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On the postbuckling and free vibrations of FG Timoshenko beams
Composite Structures 95 (2013) 247–253
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On the postbuckling and free vibrations of FG Timoshenko beams G.H. Rahimi a, M.S. Gazor a, M. Hemmatnezhad b,⇑, H. Toorani a a b
Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran Faculty of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran
a r t i c l e
i n f o
Article history: Available online 7 August 2012 Keywords: Postbuckling behavior Free vibration Functionally graded Timoshenko beam theory Exact solution
a b s t r a c t The postbuckling behavior of functionally graded beams is investigated by means of an exact solution method. The Von-Karman type nonlinear strain–displacement relationships are employed. The effects of the transverse shear deformation and rotary inertia are also included based upon the Timoshenko beam theory. After writing the kinetic and potential energy functionals, the governing equations of motion including the axial, transverse deflections and also the cross sectional rotation are derived using the Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. Neglecting the inplane inertia, the three equations of motion are reduced to two nonlinear partial–integral–differential equations in terms of the transverse mid-plane deflection and the cross sectional rotation. FG beams are considered to have fixed–fixed, fixed–hinged, and hinged–hinged end conditions. A closed-form solution is achieved for the postbuckling deformation as a function of the exerted axial load which is beyond the critical buckling load. In order to study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first buckled configuration of a hinged–hinged FG beam. This leads to a characteristic equation whose eigenvalues are the natural frequencies and the corresponding eigenvectors also determine the mode shapes. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via Euler–Bernoulli beam theory clarifies the overestimation of the frequencies by the later one. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
1. Introduction Buckling is a static instability of structures occurring in the presence of inplane loading. During the past decades buckling analysis of beam structures has attracted more attention from the scientific community as reflected by increasing number of publications devoted to that. Most of the investigations performed on the buckling problem are concerned with determining the critical buckling loads and their associated mode shapes. To investigate the postbuckling problem, one should enter the geometric nonlinearity terms due to the midplane stretching into the formulation until reaches to a nonlinear buckling problem. Solving this nonlinear problem for a given axial load yields the postbuckling configurations. Based on the Euler–Bernoulli beam theory (EBT), Nayfeh et al. [1] obtained an exact solution for the linear free vibration problem of buckled beams. They arrived at results in an excellent agreement with the experimental data. Using the dynamic stiffness matrix method, Benerjee [2] investigated the problem of free vibrations ⇑ Corresponding author. Tel./fax: +98 282 5270131. E-mail address: [email protected] (M. Hemmatnezhad).
of axially loaded composite beams. He took into consideration the effects of shear deformation and rotary inertia based on the Timoshenko beam theory (TBT). He studied the free vibration characteristics of composite beams in the prebuckling domain and did not consider the postbuckling behavior. Lee and Choi [3] studied the thermal buckling as well as postbuckling behaviors of composite beams embedded with shape-memory alloy wires using analytical technique. They also constructed a finite element model for the structure using ABAQUS software. They concluded that using shape-memory alloy wires can significantly enhance the critical buckling temperature and reduce the lateral deflection of a laminated composite beam. Matsunaga [4] obtained the natural frequencies and buckling stresses of simply supported laminated composite beams based upon the higher-order shear deformation beam theory. They showed that the higher-order theories provide us with accurate results for the natural frequencies, buckling stresses and interlaminar stresses of multilayered composite beams. This is because of the coupling effects of transverse shear and rotary inertia which taken into account in this theory. Emam [5] theoretically and experimentally investigated the nonlinear forced vibrations of a clamped– clamped isotropic buckled beam. In his analysis the beam was
0263-8223/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.07.034
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modeled according to the EBT. Applying the Hamilton’s principle, the governing equations of motion were derived which then solved using perturbation methods. Taking the geometric nonlinearity effects into consideration, Li and Zhou [6] investigated the free vibrations of a thermally buckled beam in the prebuckling and postbuckling domains. They used a shooting method for solving the postbuckling configurations and the linear vibration modes of prebuckled and postbuckled beams. Based on a three-degree-offreedom shear deformable beam theory, Aydogdu [7] investigated the thermal buckling of cross-ply laminated beams under different boundary conditions using the Ritz method. Several other researches have been published in the literature concerning to the buckling and stability of composite structures [8–10]. Also, a number of publications can be found in the literature concerning with the transverse shear deformation effect on the buckling and free vibration of composite beams using classical, first-order and higher-order theories [11–13]. Literature review illustrates that the number of researches performed on the postbuckling of beams are scarce. Nayfeh and Emam [14] presented a closed form solution for the postbuckling analysis of isotropic beams based on the EBT. They studied critical buckling loads and the associated mode shapes. They also studied the free vibration behavior of the buckled isotropic beams in the postbuckling domain. This type of vibration analysis means investigating the vibration characteristics which takes place in the vicinity of a buckled configuration. Afterwards, they extended their work and found an exact solution for the postbuckling behavior of symmetrically laminated composite beams [15]. They investigated the critical buckling load and free vibration in the postbuckling region. They arrived at this point that the lay-up of the laminate can be manipulated to control both the static and dynamic response in the postbuckling domain. Emam [16] extended the previous model proposed in [15] to study the contribution of the transverse shear deformation. He investigated the significant effect of the shear deformation on the static postbuckling of symmetrically laminated composite beams. He did no attempt on the nonlinear vibration analysis in the postbuckling domain with considering the shear deformation effect. Recently, Fallah and Aghdam [17] studied the large amplitude free vibration and postbuckling of functionally graded (FG) beams rested on nonlinear elastic foundation and subjected to axial load by means of an analytical method based on the variational approach. Their analysis is based upon the EBT assumptions together with Von-Karman’s strain–displacement relations. Using the same approach and model, they also studied thermomechanical buckling and nonlinear free vibration of FG beams [18]. In the present work, a closed form solution is presented for postbuckling configuration of FG beams. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. The effects of the transverse shear deformation and rotary inertia are also included within the framework of the TBT. The Von-Karman type nonlinear strain–displacement relationships are employed. Free vibration of the beam in the vicinity of the buckled configuration is also investigated. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via EBT clarifies the overestimation of the frequencies by the later one. This means that TBT provides more accurate as well as reliable results in predicting the postbuckling configuration and natural frequencies.
