Vibration and buckling of composite beams using refined shear deformation theory

International Journal of Mechanical Sciences 62 (2012) 67–76

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Vibration and buckling of composite beams using refined shear deformation theory Thuc P. Vo a,b,n, Huu-Tai Thai c a

ˆ r University, Mold Road, Wrexham LL11 2AW, UK School of Mechanical, Aeronautical and Electrical Engineering, Glyndw Advanced Composite Training and Development Centre, Unit 5, Hawarden Industrial Park Deeside, Flintshire CH5 3US, UK c Department of Civil and Environmental Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea b

a r t i c l e i n f o

abstract

Article history: Received 7 March 2012 Received in revised form 11 May 2012 Accepted 1 June 2012 Available online 12 June 2012

Vibration and buckling analysis of composite beams with arbitrary lay-ups using refined shear deformation theory is presented. The theory accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from Hamilton’s principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are obtained for composite beams to investigate effects of fibre orientation and modulus ratio on the natural frequencies, critical buckling loads and corresponding mode shapes. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Composite beams Refined shear deformation theory Triply coupled vibration and buckling

1. Introduction Structural components made with composite materials are increasingly being used in various engineering applications due to their attractive properties in strength, stiffness, and lightness. Understanding their dynamic and buckling behaviour is of increasing importance. The classical beam theory (CBT) known as Euler–Bernoulli beam theory is the simplest one and is applicable to slender beams only. For moderately deep beams, it overestimates buckling loads and natural frequencies due to ignoring the transverse shear effects. The first-order beam theory (FOBT) known as Timoshenko beam theory is proposed to overcome the limitations of the CBT by accounting for the transverse shear effects. Since the FOBT violates the zero shear stress conditions on the top and bottom surfaces of the beam, a shear correction factor is required to account for the discrepancy between the actual stress state and the assumed constant stress state. To remove the discrepancies in the CBT and FOBT, the higher-order beam theory (HOBT) is developed to avoid the use of shear correction factor and has a better prediction of response of laminated beams. The HOBTs can be developed based on the assumption of higher-order variations of in-plane displacement or

n Corresponding author at: School of Mechanical, Aeronautical and Electrical ˆ r University, Mold Road, Wrexham LL11 2AW, UK. Engineering, Glyndw Tel.: þ44 1978 293979. E-mail address: [email protected] (T.P. Vo).

0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.06.001

both in-plane and transverse displacements through the depth of the beam. Many numerical techniques have been used to solve the dynamic and/or buckling analysis of composite beams using HOBTs. Some researchers studied the free vibration characteristics of composite beams by using finite element [1–7]. Khdeir and Reddy [8,9] developed analytical solutions for free vibration and buckling of cross-ply composite beams with arbitrary boundary conditions in conjunction with the state space approach. Analytical solutions were also derived by Kant et al. [10,11] and Zhen and Wanji [12] to study vibration and buckling of composite beams. By using the method of power series expansion of displacement components, Matsunaga [13] analysed the natural frequencies and buckling stresses of composite beams. Aydogdu [14–16] carried out the vibration and buckling analysis of crossply and angle-ply with different sets of boundary conditions by using Ritz method. Jun et al. [17,18] introduced the dynamic stiffness matrix method to solve the free vibration and buckling problems of axially loaded composite beams with arbitrary lay-ups. In this paper, which is extended from previous research [19], vibration and buckling analysis of composite beams using refined shear deformation theory is presented. The displacement field is reduced from the so-called Refined Plate Theory developed by Shimpi [20,21] and based on the following assumptions: (1) the axial and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments; (2) the bending component

68

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

of axial displacement is similar to that given by the CBT; and (3) the shear component of axial displacement gives rise to the higher-order variation of shear strain and hence to shear stress through the depth of the beam in such a way that shear stress vanishes on the top and bottom surfaces. The most interesting feature of this theory is that it satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. The three governing equations of motion are derived from Hamilton’s principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom (DOF) per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are obtained for composite beams to investigate effects of fibre orientation and modulus ratio on the natural frequencies, critical buckling loads and corresponding mode shapes.

ksx ¼ w00s

ð4dÞ

where differentiation with respect to the x-axis is denoted by primes (0 ).

