Free vibration analysis of generally laminated beams via state-space-based differential quadrature

Composite Structures 63 (2004) 417–425 www.elsevier.com/locate/compstruct

Free vibration analysis of generally laminated beams via state-space-based differential quadrature W.Q. Chen *, C.F. Lv, Z.G. Bian Department of Civil Engineering, Zhejiang University, Hangzhou 310027, PR China

Abstract A new method of state-space-based differential quadrature is presented for free vibration of generally laminated beams. By discretizing the state space formulations along the axial direction using the technique of differential quadrature, new state equations at discrete points are established. Applying end conditions and using matrix theory, the general solution is derived. Taking account of the boundary conditions at the top and bottom planes, frequency equation governing the free vibration of generally laminated beams is then formulated. The method is validated by comparing numerical results with that available in the literature.
2003 Elsevier Ltd. All rights reserved. Keywords: State-space-based differential quadrature; Generally laminated beams; Free vibration

1. Introduction Composite materials are finding increasing applications in civil engineering, transportation vehicles, aerospace, marine, aviation, and chemical industries in recent decades. This is due to their excellent features, such as high strength-to-weight and stiffness-to-weight ratios [1], the ability of being different strengths in different directions [2] and the nature of being tailored to satisfy the strength and stiffness requirements in practical designs. In the dynamic analyses, it is quite essential to consider an overview of the free vibration characteristics, including the natural frequencies of these composite structures. The present work investigates the free vibration problems of generally laminated beams. Dynamic analysis of laminated composite beams with cross-ply laminas has received considerable attention in recent decades [3–7] and continues to provide great interest in the beginning of the new millennium [8–11]. Whereas the literature available for beams with arbitrary angle-ply laminas, to the authorÕs knowledge, is rather small in number. Chandrashekhara and Bangera [12] put forward a finite element model based on a

*

Corresponding author. Tel.: +86-571-8795-2284; fax: +86-5718795-2165. E-mail address: [email protected] (W.Q. Chen). 0263-8223/$ - see front matter
2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00190-9

higher-order shear deformation theory to study the free vibration of generally layered composite beams. The Poisson effect, which is often neglected in one-dimensional laminated beam analysis, was incorporated in the formulation of the constitutive equation. The in-plane and rotary inertia were accounted for in the formulation of mass matrix. Krishnaswamy et al. [13] used HamiltonÕs principle to formulate the dynamic equations governing the free vibration of laminated composite beams. The influences of transverse shear deformation and inertia were also considered, and analytical solutions for un-symmetric laminated beams were obtained by applying the Lagrange multipliers method. Symbolic computation technique was reported by Teboub and Hajela [14] on the basis of a first order shear deformation theory. Little effort has been done on analysis of generally laminated beams directly based on elasticity theory. Recently, a semi-analytical method, i.e. the statespace-based differential quadrature (SSDQM) was proposed by the authors [15], which has been demonstrated very efficient for determination of natural frequencies of cross-ply laminated beams or of that with more special material properties. This new method was directly based on the orthotropic elasticity equations for plane stress problems and enabled to dispose the problems of cross-ply laminated beams fluently by applying the technique of differential quadrature (DQ). In this paper, the same idea is exploited to investigate

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W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

the free vibration problems of general anisotropic beams as well as laminated beams composed of arbitrary angle plies. It is noted that, unlike the cross-ply laminates, there is no exact elasticity solutions for angle-ply laminates, even for simply supported end conditions, because of the coupling between normal (shear) stresses and shear (normal) strains. In fact, the cross-ply laminate is just a special case of the generally laminated beams. Numerical experiments for various types of lamina are conducted. Results of natural frequencies are compared to the data available in existing publications to validate the proficiency of SSDQM. Relative mode shapes are also presented for illustration. Effects of some parameters, such as the symmetry feature, number of layers, lamina scheme, etc., on the natural frequencies are discussed.