2. Material properties of FG material A rectangular FG beam of length L, width b and thickness h is shown in Fig. 1. It is assumed that the material properties of the
Fig. 1. An element of a FG beam.
beam such as, Young’s modulus E, mass density q and Poisson’s ratio m vary continuously across the thickness based upon the following relations
EðzÞ ¼ Es þ Eas V f ðzÞ;
Eas ¼ Ea Es
mðzÞ ¼ ms þ mas V f ðzÞ; mas ¼ ma ms qðzÞ ¼ qs þ qas V f ðzÞ; qas ¼ qa qs
ð1Þ
where subscripts s and a refer to properties of steel and alumina respectively and Vf(z) denotes the volume fraction function which can be defined as following for power-law FGMs
z 1 þ h 2
V f ðzÞ ¼
N ð2Þ
in which N denotes the power exponent standing for the material variation profile through the thickness of the beam. 3. Problem formulation According to the TBT, the axial and transverse displacement components can be written as
U ¼ uðx; tÞ þ zwðx; tÞ;
V ¼ 0;
W ¼ wðx; tÞ
ð3Þ
in which u and w are the axial and transverse displacements of the mid-plane in the x and z directions, respectively and w is the cross sectional rotation. Therefore, the Von-Karman type nonlinear strain–displacement relations of the beam at a distance z from the mid-plane can be written as following for the axial and shear strains
ex ¼
2 @u 1 @w @w þ þz x; @x 2 @x @x
cxz ¼ w þ
@w @x
ð4Þ
The potential and kinetic energies for a beam are given as below
R L 2 ðrx ex þ sxz cxz ÞdA dx þ 12 N 0 @w dx @x h i R R L 2 2 2 _ dA dx _ þ ðwÞ _ þ ðzwÞ T ¼ 12 0 A qðzÞ ðuÞ
V ¼ 12
RL R 0
A
ð5Þ
where () denotes differentiation with respect to time and A is the cross-sectional area of the beam. Using Eq. (4), the potential energy can be rewritten in the following form
2 ! ! @u 1 @w @wx @w þ rx z þ dAdx rx þ sxy w þ @x 2 @x @x @x A 0 2 ! Z Z L 2 1 @w 1 L @u 1 @w þ N þ dx ¼ Nx 2 @x 2 0 @x 2 @x 0 Z L 2 @w @w 1 @w dx þ N dx ð6Þ þ Mx x þ Q x w þ @x 2 @x @x 0
1 V¼ 2
Z
L
Z
where N ¼ P denotes the applied axial force. In the above equation, the stress resultants Nx, Mx and Qx represent the axial force, bending moment and shear force per unit length of the beam’s width acting at the midplane and they are given as
Nx ¼ A11 ex;0 þ B11 kx Mx ¼ B11 ex;0 þ D11 kx Q x ¼ KA44 cxz
ð7Þ
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Here, ex,0 and kx are the axial strain and curvature of the midplane of the beam. The extensional, coupling and bending stiffness coefficients are defined as
ðAij ; Bij ; Dij Þ ¼
Z
ð17Þ
h=2
Q ij ð1; z; z2 Þb dz for i; j ¼ 1; 4
ð8Þ
h=2
where Qij in the above relation are the reduced stiffness coefficients which can be defined as following for the FG beam
EðzÞ
Q 11 ¼
1 mðzÞ2
;
Q 44
EðzÞ ¼ 2ð1 þ mðzÞÞ
ð9Þ
and K stands for the shear correction factor and it is taken as 56. Substituting the kinetic and potential energy expressions from Eqs. (5) and (6) together with applying Hamilton’s principle as
Z d
2 R 2 € þ P bA2L11 0L @w I0 w dx bKA66 bBL11 ðwðLÞ wð0ÞÞ @@xw2 bKA44 @w ¼0 @x @x 2 2 € þ b B11 bD11 @ w2 þ bKA44 w þ bKA44 @w ¼ 0 I1 w @x A11 @x
Introducing the following non-dimensional variables
^ ¼ w
w ; r
sffiffiffiffi I1 ; r¼ I0
x ^x ¼ ; L
^t ¼ t
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 ub D11 B11 t A11
ð18Þ
I0 L4
Eq. (17) can be rewritten in the following form ^ @2 w @^t 2
2 R 2 ^ ^ @ w b g1 1 @w^ 2 d^x g ðwðLÞ ^ ^ þ P wð0ÞÞ g3 @@^xw2^ g4 @@w^x ¼ 0 2 2 0 @ ^x @ ^x2
^ @2 w @^t 2
^ þ g @w^ ¼ 0 þ ðg5 g6 Þ @@^xw2 þ g7 w 4 @ ^x
2^
ð19Þ
t
ðT VÞdt ¼ 0
ð10Þ
0
results into the following differential equations of motion
b are given as in which g1–g7 and P
g1 ¼
x € @N I0 u ¼0 @x 2
x € Nx @@xw2 @Q þ P @@xw2 ¼ 0 I0 w @x @M € x þ Qx ¼ 0 I1 w
ð11Þ
@x
R 2h
2
qðzÞz2 dA
2 @ € b @x I0 u A11 @u þ A211 @w þ B11 @w ¼0 @x @x @x 2 2 2 A @ w @ € b A11 @u þ 211 @w þ B11 @w bKA44 @x w þ @w þ P @@xw2 ¼ 0 I0 w @x @x2 @x @x @x € b @ B11 @u þ B11 @w 2 þ D11 @w þ bKA44 w þ @w ¼ 0 I1 w @x @x @x @x 2 @x ð12Þ Assuming that the inplane inertia is negligible, the first relation in Eq. (12) gives
! 2 @ @u 1 @w @w ¼0 A11 þ A11 þ B11 @x @x 2 @x @x
ð13Þ
which can be integrated twice with respect to the spatial coordinate x, to give
uðx; tÞ ¼
B11 1 wðx; tÞ 2 A11
Z
x 0
2 @w dx þ C 1 x þ C 2 @x
ð14Þ
1 C 1 ¼ 2L
wðLÞ R L @w2 0
@x
B11 dx þ LA ðwðLÞ wð0ÞÞ 11
ð15Þ
Substituting Eq. (15) into Eq. (14) and differentiating with respect to x, we arrive at
2 Z L 2 @u 1 @w B11 @w 1 @w B11 ¼ þ dx þ ðwðLÞ @x 2 @x @x A11 @x 2L 0 LA11 wð0ÞÞ
2
g3 ¼ KA44BL2
;
D11 A11
;
B2
r D11 A11
g5 ¼
11
B211 L2
B2
A11 r D11 A11
;
2 g6 ¼ D11 L B2
r 2 D11 A11
11
D11 A11 11
b¼ P
PL2 B2
b D11 A11
ð20Þ
11
11
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ tÞ ¼ wð1; ^ tÞ ¼ 0 wð0;
ð16Þ
Substituting the above relation into the second and third governing differential equations, one reaches to
ð21Þ
Fixed–hinged boundary condition
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ tÞ ¼ 0; wð0;
^ @w ð1; tÞ ¼ 0 @ ^x
ð22Þ
Hinged–hinged boundary condition
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ ^ @w @w ð0; tÞ ¼ ð1; tÞ ¼ 0 @ ^x @ ^x
ð23Þ
4. Buckling analysis The governing equation of the linear buckling problem of a FG beam can be achieved from Eq. (19) by dropping the nonlinear and inertia terms. The result is 2
^
2
k2 @@^xw2^ g3 @@^xw2^ g4 @@w^x ¼ 0 2^
Assume that the beam is constrained to move axially at both ends, i.e. u(0, t) = u(L, t) = 0. Applying these boundary conditions, Eq. (14) yields B11 A11
B2
