Free vibration of composite beams using a refined shear flexible beam element

Computers& SlrucfuresVol. 43. No. 4. pp. 719-727, 1992

WS-7949192SS.oO+O.OO 0 1992PergamonPress Ltd

Printed inGreatBritain.

FREE VIBRATION OF COMPOSITE BEAMS USING A REFINED SHEAR FLEXIBLE BEAM ELEMENT K.

and K. M.

CHANDRASHEKHARA

BANGERAt

Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, MO 65401-0249, U.S.A. (Received 4 April 1991)

Abstract-A finite element model based on a higher-order shear deformation theory is developed to study the free vibration characteristics of laminated composite beams. The Poisson effect, which is often neglected in one-dimensional laminated beam analysis, is incorporated in the formulation of the beam constitutive equation. Also, the effects of in-plane inertia and rotary inertia are considered in the formulation of the mass matrix. Numerical results for symmetrically laminated composite beams are obtained as special cases and are compared with other exact solutions available in the literature. A variety of parametric studies are conducted to demonstrate the influence of beam geometry, Poisson effect, ply orientation, number of layers and boundary conditions on the frequencies and mode shapes of generally layered composite beams.

mation theory when compared with the 3-D elasticity solution. A survey of recent developments in the vibration analysis of laminated composite beams has been compiled by Kapania and Raciti [1 11. Exact solutions are available in [12] for the free vibration of simply supported composite beams based on the classical theory which neglects the effects of transverse shear deformation and rotary inertia. Free vibration analysis of symmetrically-laminated beams based on the first-order shear deformation theory was reported by Chen and Yang [ 131,and Chandrashekhara et al. [ 141. The results reported in [13] were based on the finite

INTRODUCTION

Fiber-reinforced composite beams are finding increasing applications in a variety of structural components such as helicopter blades, turbine blades and robot arms. Because of their high ratio of extensional modulus to transverse shear modulus, transverse shear deformation effects are pronounced in composite laminates and need to be included in the analysis for accurate prediction of their natural frequencies. Until recently, shear deformation was incorporated in the analysis of composite beams, plates and shells using the first-order shear deformation theory. This theory has some limitations such as inaccurate constant transverse shear distribution through the thickness necessitating the need for a shear correction factor. The value of this factor has been the subject of considerable discussion and it has been shown that this factor is not a constant as assumed by most works, but changes with mode number [ 1,2], Poisson’s ratio [3] and lamination scheme [4]. Also, the isoparametric finite element model based on the first-order shear deformation theory requires reduced integration of the shear stiffness terms in order to avoid the ‘shear locking’ phenomenon [S]. To avoid these inconsistencies, several higher-order theories have been proposed for isotropic beams [6-81 and composite plates [9]. Reddy [9] has shown that, for laminated plates, the results based on the higher-order shear deformation theory are more accurate than those obtained using the first-order shear defor-

TPresently at EGS Inc., Oak Park, MI 48237, U.S.A.

element approach whereas exact solutions for beams with various boundary conditions were presented in

[14]. It is noted that the laminated beam equations presented in [12, 141neglect the Poisson effect. In the present work, equations of motion are derived for laminated composite beams based on a higher-order plate theory [9] and the Poisson effect is incorporated in the one-dimensional model. The finite element method is used to solve the free vibration problem since exact solutions are not tractable for generally layered composite beams. Natural frequencies and mode shapes for symmetric and unsymmetric laminated beams under various boundary conditions are presented. MATHEMATICAL

FORMULATION

A generally layered laminated beam, as shown in Fig. 1, is considered. The one-dimensional laminated beam equations that account for Poisson effect are derived based on a higher-order plate theory as follows.

719

K.

720

CHANDRASHEKHARA and

BANGERA

K. M.

where

and A,, PI

=

A,6

42

46

42

46

h2

D,,

~512

-46

F12

F16

42 F,2 H,2

46 F,6

.

H,6 I

The strains {c$~(~,K~K&K:K&)~ can be solved in terms of {~~K~K~}’ using eqn (1) and substituting in eqn (5) results in Fig. I. Geometry of a laminated beam.

