Abstract Nonlinear vibration of beams made of func-tionally graded materials (FGMs) is studied in thispaper based on Euler-Bernoulli beam theory and vonKármán geometric nonlinearity. It is assumed that ma-terial properties follow either exponential or powerlaw distributions through thickness direction. Galerkinprocedure is used to obtain a second order nonlinearordinary equation with quadratic and cubic nonlinearterms. The direct numerical integration method andRunge-Kutta method are employed to find the nonlin-ear vibration response of FGM beams with differentend supports. The effects of material property distrib-ution and end supports on the nonlinear dynamic be-havior of FGM beams are discussed. It is found thatunlike homogeneous beams, FGM beams show differ-ent vibration behavior at positive and negative ampli-tudes due to the presence of quadratic nonlinear termarising from bending-stretching coupling effect.
L.-L. Ke · S. KitipornchaiDepartment of Building and Construction, City Universityof Hong Kong, Kowloon, Hong Kong
J. Yang (�)School of Aerospace, Mechanical and ManufacturingEngineering, RMIT University, P.O. Box 71, Bundoora,VIC 3083 Australiae-mail: [email protected]
L.-L. KeInstitute of Engineering Mechanics, Beijing JiaotongUniversity, Beijing, 100044, P.R. China
Functionally graded materials (FGMs) are inhomoge-neous composites characterized by smooth and contin-uous variations in both compositional profile and ma-terial properties and have found a wide range of appli-cations in many industries. In recent years, the largeamplitude vibration and nonlinear analysis of FGMstructures have attracted increasing research efforts.Praveen and Reddy [1] analyzed the nonlinear tran-sient response of FGM plates subjected to a steadytemperature field and lateral dynamic loads by us-ing the first-order shear deformation plate theory andthe finite element method. Reddy [2] developed boththeoretical and finite element formulations for thickFGM plates according to higher-order shear deforma-tion plate theory. Shen [3] analyzed nonlinear bend-ing of a simply supported, FGM rectangular plate sub-jected to a transverse uniform or sinusoidal load andin thermal environments. Woo et al. [4] reported ananalytical solution for the nonlinear free vibration be-havior of thin rectangular functionally graded plates.Yang et al. [5] presented a large amplitude vibrationanalysis of pre-stressed FGM laminated plates consist-ing of a shear deformable functionally graded layerand two surface-mounted piezoelectric actuator lay-ers. Huang and Shen [6] discussed the nonlinear vi-bration and dynamic response of functionally graded
plates in a thermal environment by using improvedperturbation technique. Kitipornchai et al. [7] exam-ined the effects of geometric imperfections on the non-linear vibration of laminated FGM plates and foundthat the nonlinear vibration frequencies are very sen-sitive to the imperfection mode. Agarwal et al. [8]used the existing statically exact beam finite elementto study the geometric nonlinear effects on static anddynamic responses in isotropic, composite and FGMbeams. Navazi et al. [9] and Navazi and Haddad-pour [10] presented the exact solution for the non-linear cylindrical bending and postbuckling of func-tionally graded plates. Chen and his co-authors con-sidered nonlinear vibration of shear deformable FGMplates [11], initially stressed FGM plates [12] and ini-tially stressed FGM plates with geometric imperfec-tion [13]. More recently, Yang and Huang [14] studiedthe nonlinear transient response of simply supportedimperfect functionally graded plates in thermal envi-ronments. The asymptotic solution is obtained by us-ing an improved perturbation approach, Galerkin tech-nique, and Runge–Kutta iteration process. Haddad-pour et al. [15] examined the nonlinear aeroelastic be-havior of functionally graded plates in supersonic flow.Allahverdizadeh et al. [16, 17] presented the nonlineardynamic analysis of thin circular functionally gradedplates. Hao et al. [18] used the asymptotic perturbationmethod to analyze the nonlinear oscillations, bifurca-tions and chaos of functionally graded materials plate.
It should be noted that the above investigations arefor FGM plates and shells only. Although many stud-ies [19–27] on the linear dynamic analyses of FGMbeams have been reported, to the best of authors’knowledge, no previous work has been done on thenonlinear free vibration of FGM beams. The objec-tive of this paper is to study the nonlinear free vibra-tion of FGM beams with different end supports basedon Euler-Bernoulli beam theory and von Karman non-linear strain-displacement relationship and examinethe effects of material property distribution, boundaryconditions, and vibration amplitude on the nonlineardynamic behavior of FGM beams.
