þ D11;1 Wx4m þ D16;1 Wy4m Þ

m¼0

Wy4m ¼ 0;

Clamped–Free beam (C–F):

nt X ½Wy1m  ð1Þm Wy3m  ¼ 0;

m¼0

m¼0

b c nt nt X X 2h3 X ½U 1m  ð1Þm U 3m  þ W 1m ð2m  4k1  1Þ ¼ 0; L1 m¼1 m¼0 k ¼0 m1 2

m1 bX ng 2 c 2X 2 ðA11;1 U 1m þ B11;1 Wx1m þ B16;1 Wy1m Þ ð2m  4k1  1Þ  L1 m¼1 L 2 k ¼0 1

ng h X

A11;2 U 2m þ B11;2 Wx2m þ B16;2 Wy2m



m¼1

m1 bX 2 c ð2m  4k1  1Þ ¼ 0;

þðA11;3 U 3m þ B11;3 Wx3m þ B16;3 Wy3m Þ

k1 ¼0

m¼0

m¼0

nt X ½Wy1m  ð1Þm Wy2m  ¼ 0;

k1 ¼0

3 m1 bX 2 c ðA11;1 U 4m þ B11;1 Wx4m þ B16;1 Wy4m Þ ð2m  4k1  1Þ ¼ 05;

nt X W 4m ¼ 0;

m¼0

m1 bX 2 c ð2m  4k1  1Þð1Þm2k1 1 ¼ 0;

m¼0

nt X

nt X ½Wx1m  ð1Þm Wx3m  ¼ 0;

1

m¼0

m¼1

m¼0

nt X ½Wx1m  ð1Þm Wx2m  ¼ 0;

m1 bX nt 2 c 2X ð2m  4k1  1Þ  ½A55;2 W 2m þ A55;3 W 3m  L2 m¼0 k ¼0

nt X ð1Þm Wy1m ¼ 0;

nt X ð1Þm W 1m ¼ 0; m¼0 " nt X I : H—H : U 4m ¼ 0 and M : H-H :

nt X

m¼0

nt nt X X 2 ½A55;1 Wx1m  A55;2 Wx2m ð1Þm  A55;3 Wx3m ð1Þm  þ A55;1 W 1m L m¼0 m¼1 1

ðB11;1 U 1m þ D11;1 Wx1m þ D16;1 Wy1m Þ

m¼1

nt X ½W 1m  ð1Þm W 3m  ¼ 0;

1

k1 ¼0

nt X

nt X ½W 1m  ð1Þm W 2m  ¼ 0;

1

ðA:2Þ

m¼0

nt X ð1Þm U 1m ¼ 0;

ðA:4Þ

m1 bX nt nt 2 c X 2h2 X ½U 1m  ð1Þm U 2m   W 1m ð2m  4k1  1Þ ¼ 0; L1 m¼1 m¼0 k ¼0

k1 ¼0

nt X

m¼1

k1 ¼0

nt 2X W 4m L m¼1

Also, by substituting Eq. (17) in Eqs. (19) and (20), the C:C:s can be expressed as: C:C:s at left end of sub-beams 2 and 3:

3 m1 nt  2 c bX X A11;1 U 4m þ B11;1 Wx4m þ B16;1 Wy4m ð2m  4k1  1Þ ¼ 05;

m¼1

Wx4m þ

k1 ¼0

m¼0

ðB11;1 U 4m þ D11;1 Wx4m þ D16;1 Wy4m Þ

nt X

bXc ð2m  4k1  1Þ ¼ 0

nt X I : C—H : U 4m ¼ 0 and M : C—H :

nt X

k1 ¼0

m1 2

"

m¼1

bXc ð2m  4k1  1Þ ¼ 0; m1 2

ðA:3Þ

i

m1 bX 2 c ð2m  4k1  1Þð1Þm2k1 1 ¼ 0;

k1 ¼0 m1 bX ng 2 c 2X 2 ðB16;1 U 1m þ D16;1 Wx1m þ D66;1 Wy1m Þ ð2m  4k1  1Þ  L1 m¼1 L 2 k ¼0 1