2. Theoretical formulations A multi-layered laminated beam of length L and depth H is considered in a referred system of Cartesian coordinates originating at the bottom plane of the beam with the x axis coincident with the beam axial direction, Fig. 1(a). It is assumed that the laminate is made up of m orthotropic plies whose principal material axes are not necessarily coincident with the reference axes. From this point of view, the plies are then said to be generally orthotropic, and the beam is referred to be generally laminated. To develop the equations of motion for free vibration, the following hypotheses are adopted:

2.1. State space equations Considering the anisotropy of material in the reference axes (x–y) system, the constitutive equations for an arbitrary lamina read
 ou ov ov ou rx ¼ c11 þ c12 þ c16 þ ; ox oy ox oy
 ou ov ov ou ð1Þ þ ; ry ¼ c12 þ c22 þ c26 ox oy ox oy
 ou ov ov ou þ ; sxy ¼ c16 þ c26 þ c66 ox oy ox oy where u, v are the displacement components, rx , ry the normal stress components in x and y directions respectively, sxy the shear stress component, and cij denote the transformed reduced stiffness terms from the material axes (1–2) to the reference axes (x–y). Consider the generally orthotropic ply oriented at a positive angle h relative to the reference x-axis, see Fig. 1(b). As mentioned in Ref. [16], cij can be obtained by transforming the reduced stiffnesses Qij from the material axes (1–2) to the references axes (x–y) as follows: c11 ¼ Q11 cos4 h þ 2ðQ12 þ 2Q33 Þ sin2 h cos2 h þ Q22 sin4 h; c12 ¼ ðQ11 þ Q22
4Q33 Þ sin2 h cos2 h þ Q12 ðsin4 h þ cos4 hÞ; c22 ¼ Q11 sin4 h þ 2ðQ12 þ 2Q33 Þ sin2 h cos2 h þ Q22 cos4 h; c16 ¼ ðQ11
Q12
2Q33 Þ sin h cos3 h þ ðQ12
Q22 þ 2Q33 Þ sin3 h cos h; c26 ¼ ðQ11
Q12
2Q33 Þ sin3 h cos h

(1) each laminate behaves as an anisotropic elastic body in plane stress state; (2) perfect bonding exists between arbitrary adjacent plies; (3) the deformation is assumed to be adequately small.

þ ðQ12
Q22 þ 2Q33 Þ sin h cos3 h; c66 ¼ ðQ11 þ Q22
2Q12
2Q33 Þ sin2 h cos2 h þ Q33 ðsin4 h þ cos4 hÞ; ð2Þ

y ym

m

θ

yi

i

y i −1

H

2

1

x

2 1

y0

L

x

Angle-ply notation

(a)

(b)

x, y – Beam’s reference axes, 1, 2 – Material axes Fig. 1. Geometry of a laminated beam in Cartesian coordinates.

W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

where, Q11 ¼

E12 m212 E2

;

E1
E 1 E2 Q22 ¼ ; E1
m212 E2

Q12 ¼

m12 E1 E2 ; E1
m212 E2

ð3Þ

Q33 ¼ G12 ;

in which E1 and E2 are the YoungÕs moduli along the material axes 1 and 2 respectively, m12 is called the major PoissonÕs ratio, and G12 is the in-plane shear modulus. In the absence of body forces, the equilibrium equations for free vibration of an elastic body with a circular frequency x are expressed as orx osxy þ ¼
qx2 u; ox oy osxy ory þ ¼
qx2 v; ox oy

ð4Þ

where q denotes the mass density of the lamina. For the convenience of algebraic manipulation, the stress and displacement components are nondimensionlaized as that carried out in Ref. [15]. Utilization of Eqs. (1) and (4) leads to the following so-called state space equations: o
u o
u o
v ¼ a1 k þ a2 rg
k
a3 s; og on on org q 2 os ¼
X
v
k ; q1 on og o
v o
u ¼ a4 k
a5 rg þ a2 s; og on os q o2
org os u ¼
X2
þ a1 k ; u
a6 k2 2 þ a4 k og q1 on on on