N, M,

The higher-order laminated plate equation can be expressed as [9]

constitutive

(6)

=

ii p, where A,,, B,,, I:t c., are the coefficients of the matrix ([a] - [b][c]-‘[b]r) and the matrix [c] is given by

[cl = where {N, M, P, Q, R} are the resultant vectors, {co, e*} are the midplane strain vectors and {K ‘, K’, K*} are the The laminate stiffness coefficients A,,, AZ, are by

M,, B,,De, E,,,F,, H,,) z, z2, z’, z4, z6) dz

A22

A26

B22

46

E22

E26

A26

A66

B26

&

E26

J%

B22

B26

D22

D26

F22

f-26

B26

B66

O26

D,

F26

F66

E22

E26

F22

F26

H22

H26

~36

&

5-26

F66

H26

H66

L

Using eqn (2), the transverse shear forcestrain relation for the laminated beam can be expressed as

(i, j = I, 2,6)

(3) Equations (6) and (7) are the one-dimensional laminated beam equations that account for transverse shear deformation and Poisson effect. The strain-displacement relation can be expressed as

and

s h/2

(A,:,D,r,F,:)=

-h/2

P,(l,

z2, z4) dz (i, j

=

4, 5).

For a laminated beam the width or y-direction free of stresses and eqn (1) can be written as

(4)

is

where u(x, t) and w(x, t) are the midplane strains and 4(x, t) is the normal rotation.

721

Free vibration of composite beams

The dynamic version of the principle of virtual work for the present problem can be written as

(9)

where p(l,zZ,z4,z6)dz

(&,I,,Lf,)=

a# a2w ax ax2 =-3-z

(

x -+-

hi2 (IO)

)

a*

16 I

a2w

7s

z (

>

1 -h/2 and p is the mass density. The equations of motion for the laminated beam are obtained by substituting eqn (9) and integrating the resulting equation by parts. The governing equations are given by

--

16 9h4

The appropriate

(1w

boundary

Essential boundary conditions

conditions

are as follows:

Natural boundary

conditions

K. CHANDRASHEKHARA

122

Table 1. Comparison of natural frequencies of a simply supported orthotropic (0”) graphite/epoxy beam

Table 2. Comparision of non-dimensional natural frequencies of symmetrically laminated [O/90/90/0] and [45/ -45/-451451 beams under various boundary conditions

f W-W t

Mode No.

h

I 120 (L = 762 mm)

2 i

*r

2 mm)

5

0.051 0.203 0.454 0.804 1.262 0.755

0.202 0.453 0.799 1.238 0.756

3.250 7.314 13.002 20.316

2.548 4.716 6.960 9.194

2.554 4.742 7.032 9.355

Boundary conditions ss cc cs CF

t CLT-Classical lamination theory. $ FSDT-First order shear deformation theory. $ HSDT-Higher order shear deformation theory.

FSDT exact ]I41

Present HSDT

FSDT exact P41

Present HSDT

2.4918 4.6602 3.5446 0.9231

2.5023 4.5940 3.5254 0.9241

1.5368 3.1843 2.3032 0.5551

0.8295 1.8472 1.2855 0.2965

u(x, t) = ;

Uj(l)Nj(X)

j=I

dtx9 r,=

j=l

FINITE ELEMENT MODEL

The generalized displacements expressions of the form

[45/-45/-451451

W/96/9Wl HSDTg FEM 0.051

0.203 0.457 0.812 1.269 0.813

5 1 (L = 3:;

FSDTj exact

CLTt ]I21 0.051

and K. M. BANGERA

are interpolated

by

4ji(r)Nj(x)

(13)

w(x9f, = i Wj(t)$j(X), j-l

LLI ----------------_

;.:$ 01483:

0.482: 0.481

_ o

:*:;: Ok 0.477 0.476 0.475 0.474 0.473 0.4720.471 0.470: 0.469: 0.4681 0.467: 0.466; 0.465 0.464; 0.463 0.462 0.461 0.460: 0.459

: *

I I

:

I I I I I I I I

-

I I I I : I , I I I I I I I I

-

0.458:

0.457: 0.456: :*:z: ;::5”;: 0:451

;

0.4500.449; 0.448 0.447

,

:::::$ 0

, 10

,

, 20

‘

, 30

,

, 40

(

, SO

*

, 60

,

, 70

,

, 80

,

, 90

.

, 100

,

,

,

110

120

frequencies of a [O/90] clamped-free

beam.

L/h Fig. 2. Effect of shear deformation on non-dimensional

Free vibration of composite beams where Nj are the Lagrange interpolation functions and tJj are the Hermite cubic interpolation functions. It is noted that linear interpolation functions are used for u and 4, and Hermite cubic interpolation functions are used for w in order to meet the continuity requirements of the variational statement (9). Substituting eqn (13) in eqn (9) gives the following element equations [Klp{A}’ + [Mr{a}e = {O},

and [K]’ and [M]’ are the element stiffness and mass matrices. For free vibration problems, the element equations (14) after assembly of the elements take the following eigenvalue form [with A(t) = &eio’]

d[A4l){iQ = 0,

(15)

where o is the circular frequency and {a} is the mode shape. Equation (15) can be solved after imposing the boundary conditions.

