Shear deformation and rotary inertia effects of vibrating composite beams

Composite Structures 20 (1992) 165-173

Shear deformation and rotary inertia effects of vibrating composite beams Haim Abramovich Faculty of Aerospace Engineering, Technion -- Israel Institute of Technology, Haifa 32000, Israel Free vibration of symmetrically laminated composite beams is studied based on Timoshenko type equations. Shear deformation and rotary inertia are included in the analysis, but with the term representing the joint action of these effects omitted in the Timoshenko equations. Detailed analytical analysis is performed for the natural frequencies of laminated beams with different boundary conditions at their ends, with numerical examples being calculated for a hinged-hinged beam.

INTRODUCTION The great use of laminated composite beams as rotating blades or robot arms requires a deeper understanding of vibration characteristics of these beams. Most of the work done was focused on natural frequencies of laminated plates 1,2although some of the works dealt also with beams (see Ref. 3). The classical lamination theory (which neglects shear deformation and rotary inertia) fails to predict the natural frequencies of laminated beams, as the transverse shear deformation is more pronouneed for composites because of the relatively high ratio of extensional modulus to the transverse shearing modulus. Teoh and Huang 4 introduced the effects of shear deformation and rotary inertia on the free vibrations of orthotropic cantilever beam based on an energy approach. Chen and yang.S introduced a fmite element method to predict bending and free vibration of laminated beams including shear deformation. Chandrashekhara and Krislmamurthy6 present exact solutions for the free vibration of symmetrically laminated composite beam including shear deformation and rotary inertia for some arbitrary boundary conditions. The present work follows the approach adopted by Abramovich and Elishak o f f 7,s w h o , following Timoshenko's 9 suggestion, omitted the term representing the joint action of rotary inertia and shear deformation (whose contribution they found to be negligible) from the differential equations of motion, but retaining their independent contribution.

In the present paper, the above approach is applied to laminated composite beams and put to further use. Its results are compared with those of Ref. 6. Ten different combinations of boundary conditions are considered, with the relevant frequency equations. Numerical examples for hinged-hinged boundary conditions and layup configurations of the beams are presented in graphical form. FORMULATION OF PROBLEM

Figure 1 shows a laminated composite beam referred to a system of Cartesian co-ordinates with the origin on the midplane of the beam and x-axis being coincident with the beam axis. Applying the theory developed by Yang et al. I° to a composite beam yields the beam displacements:

u(x, z, t) = u°(x, t) + zip(x, t)

(1)

w(x, Z, t)= W°(X, t)

Z

"/"

I I,,'-._.__.~

~h

Fig. 1. Geometry of a laminated composite beam. 165 CompositeStructures0263-8223/92/S05.00 @ 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain

x

166

H. Abramovich

where u ° and w ° are the axial and vertical displacements of a point on the midplane and g, is the rotation of the normal to the midplane. The strain-displacement relations are given by 11 0u

EQUATIONS OF MOTION

0~p z 0x

e~z=2 ~ ez =

constants Qn, Q22, Q12 and 066 Can be found in Ref. 12.

Following the procedure presented in Ref. 6 the virtual work of the present problem is

(2)

Ox ]

0

The beam constitutive equations are written as 12

+Qxz6

LB. o.Jtou,/Oxl

Ox]

+ (l, ti + I2~)) &i + l,}06}P +(13¢+ Izti) 6q~} dx dt=O

(8)

where

f h/2 cox dx N,, = .I-h/2

where M,, =

f h/2

- h/2

co,,z dx

(4) 11 =

c

o~being the normal stress, c the beam width. The constants A . , Bit, DH are given by 13

Dn = c

J-h~2

13 =

pZ dz

(9)

C

f h/2

]OZ2

dz

J-h~2

(~nZ dz

(5)

Q n z 2 dz

and A55 is the shear coefficient defined by 13

I h/2 A55 = ck

dz

(~11 dz

3-h/2 J-h~2 I h/2

p

J-h/2

h/2

BH = c

-hiE I hi2

12 = C

h/2 All = C

f h/2

J-h/2

Q55 dz

(6)

where k is a shear correction factor and Q~ and Q55 are the transformed material constants given bym ~3 QH = Q, cos40+ 022 Sin40 + 2(QI2 + 2066) Sin20 COS20