599

M.H. Kargarnovin et al. / Composites: Part B 45 (2013) 587–600 ng X

  B16;2 U 2m þ D16;2 Wx2m þ D66;2 Wy2m

nt X

ðB16;2 U 2m þ D16;2 Wx2m þ D66;2 Wy2m Þ

m¼1

m¼1

 þðB16;3 U 3m þ D16;3 Wx3m þ D66;3 Wy3m Þ

m1 bX 2 c

ð2m  4k1  1Þð1Þ

m2k1 1

þðB16;3 U 3m þ D16;3 Wx3m þ D66;3 Wy3m Þ m1 bX 2 c ð2m  4k1  1Þ ¼ 0

¼ 0;

k1 ¼0

k1 ¼0

bXc nt 2X 2 ðB11;1 U 1m þ D11;1 Wx1m þ D16;1 Wy1m Þ ð2m  4k1  1Þ  L1 m¼1 L 2 k ¼0 m1 2

1

nt X

ðB11;2 U 2m þ D11;2 Wx2m þ D16;2 Wy2m Þ

m¼1

m1 bX 2 c

ð2m  4k1  1Þð1Þm2k1 1 þ

m1 2

ð2m  4k1  1Þð1Þm2k1 1 ¼ 0

ðA:5Þ

k1 ¼0

C:C:s at right end of sub-beams 2 and 3:

½ð1Þm W 4m  W 2m  ¼ 0;

m¼0 nt X

m

½ð1Þ Wx4m  Wx2m  ¼ 0;

nt X

½ð1Þm W 4m  W 3m  ¼ 0;

m¼0 nt X

m¼0

½ð1Þm Wy4m  Wy2m  ¼ 0; ½ð1Þm U 4m  U 2m  

m¼0

½ð1Þm Wy4m  Wy3m  ¼ 0;

nt 2h2 X W 4m L4 m¼1

ð2m  4k1  1Þð1Þm2k1 1 ¼ 0;

k1 ¼0

W 4m

m¼1

nt X

ng X 2h3 ½ð1Þm U 4m  U 3m  þ L4 m¼0

ð2m  4k1  1Þð1Þm2k1 1 ¼ 0;

k1 ¼0

nt nt X X 2 ½A55;1 Wx4m ð1Þm  A55;2 Wx2m  A55;3 Wx3m  þ A55;1 W 4m L m¼0 m¼1 4

ð2m  4k1  1Þð1Þm2k1 1 

k1 ¼0 m1 bX 2 c

nt 2X ½A55;2 W 2m þ A55;3 W 3m  L2 m¼0

ð2m  4k1  1Þ ¼ 0;

k1 ¼0 nt X

2 ðA11;1 U 4m þ B11;1 Wx4m þ B16;1 Wy4m Þ L4 m¼1 m1 bX 2 c

ð2m  4k1  1Þð1Þm2k1 1 

k1 ¼0

2 L2

nt h X

ðA11;2 U 2m þ B11;2 Wx2m þ B16;2 Wy2m Þ i þðA11;3 U 3m þ B11;3 Wx3m þ B16;3 Wy3m Þ m1 bX 2 c ð2m  4k1  1Þ ¼ 0 m¼1

k1 ¼0 nt 2X ðB16;1 U 4m þ D16;1 Wx4m þ D66;1 Wy4m Þ L4 m¼1 m1 bX 2 c

1

nt h X

h2 ðA11;2 U 2m þ B11;2 Wx2m þ B16;2 Wy2m Þ

m¼1

m1 bX 2 c ð2m  4k1  1Þ ¼ 0

i

ðA:6Þ

k1 ¼0

½ð1Þm Wx4m  Wx3m  ¼ 0;

m¼0 nt X

m1 bX 2 c

m1 2 c bX 2 ð2m  4k1  1Þ þ þðB11;3 U 3m þ D11;3 Wx3m þ D16;3 Wy3m Þ L 2 k ¼0

m¼0 nt X

m¼0 nt X

m1 bX 2 c

m¼1

h3 ðA11;3 U 3m þ B11;3 Wx3m þ B16;3 Wy3m Þ

nt X

m1 bX 2 c

m1 bX 2 c 2 ð2m  4k1  1Þð1Þm2k1 1  L 2 k ¼0

nt X

ðB11;2 U 2m þ D11;2 Wx2m þ D16;2 Wy2m Þ

nt h 2X h2 ðA11;2 U 2m þ B11;2 Wx2m L2 m¼1 k1 ¼0 i þB16;2 Wy2m Þ  h3 ðA11;3 U 3m þ B11;3 Wx3m þ B16;3 Wy3m Þ

bXc

nt 2X ðB11;1 U 4m þ D11;1 Wx4m þ D16;1 Wy4m Þ L4 m¼1

1

 þðB11;3 U 3m þ D11;3 Wx3m þ D16;3 Wy3m Þ

k1 ¼0



ð2m  4k1  1Þð1Þm2k1 1 

2 L2

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