NUMERICAL

RESULTS

Unless mentioned otherwise, the material properties of AS4/3501-6 graphite/epoxy composite used in the analysis are: E, = 144.80 GPa, E, = 9.65 GPa, G23= 3.45 GPa, G,* = G,, = 4.14 GPa, v,* = 0.3, p = 1389.23 kg/m’. The following boundary conditions of the beam are considered: Clamped edge: u = 0, Simply-supporded

w = 0,

4 = 0;

edge: u = 0,

w = 0.

Unless mentioned as otherwise, the frequencies are non-dimensionalized as (3 = oL*,/(p/E, h*) and the L/h ratio is taken as 15. In Table 1, the present higher-order theory results are compared with the exact solutions based on classical and first order shear deformation theories for an orthotropic (0”) beam. The comparison with

Table 3. Effect of ply orientation on the non-dimensional frequencies (6) of an unsymmetric clamped-clamped beam Mode : 3 4 5 CAS 43,4-H

[O/90/0/90] [45/- 45/45/-45) 8.9275 3.7244 15.3408 22.3940 24.3155

5.2165 1.9807 9.6912 10.5345 15.0981

Table 4. Effect of number of layers on the non-dimensional frequencies (6 = oL *,/b/E, h 2)) of angle-ply and cross-ply beams with clamped-free edges No. of layers

[45/- 451451. .]

[O/90/0/. .]

2 4 6 8 10

0.3031 0.3223 0.3242 0.3247 0.3249

0.4800 0.6748 0.7047 0.7148 0.7195

(14)

where

([K] -

723

[30/50/30/50] 5.8624 2.2526 10.7609 11.9506 16.5747

the exact results are quite good. For thick beams the classical theory overpredicts the natural frequencies. The effect of shear deformation is more pronounced for the higher modes. The non-dimensional natural frequencies of symmetrically laminated cross-ply and angle-ply beams with various boundary conditions are compared in Table 2. It is noted that the cross-ply results are in good agreement with the exact solutions presented in [14]. However, the angle-ply results deviate significantly from the exact solutions. This is due to the fact that the laminated beam theory considered in [14] neglects the Poisson effect. Figure 2 shows the effect of L/h ratio on the non-dimensionalized fundamental frequency of [O/90] cross-ply beams. For L/h c 30, the difference between the classical and the shear deformation theories is quite significant. The effect of shear deformation is to decrease the natural frequency of the beam. Table 3 shows the first five natural frequencies for different unsymmetric lamination schemes. The effect of number of layers on the frequencies is shown in Table 4. It is seen that for both cross-ply and angle-ply beams considered, the frequency increases with the number of layers but the percentage increase decreases. The reason for this is that as the number of layers increases, the magnitude of the coupling stiffnesses B,, and i?,, decrease. In fact, when the number of layers tends to infinity, B,, , and E,, become zero. Table 5 shows the effect of boundary conditions on the first five natural frequencies of antisymmetric angle ply beams. Of the five boundary conditions considered, the clamped-free (CF) boundary condition exhibits the lowest frequency. Figure 3

Table 5. Effect of various boundary conditions on the non-dimensional frequencies (fi) of a [45/-45/45/-451 antisymmetric angle-ply beam Mode No. Beam type 1 CF 0.2962 ss 0.8278 SC 1.2786 1.2883 SF cc 1.8298

2

3

4

5

1.8156 3.2334 4.0139 4.0653 4.8472

4.9163 7.0148 8.0261 5.3660 9.0601

5.3660 10.7449 10.7449 8.1608 10.7449

9.2162 11.9145 13.0579 13.3136 14.1999

K.