2 Formulations
Figure 1 shows an FGM beam of length L and thick-ness h. Young’s modulus E(z) and mass density ρ(z)
Fig. 1 Geometry of an FGM beam
are assumed to change continuously along the thick-ness according to power law distribution
E(z) = (E2 − E1)
(2z + h
2h
)n
+ E1,
ρ(z) = (ρ2 − ρ1)
(2z + h
2h
)n
+ ρ1
(1)
or exponential function
E(z) = E0eβz, ρ(z) = ρ0e
βz, (2)
where the subscripts 0, 1 and 2 denote midplane(z = 0), the top surface (z = −h/2) and bottom sur-face (z = h/2), respectively; β and n are the constantscharacterizing the distributions of material properties.n = 0 or β = 0 correspond to an isotropic homoge-neous beam. Poisson’s ratio v(z) is assumed to be aconstant.
Based on Euler-Bernoulli beam theory, the dis-placements of an arbitrary point along the x- andz-axes, denoted by U (x, z, t) and W (x, z, t) respec-tively, are
U (x, z, t) = U(x, t) − z∂W
∂x,
W (x, z, t) = W(x, t),
(3)
where t is time, U(x, t) and W(x, t) are displacementcomponents in the midplane. The von Kármán typenonlinear strain-displacement relationship gives
εx = ∂U
∂x− z
∂2W
∂x2+ 1
2
(∂W
∂x
)2
. (4)
The normal stress σxx is given by linear elastic consti-tutive law as
σxx = E(z)
1 − ν2
[∂U
∂x− z
∂2W
∂x2+ 1
2
(∂W
∂x
)2]. (5)
Meccanica (2010) 45: 743–752 745
By using Hamilton’s principle, the equations ofmotion can be derived as
∂Nx
∂x= I1
∂2U
∂t2, (6)
∂2Mx
∂x2+ ∂
∂x
(Nx
∂W
∂x
)= I1
∂2W
∂t2, (7)
where the force and bending moment resultants are
Nx = A11
[∂U
∂x+ 1
2
(∂W
∂x
)2]− B11
∂2W
∂x2, (8a)
Mx = B11
[∂U
∂x+ 1
2
(∂W
∂x
)2]− D11
∂2W
∂x2, (8b)
The stiffness components and inertia related termare defined as
{A11,B11,D11} =∫ h/2
−h/2
E(z)
1 − ν2{1, z, z2}dz,
I1 =∫ h/2
−h/2ρ(z)dz.
(9)
If the axial inertia is neglected, (6) gives
Nx = Nx0 or A11
[∂U
∂x+ 1
2
(∂W
∂x
)2]
− B11∂2W
∂x2= Nx0. (10)
For beams with immovable ends (i.e. U = 0, at x =0 and L), integrating (10) with respect to x leads to
0 = [U ]x=Lx=0 =
∫ L
0
[A11
(Nx0 + B11
∂2W
∂x2
)
− 1
2
(∂W
∂x
)2]dx. (11)
Hence,
Nx0 = A11
L
∫ L
0
[1
2
(∂W
∂x
)2
− B11
A11
∂2W
∂x2
]dx. (12)
From (8b) and (10), bending moments can be re-expressed in terms of deflection as
Mx = B11
A11
[Nx0 + B11
∂2W
∂x2
]− D11
∂2W
∂x2. (13)
With (10) and (13) in mind and axial inertia being ne-glected, (6) and (7) can be reduced to
(B2
11
A11− D11
)∂4W
∂x4+ Nx0
∂2W
∂x2= I1
∂2W
∂t2. (14)
It is should be noted that the above equation is nonlin-ear due to the fact that Nx0 is nonlinear in W .