(7)

055 = GI3 cos20 + G23 sinE0 The angle 0 is the angle between the fiber direction and longitudinal axis of the beam and the

and p being the mass density of beam material. Substituting the Nx, Mx and Q= from eqns (3) into eqn (8) and carrying out the variational operations yields the governing equations of motion for the laminated composite beam. Assuming midplane symmetry (thus Bll=0) and neglecting the in-plane displacement, u, compared to the flexural displacement, w, the equations of motion become: Dn 027) A55 ( 0 w + q,] ~2~1) "0X -'T~ 0X / -13 - 0 7 = 0 [a2._ ~.,.\ 02W I, As5 |u w + u v / I - - o 0t 2 ~ OX2 OX]

(10)

It is remarkable to notice that eqns (10) are identical to the Timoshenko coupled equations (developed for an isotropic beam), if one replaces D~ l, A55, 11 and 13 by El, k A G , p A and pI, respectively (see Refs 7 and 8).

Free vibration of composite beams

SOLUTION OF EQUATIONS OF MOTION Equations (10) can be de.coupled as follows

~4W .~ 02W [ Dll / a4W Dn Ox4 11 i~t 2 -- ~13 +11 A551 Ox-~t 2 1,13 O'w +-- --=0 A55 0t 4

04'~

02~]) (

W(~)= C, cosh(pSl~)+ C2 sinh(PSl ~)

Dill 04'l,D

(11)

D,i Ox4

Dll / 0'w DH/

04"1])

w(x, t)= W(x)e i°'t

(13)

vibrational mode; W(x) = rotational vibrational mode; L = length of beam; ~ = non-dimensional span of beam; w -- angular frequency. Substitution of eqns (13) in eqns (12) yields: W Iv + p2(r2 + b 2) W" _p2 W-- 0 (14)

where

Dn

b2= D11 A.L 2

SI ]

_ P (s~ + b2) B '

c -z

,--U/

(18)

_ P (s~-b21B 4

C 3 - L ~ S--~]

b21B 3 c4---ZP ( s ~ -s---f-/ Hinged end: W--0 and W' = 0

(19)

Clamped end: W-- 0 and W = 0

(20)

1

Free end: W = 0 and - W' - ~ = 0

(21)

Guided end: ~

(22)

L

= 0 and 1 1 4 / - tIJ = 0

L

Compliance with the boundary conditions yields characteristic equations, which are tabulated in Table 1 for 10 various sets.

p2=llto2L 4 6L~

~j

The boundary conditions read as follows:

~ = x/L, i= f - ~ where W(x) = transverse

Wlv + p2(r 2 + b 2)W,, _ p2W = 0

2 ( r 2 + f f ) 2 + ~- ~. j|

The relations between the coefficients in eqns (16) are obtained by substituting the latter two in the governing set, eqns (10), bearing in mind eqns (14) and (15). The resulting relationships read as follows:

CI-~L\

-~t z - \13 + 11 Ass] Ox20t 2=0

~O(x, t)=ltt(x)ei°n

2

(12)

which, as Prof. M. Sayir (pers. comm. 1988) informs the authors on the basis of his asymptotic analysis, are more consistent than the original Timoshenko equations, 9 presented here in eqns (10). Let

g2 ~ 13

4 ]1'211'2

(r2 +b 2)+1 [

Dll OX--'-~+ 11 -~t 2 - 13 + 11 Ass] 0x-~t ~=0 (

+ C 3 cos(ps2~)+ C 4 ( s i n p s 2 ~ )

si--

It should be noted that the coupling is retained via the boundary conditions. As explained in the Introduction, the last term of eqns (11) can be omitted due to its negligible contribution to yield

0av"+ I, 0zT'

(16)

with

Ii13 04~) +-------0 Ass i~t4

0~w (

and the prime denotes differentiation with respect to~. The general solutiom of eqm (14) read

W(~) -- B 1 cosh(ps I ~) + B 2 sinh(psl ~) + B3 cos(ps2~)+B 4 sin(PS2~)

Oil b-~x"+ 11 ~ t ~ - /3 + I, A./Ox~Ot 2

O'w

167

(15)