ð5Þ

ð7Þ

ðnÞ

in which Wik are the weight coefficients whose expressions can be found in Ref. [17]. Applying Eq. (7) to Eqs. (5) and (6), the following state equations about the state variables at an arbitrary sampling point ni in any given lamina are then obtained: N N X X d
ui ð1Þ ð1Þ ¼ a1 k Wik
uk þ a2 rgi
k Wik
vk
a3 si ; dg k¼1 k¼1 N X drgi q ð1Þ ¼
X2
vi
k Wik sk ; q1 dg k¼1 N X d
vi ð1Þ ¼ a4 k Wik
uk
a5 rgi þ a2 si ; dg k¼1

ð8Þ

þ a1 k

N X

ð1Þ

Wik sk ;

and the induced variables rni are

ð1Þ

ð1Þ

and c66 denoting the mass density and transformed stiffness term of the first lamina, respectively.
u, rg ,
v and s are named as the non-dimensional state variables [15], and the induced variable rn is determined by o
u
a4 rg
a1 s: on

n ¼ 1; 2; . . . ; N
1; i ¼ 1; 2; . . . ; N ;

k¼1

q1 =c66 is the non-dimensional frequency with q1

rn ¼ a6 k

respect to n at a given point ni can be approximated by a linear sum of weighted function values at all discrete points. Assuming that N discrete points are sampled in the domain of n, the nth-order partial differential should be  N X on f ðn; gÞ  ðnÞ ¼ Wik f ðnk ; gÞ; n  on n¼ni k¼1

N N X X dsi q ð2Þ ð1Þ ¼
X2
ui
a6 k2 Wik
uk þ a4 k Wik rgk q1 dg k¼1 k¼1

where k ¼ H =L is the depth-to-length ratio, ai are the coefficients qffiffiffiffiffiffiffiffiffiffiffiffiffiffidetermined by cij (see Appendix A), X ¼

xH

419

ð6Þ

2.2. State equations at discrete points It is impossible, to some extent, to seek a series of continuous functions which can exactly satisfy Eq. (5) and capture the end conditions simultaneously for an anisotropic beam. Fortunately, one can resort to approximate solutions in conjunction with the application of differential quadrature (DQ) to study the free vibration characteristics of generally laminated beams, just as that done for cross-ply laminates [15]. According to the principle of DQ, an arbitrary order differentiation of a continuous function f ðn; gÞ with

rni ¼ a6 k

N X

ð1Þ

Wik
uk
a4 rgi
a1 si ;

ð9Þ

k¼1

for i ¼ 1; 2; . . . ; N , where u
i ¼
uðni ; gÞ, rgi ¼ rg ðni ; gÞ,
vi ¼
vðni ; gÞ, si ¼ sðni ; gÞ, and rni ¼ rn ðni ; gÞ for brevity. Note that Eq. (8) degenerates identically to that obtained in Ref. [15] for a cross-ply laminated beam. 2.3. Frequency equation The main advantage of the DQ technique is that the discretization in the domain of n makes it realistic for the state variables to satisfy arbitrary end conditions precisely. In an effort to illustrate the solution procedure of Eq. (8), simply supported–simply supported (S–S), clamped–clamped (C–C) and clamped–free (C–F) laminated beams are considered for instances. In the case of zero initial displacement and stress conditions, the end conditions of the above-mentioned three beam types are outlined as follows. S–S beams:
v1 ¼ rn1 ¼ 0;
vN ¼ rnN ¼ 0;

at n ¼ 0; at n ¼ 1:

ð10aÞ ð10bÞ

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W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

To account for all the end conditions in the state equations, Eq. (8), it is essential to express the end stress conditions in terms of state variables. For this purpose, the stress conditions in Eq. (10) should be substituted into Eq. (9), and elimination of rni from Eq. (9) gives rgi ¼

N a6 X a1 ð1Þ k W
uk
si ; a4 k¼1 ik a4

at n ¼ 0;

ð12aÞ

at n ¼ 1:

ð12bÞ

C–F beams:
u1 ¼
v1 ¼ 0; rnN ¼ sN ¼ 0;

at n ¼ 0;

ð13aÞ

at n ¼ 1:

ð13bÞ

Similar to S–S beams, the stress end condition rnN ¼ 0 should be replaced by that in terms of the state variable rgN as rgN ¼