It should be said that for isotropic beams or symmetrically laminated ones, the coupling stiffness B11 vanishes. The geometric boundary conditions can be expressed as Fixed–fixed boundary condition
Substituting Eqs. (4) and (7) into the governing equations (Eq. (11)), we arrive at
C2 ¼
g4 ¼
B11 L
D11 A11
4 g7 ¼ KA66 LB2 ;
qðzÞdA h h2
g2 ¼
11
3 KA66 L
r2
I1 ¼
;
11
where
R 2h
B2
11
2
I0 ¼
A11 r2
D11 A11
^ þ g @ w^ ¼ 0 ðg5 g6 Þ @@^xw2 þ g7 w 4 @ ^x
ð24Þ
subjected to the boundary conditions given by Eqs. (21)–(23). In the above equation, k2 denotes the critical buckling load given as
b g1 k2 ¼ P 2
Z 0
1
2 ^ @w ^ ^ d^x g2 ðwð1Þ wð0ÞÞ @ ^x
ð25Þ
^ by sim^ or w The system in Eq. (24) can be written in terms of w ple manipulation. Since the definite integral in Eq. (25) yields a ^ s , it is posconstant value for any type of buckling configuration w sible to obtain a closed-form solution for the buckling configuration (for further studies the reader is referred to [14]). The closed-form solution for the buckled configurations of FG beams gets the following form
^ s ð^xÞ ¼ C 1 þ C 2 sinðc^xÞ þ C 3 cosðc^xÞ w ^ s ð^xÞ ¼ K 1 C 1 ^x þ C 2 K 2 cosðc^xÞ C 3 K 2 sinðc^xÞ þ C 4 w
ð26Þ
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in which
K2 ¼
ðg6 g5 Þc
g4
fð^x; ^tÞ ¼
þ
g7 g k2 g 7 ; K 1 ¼ 7 ; c2 ¼ g4 c g4 ðg5 g6 Þcðk2 g3 Þ
1 X ^ an eixt sinðnp^xÞ 1
uð^x; ^tÞ ¼
1 X
ð27Þ After imposing the associated boundary conditions, the closedform solutions for FG beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are respectively obtained as
cosc 1 ^ s ð^xÞ ¼ C K 1 ^x þ K 2 cosðc^xÞ þ K 2 sinðc^xÞ w sin c
1 cosc þK 2 sin c ^ s ð^xÞ ¼ C ½K 1 ^x K 2 tan c cosðc^xÞ þ K 2 sinðc^xÞ þ K 2 tan c w ^ s ð^xÞ ¼ CK 2 sinðc^xÞ w
K1 sin c ¼ 0 K2
K1 cos c ¼ 0 K2 sin c ¼ 0
sin c
"
ðg3 k2 ÞðnpÞ2 þ C
2
g1 ðnpÞ4 2
g np
¼
4 0
x2
g4 np 2p2 g2 C ðg6 g5 ÞðnpÞ2 þ g7 x2
#
0
ð28Þ
ð39Þ
ð29Þ
For a non-trivial solution of Eq. (39), the determinant of the coefficient matrix must vanish, resulting in a characteristic equation whose eigenvalues are the natural frequencies of FG beams.
ð30Þ
ð31Þ ð32Þ ð33Þ
In order to investigate the vibrational behavior in the postbuckling domain, we consider the vibrations that take place around a buckled configuration which obtained from the previous section. To this aim assume that the solutions for transverse deflection and rotation be in the following form
ð34Þ
s
where fð^ x; ^tÞ and uð^x; ^tÞ define the dynamic response around the buckled configuration. Substituting Eq. (34) into Eq. (19), we arrive at
R b g1 1 @ w^ 2 d^x g wð1Þ €f þ P ^ ^ wð0Þ f00 2 2 0 @ ^x R1 g3 f00 g4 u0 g1 w00s 0 f0 w0s d^x R1 g2 ½uð1Þ uð0Þw00s þ f ðws ; ws Þ ¼ g1 f00 0 f0 w0s d^x R1 R1 þ g21 w00s 0 f02 d^x þ g21 f00 0 ðf0 Þ2 d^x € þ ðg5 g6 Þu00 þ g7 u0 þ g4 w0 þ gðws ; ws Þ ¼ 0 u
and put them into Eq. (37) and rearrange them in a matrix form as
b
5. Free vibration analysis in the postbuckling domain
^ x^; ^tÞ ¼ ws ð^xÞ þ fð^x; ^tÞ wð ^ ^x; ^tÞ ¼ w ð^xÞ þ uð^x; ^tÞ wð
1
a
in which C is a constant. The characteristic equation of FG beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are respectively given by
2 2 cos c
ð38Þ ^
bn eixt cosðnp^xÞ
ð35Þ
6. Results and discussion In order to validate the present analysis, a comparison is made for midspan deflection of fixed–hinged composite beams versus the axial load with those reported via EBT [15] in Fig. 2. Fig. 3 shows a comparison of the fundamental natural frequency of hinged–hinged composite beams against the axial load with those reported in [15]. The material and geometrical properties for these graphs are taken from [15]. As can be seen there is an excellent agreement between two theories. The FG beam considered here is composed of steel and alumina whose properties vary according to the power-law volume fraction function. The material properties of each constituent material are given in Table 1. Table 2 gives the first three critical buckling loads for FG beams with different boundary conditions. From this table it is concluded that the critical buckling load decreases by an increment in the power index N. This is due to the conversion in the material properties from pure alumina to pure steel as N increases from zero to infinity. Also, a beam with hinged–hinged end conditions has the lowest critical buckling load while the fixed–fixed beam has the highest one. Fig. 4 gives the variation of the midspan deflection of a fixed–fixed FG beam with the axial load for various values of the power index. It can be observed that for a given axial load the static deflection increases as the power index increases. The effect of length to the thickness ratio on the nondimensional static deflection of fixed–hinged FG beam is illustrated in Fig. 5. The nondimensional midspan deflection decreases as the slender-
in which
R b g1 1 @ w^ 2 d^x g wð1Þ ^ ^ f ðws ; ws Þ ¼ P wð0Þ w00s g3 w00s g4 w0s 2 2 0 @ ^x ^ þ g w0 gðws ; ws Þ ¼ ðg5 g6 Þw00s þ g7 w 4 s ð36Þ which in view of Eq. (24) are equal to zero. We only consider the linear vibration analysis here. Discarding the nonlinear terms from the right-hand side of Eq. (35), we reach to
b g1 R 1 @ w^ 2 d^x g ðwðLÞ €f þ P ^ ^ wð0ÞÞ f00 2 2 0 @ ^x R 1 g3 f00 g4 u0 g1 w00s 0 f0 w0s d^x
ð37Þ
g2 ½uð1Þ uð0Þw00s ¼ 0 € ðg5 g6 Þu00 þ g7 u0 þ g4 w0 ¼ 0 u To solve the above system for hinged–hinged beams, we assume the following harmonic dynamic responses
Fig. 2. Variation of the beam’s midspan deflection with the applied axial load for a fixed–hinged composite beam with unidirectional layup (L/h = 250).
