Analysis of Free Vibration of Sandwich Beams Consider Normal Deformation Effect
R. Bennai
Department of Technology University of Djilali BounaamaKhemis Meliana, Algeria r.bennai@ yahoo.fr
H. Ait Atmane
Material and Hydrology Laboratory,
University of Sidi Bel Abbes,Faculty of Technology, Civil Engineering Department , Algeria
Abstract—A new refined hyperbolic shear and normal
deformation beam theory is developed to study the free vibration of functionally graded (FG) sandwich beams under various boundary conditions.
The effects of transverse shear strains as well as the transverse normal strain are taken into account. Material properties of the sandwich beam faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. Equations of motion are derived from Hamilton’s principle.
Analytical solutions for the bending, free vibration and buckling analyses are obtained for simply supported sandwich beams. Illustrative examples are given to show the effects of varying gradients, thickness stretching, boundary conditions, and thickness to length ratios on the bending, free vibration of functionally graded sandwich beams.
Keywords— functionally graded sandwich beam; refined shear deformation theory; Stretching effect; free vibration; boundary conditions.
I. INTRODUCTION
Hyperbolic shear deformation theories have been applied to various problems in literature such as bending and vibration of beams (Ghugal and Sharma 2009, Li et al. 2013, Sayyad and Ghugal 2011, Berrabah et al. 2013, Ould Larbi et al. 2013);
bending, vibration and buckling of plates (Ghugal and Pawar 2011, Ghugal 2011); bending, vibration and buckling of laminated composite
plates (Nedri et al. 2014, Grover et al. 2013, Akavci and Tanrikulu 2008, Akavci 2010) and bending, vibration and buckling of FG plates (Akavci 2014a, b, Hebali et al. 2014, Mahi et al. 2015, Belkorissat et al. 2015, El Meiche et al.
2011). Noting the fact that hyperbolic shear deformation theories have been utilized earlier in flexure, vibration and buckling analysis of beams and plates; and also taking a cue from exact three dimensional theory of elasticity solutions of plate, hyperbolic functions are used in the present work, for describing displacement variation across plate thickness. In
this work, a new hyperbolic shear and normal deformation beam theory is presented to study dividing the transverse displacement into bending, shear and thickness stretching parts, the motion equations of the functionally graded sandwich beams are obtained together with Hamilton’s principle. Material properties of the sandwich beam faces are assumed to vary in the thickness direction only according to power-law form distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. Numerical examples are given to show the effects of varying gradients, thickness stretching, boundary conditions, and thickness to length ratios on the free vibration a of FG sandwich beams.
II. REFINED THEORY FOR FUNCTIONALLY GRADED SANDWICH BEAMS
A. Problem formulation
Consider a sandwich beam with homogeneous core and FG face layers, with total height (h), length (L), and width (b) referred to the Cartesian coordinates (x, y, z) as shown in Fig.
1. The top and bottom faces of the beam are at z = ±h / 2, and the horizontal edges of the beam are parallel to axes x and y.
The sandwich beam is composed of three layers, namely,
“Layer 1”, “Layer 2”, and “Layer 3”, from bottom to top of the beam. The vertical ordinates of the bottom, the two interfaces, and the top are denoted by h1
The face layers of the sandwich beam are made of an isotropic material with material properties
varying smoothly in the z-direction only. The core layer is made of an isotropic homogeneous material. For the brevity, the ratio of the height of each layer from bottom to top is denoted by the combination of three numbers, i.e., ‘‘1-0-1’’,
‘‘2-1-2’’ and so on. As shown in Fig. 1.
Fig. 1 Coordinate and geometry of a FG sandwich beam in the rectangular Cartesian coordinates
B. Material properties
The propretés of FGM vary continuously due to gradually changing the volume fraction of the constituent materials, usually in the height direction only. Power-law function is used to describe these variations of materials properties:
k
h h
h V z
1 2 ) 1 1
( , z[h1,h2]
) 1
2 (
V , z[h2,h3]
k
h h
h V z
4 3
4 )
3
( , z[h3,h4]
where V(n), (n ,12,3) denotes the volume fraction function of layer n; k is the volume fraction index (0k), which dictates the material variation profile through the height of beam.
The effective material properties, like Young’s modulusE, Poisson’s ratio, and mass density, then can be expressed by the rule of mixture (Benachour et al, 2011; Bachir Bouiadjra et al, 2012; Tounsi et al, 2013; Bouderba et al, 2013; Bachir Bouiadjra et al, 2013; Zidi et al., 2014) as follows:
where P(n) is the effective material property of FGM of layer n. Where,P1 and P2 are the properties of the top and bottom faces of layer 1, respectively, and vice versa
for layer 3 depending on the volume fraction
V(n), (
n ,12,3).
For simplicity, Poisson’s ratio of plate is assumed to be constant in this study for that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus (Dellal and Erdogan, 1983).
C. Basic assumptions
The assumptions of the present theory are as follows:
(i) The displacements are small in comparison with the beam thickness and, therefore, strains involved are infinitesimal.