3. Variational formulation In order to derive the equations of motion, Hamilton’s principle is used: Z t2 d ðKUVÞ dt ¼ 0 ð5Þ t1

where U, V and K denote the strain energy, potential energy, and kinetic energy, respectively. The variation of the strain energy can be stated as Z dU ¼ ðsx dEx þ sxz dgxz Þ dv v

Z

2. Kinematics

l

¼ 0

A laminated composite beam made of many plies of orthotropic materials in different orientations with respect to the x-axis, as shown in Fig. 1, is considered. Based on the assumptions made in the preceding section, the displacement field of the present theory can be obtained as   @wb ðx,tÞ 1 5  z 2 @ws ðx,tÞ Uðx,z,tÞ ¼ uðx,tÞz þz  ð1aÞ @x 4 3 h @x

A

Z

@U ¼ E0x þ zkbx þ f ksx Ex ¼ @x

gxz ¼

ð2aÞ

@W @U 0 þ ¼ ð1f Þg0xz ¼ g g0xz @x @z

ð2bÞ

where   1 5  z 2 f ¼z  þ 4 3 h 0

g ¼ 1f ¼

sx z dA

ð7bÞ

sx f dA

ð7cÞ

sxz g dA

ð7dÞ

A

ð1bÞ

where u is the axial displacement along the mid-plane of the beam, wb and ws are the bending and shear components of transverse displacement along the mid-plane of the beam, respectively. The non-zero strains are given by

Z

Msx ¼

A

Q xz ¼

Z A

The variation of the potential energy of the axial force P0, which is applied through the centroid, can be expressed as Z l dV ¼  P 0 ½dw0b ðw0b þ w0s Þ þ dw0s ðw0b þ w0s Þ dx ð8Þ 0

ð3aÞ

  z 2  5 14 4 h

The variation of the kinetic energy is obtained as Z _ dW _ Þ dv dK ¼ rk ðU_ dU_ þ W v

Z

ð3bÞ

l

¼ 0

_ 0 um _ _ 0b mf w _ 0s Þ þ dw_b m0 ðw_b þ w_ s Þ ½duðm 1w

and E0x , g0xz , kbx and ksx are the axial strain, shear strains and curvatures in the beam, respectively, defined as

_ 0b ðm1 u_ þ m2 w _ 0b þ mfz w _ 0s Þ þ dw

E0x ¼ u0

ð4aÞ

_ 0s ðmf u_ þmfz w _ 0b þmf 2 w _ 0s Þ dx þ dw_ s m0 ðw_b þ w_ s Þ þ dw

g0xz ¼ w0s

ð4bÞ

kbx ¼ w00b

ð4cÞ

z

ð6Þ

where Nx ,M bx ,M sx and Qxz are the axial force, bending moments and shear force, respectively, defined by integrating over the crosssectional area A as Z ð7aÞ Nx ¼ sx dA

Mbx ¼ Wðx,z,tÞ ¼ wb ðx,tÞ þws ðx,tÞ

ðNx dE0z þM bx dkbx þ M sx dksx þQ xz dg0xz Þ dx

where the differentiation with respect to the time t is denoted by dot-superscript convention and rk is the density of a kth layer and m0 ,m1 ,m2 ,mf ,mfz and mf 2 are the inertia coefficients, defined by mf ¼ 

y

m1 5 þ 2 m3 4 3h

ð10aÞ

m2 5 þ 2 m4 4 3h

ð10bÞ

mfz ¼ 

x mf 2 ¼

h b L Fig. 1. Geometry of a laminated composite beam.

ð9Þ

m2 5 25  m4 þ 4 m6 16 6h2 9h

ð10cÞ

where ðm0 ,m1 ,m2 ,m3 ,m4 ,m6 Þ ¼

Z A

rk ð1,z,z2 ,z3 ,z4 ,z6 Þ dA

ð11Þ

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

Z

69

By substituting Eqs. (6), (8) and (9) into Eq. (5), the following weak statement is obtained: Z t2 Z l _ 0 um _ _ 0b mf w _ 0s Þ þ dw_b m0 ðw_b þ w_ s Þ 0¼ ½duðm 1w

R23 ¼

ð15eÞ

_ 0b ðm1 u_ þ m2 w _ 0b þ mfz w _ 0s Þ þ dw_ s m0 ðw_b þ w_ s Þ þ dw 0 0 _ s ðmf u_ þ mfz w _ b þ mf 2 w _ 0s Þ þP 0 ½dw0b ðw0b þ w0s Þ þ dw0s ðw0b þ w0s Þ þ dw