G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
Fig. 3. Variation of the fundamental natural frequency of the vibration around the first buckled configuration with the axial load for a hinged–hinged composite beam with unidirectional layup (L/h = 250).
251
Fig. 4. Variation of the midspan deflection of fixed–fixed beam with the applied axial load for different values of N (b = 0.002, L/h = 10).
Table 1 Material properties of FGM constituents. Properties
Unit
Steel
Alumina (Al2O3)
E
GPa kg/m3
210 7800 0.29
390 3960 0.22
q t
Table 2 The first three critical buckling loads for FG beams under various boundary conditions (L = 0.4, h = 0.04, b = 0.08). Buckling load (104 KN)
N=0
N=2
N=4
N=8
N = 10
Fixed–fixed P1 P2 P3
3.96 7.91 10.58
2.80 5.59 10.12
2.66 5.31 10.65
2.53 5.05 10.13
2.49 4.97 9.96
Fixed–hinged P1 P2 P3
2.12 6.28 10.25
1.50 4.44 8.85
1.43 4.24 8.44
1.37 4.04 8.04
1.34 3.97 7.91
1.70 4.27 9.60
0.75 3.03 6.81
0.72 2.89 6.50
0.69 2.75 6.19
0.68 2.71 6.10
Hinged–hinged P1 P2 P3
ness ratio increases. However for higher ratios of L/h the graphs converge. Fig. 6 shows the variation of the static deflection at the midspan of the FG beam with the applied axial load in the postbuckling domain for three common boundary conditions. As shown in this figure the static deflection decreases as one travels from hinged–hinged end condition to the fixed–fixed one. Fig. 7 clarifies the nondimensional midspan deflection variation of a fixed–hinged FG beam over the axial load for two beam theories and L/h = 10. The similar graph for L/h = 40 is depicted in Fig. 8. As can be seen the Timoshenko beam goes sooner under buckling and this is due to the presence of shear deformation effects. Also, the difference between EBT and TBT is more significant for lower values of the slenderness ratio and for higher ones this difference fades away. The variation of the nondimensional fundamental natural frequency of H–H FG beams in both the prebuckling and postbuckling domains is illustrated in Fig. 9. The results are also compared with those obtained via the analysis based upon EBT [15] which clarifies
Fig. 5. Variation of the nondimensional midspan deflection of fixed–hinged beam with the applied axial load for different values of slenderness ratios (N = 2).
Fig. 6. Variation of the nondimensional midspan deflection with the applied axial load for different boundary conditions of a FG beam (N = 2).
the overestimation of the results caused by the later theory. This is mainly due to the presence of the effects of the shear deformation
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Fig. 7. Variation of the nondimensional midspan deflection with the applied axial load for a fixed–hinged FG beam (N = 2, L/h = 10).
Fig. 10. Variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams with different values of power index N (b = 0.002, L/h = 10).
Fig. 8. Variation of the nondimensional midspan deflection with the applied axial load for a fixed–hinged FG beam (N = 2, L/h = 40). Fig. 11. Variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams for different length to thickness ratios (b = 0.002, N = 2).
ling. However, in the postbuckling domain the trend becomes reverse and the fundamental natural frequency increases as the applied load increases. The influence of the power index on the fundamental frequency response in the prebuckling and postbuckling domains is depicted in Fig. 10. It can be noted from this figure that N significantly affects the resulting vibration in both domains. Fig. 11 shows the variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams with different length to thickness ratios. This graph verifies the fact that a beam with the lowest L/h ratio gets the highest critical buckling load. 7. Conclusion
Fig. 9. Variation of the nondimensional natural frequency of the vibration around the first buckled configuration against the axial load of FG beams (N = 2).
and rotary inertia which taken into consideration in TBT. As can be seen in the prebuckling domain the natural frequency decreases as the applied load increases until it meets zero at the onset of buck-
An exact solution is obtained for the postbuckling behavior of functionally graded beams. First, the buckling problem was solved to find the buckling configuration of the beams in terms of the applied axial load for fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. Then the linear vibration problem is investigated around the first buckled configuration and the natural frequencies are obtained for hinged–hinged beams. The effects of shear deformation and rotary inertia are included in the framework of Timoshenko beam theory. A functionally graded
G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
beam composed of steel and alumina is considered for our analysis whose material properties are assumed to be graded in the thickness direction according to the power-law distribution. Results show that in the prebuckling domain the natural frequency decreases as the applied load increases until it meets zero at the onset of buckling. However, in the postbuckling domain the trend becomes reverse and the fundamental natural frequency increases as the applied load increases. Results obtained are compared to those achieved via Euler–Bernoulli beam theory. This comparison illustrate that the Euler–Bernoulli beam theory predicts the natural frequencies with overestimation. The results obtained show the significant dependency of the fundamental natural frequency and midspan deflection to the power index and length to the thickness ratio of the beam. The present achievements can be used as a benchmark for further studies. References [1] Nayfeh AH, Kreider W, Anderson TJ. Investigation of natural frequencies and mode shapes of buckled beams. AIAA J 1995;33:1121–6. [2] Benerjee JR. Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Comput Struct 1998;69:197–208. [3] Lee JJ, Choi S. Thermal buckling and postbuckling analysis of a laminated composite beam with embedded SMA actuators. Compos Struct 1999;47:695–703. [4] Matsunaga H. Vibration and buckling of multilayered composite beams according to higher order deformation theories. J Sound Vib 2001;246:47–62.