NUMERICAL RF~ULTS

For a given beam with r and b known, the frequency coefficients Pn (n-- 1, 2, 3) can be found

H. Abramovich

168 %

o

+ ~J v

+

°°i

i

I

~

II

~ll

i

~~

~

m.~

8

"i

II

il

U

H

U

-Iv

i

v

+ v

u +

~i- •

kl

q-

×

~_

~-~

I

v

o

c~

cp

If

.o °..q

H

il

v

v

v

I c~

li I[ v

11

"~

U

If II

~'~

v

11

o ~

v

~L

I

v

I

~

I

v

v

v

If v

~

I

I

Free vibration o[ composite beams

+

v

+

~

li

II

II

+

e,i

+ e~

i U

II e,i

~

v

+

×

~

I

+

%U

II

g

g

I

i

~l~ II

II

II

II

U

I

I

I

II

II

I

N I

~I.~I ~ II

U

U

I

v

II

~I~ II

II

II

fl

II I

II

169

H. Abramovich

170

from the appropriate characteristic equations and the corresponding natural frequencies are then calculated from the first of eqns (15). These frequency equations are highly transcendental and capable of closed form solution only in the simplest cases. This dictates recourse to frequency charts representing different combinations of r and b. It should be pointed out that for a few boundary conditions (hinged-hinged, hingedguided and guided-guided) the frequency coefficients p , ( n = 1, 2, 3) can be calculated from a closed-form expression (see Table 1). The present paper shows numerical results of a simply supported (hinged-hinged) laminated composite beam. Other cases presenting various boundary conditions, can be calculated from the frequency equations appearing in Table 1. The AS/3501-6 graphite epoxy material properties, 6 used in the numerical results, are: E i=

21"0 x

106

G23 = 0"5

X

psi,

E: = 1"4 x

106

Laminated Composite Beam, 0,0,0,0 1 o.*

X

_

0.7 0.6 ,~

o.5

0.4 0.3 0.2

o.1 o -10

10

30

80

70

SO

0 (0) Fig. 2. Non-dimensional frequency (fifo) versus fiBer orientation k = 5/6; r=0"01925, L = 15 in, h = 1 in, c-- 1 in;

[01- OI - 010].

Laminated Composite Beam, 0,0,0,0 L==15",h= 1" o=1",t= 0.0192S,k= 2/3

psi,

1 -r---'-'F

-

T

-

0.9

106 psi

7,

oi

106 psi,

GI2 = GI3 = 0-6

~~I'=IS", h - I", 0=1", r= 0.01925 k = 5/6

;

-

Vl2 = 0 " 3 0.7

p = 0"13 x 10 -3 lb s2/in 4

0.6

The shear correction factor k is taken as 5/6 and as 2/3 to show its influence on the results. 13Table 2 shows a comparison between the present results and the ones of Ref. 6, presenting a very good agreement. It is also shown that taking a low value for k(k=3) yields a lower value for the natural frequency of a laminated composite beam, as compared to k = 5/6 (see Table 2). The influence of the fiber orientation on the non-dimensional frequency (fifo) (where f0 is the classical frequency, calculated for r= b =0) for a [ 0 / - 0 / - 0 / 0 ] configuration, is shown in Figs 2 and 3, for k = 5/6 and k= 2/3, respectively. For both values of k, the frequencies decrease with

0.4 0.3 O.l 0.1 0 -lo

lO

30

SO

70

S)o

O (o)

Fig. 3. Non-dimensional frequency (fifo) versus fiBer orientation k = 2/3; r--0.01925, L-- 15 in, h--- 1 in, c = 1 in;

[0/- 0/- 0/0]. Laminated Composite Beam, h=

1

1", c=1", k=

0,90,90,0

5/6

0.9

0.8

Table 2. Natural frequencies of a simply supported orthotropic (0") graphite-epoxy beam (L--15 in, h : 1 in, c = 1 in)

f(kHz)

Mode

Present results

0.7 0.6 ,~o

O.S O.4

Ref. 6

O.3 nn2

O.2

1 2 3 4 5

k = 5/6

k = 213

k = 5/6

0-755 2"543 4-697 6"919 9"127

0"743 2'431 4"393 6"383 8-349

0"755 2"548 4"716 6"960 9"194

nn3

=t

0.1 0

I

0

I

0,02

I

I

0.04

I

I

0.06

I

I

O.OI

)

i

0.1

I

0.12

Fig. 4. Non-dimensional frequency (fifo) versus nondimensional radius of gyration (r) for a [00/90°/90°[0°] crossply Beam k = 5/6 (h = 1 in, c = 1 in).