 Z  D11 5 þ 2 F 11 dy  4 y 3h

R33 ¼

 Z  D11 5 25  2 F 11 þ 4 H11 dy 16 6h y 9h

ð15fÞ

R44 ¼

 Z  25 25 25 A55  2 D55 þ 4 F 55 dy y 16 2h h

ð15gÞ

t1

0

0

Nx du

þ M bx dw00b þ Msx dw00s Q xz dw0s 

dx dt

The stress–strain relations for the kth lamina are given by

sx ¼ Q 11 Ex

ð13aÞ

sxz ¼ Q 55 gxz

ð13bÞ

where Q 11 and Q 55 are the elastic stiffnesses transformed to the x direction. More detailed explanation can be found in Ref. [22]. The constitutive equations for bar forces and bar strains are obtained by using Eqs. (2), (7) and (13): 9 2 8 38 0 9 R Nx > R12 R13 0 > E > > > > > 6 11 > > b> > x > > > 7> b > = > > > k 33 > 5> > 4 > > x> > x> > > > > :Q ; sym: R44 : g0 ; xz

where Rij are the laminate stiffnesses of general composite beams and given by Z R11 ¼ A11 dy ð15aÞ y

R12 ¼

Z

B11 dy

ð15bÞ

y

 Z  B11 5 þ 2 E11 dy  R13 ¼ 4 y 3h

ð15cÞ

D11 dy

ð15dÞ

y

ð12Þ

4. Constitutive equations

xz

R22 ¼

where Aij ,Bij and Dij matrices are the extensional, coupling and bending stiffness and Eij ,F ij ,Hij matrices are the higher-order stiffnesses, respectively, defined by Z ðAij ,Bij ,Dij ,Eij ,F ij ,Hij Þ ¼ Q ij ð1,z,z2 ,z3 ,z4 ,z6 Þ dz ð16Þ z

5. Governing equations of motion The equilibrium equations of the present study can be obtained by integrating the derivatives of the varied quantities by parts and collecting the coefficients of du, dwb and dws : € € 0b mf w € 0s N0x ¼ m0 um 1w

ð17aÞ

00 00 € € €0 € 00 € 00 Mb00 x P 0 ðwb þ ws Þ ¼ m0 ðwb þ ws Þ þ m1 u m2 w b mfz w s

ð17bÞ

0 00 00 € € 00 € 00 € € Ms00 x þ Q xz P 0 ðwb þ ws Þ ¼ m0 ðwb þ ws Þ þ mf u mfz w b mf 2 w s

ð17cÞ

0

The natural boundary conditions are of the form:

du : N x

ð18aÞ

0 0 € €0 €0 dwb : Mb0 x P 0 ðwb þws Þm1 u þ m2 w b þ mfz w s

ð18bÞ

dw0b : Mbx

ð18cÞ

Table 1 The first five natural frequencies (Hz) of simply-supported beams with a symmetric cross-ply [901/01/01/901] lay-up (Material I with L=h ¼ 2:273 and 22.73). Mode

1 2 3 4 5

L=h ¼ 2:273

L=h ¼ 22:73

ABAQUS [3]

Ref. [3]

Ref. [7]

Present

ABAQUS [3]

Ref. [3]

Ref. [7]

Ref. [18]

Present

82.90 200.60 324.30 450.10 576.40

83.70 195.80 313.40 441.80 583.80

82.81 195.62 319.36 460.18 515.41

82.42 195.20 315.88 449.83 578.65

14.95 57.60 122.80 204.20 296.60

14.96 57.90 123.70 206.40 300.60

14.97 57.85 123.55 206.18 300.71

14.97 57.87 123.58 206.01 299.68

14.42 55.88 119.76 200.44 292.73

Table 2 Effect of span-to-height ratios on the non-dimensional fundamental natural frequencies of a symmetric and an anti-symmetric cross-ply composite beam with simply-supported boundary condition (Material II with E1/E2 ¼40). Lay-ups

[01/901/01]

Theory

FOBT HOBT

[01/901]

FOBT HOBT

Reference

L=h 5

10

20

50

Khdeir and Reddy [8] Present Murthy et al. [5] Khdeir and Reddy [8] Aydogdu [14] Present

9.205 9.205 9.207 9.208 9.207 9.206

13.670 13.665 13.614 13.614 – 13.607

– 16.359 – – 16.337 16.327

– 17.456 – – – 17.449

Khdeir and Reddy [8] Present Murthy et al. [5] Khdeir and Reddy [8] Aydogdu [14] Present

5.953 5.886 6.045 6.128 6.144 6.058

6.886 6.848 6.908 6.945 – 6.909

– 7.187 – – 7.218 7.204

– 7.294 – – – 7.296

70

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

€ 0b þ mf 2 w € 0s dws : Ms0x þQ xz P 0 ðwb0 þ ws0 Þmf u€ þ mfz w

ð18dÞ

dw0s : Msx

ð18eÞ

Substituting these expressions in Eq. (20) into the corresponding weak statement in Eq. (12), the finite element model of a typical element can be expressed as the standard eigenvalue