Introducing the following dimensionless quantities[28, 29]
ζ = x
L, (u,w) = (U,W)
h,
I = I1
I10, η = h
L,
(15a)
(a11, b11, d11) =(
A11
A110,
B11
A110h,
D11
A110h2
),
τ = t
L
√A110
I10,
(15b)
where A110 and I10 are taken as the values of A11
and I1 of a homogeneous beam. Equation (14) can berewritten in dimensionless form as
d0η2 ∂4w
∂ζ 4+ Nx0
∂2w
∂ζ 2= I1
∂2w
∂τ 2, (16)
where
d0 = b211
a11− d11,
Nx0 = a11η2∫ 1
0
[1
2
(∂w
∂ζ
)2
− b11
a11
∂2w
∂ζ 2
]dζ.
(17)
3 Linear vibration analysis
The dimensionless governing equation for linear freevibration of FGM beams can be obtained by substi-tuting the linear force and bending moment resultantswithout nonlinear components into (6) and (7)
a11∂2u
∂ζ 2− b11η
∂3w
∂ζ 3= 0, (18)
d0η2 ∂4w
∂ζ 4− I1
∂2w
∂τ 2= 0. (19)
For harmonic vibration, let � be the natural frequencyof the beam and express the displacements as
I10/A110 is the dimensionless linearnatural frequency. Substituting (20) into (18) and (19)and then solving the resulting differential equations,one has
w∗(ζ ) = f1 sin(λζ ) + f2 cos(λζ )
+ f3 sinh(λζ ) + f4 cosh(λζ ), (21)
u∗(ζ ) = b11ηλ
a11[f1 cos(λζ ) − f2 sin(λζ )f3 cosh(λζ )
+ f4 sinh(λζ )] + f6ζ + f7, (22)
where λ4 = −ωlI21 /d0, f1 to f7 are unknown con-
stants to be determined from boundary conditions.The present study considers three different end sup-
ports, i.e., hinged at both ends (H–H), clamped at bothends (C–C), or clamped at left end and hinged at rightend (C–H), with the following boundary conditions:
(i) hinged–hinged (H–H)
u∗ = 0, w∗ = 0,
Mx = 0, at ζ = 0,1,(23)
(ii) clamped–clamped (C–C)
u∗ = 0, w∗ = 0,
dw∗
dζ= 0, at ζ = 0,1,
(24)
(iii) clamped–hinged (C–H)
u∗ = 0, w∗ = 0,dw∗
dζ= 0,
at ζ = 0, (25a)
u∗ = 0, w∗ = 0, Mx = 0,
at ζ = 1. (25b)
Applying the associated boundary conditions resultsin a nonlinear algebraic equation
[H(ωl)]{χ} = {0}, (26)
where H(ω) is a matrix nonlinearly dependent on thelinear natural frequency ωl,χ is a vector composed ofseven unknown coefficients f1 to f7. For non-trivialsolutions of χ , the determinant of H(ωl) must be zero
det[H(ωl)] = 0. (27)
Linear natural frequencies and associated mode shapescan be solved from (27).
4 Nonlinear vibration analysis
For nonlinear vibration analysis, the transverse dis-placement is expressed as
w(ζ, τ ) = w1(τ )ψ(ζ ), (28)
where w1(τ ) is the time dependent function to be de-termined and ψ(ζ ) is the linear fundamental vibra-tion mode. Inserting w(ζ, τ ) into (16) and applyingGalerkin’s procedure yields a second order nonlinearordinary differential equation
w1 + γaw1 + γbw21 + γcw
31 = 0, (29)
where a super dot denotes differentiation with respectto time, and
γa = −d0η2
�0
∫ 1
0ψ
d4ψ
dζ 4dζ,
γb = −b11η2
�0
(∫ 1
0
d2ψ
dζ 2dζ
)∫ 1
0ψ
d2ψ
dζ 2dζ,
γc = a11η2
2�0
(∫ 1
0ψ
d2ψ
dζ 2dζ
)∫ 1
0
(dψ
dζ
)2
dζ,
�0 = I
∫ 1
0ψψdζ.