N a6 X ð1Þ k W
uk : a4 k¼1 Nk

ð14Þ

Incorporating the end conditions corresponding to a considered beam into Eq. (8), solutions for state variables can be obtained. The explicit formulae of the endconditions-involved state equations are presented in Appendix B, and for simplicity they can be written in a uniform matrix notation as d di ¼ M i di ; dg

ð15Þ

uT rTg
vT sT Ti is named the state vector, where di ¼ ½
in which
u,
v and rg , s are column vectors of the unknown displacement and stress components at all discrete points in the ith layer. The coefficient matrix Mi can be obtained directly from the state equations for the corresponding beams with the explicit expressions provided in Appendix C, and the subscript i denotes the ith layer. According to the matrix theory, the general solution to Eq. (15) is di ðgÞ ¼ exp½Mi ðg
gi
1 Þdi ðgi
1 Þ;

ð18Þ

where the global transfer matrix S is defined as 1 Y

exp½Mk ðgk
gk
1 Þ:

ð19Þ

k¼m

C–C beams:
uN ¼
vN ¼ 0;

dm ð1Þ ¼ Sd1 ð0Þ;

S¼

for i ¼ 1 and i ¼ N : ð11Þ


u1 ¼
v1 ¼ 0;

Chen et al. [15], the following relation of the state vectors at the top and bottom planes is derived,

gi
1 6 g 6 gi ;

ð16Þ

where di ðgÞ is the state vector for an arbitrary nondimensional vertical coordinate g, and 8 0; i ¼ 0; yi < P ð17Þ gi ¼ ¼ 1 i H : H hk ; i ¼ 1; 2; . . . ; m; k¼1

in which hk ¼ yk
yk
1 is the thickness of the kth lamina. By imposing the continuity conditions at the interface of arbitrary two adjacent layers into Eq. (16) through all the m layers, just similar to the procedure outlined by

Partitioning matrix S into rewritten as 9 8 2
uð1Þ > S11 S12 > > > = < 6 S21 S22 rg ð1Þ ¼6 4 S31 S32
vð1Þ > > > > ; : S41 S42 sð1Þ m

sub-matrices, Eq. (18) can be S13 S23 S33 S43

9 38
uð0Þ > S14 > > > = < S24 7 7 rg ð0Þ : S34 5> >
vð0Þ > > ; : S44 sð0Þ 1

ð20Þ

In the state of free vibration, both the top and bottom planes of the laminated beam are subjected to no force, which leads to rg ð1Þ ¼ sð1Þ ¼ 0;

rg ð0Þ ¼ sð0Þ ¼ 0;

ð21Þ

Substitution of the above expressions into Eq. (20) yields     
uð1Þ
uð0Þ S11 S13 ¼ ; ð22Þ
vð1Þ m
vð0Þ 1 S31 S33     
uð0Þ S21 S23 u
ð0Þ ¼T ¼ 0: ð23Þ
vð0Þ 1
vð0Þ 1 S41 S43 The fact that Eq. (23) has non-trivial solutions for free vibrating beams requests the determinant of coefficient matrix T to vanish, that is jTj ¼ 0:

ð24Þ

The frequencies can be calculated from Eq. (24) using the bisection method, and the mode shapes then can be determined from Eq. (23).

3. Numerical results and discussions In order to demonstrate the efficiency of the present solution methodology, free vibrations of several generally laminated beams are investigated. Unequally spaced sampling points, i.e. the so-called Chebyshev–Gauss– Lobatto points, are adopted [18,19] 1
cos½ði
1Þp=ðN
1Þ ; i ¼ 1; 2; . . . ; N : ð25Þ 2 To identify the convergence of SSDQM for generally laminated beams, free vibration of a laminated C–C beam with lay-up of 0/45/0/45 with material properties from Ref. [4] is investigated. The geometry and material properties are listed in Table 1. Results tabulated in Table 2 indicate that SSDQM is rapidly convergent for the lowest three frequencies and shows relatively slow convergence for higher frequencies. The discrete points number is taken to be N ¼ 15 in the

ni ¼

W.Q. Chen et al. / Composite Structures 63 (2004) 417–425 Table 1 Material and geometry properties Descriptions

Ref. [4]