By substituting Eqs. (4) and (14) into Eq. (17), the explicit form of the governing equations of motion can be expressed with respect to the laminate stiffnesses Rij: 000 € €0 €0 R11 u00 R12 w000 b R13 ws ¼ m0 um1 w b mf w s

ð19aÞ 0

ð19cÞ

3

-5

uj Cj

4 X

5

10

15

20

25

30

L/h = 5 L/h = 10 L/h = 20

ω

6

4

2

-1

0

0

1

2

3

4

5

6

P c wbj c j

ð20bÞ

c wsj c j

ð20cÞ

j¼1

ws ¼

0

P

ð20aÞ

j¼1 4 X

0

8

The present theory for composite beams described in the previous section is implemented via a displacement based finite element method. The variational statement in Eq. (12) requires that the bending and shear components of transverse displacement wb and ws be twice differentiable and C1-continuous, whereas the axial displacement u must be only once differentiable and C0-continuous. The generalized displacements are expressed over each element as a combination of the linear interpolation c for function Cj for u and Hermite-cubic interpolation function c j wb and ws associated with node j and the nodal values:

wb ¼

9 6

6. Finite element formulation

2 X

L/h = 20

12

0

Eq. (19) is the most general form for vibration and buckling of composite beams of composite beams, and the dependent variables, u, wb and ws are fully coupled. The resulting coupling is referred to as triply axial–flexural coupled vibration and buckling. It can be seen that the explicit solutions for vibration and buckling of composite beams become complicated due to this triply coupling effect.

u¼

L/h = 10

15

iv 00 00 00 € € € R13 u000 R23 wiv b R33 ws þ R44 ws P 0 ðwb þws Þ ¼ m0 ðwb þ ws Þ þ mf u

€ 00b mf 2 w € 00s mfz w

L/h = 5

ð19bÞ

ω

iv 00 00 € € € R12 u000 R22 wiv b R23 ws P 0 ðwb þ ws Þ ¼ m0 ðwb þ ws Þ þ m1 u 00 00 € b mfz w €s m2 w

18

Fig. 2. The interaction diagram between non-dimensional critical buckling load and fundamental natural frequency of a symmetric and an anti-symmetric crossply composite beam with simply-supported boundary condition (Material II with L=h ¼ 5, 10 and 20). (a) Symmetric cross-ply lay-up ([01/901/01]), (b) antisymmetric cross-ply lay-up ([01/901]).

j¼1

Table 3 Effect of span-to-height ratios on the non-dimensional critical buckling loads of a symmetric and an anti-symmetric cross-ply composite beam with simply-supported boundary condition (Materials II and III with E1/E2 ¼ 10). Lay-ups

Material II [01/901/01]

[01/901]

Material III [01/901/01]

[01/901]

Theory

FOBT HOBT FOBT HOBT

FOBT HOBT FOBT HOBT

Reference

L=h 5

10

20

50

Present Aydogdu [15] Present Present Aydogdu [15] Present

4.752 4.726 4.709 1.883 1.919 1.910

6.805 – 6.778 2.148 – 2.156

7.630 7.666 7.620 2.226 2.241 2.228

7.897 – 7.896 2.249 – 2.249

Present Aydogdu [15] Present Present Aydogdu [15] Present

4.069 3.728 3.717 1.605 1.765 1.758

6.420 – 6.176 1.876 – 2.104

7.503 7.459 7.416 1.958 2.226 2.214

7.875 – 7.860 1.983 – 2.247

ω2

40

ω3

35

ω4

ω

30 25 20 15 10 5 0

0

10

20

30

40

30

40

50

E1/E2 35

ω1 ω2

30

ω3 ω4

ω

25 20 15 10 5 0

0

10

20

50

E1/E2 Fig. 4. Effect of material anisotropy on the first four non-dimensional natural frequencies of a symmetric and an anti-symmetric cross-ply composite beam with simply-supported boundary condition (Material II with L=h ¼ 5). (a) Symmetric cross-ply lay-up ([01/901/01]), (b) anti-symmetric cross-ply lay-up ([01/901]).

30.549 – 30.319 5.381 – 5.387 25.620 24.685 24.655 5.180 5.225 5.219

50

– 30.882 – – 30.859 5.395 – 5.399 – 27.154 – 27.084 27.050 5.265 5.296 5.290 18.989 18.974 18.832 – 18.814 4.848 – 4.936

16.253 – 14.857 4.571 – 4.697 FOBT HOBT

ω1

45

[01/901]

50

6.600 5.896 5.895 3.110 3.376 3.373

Fig. 3. Effect of material anisotropy on the non-dimensional critical buckling loads of a symmetric and an anti-symmetric cross-ply composite beam with simplysupported boundary condition (Material II with L=h ¼ 5).