(30)
Note that γa = ω2l . The values of γa, γb and γc for
different FGM beams are listed in Table 1.Equation (29) contains a quadratic nonlinear term
due to the presence of bending-extension coupling ef-fect in FGM beams (i.e. B11 �= 0). This term, however,vanishes and (29) reduces to a Duffing equation for ho-mogeneous beams and clamped–clamped FGM beamsbecause γb = 0 in both cases, as can be seen from Ta-ble 1. This clearly indicates that FGM beams, exceptin these two cases, have different vibration behaviorat positive and negative amplitudes. When γb = 0, theexact solution of (29) has been obtained by Lestari andHanagud [30], and the nonlinear frequency (ωnl) maybe expressed as
ωnl = π√
γa + a2γc
2K, (31)
where a is the maximum vibration amplitude; K is thecomplete elliptic integral,
K =∫ π/2
0
dθ√1 − k2 sin2 θ
, k2 = γa + a2γc. (32)
Meccanica (2010) 45: 743–752 747
Table 1 Dimensionless coefficients γa (×10−2), γb (×10−2) and γc (×10−2) in (29)
By multiplying (29) by w1 and integrating with re-spect to time, the following energy balance equation isobtained
(w1)2 + γaw
21 + 2
3γbw
31 + 1
2γcw
41
= H = constant. (33)
The constant H is evaluated from initial conditions.By assuming w1 = wmax and w1 = 0 at τ = 0, onehas,
H = γaw2max + 2
3γbw
3max + 1
2γcw
4max. (34)
Putting H into (33) leads to
(w1)2 = γa(w
2max − w2
1) + 2
3γb(w
3max − w3
1)
+ 1
2γc(w
4max − w4
1). (35)
In the absence of bending-stretching coupling ef-fect, i.e. γb = 0, (35) has two real, equal and opposite
roots at w1 = 0. The third root is a complex number.When the bending-stretching coupling effect is present(γb �= 0), (35) has two real roots at w1 = 0 with onebeing wmax and the other one being wmin (wmax �=−wmin). The magnitude of wmin and wmax dependson the sign and magnitude of γb. Hence, the nonlin-ear vibration characteristics of FGM beams at posi-tive amplitudes are different from those at negativeamplitudes. The similar phenomenon was observedfor asymmetric cross-ply composite beams as well bySingh and Rao [31, 32]. In fact, FGM beams are quitesimilar to asymmetric cross-ply composite beams inthe sense that they all have unsymmetrical through-thickness material property distribution but the mater-ial properties of an FGM beam vary continuously andsmoothly.
By employing direct numerical integration method(DNIM), the nonlinear period (frequency) of the FGMbeam is determined from
Tnl
2= π
ωnl
=∫ wmax
wmin
dw1√ω2
l (w2max − w2
1) + 23γb(w3
max − w31) + 1
2γc(w4max − w4
1)
, (36)
where Tnl and ωnl are the fundamental nonlinear pe-riod and frequency. This integral is calculated by usingGauss-Legendre integration technique.
Solving (29) by using the 4th-order Runge–Kuttamethod, the associated vibration mode and phaseplane can be obtained. The time step used in the com-putation is τ = 0.002. In each case, the beam is as-sumed to be initially at rest, that is,
w1(0) = wmax, w1(0) = 0. (37)
5 Numerical results
Table 2 gives the nonlinear frequency ratio ωnl/ωl ofan isotropic homogeneous hinged–hinged beam withL/h = 20 and h = 0.1 m at different vibration am-
748 Meccanica (2010) 45: 743–752
plitudes Wmax/� (� = √I/A where I the area mo-
ment of inertia and A the area of cross-section). Theexact solutions given by Lestari and Hanagud [28] andRitz-Galerkin solutions given by Singh et al. [33] arealso provided for direct comparison. It is observed thatthe present results agree very well with those given byLestari and Hanagud [30] and Singh et al. [33].
Table 3 examines the effect of the number ofthe modes in the Galerkin procedure on the nonlin-ear frequency ratio ωnl/ωl of the isotropic homoge-neous hinged–hinged and clamped–clamped beamswith L/h = 20 and h = 0.1 m. Following Chandraand Raju [34], it is assumed that W1 = Wmax andWm = 0.1W1 at t = 0 where Wm is the maximumamplitude of the mth mode (m = 2,3, . . . ,N). It isfound that the number of the modes has a slight ef-fect on the nonlinear frequency ratio of hinged–hingedbeam. For clamped–clamped beam, however, the dif-ference between the single mode and multi-mode so-lutions increases as the vibration amplitude increases.It is should be noted that the number of the modeshas a minor effect on the nonlinear frequency ratioof both hinged–hinged and clamped–clamped beamswhen Wmax/� ≤ 2. Therefore, Wmax ≤ 2� is used inall of the following examples.