(ii) The transverse displacement w includes three components of bendingwb, shearws, and stretching effectwst. The two first components are functions of coordinate x only and the third one is function of x and
z
.) , , ( ) , ( ) , ( ) , ,
(xzt w xt w xt w x zt
w b s st
(iii) The displacements u in x-direction consist of extension, bending, and shear components.
s
b u
u u
u 0 ,
The bending component ubis assumed to be similar to the displacement given by the classical beam theory. Therefore, the expression for ub can be given as
x z w ub b
,
The shear component us gives rise, in conjunction with ws, to the sinusoidal variations of shear strain xzand hence to shear stress xz through the thickness of the beam in such a way that shear stressxzis zero at the top and bottom faces of the beam. Consequently, the expression for us can be given as
x z w f
us s
( ) ,
where
2 1 cosh
sinh )
(
h z z
h z f
The component due to the stretching effect wstcan be given as:wst(x,z,t)g(z)(x,t)
The additional displacement accounts for the effect of normal stress is included and
g (z )
is given as follows2 1 cosh
1 1 cosh
) (
z z h
g ;
(1a)
(1b)
(1c)
1 2
( )) 2
(n (z) P P P V n
P (2)
(3) (4)
(5)
(6)
(7)
(8)
(9)
III. KINEMATICSAND CONSTITUTIVE EQUATIONS Based on the assumptions made in the preceding section, the displacement field can be obtained using Eqs. (3) – (9) as
) , ( ) ( ) , ( ) , ( ) , , (
) ( )
, ( ) , ,
( 0
t x z g t x w t x w t z x w
x z w x f z w t x u t z x u
s b
s b
The strains associated with the displacements in Eq. (10) are
xs xb
x
x0zk f(z)k
) 0
( xz
xz g z
) 0
(' z
z g z
where
x u
x
0
0 ,
2 2
x kbx wb
,
2 2
x kxs ws
,
x x ws
xz
0 ,0z
and g('z)dgdz(z)
By assuming that the material of FG beam obeys Hooke’s law, the stresses in the beam become:
z x
x Q z
Q z
11( )
13( ) ,
xzQ55(z)xz, and
z x
z Q z
Q z
13( )
33( )
where
1
,) ) ( ( )
( 33 2
11 z Q z Ez
Q Q13(z)Q11(z),and
21 ) ) (
55(
z z E Q
A. Equations of motion
Hamilton’s principle is used here in to derive the equations of motion. The principle can be stated in analytical form as (Reddy, 2002)
02
1
t t
dt K V
U
where t is the time; t1 and t2 are the initial and end time, respectively; U is the virtual variation of the strain energy;
V
is the variation of work done by external forces; and K is the virtual variation of the kinetic energy
.
Substituting the expressions for U, V , and K and integrating by parts, and collecting the coefficients of u0,
wb
,ws and , the following equations of motion of the FG beam are obtained
0
2 0 0
2 0
2 2 2 2 2 0 2 1 0
2 0 0
2 2
2
2 2 2 2 2 0 2 1 0
2 0 0
2 2
2
1 1 0 0 0
: : : :
K w w dx J
g w w g d N dx N dQ
dx w K d dx
w J d dxu J d J w w dx I
g w w N d x q dQ dx
M w d
dx w J d dx
w I d dx
u I d J w w dx I
g w w N d dx q
M w d
dxw J d dxw I d u dx I u dN
s s b
z b
s s b
s b b s s
s s b
s b b b b
s b
B. Analytical solution
The equations of motion admit the Navier solutions for simply supported beams. The variablesu0, wb, ws and can be written by assuming the following variations
1 0
) sin(
) sin(
) sin(
) cos(
m
t stm i
t sm i
t bm i
t i m
s b
e x
e x W
e x W
e x U
w w u
where Um, Wbm, Wsm and stm are arbitrary parameters to be determined, is the frequency associated with mth eigenmode, and m /L.The transverse load q is also expanded in Fourier series as
1
) sin(
) (
m Qm x
x
q
where Qm is the load amplitude calculated from
L
m qx xdx
Q L
0
) sin(
)
2 ( The coefficients Qm are given below for some typical loads. For the case of a sinusoidally distributed
load, we have 1
m and Q1q0
and for the case of uniform distributed load, we have
....) 5 , 3 , 1 (
4 0,
m
m Qm q
for free vibration problem the load parameters vanish and the free vibrating solution can be obtained as eigenvalue problem for any fixed value of m, as:
K ²M
0 Where
44 34 24 14
34 33 23 13
24 23 22 12
14 13 12 11
a a a a
a a a a
a a a a
a a a a
K
,
44 34 24
34 33 23 13
24 23 22 12
13 12 11
0
0
m m m
m m m m
m m m m
m m m M (10a)
(10b)
(11a) (11b)
(11c)
(11d)
(11e)
(12a)
(12b)
(13)
(13)
(23)
For non-trivial solutions of Equation (13), the following determinants should be zero:
K ²M 0C. Results and discussion
In this section, various numerical examples are presented and discussed to check the accuracy of the present theory. For verification purpose, the fundamental natural frequencies of FG sandwich beams predicted by the present model are compared in Tables 1–4 with those Vo et al (2014). The material properties adopted here are as follows.