Present Aydogdu [15] Present Present Aydogdu [15] Present

50

FOBT HOBT

40

Material III [01/901/01]

30 E1/E2

FOBT HOBT

20

[01/901]

10

HOBT

0

8.606 8.604 8.613 8.613 8.609 3.680 3.906 3.903

0

71

Khdeir and Reddy [9] Present Khdeir and Reddy [9] Aydogdu [15] Present Present Aydogdu [15] Present

2

FOBT

4

Material II [01/901/01]

Pcr

6

20

Anti-symmetric cross-ply

8

10

Symmetric cross-ply

5

10

L=h

where ½K,½G and ½M are the element stiffness matrix, the element geometric stiffness matrix and the element mass matrix, respectively. The explicit forms of ½K can be found in Ref. [19] and of ½G

Reference

ð21Þ

Theory

ð½KP 0 ½Go2 ½MÞfDg ¼ f0g

Lay-ups

problem:

Table 4 Effect of span-to-height ratios on the non-dimensional critical buckling loads of a symmetric and an anti-symmetric cross-ply composite beam with simply-supported boundary condition (Materials II and III with E1/E2 ¼ 40).

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

and ½M are given by Z l c0i c0j dz G22 ij ¼

ð22aÞ

1.6190 1.6200 1.6190 1.6237 1.6152 0.7290 0.7320 0.7295 0.2600 0.2619 0.2611 1.1290 1.1360 1.1310 1.1302

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

901

72

0

0

Z

m0 Ci Cj dz

2.1

Pcr (CC) Pcr (CS)

1.8

Pcr (SS)

1.5

Pcr (CF)

1.2 0.9 0.6 0.3 0

0

15

30

45

60

75

90

θ Fig. 5. Variation of the non-dimensional critical buckling loads of symmetric angle-ply ½y=ys composite beams with respect to the fibre angle change (Material IV with L=h ¼ 15).

1.6690 2.1984 1.6690 1.6711 1.6309 0.7360 1.0124 0.7361 0.2620 0.3631 0.2634 1.1500 1.5510 1.1460 1.1407 et al. [23] al. [24]

In this section, a number of numerical examples are presented and analysed for verification the accuracy of the present theory and investigation the natural frequencies, critical buckling loads and corresponding mode shapes of composite beams with arbitrary layups. The boundary conditions of beam are presented by C for clamped edge: u ¼ wb ¼ w0b ¼ ws ¼ w0s ¼ 0, S for simply-supported edge: u ¼ wb ¼ ws ¼ 0 and F for free edge. All laminate are of equal thickness and made of the same orthotropic material, whose properties are as follows:

CS

7. Numerical examples

et al. [23]

ð23Þ

CF

fDg ¼ fu wb ws gT

et al. [23]

ð22iÞ

SS

m0 ci cj þ mf 2 c0i c0j dz

l

1.9290 3.1843 1.9470 1.8383 1.9918 0.8040 1.5368 0.9078 0.2880 0.5551 0.3253 1.1780 2.3030 1.3880 1.4019

ð22hÞ

2.1950 4.0981 2.8780 2.3445 3.2355 1.1410 2.1032 1.5540 0.4140 0.7678 0.5614 1.6710 3.0570 2.2130 2.3538

m0 ci cj þ mfz c0i c0j dz

l

4.2940 4.6635 3.9880 3.6484 4.5695 1.8960 2.5105 2.4039 0.6760 0.9249 0.8836 2.9600 3.5590 3.2430 3.5079

ð22gÞ

et al. [23] al. [24]

0

m0 ci cj þ m2 c0i c0j dz

Aydogdu [16] Chandrashekhara Krishnaswamy et Chen et al. [25] Present Aydogdu [16] Chandrashekhara Present Aydogdu [16] Chandrashekhara Present Aydogdu [16] Chandrashekhara Krishnaswamy et Present

l

0

Z

ð22fÞ

CC

0

0

Z

mf Ci c0j dz

l

451

Z

ð22eÞ

301

Z

m1 Ci c0j dz

151

0

601

l

4.9730 4.8487 4.8690 4.8575 4.8969 2.6510 2.6560 2.6494 0.9810 0.9820 0.9801 3.7750 3.7310 3.8370 3.8183