Table 2 Comparison of nonlinear frequency ratio ωnl/ωl for ahinged-hinged isotropic homogeneous beam
Wmax/� Present Lestari and Hanagud [30] Singh et al. [33]
1.0 1.0892 1.0892 1.0897
2.0 1.3178 1.3178 1.3229
3.0 1.6257 1.6257 1.6394
4.0 1.9760 1.9760 2.0000
5.0 2.3501 2.3502 2.3848
Table 4 lists the dimensionless linear natural fre-quency ωl = �
√D0/I10 for hinged–hinged and clam-
ped–clamped FGM beams whose material proper-ties change exponentially along beam thickness asdescribed in (1) with E1 = 70 GPa, v1 = 0.33,ρ1 = 2780 kg/m3, L/h = 20. Here D0 = D110 −B2
110/A110. A110, B110, D110 and I10 denote the val-ues of A11,B11,D11 and I1 of an isotropic homoge-neous beam. The present results agree very well withthe results obtained by Yang and Chen [22] based onEuler-Bernoulli beam.
Table 5 and Figs. 2–5 present results for hinged–hinged (H–H), clamped–clamped (C–C) and clamped–hinged (C–H) FGM beams. Unless otherwise stated,it is assumed that the beam thickness h = 0.1 m,slenderness ratio L/h = 16. For beams whose ma-terial properties vary exponentially, the top surface ofthe beam is 100% aluminum with material parame-ters E1 = 70 GPa, v1 = 0.33, ρ1 = 2780 kg/m3, andYoung’s modulus ratio is E2/E1 = 0.2, 1.0, 5.0. Notethat E2/E1 = 1.0 corresponds to an isotropic homoge-neous beam. For beams whose material properties fol-low power-law distributions, the top surface is 100%silicon nitride (E1 = 322.2 GPa, v = 0.24, ρ1 =2370 kg/m3) while the bottom surface is 100% stain-
Table 4 Comparison of fundamental frequencies of FGMbeams
E2/E1 H–H C–C
Present Yang and Present Yang and
Chen [22] Chen [22]
0.2 2.5127 2.51 5.2550 5.25
1.0 2.4674 2.47 5.5933 5.59
5.0 2.5127 2.51 5.2550 5.25
Table 3 Effect of the number of the modes in the Galerkin method on the nonlinear frequency ratio
Wmax/� H–H C–C
1 term 2 term 3-term 1 term 2 term 3-term
0.5 1.0231 1.0240 1.0268 1.0056 1.0058 1.0081
1.0 1.0892 1.0926 1.1002 1.0222 1.0232 1.0319
2.0 1.3178 1.3288 1.3532 1.0857 1.0894 1.1214
3.0 1.6257 1.6456 1.6894 1.1831 1.1907 1.2553
4.0 1.9760 2.0050 2.0686 1.3064 1.3185 1.4203
5.0 2.3501 2.3881 2.4713 1.4488 1.4656 1.6068
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Table 5 Dimensionless linear natural frequencies of FGM beams (ω = �L√
I10/A110)
Boundary condition Exponential Power–law
E2/E1 = 0.2 E2/E1 = 1.0 E2/E1 = 5.0 n = 0.3 n = 1.0 n = 3.0
H–H 0.1813 0.1781 0.1813 0.0918 0.1093 0.1320
C–C 0.3792 0.4037 0.3792 0.2075 0.2464 0.2983
C–H 0.2653 0.2782 0.2653 0.1431 0.1700 0.2057
less steel (E2 = 207.7 GPa, ρ2 = 8166 kg/m3). Thegradient index n = 0.3, 1.0, 3.0 are considered.
The dimensionless linear fundamental frequenciesωl = �L
√I10/A10 of FGM beams are listed in Ta-
ble 5. For exponentially varying FGM beams, the lin-ear frequencies of graded beams with E2/E1 = 0.2and 5.0 are the same because their values of I10/A10
are almost identical. The homogeneous beams, exceptthe hinged–hinged one, have higher frequencies thanthe graded beams. In case of power law distribution,the linear frequencies increase with an increase in gra-dient index n.