Ref. [20]

L (m) b (m) H (m) E1 (N m
2 ) E1 (N m
2 ) m12 G12 (N m
2 ) q (kg m
3 )

0.381 0.0254 0.0254 14.48 · 1010 0.965 · 1010 0.3 0.414 · 1010 1389.23

0.1905 12.7 · 10
3 3.175 · 10
3 12.911 · 1010 0.9408 · 1010 0.3 0.51568 · 1010 1550.1

subsequent numerical examples, of which the relative error is very small when compared to that for N ¼ 19, as shown in Table 2. Two single-layered C–F beams of 15 ply and 30 ply, respectively are now considered. Note that experimental results for anisotropic beams can be found in Refs. [20,21]. For comparison with the experimental results (Exptl) in Ref. [20], the non-dimensional natural frequencies X are converted into f in Hz (Table 3). The material and geometry properties are the same as those in Ref. [20] (see Table 1). The comparison indicates that the results obtained by SSDQM agree well with the experimental results, with a maximum relative error of 3.12% and 2.04% for 15 and 30 angle schemes respectively. It also should be noted that the SSDQM does not predict the torsional modes, because only planestress problem is considered in this paper. For further demonstration of the proficiency of SSDQM, more numerical calculations for multi-layered generally laminated beams are preformed. Results are compared to that obtained by other numerical methods

421

in the published reports. Unless mentioned otherwise, the material properties of AS4/3501-6 graphite–epoxy composite adopted from Ref. [4] (see Table 1) are used for the analysis hereafter. Note that the non-dimensional frequency X is transformed into pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi another fre
¼ xL2 q1 =ðE1 H 2 Þ for the sake of quency parameter x comparison. The effect of layer number on the non-dimensional
) of un-symmetric angle-ply C–C lamifrequencies (x nated beams of the same total thickness is exhibited in Fig. 2. It is obvious that the frequency increases with the number of layers for each vibration mode, but the increase slower when the layer number becomes larger. Further study shows that when the layer number tends to infinity, the natural frequency almost remains invariable. In Table 4 are shown the non-dimensional frequen
of angle-ply laminated beams. Numerical results cies x of anti-symmetric laminated beams with the lamina scheme of 45/)45/45/)45 expose excellent agreement with that in Ref. [12]. Table 4 also lists the results of
of multi-layered SSDQM for the frequency parameter x beams with the 30/)50/50/)30 scheme. It can be seen that the worse the symmetrical characteristic, the higher the natural frequencies are obtained. The mode shapes corresponding to the first six frequencies of the 45/)45/)45/45 schemed C–C and C– F beams are presented in Figs. 3 and 4, respectively. The curves correspond to the displacements at the bottom plane of the beam. It is obvious that for the lower modes, the amplitude of longitudinal displacement is quite small compared to that of the transverse displacement for both types of beams. But for higher

Table 2 Convergence of SSDQM solution (X) for a 0/45/0/45 schemed laminated C–C beam N 7 9 11 13 15 19

Mode number 1

2

3

4

5

6

0.1053 0.1041 0.1035 0.1031 0.1022 0.1020

0.2638 0.2552 0.2541 0.2530 0.2506 0.2507

0.4747 0.4528 0.4443 0.4419 0.4367 0.4327

– 0.7191 0.6604 0.6516 0.6435 0.6479

– – 0.8822 0.8739 0.8633 0.8577

– – 0.9100 0.9094 0.9072 0.9092

Note: Ô–Õ denotes the result not available.

Table 3 Comparison of natural frequencies (Hz) of a C–F beam with the experimental results h

Mode

1

2

3

4

5

6

7

15

SSDQM Exptl

82.55 82.5

515.68 511.3

1437.02 1423.4

– 1526.9*

2797.14 2783.6

– 4364.6*

4583.87 4731.6

30

SSDQM Exptl

52.73 52.7

330.04 331.8

922.45 924.7

1803.01 1766.9

– 1827.5*

2970.36 2984.0

4420.93 4432.4

Note: ÔÕ denotes the torsional mode, and Ô–Õ denotes the result not available.