Z

ð22dÞ

All other components are zero. In Eq. (21), fDg is the eigenvector of nodal displacements corresponding to an eigenvalue:

Pcr

751

l

Fibre angle y

M 33 ij ¼

ð22cÞ

0

M 13 ij ¼ 

M 23 ij ¼

c0i c0j dz

l

M 12 ij ¼ 

M 22 ij ¼

ð22bÞ

01

Z

c0i c0j dz

1.6120 1.6815 1.6120 1.6161 1.6056 0.7250 0.7611 0.7247 0.2580 0.2723 0.2593 1.1220 1.1750 1.1290 1.1231

0

Reference

M 11 ij ¼

l

Boundary conditions

G33 ij ¼

Z

Table 5 The non-dimensional fundamental natural frequencies of symmetric angle-ply ½y=ys composite beams with respect to the fibre angle change (Material IV with L=h ¼ 15).

G23 ij ¼

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

Material I [3]: E1 ¼ 241:5 GPa,

E2 ¼ 18:98 GPa,

G12 ¼ G13 ¼ 5:18 GPa,

G23 ¼ 3:45 GPa,

n12 ¼ 0:24, r ¼ 2015 kg=m3

ð24Þ

73

of the FOBT, a value of 5/6 is used for the shear correction factor. An excellent agreement between the predictions of the present model and the results of the other models mentioned (FOBT and HOBT) can be observed. Material II with E1 =E2 ¼ 40 is chosen to

Material II [8,9,14,15]: E1 =E2 ¼ open,

G12 ¼ G13 ¼ 0:6E2 ,

G23 ¼ 0:5E2 ,

n12 ¼ 0:25 ð25Þ 1

Material III [14,15]: E1 =E2 ¼ open,

G12 ¼ G13 ¼ 0:5E2 ,

G23 ¼ 0:2E2 ,

0.75

Material IV [23]: E1 ¼ 144:9 GPa,

E2 ¼ 9:65 GPa,

G23 ¼ 3:45 GPa,

n12 ¼ 0:3, r ¼ 1389 kg=m3

G12 ¼ G13 ¼ 4:14 GPa, ð27Þ

For convenience, the following non-dimensional terms are used in presenting the numerical results: 8 P cr L2 > > > for Materials II and III > 3 < E2 bh ð28aÞ P cr ¼ 2 > > P cr L > for Material IV > : 3 E1 bh 8 rffiffiffiffiffi > oL2 r > > > < h E2 o¼ 2 rffiffiffiffiffi > o L r > > > : h E1

u wb ws

n12 ¼ 0:25 ð26Þ

0.5

0.25 0 0

0.25

0.5

0.75

1 u wb ws

0.75

for Materials II and III ð28bÞ for Material IV

1

x/L

0.5 0.25

As the first example, simply-supported symmetric cross-ply ½901=01=01=901 composite beams with two span-to-height ratios (L=h¼2.273 and 22.73) are considered. The material properties are assumed to be Material I. The first five natural frequencies are tabulated in Table 1 along with numerical results of previous studies [3,7,18]. The ABAQUS solutions given in Ref. [3] were obtained by using the plane stress element type CPS8 (quadrilateral element of eight node, 16 DOF per element). The differences between the natural frequencies calculated by the present formulation and those using different higher-order beam theories are very small. In the next example, vibration and buckling analysis of simplysupported composite beams with symmetric cross-ply ½01=901=01 and anti-symmetric cross-ply ½01=901 lay-ups is performed. Materials II and III with E1 =E2 ¼ 10 and 40 are used. The fundamental natural frequencies and critical buckling loads for different span-to-height ratios are compared with exact solutions [8,9] and the finite element results [5,14,15] in Tables 2–4. In the case

0 -0.25

0

0.25

0.5

0.75

1

0.75

1

0.75

1

x/L

-0.5 -0.75 -1

1

u wb ws

0.75 0.5 0.25 0 -0.25

0

0.25

0.5 x/L

-0.5 -0.75 -1

0.8 Pcr (with coupling)

Pcr

0.7

Pcr (without coupling)

1

0.6

0.75

0.5

0.5

0.4

0.25

0.3

0

0.2

-0.25

0

0.25

0.5 x/L

-0.5

0.1 0

u wb ws

-0.75

0

15

30

45 θ

60

75

90

Fig. 6. Variation of the non-dimensional critical buckling loads of a simplysupported anti-symmetric angle-ply ½y=y composite beam with respect to the fibre angle change (Material IV with L=h ¼ 15).