Free vibration of composite beams

171

Laminated Composite Beam, 0,90,90,0 h - 1", o,1", k- 2/3 I

0.9. U0.70.8-

Laminated Composite Beam Third

1", c=l", k : 5/6

h:

mode,

1 0.4 -

m,l

0.9 0.8 0.? 0.6 !

1

0

!

~

p

,

i

0.D4

,

i

0.aS

i

i

~

=

0.1

0.12

~'~

0.5

r 0.4

Rg. 5. Non-dimensional frequency ([]~)) versus nondimensional radius of gyration (r) for a [0°/90°/90o/0 °] crossply beam k-- 2/3 (h -- 1 in, c-- 1 in).

0.3 0.2

30*

0.1 r

0

i

!

1 0.04

i

= 0.0(S

(c/ mode,

h - 1", o , l " ,

= 0.08

i

i 0.1

0.12

Laminated Composite Beam

Laminated Composite Beam EImt

i

r

k,, 5/6

Fourth

h:

mode,

1", o , 1 " , k = 5/6

1

OJ Q

~8

OJl-

0.7

0.7-

0.S

0.6-

~,

30"

0.S-

0.4

0.4

R.

0.3

O.3

8O*

0.2

46*

0.'1 0

0

0.1 i

i 0.02

w

i 004

= 0.06

=

(a)

|

w

w

~

0

w 0.1

!

!

=

i

i

B

O.O4

O~

(d)

r

=

I

O~

I

I

0.1

0.12

r

Laminated Composite Beam

Laminated Composite Beam

S e e o a d m o d e , h - 1", o , 1 " , k - 5/6

t

i

O.O2

0.12

F i f t h m o d e , h = 1", o , 1 " , k = 5 / 6 1

OJI

0.9.

OJI

0.8-

0.7

0.7.

0.6-

0.6.

0.s. 0.4-

0.4.

46*

0.3-

0.3.

0.2-

0.2.

g. m" 48=

0.1 0

(b)

0.1 ,

w

w

0.04

|

|

~

;

i

~

i

i

0.1

w

0

0.12

i

,

0.02

~

i

0.04

r

i

~

i

i

0.00

i

i

0.1

|

0.19

(e)

Fig. b. Non-dimensional frequency (fifo) versus non-dnnensional radius of gyration (r) for a [ ( 0 / - O)i/sym] laminated beam k -- 5/6 (0 = 0°+ 90 °, h = 1 in, c = 1 in). (a) First mode; (b) second mode; (c) third mode; (d) fourth mode; (e) fifth mode.

H. Abram.rich

172

increase in fiber orientation angle, 0, with lower values for k--- 2/3. For a given configuration of the beam, [0*/90*/ 90°/0°], the shear deformation and rotary inertia lower the natural frequencies with increasing r

(non-dimensional radius of gyration, defined in eqns (15)) with higher influence on higher modes of vibrations (see Figs 4 and 5). The shear deformation and rotary inertia have a higher influence with decrease in fiber orienta-

Laminated Composite Beam mode,

First

h = 1% c - 1 " , k = 2/3

i

15 °

0°

0.3

-

0.2 0.1 0

i

i

I

i

0.~

i

I

0.O4

I

i

0.~

(a)

i

r

0.~

Second

Laminated Composite Beam

h = 1% o - 1 " , k = 2/3

mode,

0.12

r

Laminated Composite Beam 1

I

0.1

Fourth

mode,

h = 1", c - 1 " , k - 2/a

1

0.0

0.9.