Figures 2 and 3 plot the nonlinear fundamental fre-quency ratio versus dimensionless amplitude (wmin
and wmax) curves for FGM beams with material prop-erties following exponential and power law distrib-utions, respectively. All beams exhibit typical hard-ening behavior, i.e., the nonlinear frequency ratio in-creases as the vibration amplitude is increased. It isseen in Fig. 2 that for hinged–hinged and clamped–hinged graded beams (E2/E1 = 0.2,5.0), nonlinearfrequency ratios are dependent on not only the magni-tude but also the sign of the vibration amplitude. Simi-lar results can be observed from Fig. 3 as well althoughthe difference here is not significant. This is due to thebending-stretching coupling effect term in (29). Forclamped–clamped beam and homogeneous beams, thenonlinear frequency ratio is independent of the signof vibration amplitude because the quadratic nonlinearterm representing this coupling effect in (29) vanishes.
Figure 4 gives dimensionless vibration amplitudeas a function of dimensionless time for FGM beamswith exponentially distributed material properties.The positive and negative maximum amplitudes arethe same for homogeneous beams (E2/E1 = 1.0)
and clamped–clamped beams but are different forhinged–hinged and clamped–hinged graded beamswhose positive maximum amplitude is larger (smaller)than the negative maximum amplitude at E2/E1 =
0.2 (E2/E1 = 5.0). These results imply that the non-linear period at positive deflection cycle is not equal tothat at the negative deflection cycle for hinged–hingedand clamped–hinged graded beams.
Figure 5 displays the phase plane diagrams (w1
versus w1) for FGM beams with exponentially vary-ing material properties. For homogeneous beamsand clamped-clamped FGM beams, the phase planediagrams are symmetric ellipses. The diagrams atE2/E1 = 0.2 and E2/E1 = 5.0 are the same when thebeam is clamped at both ends. For hinged-hinged andclamped-hinged FGM beams with bending-stretchingcoupling effect, the diagrams are seen to be unsym-metric about the w1 = 0 axis. It is found that under aninitial excitation with an amplitude of 1.0, a hinged–hinged graded beam oscillates with an amplitude of+1.0 in the direction of initial excitation, −0.8483 inthe opposite direction at E2/E1 = 0.2 and −1.1642at E2/E1 = 5.0, respectively. Also, the area of thediagram increases as Young’s modulus ratio E2/E1
increases. The diagrams of clamped–hinged gradedbeams are similar to those of hinged–hinged gradedbeams.
6 Conclusions
The nonlinear vibration behavior of FGM beams isstudied within the framework of Euler-Bernoulli beamtheory and von Kármán type displacement-strain rela-tionship. The materials properties are assumed to ei-ther vary exponentially or follow a power law distri-bution along thickness direction. The effects of ma-terial property gradient, end supports, and amplitudeon the nonlinear dynamic behavior of the FGM beamsare discussed in detail. Numerical results show that(1) all FGM beams exhibit typical ‘hardening’ be-havior; (2) The nonlinear frequencies of homogenousbeams and clamped–clamped graded beams are notaffected by the sign of the vibration amplitude; and
750 Meccanica (2010) 45: 743–752
Fig. 2 Nonlinear frequency ratio versus dimensionless ampli-tude curves for FGM beams with exponentially varying mate-rial properties: (a) hinged–hinged, (b) clamped–clamped, and(c) clamped–hinged
Fig. 3 Nonlinear frequency ratio versus dimensionless ampli-tude curves for FGM beams with power-law material prop-erty distributions: (a) hinged–hinged, (b) clamped–clamped,and (c) clamped–hinged
Meccanica (2010) 45: 743–752 751
Fig. 4 Time history of dimensionless amplitudes for FGMbeams with exponentially varying material properties:(a) hinged–hinged beam, (b) clamped–clamped beam, and(c) clamped–hinged beam
(3) the nonlinear frequencies of hinged–hinged andclamped–hinged graded beams are dependent on thesign of the vibration amplitudes.
Fig. 5 Phase plane diagram for FGM beams with exponen-tially varying material properties: (a) hinged–hinged beam,(b) clamped–clamped beam, and (c) clamped–hinged beam
Acknowledgements The work described in this paper wasfunded by a grant from City University of Hong Kong (ProjectNo. 7002211). The authors are grateful for this financial sup-port.
752 Meccanica (2010) 45: 743–752
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