422

W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

ω n=5

35

non-dimensional frequencies

30 n=4 25 20

ω 1 = 1.8383

ω 2 = 4.9753

ω3 = 9.5188

ω 4 = 13.5089

n=3

15 n=2

10

n=1

5 0

2

4

6

8

10

12

14

layer number

‘n’denotes the mode number Fig. 2. Effect of layer number on the first five non-dimensional fre
) of un-symmetric angle-ply C–C beams (0/45/. . .). quencies (x 0

0.5

1

0

ω 5 = 15.2979

modes, except for the fourth mode which is exactly the longitudinal vibration, the amplitudes of the two displacements are getting increasingly comparable. This fact shows that coupling of deformations in depth and longitudinal directions becomes significant. From this point of view, any numerical methods based on the assumption, that the longitudinal displacement is neglected, may yields considerable inaccuracy for higher modes prediction. Additionally, as shown in Table 4 and Figs. 3 and 4, the 45/)45/)45/45 and 45/)45/45/ )45 schemed beams have almost the same natural frequencies. Fig. 5 shows the effect of length-to-depth ratio (L=H ), where L varies while H keeps constant as defined in Table 1, on the non-dimensional fundamental frequency of a C–C beam with a lamina scheme of 45/)45. It is

0.5

Fig. 3. First six mode shapes of a C–C beam with 45/)45/)45/45 lamina (––,
v; - - -,
u).

clear that for L=H < 40, the variation of L=H has drastic effect on X. While for L=H P 40, the fundamental frequency tends to be constant, that is, the influence of L=H is practically negligible. As expected, the curve shows clearly that the smaller the length-to-depth ratio is, the higher the frequencies will be.
of a The non-dimensional fundamental frequencies x C–C laminated beams with lamina of h/)h/)h/h, 0/h and h/)h schemes are calculated in Table 5. It is seen that the results for the first scheme type are closer to that presented in Ref. [13] than that in Ref. [4] in which the Poisson effect was neglected. Generally speaking, the

Table 4
) of angle-ply laminated beams Non-dimensional frequencies (x Angle notation

End conditions

Natural frequencies 1

2

3

4

5

45/)45/45/)45

C–C

1.8446 (1.9807) 0.7998 (0.8278) 0.2969 (0.2962)

4.9871 (5.2165) 3.1638 (3.2334) 1.7778 (1.8156)

9.5395 (9.6912) 6.9939 (7.0148) 4.8953 (4.9163)

13.4736 (10.5345) 12.1471 (10.7449) 6.6557 (5.3660)

15.2920 (15.0981) 13.2745 (11.9145) 9.3750 (9.2162)

2.2640 0.9790 0.3572

6.0764 3.8585 2.2061

11.5279 8.4823 6.0394

15.1479 14.6309 7.4840

18.3213 14.8575 11.4767

S–S C–F 30/)50/50/)30

C–C S–S C–F

Note: results in parentheses are presented in Ref. [12].

1

ω6 = 19.3612

W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

ω1 = 0.2860

423


tends to decrease with the infrequency parameter x creasing of the ply angle h except that the value corresponding to h ¼ 90 has a minute rebound. This phenomenon was also predicted by the beam theory [13]. It is worth noting that, for the same total thickness, the algebraic lamination has no influence on the frequencies of beams with the same angle scheme, see the columns corresponding to h ¼ 0 and h ¼ 90 in Table 5.

ω 2 = 1.7724

4. Conclusions ω 3 = 4.8791

0

0.5

ω 4 = 6.6570

1

0

0.5

ω 5 = 9.3430

1

ω 6 = 15.0132

Fig. 4. First six mode shapes of a C–F beam with 45/)45/)45/45 lamina (––,
v; - - -,
u).