0.8

0.8 -~

0.7

0.7-

0.8

0.0-

~= o~

"~

0.1~-

0.4

o.4-

46° 0.3

0.360"

,~:

0.2

0.2

0.1

45"

U: o•

o.1

0

l

0

0.02

I

1

I

0.04

I

0.81

l

l

0.08

1

o

1

0.1

i 0.~

0.12

i

h,= 1", o=1", k=

! 0.0a

!

i 0.01

i

i 0.1

1 0.12

[

Laminated Composite Beam mode,

i 0.04

(d/

I["

Thi.'d

i

Laminated Composite Beam

2./3

Fifth

0.9-

0.9

0.8-

0.8

0.7

0.7

0.6

0.8

~= o~

~

mode,

h - 1 " , o , 1 " , k - 2/3

0.8

"-..

0.4 80"

0.3 45 °

0.2

0; 1

30"

0.1 0

1

(c)

i 0.02

=

, 0.O4

=

= ~ £

Fig. 7.

i

, 0.08

i

1 0. I

o|,,,

1 0.12

0

(e)

~

i

0.02

i

!

v

i

0.04

0.00

!

=

0.IN

|

i

0.1

0.12

r

Non-dimensional frequency (f/f,) versus non-dimensional radius of gyration (r) for a [ ( 0 ] - 0/)j/sym] laminated beam • * • • • - 2/3 ( 0 -_ _ 0 o -= 9 0 ,o h -_ l m, c = = 1 m). (a) First mode,. (b) second mode,. (c) third mode; (d) fourth mode; (e) frith mode.

k - -

Free vibration of composite beams

tion for a [(0/-0/)i/sym] configuration. This is shown in Figs 6 and 7 for k ffi 5/6 and k ffi 2/3, respectively, for the five first modes. It should be pointed out that this influence is lowered with increase in mode number, n. The value of the shear correction factor, k, influences, the natural frequency of the laminated composite beam. A high value of k, k--~, yields higher values for the frequencies as compared to the ones obtained with k ffi 2/3 (see Figs 6 and 7). CONCLUSIONS Characteristic equations of symmetrically laminated composite beams have been derived under 10 various sets of boundary conditions based on first-order shear deformation theory, which take into account the individual contributions of shear deformation and rotary inertia, but omits their joint contribution. The resulting set of differential equations represents the first set of natural frequencies of the beam. Numerical results are reported to illustrate the new approach and its close agreement with other results presented in the literature.

REFERENCES 1. Leissa, A. W., Recent studies in plate vibrations: 1981-85. Part I: Classical theory, Shock Vib. Dig., 19 (2) (1987) 11-18.

17 3

2. Leissa, A. W., Recent studies in plate vibrations: 1981-85. Part !1: Complicating effects, Shock Vib. Dig., 19 (3)(1987) 10-24. 3. Vinson, J. R. & Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials. Martinus Nijhoff, Dordrecht, The Netherlands, 1986, p. 141. 4. Teoh, L. S. & Huang, C. C., The vibration of beams of fiber reinforced material. Z Sound Vib., 51 (1977) 467-73. 5. Chert, A. T. & Yang, T. Y., Static and dynamic formulation of symmetrically laminated beam finite element for a microcomputer. J. Composite Materials, 19 (1985) 459-75. 6. Chandrashekhara, K., Krislmamurthy, K. & Roy, S., Free vibration of composites beams including rotary inertia and shear deformation. Composite Structures, 14 (1990) 269-79. 7. Abramovich, H. & Elishakoff, I., Appfication of the Krein's method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations. Acta Mechanica, 66 (1987) 39-59. 8. Abramovich, H. & Eiishakoff, I., Influence of shear deformation and rotary inertia on vibration frequencies via Love's equations. J. Sound Vib., 137 (1990) 516-22. 9. Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, series 6, 41 (1921) 744-6. 10. Yang, P. C., Norris, C. H. & Stavsky, Y., Elastic wave propagation in heterogeneous plates. Int. J. Solids & Structures, 2 (1966)665-84. 11. Brunelle, E. J., Elastic instability of transversely isotropic Timoshenko beams. AIAA J., 8 (1970) 2271-3. 12. Ashton, J. E., Halpin, J. C. & Petit, P. H., Primer on Composite Materials: Analysis. Technical Publishing Co., Stamford, CT, 1969. 13. Whitney, J. M. & Pagano, N. J., Shear deformation in heterogeneous anisotropic plates. J. Appl. Mechanics, 37 (1970) 1031-6.