Ceramic (Pc: Alumina, Al2O3): Ec380GPa;
3 . 0
; c 3960kg/m3.
Metal (Pm: Aluminium, Al): Em70 GPa; 0.3; 2707
m
kg/m3.
Two cases of FG sandwich beams with two values of span-to- height ratio, L/h= 5 and 20, are examined:
Hardcore: homogeneous core with Al2O3
(E1Ec;
1
c;
1
c) and FG faces with topand bottom surfaces made of Al
(E2Em;2m;2m).
Softcore: homogeneous core with Al (E1Em;
1
m;
1
m) and FG faces with top and bottom surfaces made of Al2O3(E2Ec;2c;
2
c).For convenience, the following dimensionless forms are used:
m m
E h
L
2 ,
For free vibration analysis, different types of FG sandwich beams are considered. Tables 2–5 contain dimensionless fundamental natural frequencies for FG sandwich beams with both hardcore and softcore. The obtained results are compared with those of Vo et al (2014) for various values of material parameter k. It is observed that the proposed theory without the thickness stretching effect (z0) and the model of Vo et al (2014) give solutions close to each other, and these solutions are in an excellent agreement with the present theory that considers the thickness stretching effect (z 0) for moderately thick beams (L/h20). However, the present theory without the thickness stretching effect (z0) and the model of Vo et al (2014) slightly overestimate the frequency for thick beams (L/h5) due to ignoring the thickness stretching effect.
Figs. 2 and 3 depict the fundamental frequencies parameters versus the material parameter k of simply supported power- law (1-1-1) FG sandwich beams with both homogeneous hardcore and softcore, respectively. It can be seen, that as the material parameter increases, the natural frequencies decrease
for sandwich beams with hardcore and increase for sandwich beams with softcore.
TABLE2: Non-dimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous hardcore (L/h 20).
TABLE1: Non-dimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous hardcore (L/h 5).
TABLE3: Non-dimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous Softcore (L/h5).
TABLE4: Non-dimensional fundamental natural frequencies of simply supported FG sandwich beams with homogeneous Softcore (L/h 20).
TABLE5: Non-dimensional fundamental natural frequencies ofclamped – clamped FG sandwich beams with homogeneous Hardcore (L/h5).
TABLE6: Non-dimensional fundamental natural frequencies ofclamped – clamped FG sandwich beams with homogeneous Hardcore (L/h 20).
0 2 4 6 8 10
2,5 3,0 3,5 4,0 4,5 5,0 5,5
L/h=5 L/h=20
Power law index (k)
Fig. 2 Variation of fundamental frequencies ω versus the material parameter k for (1-1-1) simply supported FG sandwich beams with homogeneous hardcore
0 2 4 6 8 10 2,5
3,0 3,5 4,0 4,5 5,0 5,5 6,0
Power law index (k) L/h=5 L/h=20
Fig. 3 Variation of fundamental frequencies ω versus the material parameter k for (1-1-1) simply supported FG sandwich beams with homogeneous softcore
Conclusions
A simple hyperbolic shear deformation beam theory with thickness stretching effect for the vibration analysis of FG sandwich beams is developed. The equations of motion are obtained by utilizing the Hamilton’s principle. Results prove that the present model is able to introduce the thickness stretching effect and providing very accurate results compared with the other existing higher-order beam theories.
References
[1] Akavci, S.S. (2010), “Two new hyperbolic shear displacement models for orthotropic laminated composite plates”, Mech. Compos. Mater., 46(2), 215-226.
[2] Akavci, S.S. (2014a), “An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation”,Compos.Struct.,108, 667-676.J.
[3] Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), “An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions”,J.Sandw.Struct.Mater.,16(3), 293-318.
[4] Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015),
“Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories”, Struct. Mech., Int.J.,53(6), 1143-1165.K.
[5] Bachir Bouiadjra, M., Houari, M.S.A. and Tounsi, A. (2012), “Thermal buckling of functionally graded plates according to a four-variable refined plate theory”, J. Therm. Stress., 35(8), 677-694.
[6] Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), “A new simple shear and normal deformations theory for functionally graded beams”, Steel Compos. Struct., Int. J., 18(2), 409-423.
[7] Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), “A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates”, Meccanica, 49(4),795- 810.
[8] Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), “A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates”, Steel Compos. Struct., Int. J., 18(1), 235-253.
[9] Houari, M.S.A., Tounsi, A. and Anwar Bég, O. (2013), “Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory”, Int. J. Mech. Sci., 76, 467-479.
[10] Vo, T.P., Thai, H.T., Nguyen, T.K., Maheri, A. and Lee, J. (2014),
“Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory”, Eng.
Struct., 64, 12-22.
[11] Vo, T.P., Thai, H.T., Nguyen, T.K., Inam, F. and Lee, J. (2015), “A quasi-3D theory for vibration and buckling of functionally graded sandwich beams”, Compos. Struct., 119, 1-12.
[12] Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Bég, O. (2014), “Bending analysis of FGM plates under hygro-thermo- mechanical loading using a four variable refined plate theory”,Aerosp.
Sci.Technol.,34, 24-34.