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[5] Emam SA. A theoretical and experimental study of nonlinear dynamics of buckled beams. PhD dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA; 2002. [6] Li S, Zhou Y. Free vibration of heated Euler–Bernoulli beams with thermal postbuckling deformations. J Therm Stress 2004;27:843–56. [7] Aydogdu M. Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions. Compos Sci Technol 2007;67:1096–104. [8] De-min W, Gui-tong Y. Nonlinear dynamics stability of composite laminated plates. Appl Math Mech 2004;25(11):1215–9. [9] Sapountzakis EJ, Tsiatas GC. Elastic flexural buckling analysis of composite beams of variable cross-section by Bem. Eng Struct 2007;29:675–81. [10] Fridman Y, Abramovich H. Enhanced structural behavior of flexible laminated composite beams. Compos Struct 2008;82:140–54. [11] Khdeir AA, Reddy JN. Buckling of cross-ply laminated beams with arbitrary boundary conditions. Compos Struct 1997;37(1):1–3. [12] Song SJ, Was AM. Effect of shear deformation on buckling and free vibration of laminated composite beams. Compos Struct 1997;37(1):33–43. [13] Goyal VK, Kapania RK. A shear-deformable beam element for the analysis of laminated composites. Finite Elem Anal Des 2007;43:463–77. [14] Nayfeh AH, Emam SA. Exact solutions and stability of the postbuckling configurations of beams. Nonlinear Dynam 2008;54:395–408. [15] Emam SA, Nayfeh AH. Postbuckling and free vibration of composite beams. Compos Struct 2009;88:636–42. [16] Emam SA. Analysis of shear-deformable composite beams in postbuckling. Compos Struct 2011;94:24–30. [17] Fallah A, Aghdam MM. Nonlinear free vibration and post buckling analysis of functionally graded beams on nonlinear elastic foundation. Eur J Mech A – Solids 2011;30(4):571–83. [18] Fallah A, Aghdam MM. Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation. Compos B: Eng 2012;43(3):1523–30.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
On the postbuckling and free vibrations of FG Timoshenko beams G.H. Rahimi a, M.S. Gazor a, M. Hemmatnezhad b,⇑, H. Toorani a a b
Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran Faculty of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran
a r t i c l e
i n f o
Article history: Available online 7 August 2012 Keywords: Postbuckling behavior Free vibration Functionally graded Timoshenko beam theory Exact solution
a b s t r a c t The postbuckling behavior of functionally graded beams is investigated by means of an exact solution method. The Von-Karman type nonlinear strain–displacement relationships are employed. The effects of the transverse shear deformation and rotary inertia are also included based upon the Timoshenko beam theory. After writing the kinetic and potential energy functionals, the governing equations of motion including the axial, transverse deflections and also the cross sectional rotation are derived using the Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. Neglecting the inplane inertia, the three equations of motion are reduced to two nonlinear partial–integral–differential equations in terms of the transverse mid-plane deflection and the cross sectional rotation. FG beams are considered to have fixed–fixed, fixed–hinged, and hinged–hinged end conditions. A closed-form solution is achieved for the postbuckling deformation as a function of the exerted axial load which is beyond the critical buckling load. In order to study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first buckled configuration of a hinged–hinged FG beam. This leads to a characteristic equation whose eigenvalues are the natural frequencies and the corresponding eigenvectors also determine the mode shapes. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via Euler–Bernoulli beam theory clarifies the overestimation of the frequencies by the later one. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.
1. Introduction Buckling is a static instability of structures occurring in the presence of inplane loading. During the past decades buckling analysis of beam structures has attracted more attention from the scientific community as reflected by increasing number of publications devoted to that. Most of the investigations performed on the buckling problem are concerned with determining the critical buckling loads and their associated mode shapes. To investigate the postbuckling problem, one should enter the geometric nonlinearity terms due to the midplane stretching into the formulation until reaches to a nonlinear buckling problem. Solving this nonlinear problem for a given axial load yields the postbuckling configurations. Based on the Euler–Bernoulli beam theory (EBT), Nayfeh et al. [1] obtained an exact solution for the linear free vibration problem of buckled beams. They arrived at results in an excellent agreement with the experimental data. Using the dynamic stiffness matrix method, Benerjee [2] investigated the problem of free vibrations ⇑ Corresponding author. Tel./fax: +98 282 5270131. E-mail address: [email protected] (M. Hemmatnezhad).
of axially loaded composite beams. He took into consideration the effects of shear deformation and rotary inertia based on the Timoshenko beam theory (TBT). He studied the free vibration characteristics of composite beams in the prebuckling domain and did not consider the postbuckling behavior. Lee and Choi [3] studied the thermal buckling as well as postbuckling behaviors of composite beams embedded with shape-memory alloy wires using analytical technique. They also constructed a finite element model for the structure using ABAQUS software. They concluded that using shape-memory alloy wires can significantly enhance the critical buckling temperature and reduce the lateral deflection of a laminated composite beam. Matsunaga [4] obtained the natural frequencies and buckling stresses of simply supported laminated composite beams based upon the higher-order shear deformation beam theory. They showed that the higher-order theories provide us with accurate results for the natural frequencies, buckling stresses and interlaminar stresses of multilayered composite beams. This is because of the coupling effects of transverse shear and rotary inertia which taken into account in this theory. Emam [5] theoretically and experimentally investigated the nonlinear forced vibrations of a clamped– clamped isotropic buckled beam. In his analysis the beam was
0263-8223/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.07.034
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modeled according to the EBT. Applying the Hamilton’s principle, the governing equations of motion were derived which then solved using perturbation methods. Taking the geometric nonlinearity effects into consideration, Li and Zhou [6] investigated the free vibrations of a thermally buckled beam in the prebuckling and postbuckling domains. They used a shooting method for solving the postbuckling configurations and the linear vibration modes of prebuckled and postbuckled beams. Based on a three-degree-offreedom shear deformable beam theory, Aydogdu [7] investigated the thermal buckling of cross-ply laminated beams under different boundary conditions using the Ritz method. Several other researches have been published in the literature concerning to the buckling and stability of composite structures [8–10]. Also, a number of publications can be found in the literature concerning with the transverse shear deformation effect on the buckling and free vibration of composite beams using classical, first-order and higher-order theories [11–13]. Literature review illustrates that the number of researches performed on the postbuckling of beams are scarce. Nayfeh and Emam [14] presented a closed form solution for the postbuckling analysis of isotropic beams based on the EBT. They studied critical buckling loads and the associated mode shapes. They also studied the free vibration behavior of the buckled isotropic beams in the postbuckling domain. This type of vibration analysis means investigating the vibration characteristics which takes place in the vicinity of a buckled configuration. Afterwards, they extended their work and found an exact solution for the postbuckling behavior of symmetrically laminated composite beams [15]. They investigated the critical buckling load and free vibration in the postbuckling region. They arrived at this point that the lay-up of the laminate can be manipulated to control both the static and dynamic response in the postbuckling domain. Emam [16] extended the previous model proposed in [15] to study the contribution of the transverse shear deformation. He investigated the significant effect of the shear deformation on the static postbuckling of symmetrically laminated composite beams. He did no attempt on the nonlinear vibration analysis in the postbuckling domain with considering the shear deformation effect. Recently, Fallah and Aghdam [17] studied the large amplitude free vibration and postbuckling of functionally graded (FG) beams rested on nonlinear elastic foundation and subjected to axial load by means of an analytical method based on the variational approach. Their analysis is based upon the EBT assumptions together with Von-Karman’s strain–displacement relations. Using the same approach and model, they also studied thermomechanical buckling and nonlinear free vibration of FG beams [18]. In the present work, a closed form solution is presented for postbuckling configuration of FG beams. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. The effects of the transverse shear deformation and rotary inertia are also included within the framework of the TBT. The Von-Karman type nonlinear strain–displacement relationships are employed. Free vibration of the beam in the vicinity of the buckled configuration is also investigated. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via EBT clarifies the overestimation of the frequencies by the later one. This means that TBT provides more accurate as well as reliable results in predicting the postbuckling configuration and natural frequencies.
