The in-plane free vibration problem of symmetric cross-ply laminated beams is studied based on the ... of Timoshenko and Bernoulli-Euler theories are compared with each other for ... Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF ... by the state space approach (transfer matrix method) and.
V. Y d d m m 1 Associate Professor, Department of Mechanical Engineering, University of Oukurova, 01330 Balcah-Adana, Turkey
E. Sancaktar Professor, Department of Polymer Engineering, The University of Akron, Akron, OH 44325-0301 Fellow ASME
E. Kiral Prolessor Department of Civil Engineering, University of Qukurova, 01330 Balcah-Adana, Turkey
Comparison of the In-Plane Natural Frequencies of Symmetric Cross-Ply Laminated Beams Based on the BernoulliEuler and Timoshenko Beam Theories The in-plane free vibration problem of symmetric cross-ply laminated beams is studied based on the transfer matrix method. Distributed parameter model is used in the mathematical formulation. The rotary inertia, the shear and extensional deJormation effects, are considered for the Timoshenko's beam analysis. These effects are neglected in the Bernoulli-Euler analysis. The exact overall dynamic transfer matrix of the beam is obtained by making use of the numerical algorithm available in the literature. In order to obtain detailed knowledge about the effects of the rotary inertia, shear and axial deformations on the first six non-dimensional frequencies, the results of Timoshenko and Bernoulli-Euler theories are compared with each other for length-to-thickness ratios from ranging 3 to 20. Fixed-fixed, fixed-simple, and fixedfree boundary conditions are considered for three values of the thickness-to-width ratios of a rectangular section (2, 1, 0.5).
1
Introduction The dynamic problems of laminated composite beams have not been studied as extensively as plates and shells. Vinson and Sierakowski (1986) obtained the exact solution of a simply supported composite beam based on the classical theory, which neglects the effects of the rotary inertia and shearing deformation. Kapania and Raciti (1989a) presented a survey in the vibration analysis of laminated composite beams. It is evident from the literature survey that the free vibration problem of beams is generally studied considering the out-of-plane uniaxial bending and axial oscillations. Kapania and Raciti (1989b) studied nonlinear vibrations of unsymmetrical laminated beams. Chandrasekhara et al. (1990) derived the equation of motion of composite beams based on the first-order shear deformation theory. Hodges et al. (1991) solved the equations of motion using a mixed finite element and an exact integration method. Abramovich (1992) presented exact solutions based on the Timoshenko-type equations for symmetrically laminated composite beams with ten different boundary conditions. The rotary inertia and shear deformation effects were investigated for simply supported straight beams (Abramovich, 1992). Singh and Abdelnassar (1992) examined the forced vibration response of composite beams considering a third-order shear deformation theory. Chandrasekhara and Bangera (1992) studied the free vibration characteristics of laminated composite beams using a higher-order shear deformation theory. Krishnaswamy et al. (1992) obtained the governing equations of laminated composTo whom correspondence should be addressed. Contributed by the Applied Mechanics Division of THE AMERICANSOCIETYOF MECHANICALENGINEERSfor publication in the ASME JOURNALOF APPLIEDMECHANtCS. Discussion on the paper should be addressed to the Technical Editor, Professor Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNALOF APPLIEDMECHANICS. Manuscript received by the ASME Applied Mechanics Division, July 28, 1998; tinal revision, Jan. 19, 1999. Associate Technical Editor: W. K. Liu.
410 / Vol. 66, JUNE 1999
ite beams using the Hamilton's principle and presented analytical solutions. Abramovich and Livshits (1994) studied the in-plane free vibrations of nonsymmetrically laminated crossply composite beams based on Timoshenko-type equations. Khedeir and Reddy (1994) investigated the free vibration of cross-ply laminated beams with arbitrary boundary conditions by the state space approach (transfer matrix method) and higher-order shear deformation theory. Nabi and Ganesan (1994) developed a general finite element code based on the first-order shear deformation theory. Eisenberger et al. (1995) used the dynamic stiffness analysis of laminated beams using a first-order shear deformation theory. Abramovich et al. (1995) treated vibration of multispan nonsymmetric composite beams. Rao and Ganesan (1997) worked out the dynamical behavior of tapered composite beams by the third-order shear deformation theory. Zappe and Lesiutre (1997) presented a smeared laminate model based on the first-order shear deformation theory for the dynamic analysis of laminated beams. As it is well known, the Bernoulli-Euler beam theory omits the rotary, the shear, and the extensional deformation effects. Since the ratio of extensional stiffness to the transverse shear stiffness (E~/ E2) is high, the effect of shear in laminated beams is more significant than in homogeneous beams. These effects can be considered in the analysis by the Timoshenko beam theory. Khedeir and Reddy's (1994) study showed that the effects of the rotary inertia and shear deformation can be significant even for the fundamental frequencies of laminated beams with boundary conditions such as fixed-free, fixed-simple, and fixed-fixed. The objective of the present study is to formulate the purely in-plane free vibration problem of symmetric cross-ply laminated beams in an accurate manner and to present a detailed knowledge about the effects of rotary inertia, axial, and shear deformations. These effects are studied considering the length-to-thickness (L/h) ratios, the boundary conditions, E t/E2 ratios, and the thicknessto-width (h/b) ratios of a cross section for the first six natural in-plane frequencies.