Ω × 100 14 12 10 8 6 4 2

In this paper the SSDQM proposed in our previous paper [15] is extended to analyze the free vibration of generally laminated beams, including single-/multi-, cross-/angle-ply and symmetrical/un-symmetrical laminas. With the aid of the DQ technique, the state equations with respect to state variables at discrete points are established and the frequency equation is formulated. It is noted here that for cross-ply laminated beams, the formulations presented in this paper become the same as those in Ref. [15]. The validity of SSDQM is further demonstrated by comparing the current results to that obtained by other methods. The effects of lengthto-depth ratios, lamina schemes and the number of layers on the natural frequencies are discussed. Since SSDQM is completely based on the elasticity equations for an anisotropic body in plane-stress state, the results presented hereby are believed to be more accurate and can render a benchmark for the future numerical research. Finally, it should be emphasized that, although state space formulations for generally anisotropic materials have been established [22], their application in structural analysis is very limited. However, with the aid of DQ technique, just as in the present paper, the state space method shows a promising future in engineering, especially in the field of multi-layered structures.

0 -2

0

20

40

60

80

100

120 L H

Fig. 5. Effect of L=H on non-dimensional fundamental frequencies (X) of a C–C beam (45/)45).

Acknowledgements The work is supported by the National Natural Science Foundation of China (Project Number 10002016).

Table 5
) of C–C beams with layerups of h/)h/)h/h, 0/h and h/)h Non-dimensional fundamental frequencies (x h

0

15

30

45

60

75

90

h/)h/)h/h

4.8575 (4.869) [4.8487]

3.6484 (3.988) [4.6635]

2.3445 (2.878) [4.0981]

1.8383 (1.947) [3.1843]

1.6711 (1.644) [2.1984]

1.6161 (1.621) [1.6815]

1.6237 (1.631) [1.6200]

4.8575

4.1899 3.6113

3.3548 2.3016

2.9814 1.8145

2.9491 1.6686

2.8002 1.6200

2.8012 1.6237

0/h h/)h

Note: the results in parentheses and square brackets are presented in Refs. [13,4], respectively.

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W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

Appendix A. Coefficients ai

for i ¼ 2; . . . ; N
1, where

c22 c16
c12 c26 ; a2 ¼ c26 d; a3 ¼ c22 d; c226
c22 c66 c12 c66
c16 c26 a4 ¼ 2 ; a5 ¼ c66 d; c26
c22 c66 c22 c216 þ c66 c212
2c12 c16 c26 c11 a6 ¼ þ ð1Þ ; ð1Þ ðc226
c22 c66 Þc66 c66 where ð1Þ c d ¼ 2 66 : c26
c22 c66

Aik ¼ Wi1 W1k þ W1N WNk ; c16 c66 c11 e1 ¼ ð1Þ ; e2 ¼ ð1Þ ; e3 ¼ a6
ð1Þ : c66 c66 c66

ð1Þ

a1 ¼

ðA:1Þ

ð1Þ

ð1Þ

ð1Þ

ðB:4Þ ðB:5Þ

B.3. C–F beams

ðA:2Þ

N N X X d
ui ð1Þ ð1Þ ¼ a1 k Wik
Wik
vk
a3 si ; uk þ a2 rgi
k dg k¼2 k¼2

i ¼ 2; . . . ; N ;

Appendix B. Explicit state equations at discrete points

N
1 X drgi q ð1Þ Wik sk ; ¼
X2
vi
k q1 dg k¼1

i ¼ 1; . . . ; N
1;

N X d
vi ð1Þ ¼ a4 k Wik
uk
a5 rgi þ a2 si ; dg k¼2

B.1. S–S beams N N
1 X X d
ui ð1Þ ð1Þ ¼ a1 k Wik
Wik
vk
a3 si ; uk þ a2 rgi
k dg k¼1 k¼2

i ¼ 2; . . . ; N ;

ðB:6Þ

N X dsi q ð1Þ ð1Þ ð2Þ ¼
X2
ðWiN WNk
Wik Þ
uk ui þ a6 k2 q1 dg k¼2

i ¼ 1; . . . ; N ; N X drgi q ð1Þ ¼
X2
vi
k Wik sk ; i ¼ 2; . . . ; N
1; dg q1 k¼1 N X d
vi ð1Þ ¼ a4 k Wik
uk
a5 rgi þ a2 si ; i ¼ 2; . . . ; N
1; dg k¼1 dsi q ¼
X2
ui dg q1 N

X ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ þ a6 k2 Wi1 W1k þ WiN WNk
Wik
uk