2. Material properties of FG material A rectangular FG beam of length L, width b and thickness h is shown in Fig. 1. It is assumed that the material properties of the
Fig. 1. An element of a FG beam.
beam such as, Young’s modulus E, mass density q and Poisson’s ratio m vary continuously across the thickness based upon the following relations
EðzÞ ¼ Es þ Eas V f ðzÞ;
Eas ¼ Ea Es
mðzÞ ¼ ms þ mas V f ðzÞ; mas ¼ ma ms qðzÞ ¼ qs þ qas V f ðzÞ; qas ¼ qa qs
ð1Þ
where subscripts s and a refer to properties of steel and alumina respectively and Vf(z) denotes the volume fraction function which can be defined as following for power-law FGMs
z 1 þ h 2
V f ðzÞ ¼
N ð2Þ
in which N denotes the power exponent standing for the material variation profile through the thickness of the beam. 3. Problem formulation According to the TBT, the axial and transverse displacement components can be written as
U ¼ uðx; tÞ þ zwðx; tÞ;
V ¼ 0;
W ¼ wðx; tÞ
ð3Þ
in which u and w are the axial and transverse displacements of the mid-plane in the x and z directions, respectively and w is the cross sectional rotation. Therefore, the Von-Karman type nonlinear strain–displacement relations of the beam at a distance z from the mid-plane can be written as following for the axial and shear strains
ex ¼
2 @u 1 @w @w þ þz x; @x 2 @x @x
cxz ¼ w þ
@w @x
ð4Þ
The potential and kinetic energies for a beam are given as below
R L 2 ðrx ex þ sxz cxz ÞdA dx þ 12 N 0 @w dx @x h i R R L 2 2 2 _ dA dx _ þ ðwÞ _ þ ðzwÞ T ¼ 12 0 A qðzÞ ðuÞ
V ¼ 12
RL R 0
A
ð5Þ
where () denotes differentiation with respect to time and A is the cross-sectional area of the beam. Using Eq. (4), the potential energy can be rewritten in the following form
2 ! ! @u 1 @w @wx @w þ rx z þ dAdx rx þ sxy w þ @x 2 @x @x @x A 0 2 ! Z Z L 2 1 @w 1 L @u 1 @w þ N þ dx ¼ Nx 2 @x 2 0 @x 2 @x 0 Z L 2 @w @w 1 @w dx þ N dx ð6Þ þ Mx x þ Q x w þ @x 2 @x @x 0
1 V¼ 2
Z
L
Z
where N ¼ P denotes the applied axial force. In the above equation, the stress resultants Nx, Mx and Qx represent the axial force, bending moment and shear force per unit length of the beam’s width acting at the midplane and they are given as
Nx ¼ A11 ex;0 þ B11 kx Mx ¼ B11 ex;0 þ D11 kx Q x ¼ KA44 cxz
ð7Þ
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G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
Here, ex,0 and kx are the axial strain and curvature of the midplane of the beam. The extensional, coupling and bending stiffness coefficients are defined as
ðAij ; Bij ; Dij Þ ¼
Z
ð17Þ
h=2
Q ij ð1; z; z2 Þb dz for i; j ¼ 1; 4
ð8Þ
h=2
where Qij in the above relation are the reduced stiffness coefficients which can be defined as following for the FG beam
EðzÞ
Q 11 ¼
1 mðzÞ2
;
Q 44
EðzÞ ¼ 2ð1 þ mðzÞÞ
ð9Þ
and K stands for the shear correction factor and it is taken as 56. Substituting the kinetic and potential energy expressions from Eqs. (5) and (6) together with applying Hamilton’s principle as
Z d
2 R 2 € þ P bA2L11 0L @w I0 w dx bKA66 bBL11 ðwðLÞ wð0ÞÞ @@xw2 bKA44 @w ¼0 @x @x 2 2 € þ b B11 bD11 @ w2 þ bKA44 w þ bKA44 @w ¼ 0 I1 w @x A11 @x
Introducing the following non-dimensional variables
^ ¼ w
w ; r
sffiffiffiffi I1 ; r¼ I0
x ^x ¼ ; L
^t ¼ t
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 ub D11 B11 t A11
ð18Þ
I0 L4
Eq. (17) can be rewritten in the following form ^ @2 w @^t 2
2 R 2 ^ ^ @ w b g1 1 @w^ 2 d^x g ðwðLÞ ^ ^ þ P wð0ÞÞ g3 @@^xw2^ g4 @@w^x ¼ 0 2 2 0 @ ^x @ ^x2
^ @2 w @^t 2
^ þ g @w^ ¼ 0 þ ðg5 g6 Þ @@^xw2 þ g7 w 4 @ ^x
2^
ð19Þ
t
ðT VÞdt ¼ 0
ð10Þ
0
results into the following differential equations of motion
b are given as in which g1–g7 and P
g1 ¼
x € @N I0 u ¼0 @x 2
x € Nx @@xw2 @Q þ P @@xw2 ¼ 0 I0 w @x @M € x þ Qx ¼ 0 I1 w
ð11Þ
@x
R 2h
2
qðzÞz2 dA
2 @ € b @x I0 u A11 @u þ A211 @w þ B11 @w ¼0 @x @x @x 2 2 2 A @ w @ € b A11 @u þ 211 @w þ B11 @w bKA44 @x w þ @w þ P @@xw2 ¼ 0 I0 w @x @x2 @x @x @x € b @ B11 @u þ B11 @w 2 þ D11 @w þ bKA44 w þ @w ¼ 0 I1 w @x @x @x @x 2 @x ð12Þ Assuming that the inplane inertia is negligible, the first relation in Eq. (12) gives
! 2 @ @u 1 @w @w ¼0 A11 þ A11 þ B11 @x @x 2 @x @x
ð13Þ
which can be integrated twice with respect to the spatial coordinate x, to give
uðx; tÞ ¼
B11 1 wðx; tÞ 2 A11
Z
x 0
2 @w dx þ C 1 x þ C 2 @x
ð14Þ
1 C 1 ¼ 2L
wðLÞ R L @w2 0
@x
B11 dx þ LA ðwðLÞ wð0ÞÞ 11
ð15Þ
Substituting Eq. (15) into Eq. (14) and differentiating with respect to x, we arrive at
2 Z L 2 @u 1 @w B11 @w 1 @w B11 ¼ þ dx þ ðwðLÞ @x 2 @x @x A11 @x 2L 0 LA11 wð0ÞÞ
2
g3 ¼ KA44BL2
;
D11 A11
;
B2
r D11 A11
g5 ¼
11
B211 L2
B2
A11 r D11 A11
;
2 g6 ¼ D11 L B2
r 2 D11 A11
11
D11 A11 11
b¼ P
PL2 B2
b D11 A11
ð20Þ
11
11
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ tÞ ¼ wð1; ^ tÞ ¼ 0 wð0;
ð16Þ
Substituting the above relation into the second and third governing differential equations, one reaches to
ð21Þ
Fixed–hinged boundary condition
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ tÞ ¼ 0; wð0;
^ @w ð1; tÞ ¼ 0 @ ^x
ð22Þ
Hinged–hinged boundary condition
^ ^ wð0; tÞ ¼ wð1; tÞ ¼ 0;
^ ^ @w @w ð0; tÞ ¼ ð1; tÞ ¼ 0 @ ^x @ ^x
ð23Þ
4. Buckling analysis The governing equation of the linear buckling problem of a FG beam can be achieved from Eq. (19) by dropping the nonlinear and inertia terms. The result is 2
^
2
k2 @@^xw2^ g3 @@^xw2^ g4 @@w^x ¼ 0 2^
Assume that the beam is constrained to move axially at both ends, i.e. u(0, t) = u(L, t) = 0. Applying these boundary conditions, Eq. (14) yields B11 A11
B2