-1 Fig. 7. Vibration mode shapes with the axial and flexural components of a simplysupported composite beam with the fibre angle 451. (a) Fundamental mode shape o1 ¼ 0:9078, (b) second mode shape o2 ¼ 3:5255, (c) third mode shape o3 ¼ 7:5850, (d) fourth mode shape o4 ¼ 12:7587.

16.6431 15.7685 11.8313 7.5850 6.2616 6.1639 6.1977 8.9572 8.3223 5.7944 3.5255 2.8798 2.8352 2.8526 2.6494 2.4039 1.5540 0.9078 0.7361 0.7247 0.7295 16.6431 15.7685 11.8313 7.5850 6.2616 6.1639 6.1977 8.9572 8.3223 5.7944 3.5255 2.8798 2.8352 2.8526

24.7032 23.7045 18.8714 12.7587 10.6606 10.4930 10.5426

o3 o2 o1 oz3 oz2

oz4

With coupling

show the effect of the axial force on the fundamental natural frequencies of beam with various L=h ratios (Fig. 2). It can be seen that the change of the natural frequency due to the axial force is noticeable. The natural frequency diminishes when the axial force changes from tensile to compressive, as expected. It is obvious that the natural frequency decreases with the increase of axial force, and the decrease becomes more quickly when the axial force is close to critical buckling load. For an anti-symmetric cross-ply lay-up, with L=h ¼ 5, 10 and 20, at about P¼3.903, 4.936 and 5.290, respectively, the natural frequencies become zero which implies that at these loads, bucklings occur as a degenerate case of natural vibration at zero frequency. It also means that the buckling loads of composite beams under the axial force can be also obtained indirectly through vibration problem by increasing the axial force until the corresponding natural frequency vanishes. In order to show the effect of material anisotropy (E1 =E2 ) on the critical buckling loads and the first four natural frequencies of a symmetric and an anti-symmetric cross-ply layup, a simply-supported composite beam with L=h ¼ 5 is performed. It is observed that the critical buckling loads and natural frequencies increase with increasing orthotropy (Figs. 3 and 4). For a symmetric cross-ply lay-up, as ratio of E1 =E2 increases, the order of the second and third vibration modes as well as the third and fourth vibration modes changes each other at E1 =E2 ¼ 7 and 27, respectively (Fig. 4). To demonstrate the accuracy and validity of this study further, the fundamental natural frequencies of symmetric angle-ply ½y=ys composite beams are given in Table 5 to illustrate the effect of boundary conditions and of fibre orientation. In the following examples, Material IV with L=h ¼ 15 is used. Variation of the critical buckling loads with respect to the fibre angle change is plotted in Fig. 5. The natural frequencies and buckling loads decrease monotonically with the increase of the fibre angle for all the boundary conditions considered. As the fibre angle increases, the buckling loads decrease more quickly than natural frequencies. For instant, the ratio between the buckling load at the fibre angles 01 and 901 is 9.8 and similar value for natural frequency is 3.0 for clamped–clamped boundary condition. It is observed that the present results are in good agreement with previous studies [16,23–25] for all fibre angles. In order to investigate the effects of fibre orientation on the natural frequencies, critical buckling loads and corresponding mode shapes, a simply-supported anti-symmetric angle-ply ½y=y composite beam is considered. The first four natural frequencies and critical buckling loads with respect to the fibre angle change are shown in Table 6 and Fig. 6. The uncoupled solution, which neglects the coupling effects coming from the 2.1

Pcr (with coupling)

1.8

Pcr (without coupling)

Pcr

2.6494 2.4039 1.5540 0.9078 0.7361 0.7247 0.7295

1.2

oz1

No coupling

1.5

0.9 0.6 0.3 0

0

15

30

45

60

75

90

θ 01 151 301 451 601 751 901

Fibre angle

Table 6 The first four non-dimensional natural frequencies of a simply-supported anti-symmetric angle-ply ½y=y composite beam with respect to the fibre angle change (Material IV with L=h ¼ 15).

24.7032 23.7045 18.8714 12.7587 10.6606 10.4930 10.5426

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

o4

74

Fig. 8. Variation of the non-dimensional critical buckling loads of a clamped– clamped unsymmetric ½01=y composite beam with respect to the fibre angle change (Material IV with L=h ¼ 15).