þ a4 k

N
1 X

ð1Þ

Wik rgk þ a1 k

k¼1

N
1 X

ð1Þ

Wik sk ;

k¼1

i ¼ 1; . . . ; N
1:

When i ¼ N , use should be made of the following relation rgN ¼

N a6 X ð1Þ k W
uk : a4 k¼2 Nk

ðB:7Þ

k¼1

þ a4 k

N
1 X

ð1Þ

Wik rgk þ a1 k

k¼2

N
1 X

ð1Þ

Wik sk ;

Appendix C. Expression of Mi

k¼2

i ¼ 1; . . . ; N ; ðB:1Þ

where N a6 X a1 ð1Þ rgi ¼ k Wik
uk
si ; a4 k¼1 a4 when i ¼ 1 or i ¼ N .

ðB:2Þ

B.2. C–C beams d
ui ¼ a1 k dg

N
1 X

ð1Þ

Wik
uk þ a2 rgi
k

k¼2

N
1 X

ð1Þ Wik
vk
a3 si ;

k¼2

N
1 N
1 X X drgi q ð1Þ ¼
k2 Aik ðe1
Wik sk ; uk
e2
vk Þ
X2
vi
k dg q1 k¼2 k¼2 N
1 X d
vi ð1Þ ¼ a4 k Wik
uk
a5 rgi þ a2 si ; dg k¼2 N
1

X dsi q ð2Þ ¼
X2
e3 Aik
a6 Wik
ui þ k2 uk dg q1 k¼2 N
1 N
1 N
1 X X X ð1Þ ð1Þ þ a4 k Wik rgk
e1 k2 Aik
vk þ a1 k Wik sk ; k¼2

k¼2

k¼2

ðB:3Þ

For illustration, only the explicit expression of Mi for S–S beams is presented hereby: 2 3 m11 m12 m13 m14 6 0 0 m23 m24 7 7: ðC:1Þ Mi ¼ 6 4 m31 m32 0 m34 5 m41 m42 0 m44 i The expressions of partitioned matrices mij are presented as follows with the second column denoting the order: 2 3 a a2 a6 4 1 5 m11 ¼ a1 kw1 þ k 0 ; N  N; a4 aN 2 3 01ðN
2Þ m12 ¼ a2 4 IðN
2ÞðN
2Þ 5; N  ðN
2Þ; 01ðN
2Þ m13 ¼
k½ b2

   bN
1 ; N  ðN
2Þ; 2 3 1 0 0 a1 a2 4 0 0 0 5; N  N ; m14 ¼
a3 I
a4 0 0 1 b3

W.Q. Chen et al. / Composite Structures 63 (2004) 417–425

q 2 X I; ðN
2Þ  ðN
2Þ; q1  T m24 ¼
k aT2 aT3    aTN
1 ; ðN
2Þ  N ;  T m31 ¼ a4 k aT2 aT3    aTN
1 ; ðN
2Þ  N ; m23 ¼


m32 ¼
a5 I; ðN
2Þ  ðN
2Þ;  m34 ¼ a2 0ðN
2Þ1 IðN
2ÞðN
2Þ 0ðN
2Þ1 ; ðN
2Þ  N ; m41 ¼ a6 k2 B


q 2 X I; q1

N  N;

m42 ¼ a4 k½ b2

b3



bN
1 ;

m44 ¼ a1 k½ 0

b2



b3

bN
1

N  ðN
2Þ; 0 ;

N  N;

where I is the unity matrix, 2

ð1Þ

W12 ð1Þ W22 .. .

  .. .

ð1Þ 3 W1N ð1Þ W2N 7 7 ; .. 7 . 5

ð1Þ



WNN

ð1Þ

W11 6 W ð1Þ 6 w1 ¼ 6 21 4 ... ð1Þ

WN 1

WN 2

ðC:2Þ

ð1Þ

ai and bi are the ith row and column vectors of matrix w1 respectively, and the elements of matrix B are ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð2Þ

Bij ¼ Wi1 W1j þ WiN WNj
Wij ; ¼ 1; 2; . . . ; N :

i; j ðC:3Þ

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