It should be said that for isotropic beams or symmetrically laminated ones, the coupling stiffness B11 vanishes. The geometric boundary conditions can be expressed as Fixed–fixed boundary condition
Substituting Eqs. (4) and (7) into the governing equations (Eq. (11)), we arrive at
C2 ¼
g4 ¼
B11 L
D11 A11
4 g7 ¼ KA66 LB2 ;
qðzÞdA h h2
g2 ¼
11
3 KA66 L
r2
I1 ¼
;
11
where
R 2h
B2
11
2
I0 ¼
A11 r2
D11 A11
^ þ g @ w^ ¼ 0 ðg5 g6 Þ @@^xw2 þ g7 w 4 @ ^x
ð24Þ
subjected to the boundary conditions given by Eqs. (21)–(23). In the above equation, k2 denotes the critical buckling load given as
b g1 k2 ¼ P 2
Z 0
1
2 ^ @w ^ ^ d^x g2 ðwð1Þ wð0ÞÞ @ ^x
ð25Þ
^ by sim^ or w The system in Eq. (24) can be written in terms of w ple manipulation. Since the definite integral in Eq. (25) yields a ^ s , it is posconstant value for any type of buckling configuration w sible to obtain a closed-form solution for the buckling configuration (for further studies the reader is referred to [14]). The closed-form solution for the buckled configurations of FG beams gets the following form
^ s ð^xÞ ¼ C 1 þ C 2 sinðc^xÞ þ C 3 cosðc^xÞ w ^ s ð^xÞ ¼ K 1 C 1 ^x þ C 2 K 2 cosðc^xÞ C 3 K 2 sinðc^xÞ þ C 4 w
ð26Þ
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G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
in which
K2 ¼
ðg6 g5 Þc
g4
fð^x; ^tÞ ¼
þ
g7 g k2 g 7 ; K 1 ¼ 7 ; c2 ¼ g4 c g4 ðg5 g6 Þcðk2 g3 Þ
1 X ^ an eixt sinðnp^xÞ 1
uð^x; ^tÞ ¼
1 X
ð27Þ After imposing the associated boundary conditions, the closedform solutions for FG beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are respectively obtained as
cosc 1 ^ s ð^xÞ ¼ C K 1 ^x þ K 2 cosðc^xÞ þ K 2 sinðc^xÞ w sin c
1 cosc þK 2 sin c ^ s ð^xÞ ¼ C ½K 1 ^x K 2 tan c cosðc^xÞ þ K 2 sinðc^xÞ þ K 2 tan c w ^ s ð^xÞ ¼ CK 2 sinðc^xÞ w
K1 sin c ¼ 0 K2
K1 cos c ¼ 0 K2 sin c ¼ 0
sin c
"
ðg3 k2 ÞðnpÞ2 þ C
2
g1 ðnpÞ4 2
g np
¼
4 0
x2
g4 np 2p2 g2 C ðg6 g5 ÞðnpÞ2 þ g7 x2
#
0
ð28Þ
ð39Þ
ð29Þ
For a non-trivial solution of Eq. (39), the determinant of the coefficient matrix must vanish, resulting in a characteristic equation whose eigenvalues are the natural frequencies of FG beams.
ð30Þ
ð31Þ ð32Þ ð33Þ
In order to investigate the vibrational behavior in the postbuckling domain, we consider the vibrations that take place around a buckled configuration which obtained from the previous section. To this aim assume that the solutions for transverse deflection and rotation be in the following form
ð34Þ
s
where fð^ x; ^tÞ and uð^x; ^tÞ define the dynamic response around the buckled configuration. Substituting Eq. (34) into Eq. (19), we arrive at
R b g1 1 @ w^ 2 d^x g wð1Þ €f þ P ^ ^ wð0Þ f00 2 2 0 @ ^x R1 g3 f00 g4 u0 g1 w00s 0 f0 w0s d^x R1 g2 ½uð1Þ uð0Þw00s þ f ðws ; ws Þ ¼ g1 f00 0 f0 w0s d^x R1 R1 þ g21 w00s 0 f02 d^x þ g21 f00 0 ðf0 Þ2 d^x € þ ðg5 g6 Þu00 þ g7 u0 þ g4 w0 þ gðws ; ws Þ ¼ 0 u
and put them into Eq. (37) and rearrange them in a matrix form as
b
5. Free vibration analysis in the postbuckling domain
^ x^; ^tÞ ¼ ws ð^xÞ þ fð^x; ^tÞ wð ^ ^x; ^tÞ ¼ w ð^xÞ þ uð^x; ^tÞ wð
1
a
in which C is a constant. The characteristic equation of FG beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are respectively given by
2 2 cos c
ð38Þ ^
bn eixt cosðnp^xÞ
ð35Þ
6. Results and discussion In order to validate the present analysis, a comparison is made for midspan deflection of fixed–hinged composite beams versus the axial load with those reported via EBT [15] in Fig. 2. Fig. 3 shows a comparison of the fundamental natural frequency of hinged–hinged composite beams against the axial load with those reported in [15]. The material and geometrical properties for these graphs are taken from [15]. As can be seen there is an excellent agreement between two theories. The FG beam considered here is composed of steel and alumina whose properties vary according to the power-law volume fraction function. The material properties of each constituent material are given in Table 1. Table 2 gives the first three critical buckling loads for FG beams with different boundary conditions. From this table it is concluded that the critical buckling load decreases by an increment in the power index N. This is due to the conversion in the material properties from pure alumina to pure steel as N increases from zero to infinity. Also, a beam with hinged–hinged end conditions has the lowest critical buckling load while the fixed–fixed beam has the highest one. Fig. 4 gives the variation of the midspan deflection of a fixed–fixed FG beam with the axial load for various values of the power index. It can be observed that for a given axial load the static deflection increases as the power index increases. The effect of length to the thickness ratio on the nondimensional static deflection of fixed–hinged FG beam is illustrated in Fig. 5. The nondimensional midspan deflection decreases as the slender-
in which
R b g1 1 @ w^ 2 d^x g wð1Þ ^ ^ f ðws ; ws Þ ¼ P wð0Þ w00s g3 w00s g4 w0s 2 2 0 @ ^x ^ þ g w0 gðws ; ws Þ ¼ ðg5 g6 Þw00s þ g7 w 4 s ð36Þ which in view of Eq. (24) are equal to zero. We only consider the linear vibration analysis here. Discarding the nonlinear terms from the right-hand side of Eq. (35), we reach to
b g1 R 1 @ w^ 2 d^x g ðwðLÞ €f þ P ^ ^ wð0ÞÞ f00 2 2 0 @ ^x R 1 g3 f00 g4 u0 g1 w00s 0 f0 w0s d^x
ð37Þ
g2 ½uð1Þ uð0Þw00s ¼ 0 € ðg5 g6 Þu00 þ g7 u0 þ g4 w0 ¼ 0 u To solve the above system for hinged–hinged beams, we assume the following harmonic dynamic responses
Fig. 2. Variation of the beam’s midspan deflection with the applied axial load for a fixed–hinged composite beam with unidirectional layup (L/h = 250).