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

material anisotropy, is also given. Due to coupling effects, the uncoupled solution might not be accurate. However, as the fibre angle increases, these effects become negligible. Therefore, it can be seen in Table 6 and Fig. 6 that the results by uncoupled and coupled solutions are identical. For all fibre angles, the first four natural frequencies by the coupled solution exactly correspond

1

u wb ws

0.75

75

to the first, second, third and fourth flexural modes by the uncoupled solution, respectively. It can be explained partly by the typical vibration mode shapes with the fibre angle y ¼ 451 in Fig. 7. All the vibration modes exhibit double coupling (bending and shear components). It is indicated that the uncoupled solution is sufficiently accurate for an anti-symmetric angle-ply lay-up. To investigate the coupling effects further, a clamped–clamped unsymmetric ½01=y composite beam is chosen. As the fibre angle increases, major effects of coupling on the natural frequencies

1

u

0.5

wb ws

0.75

0.25 0 0

0.25

0.5

0.75

1

0.5

x/L

-0.25

0.25 1

u wb ws

0.75 0.5

0 0

0.25

0.25

0.5 x/L

0.75

1

0.5 x/L

0.75

1

-0.25

0 -0.25

0

0.25

0.5

0.75

1

x/L

1 u

-0.5 -0.75

wb

0.75

ws

-1

0.5

1

u wb ws

0.75

0.25

0.5 0.25

0 0

-2E-15 0

0.25

-0.25

0.5

0.75

1

x/L

0.25

-0.25

-0.5

1

-0.75

u

1 0.75

wb

0.75

u wb ws

ws

0.5

0.5 0.25

0.25

0 -0.25

0

0.25

0.5

0.75

1

x/L

-0.5

0 0

-0.75 -1 Fig. 9. Vibration mode shapes with the axial and flexural components of a clamped–clamped composite beam with the fibre angle 601. (a) Fundamental mode shape o1 ¼ 2:859, (b) second mode shape o2 ¼ 7:400, (c) third mode shape o3 ¼ 13:071, (d) fourth mode shape o4 ¼ 19:975.

-0.25

0.25

0.5

0.75

1

x/L

Fig. 10. Bucking mode shapes with the axial and flexural components of a clamped–clamped composite beam with the fibre angles 301, 601 and 901. (a) P cr ¼ 1:3028 with the fibre angle 301, (b) P cr ¼ 0:7888 with the fibre angle 451, (c) P cr ¼ 0:6585 with the fibre angle 901.

76

T.P. Vo, H.-T. Thai / International Journal of Mechanical Sciences 62 (2012) 67–76

Table 7 The first four non-dimensional natural frequencies of an unsymmetric ½0=y clamped–clamped composite beam with respect to the fibre angle change (Material IV with L=h ¼ 15). Fibre angle

01 151 301 451 601 751 901

No coupling

With coupling

oz1

oz2

oz3

oz4

o1

o2

o3

o4

4.897 4.742 4.272 4.009 3.950 3.938 3.935

11.493 11.212 10.330 9.802 9.665 9.625 9.615

18.400 18.037 16.901 16.192 15.977 15.896 15.872

26.448 26.011 24.637 23.743 23.437 23.306 23.264

4.897 4.730 3.957 3.108 2.859 2.840 2.846

11.493 11.192 9.744 7.967 7.400 7.351 7.361

18.400 18.015 16.218 13.886 13.071 12.984 12.992

26.448 25.988 23.893 21.042 19.975 19.841 19.844

and critical buckling loads are seen in Table 7 and Fig. 8. The uncoupled and coupled solutions show discrepancy indicating that the coupling effects become significant, especially at the higher fibre angles. The typical vibration mode shapes corresponding to the first four natural frequencies with the fibre angle y ¼ 601 are illustrated in Fig. 9. The buckling mode shapes with various fibre angles y ¼ 301, 601 and 901 are also given in Fig. 10. Relative measures of the axial and flexural displacements show that all the vibration and buckling modes are triply coupled mode (axial, bending and shear components). This fact explains as the fibre angle changes, the uncoupled solution disagrees with coupled solution as anisotropy of the beam gets higher. That is, the uncoupled solution is no longer valid for unsymmetrically laminated composite beams, and triply extension–bending–shear coupled vibration and buckling should be considered simultaneously for accurate analysis of composite beams.

8. Conclusions A two-noded C1 beam element of five degree-of-freedom per node is developed to study the vibration and buckling behaviour of composite beams using refined shear deformation theory. This model is capable of predicting accurately the natural frequencies, critical buckling loads and corresponding mode shapes. It accounts for the parabolical variation of shear strains through the depth of the beam, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factor. The uncoupled solution is accurate for lower degrees of material anisotropy, but, becomes inappropriate as the anisotropy of the beam gets higher, and triply extension–bending–shear coupled vibration and buckling should be considered simultaneously for accurate analysis of composite beams. The present model is found to be appropriate and efficient in analysing vibration and buckling problem of composite beams.

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