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(a) y, 30;) _
"'
-- a . )
-
M.(~,)
= - @ T , (U,)
Mt(~)t)
J
n
(b)
(e)
Fig. 1 (a) A laminated composite beam, (b) the beam and material symmetry axes, (c) stress and displacement resultants in Frenet coordinates
2
Formulation
of the Problem
0-,= Qijej
2.1 E l e m e n t s of t h e R e d u c e d a n d T r a n s f o r m e d Stiffness Matrix. The generalized Hooke's law for a linearly elastic ma-
terial can be written in the conventional contracted notation for engineering stresses (0-1 = 011, 0"2 = 0"22, 0 3 = 0"33, 0"4 = 023, 0-5 = ~r~3, 0-6 = 0"~2), and engineering strains (el = eta, e2 = e22, e3 = e33, e4 = 2e23, e5 = 2e13, e6 = 2e~2) as follows:
0"i= Cijej
ei=Sij0-j,
or
(i,j=
1,2 .....
6)
(1)
where Ctj and S~j represent the elements of the stiffness and compliance matrices, respectively. Employing the classical rod theory with o'2 = 0-3 = 0"4 = 0 (Fig. 1(a)), the strain components subscripted by 2, 3, and 4 are determined in terms of the other strain components in Eq. (1) as below:
(i,j=
1,5,6)
(4)
where
Qo= Cij+ Cit3S~kakj, ( i , j , k =
1 , 5 , 6; / 3 = 2, 3 , 4 ) .
(5)
Representing the reduced stiffness matrix of a lamina by 0 and using the following contractions, 6-~ = 01,
6-2 = 0-6,
QI1 = Q I 1 ,
012 =
022 = Q66,
63 = 0-5, Qt6,
031
=
~ = el,
~2 = e6,
e3 = es,
013 =
Q,s, 02, = Q6,, 023 = Q65,
Qs,,
033
=
Q55,
032 = Q56
(6)
the generalized Hooke's law for beams may be written as follows:
e~=Sojajk,
(j,k=
1,5,6;/3=2,3,4)
and
855866 --
326
d
811866 -
,
ass -
SllSss
a15 =
Si6856 - 815S66 d '
a16 --
d =
-
S~ 6
d
Sis
d
OL66 --
6-~
'
$15 $55 856
816 $56 866
Substitution of Eq. (2) into Eq. (1) gives the following: Journal of Applied Mechanics
1 , 2 , 3).
(7)
Qq i,
(i,j=
1 , 2 , 3)
(8)
Using Equations (5) and (6) with the transformed stiffness and compliance matrices, C' and S', the nonzero terms of Q' for cross-ply laminates are obtained as follows (in Eq. (3) a~5 = a j6 = as~ = 0 and d = SILSssS66 for /3 = 0 deg and /3 = 90 deg):
815816 - Si iS56 as6 d
8 1 5 8 5 6 - S16S55 d
811 Sis 816
(i,j=
Figure l(b) shows the relations between the material axes of elastic symmetry (1', 2', 3') and the beam principal axes (1, 2, 3) for a rotation about the 3-axis. The angle between the fiber direction and the beam axis is denoted by/3. Equation (7) is written in the following transformed form (Yddmm, 199%):
where all
6-;= 00~j,
(2)
a j k = OLkj
Q'Jl = C'lt + (C'i2S'21 + C,3S3t)/Si;' -t
!
Q33 = C55
(3)
'
t
Q22-' = C66, (9)
where m = cos/3 JUNE 1999, Vol. 66 / 411
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+ 2(m 2 - m 4 ) C 1 2 q- C22(1 -
C'll = m4Clt
2 m 2 + m 4)
where
+ 4(m 2 - m4)C66
N
Aij = E
...1.- ( m 2 - m 4 ) C 2 2 .q-- C12(1 - 2m 2 + 2 m 4)
C'12 = ( m 2 - m 4 ) C l l
Ql~ (k)ACk),
k=l
- 4(m 2 - m 4 ) C 6 6
N
B 0 = Fji = e,~jp ~ ~-~