124

CHANDRASHEKHARAand

K. M.

BANGERA

0.8

0.6

0.5

..

w 0.4

0.3

0.2

0.1

0.0 T

0

10

20

30

40

50

60

70

80

90

e Fig. 3. Effect of ply orientation on fundamental frequencies of two-layer clamped-free

shows the effect of increasing the angle of orientation 0 on the natural frequencies of two lamination schemes [0/-e] and [O/0]. The presence of the 0” layer in the [O/0] configuration reduces the degree of decrease in the frequency with increasing fI as is evident from Fig. 3. Figure 4 shows the effect of material anisotropy on fundamental frequencies. It is noted that the value of E, is varied, while the other elastic constants are the same as those of graphite/epoxy material. The effect of increasing material anisotropy is to decrease the natural frequencies of the beam. The effect of lamination scheme begins to show around E,/E, = 4. Thereafter the angle-ply configuration tends to lower the frequencies more rapidly than the cross-ply configuration. Figures 5-7 show the first three mode-shapes of a clamped-clamped [O/90]cross-ply beam. It is interesting to note the increasing difference between the bending slope dw/dx and the rotation C$with increasing mode number. This observation is very significant in light of the fact that the approximation 4 = dw /dx is made in the classical theory [12]. This assumption may lead to significant errors due to the difference

beams.

between dw/dx and 4, especially for the higher modes. This is indeed true as is reflected in Table 1 by the increasing difference between the classical and the shear deformation theory results for higher modes. Also significant is the increasing magnitude of the in-plane displacement ‘u’ for the higher modes. CONCLUSIONS

Finite element solutions have been obtained for the free vibration of laminated composite beams using a higher-order theory. The validity of the results is established by comparing with the existing solutions in the literature. The importance of Poisson effect in one-dimensional laminated beam analysis is demonstrated. Also, various results have been presented to show the effect of shear deformation, ply-orientation, number of layers, boundary conditions, and material anisotropy on the natural frequencies of generally layered fiber-reinforced composite beams. The mode shapes for cross-ply clamped-clamped beams indicate that the effects of shear deformation are greater for the higher modes.

Free vjb~tion of composite beams

1.0

0.9

0.8

0.7

0.6

G 0.5

0.4

0.3

0.2

0.1

0.0

Fig. 4. Effect of material ~isotropy

on f~dament~1 frequencies of cross-ply and angle-ply clamped-free beams.

w

Fig. 5. Fundamental mode shape of a clamped-clamped &l/90]cross-ply beam.

Fig. 6. Second mode &ape of a ~lam~~lam~a 726

~UPVJ cross-pty veam.

Free vibration of composite beams

727

f

Fig. 7. Third mode shape of a clamped-clamped

[O/90] cross-ply beam.

REFERENCES

8. M. Levinson, A new rectangular beam theory. 1. Sound

1. L. E. Goodman and J. G. Sutherland, Natural freguencies of continuous beams of uniform span length. ;, uppl. Mech. 18, 217-218 (1951). 2. R. D. Mindlin and H. Deresiewicz. Timoshenko’s shear coefficient for flexural vibrations of beams. Pro-

9. J. N. Reddy, A simple higher order theory for laminated composite plates. J. appi. Mech. 51, 745-752 (1984). 10. A. Bhimaraddi and L. K. Stevens, A higher order theory for free vibration of orthotropic, homogeneous and laminated rectangular plates. J. appi. Mech. 51, 195-198 (1984). 11. R. K. Kapania and S. Raciti, Recent advances in analysis of laminated beams and plates, part II: vibrations and wave propogation. AIAA JI. 27,

Vibr. 74, 81-87 (1981).

ceedings of the Second U.S. National Congress of Applied Mechanics, pp. 175-l 78 (1954).

3. G. R. Cowper, The shear coefficient in Timeshenko’s beam theory. J. uppl. Mech. 33, 335-340 (1966). 4. C. W. Bert, Simplified analysis of static shear factors for beams of non-homogeneous cross-section. J. Comp. Mater. 7, 525-529 (1973). 5. G. Prathap and G. R. Bhashyam, Reduced integration and the shear flexible beam element. Int. J. Numer. Meth. Engng 20, 353-367 (1984). 6. W. B. Bickford, A consistent higher order beam theory. Dwel. Theoret. appl. Mech. 11, 137-150

(1982). 7. P. R. Heyliger and J. N. Reddy, A higher order beam finite element for bending and vibration problems. J. Sound Vibr. 126, 309-326 (1988).

935-946 (1989). 12. J. R. Vinson and R. L. Sierakowski, Behuoiour of Structures Composed of Composite Materials, pp.

139-144. Martinus Nijhoff (1986). 13. A. T. Chen and T. Y. Yang, Static and dynamic

formulation of a symmetrically laminated beam finite element for a microcomputer. J. Comp. Mater. 19, 459475 (1985). 14. K. Chandrashekhara, K. Krishnamurthy and S. Roy, Free vibration of composite beams including rotary inertia and shear deformation. Compos. Struct. 14, 269-279

(1990).