G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
Fig. 3. Variation of the fundamental natural frequency of the vibration around the first buckled configuration with the axial load for a hinged–hinged composite beam with unidirectional layup (L/h = 250).
251
Fig. 4. Variation of the midspan deflection of fixed–fixed beam with the applied axial load for different values of N (b = 0.002, L/h = 10).
Table 1 Material properties of FGM constituents. Properties
Unit
Steel
Alumina (Al2O3)
E
GPa kg/m3
210 7800 0.29
390 3960 0.22
q t
Table 2 The first three critical buckling loads for FG beams under various boundary conditions (L = 0.4, h = 0.04, b = 0.08). Buckling load (104 KN)
N=0
N=2
N=4
N=8
N = 10
Fixed–fixed P1 P2 P3
3.96 7.91 10.58
2.80 5.59 10.12
2.66 5.31 10.65
2.53 5.05 10.13
2.49 4.97 9.96
Fixed–hinged P1 P2 P3
2.12 6.28 10.25
1.50 4.44 8.85
1.43 4.24 8.44
1.37 4.04 8.04
1.34 3.97 7.91
1.70 4.27 9.60
0.75 3.03 6.81
0.72 2.89 6.50
0.69 2.75 6.19
0.68 2.71 6.10
Hinged–hinged P1 P2 P3
ness ratio increases. However for higher ratios of L/h the graphs converge. Fig. 6 shows the variation of the static deflection at the midspan of the FG beam with the applied axial load in the postbuckling domain for three common boundary conditions. As shown in this figure the static deflection decreases as one travels from hinged–hinged end condition to the fixed–fixed one. Fig. 7 clarifies the nondimensional midspan deflection variation of a fixed–hinged FG beam over the axial load for two beam theories and L/h = 10. The similar graph for L/h = 40 is depicted in Fig. 8. As can be seen the Timoshenko beam goes sooner under buckling and this is due to the presence of shear deformation effects. Also, the difference between EBT and TBT is more significant for lower values of the slenderness ratio and for higher ones this difference fades away. The variation of the nondimensional fundamental natural frequency of H–H FG beams in both the prebuckling and postbuckling domains is illustrated in Fig. 9. The results are also compared with those obtained via the analysis based upon EBT [15] which clarifies
Fig. 5. Variation of the nondimensional midspan deflection of fixed–hinged beam with the applied axial load for different values of slenderness ratios (N = 2).
Fig. 6. Variation of the nondimensional midspan deflection with the applied axial load for different boundary conditions of a FG beam (N = 2).
the overestimation of the results caused by the later theory. This is mainly due to the presence of the effects of the shear deformation
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G.H. Rahimi et al. / Composite Structures 95 (2013) 247–253
Fig. 7. Variation of the nondimensional midspan deflection with the applied axial load for a fixed–hinged FG beam (N = 2, L/h = 10).
Fig. 10. Variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams with different values of power index N (b = 0.002, L/h = 10).
Fig. 8. Variation of the nondimensional midspan deflection with the applied axial load for a fixed–hinged FG beam (N = 2, L/h = 40). Fig. 11. Variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams for different length to thickness ratios (b = 0.002, N = 2).
ling. However, in the postbuckling domain the trend becomes reverse and the fundamental natural frequency increases as the applied load increases. The influence of the power index on the fundamental frequency response in the prebuckling and postbuckling domains is depicted in Fig. 10. It can be noted from this figure that N significantly affects the resulting vibration in both domains. Fig. 11 shows the variation of the fundamental natural frequency of the vibration around the first buckled configuration against the axial load of FG beams with different length to thickness ratios. This graph verifies the fact that a beam with the lowest L/h ratio gets the highest critical buckling load. 7. Conclusion
Fig. 9. Variation of the nondimensional natural frequency of the vibration around the first buckled configuration against the axial load of FG beams (N = 2).
and rotary inertia which taken into consideration in TBT. As can be seen in the prebuckling domain the natural frequency decreases as the applied load increases until it meets zero at the onset of buck-
An exact solution is obtained for the postbuckling behavior of functionally graded beams. First, the buckling problem was solved to find the buckling configuration of the beams in terms of the applied axial load for fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. Then the linear vibration problem is investigated around the first buckled configuration and the natural frequencies are obtained for hinged–hinged beams. The effects of shear deformation and rotary inertia are included in the framework of Timoshenko beam theory. A functionally graded
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beam composed of steel and alumina is considered for our analysis whose material properties are assumed to be graded in the thickness direction according to the power-law distribution. Results show that in the prebuckling domain the natural frequency decreases as the applied load increases until it meets zero at the onset of buckling. However, in the postbuckling domain the trend becomes reverse and the fundamental natural frequency increases as the applied load increases. Results obtained are compared to those achieved via Euler–Bernoulli beam theory. This comparison illustrate that the Euler–Bernoulli beam theory predicts the natural frequencies with overestimation. The results obtained show the significant dependency of the fundamental natural frequency and midspan deflection to the power index and length to the thickness ratio of the beam. The present achievements can be used as a benchmark for further studies. References [1] Nayfeh AH, Kreider W, Anderson TJ. Investigation of natural frequencies and mode shapes of buckled beams. AIAA J 1995;33:1121–6. [2] Benerjee JR. Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Comput Struct 1998;69:197–208. [3] Lee JJ, Choi S. Thermal buckling and postbuckling analysis of a laminated composite beam with embedded SMA actuators. Compos Struct 1999;47:695–703. [4] Matsunaga H. Vibration and buckling of multilayered composite beams according to higher order deformation theories. J Sound Vib 2001;246:47–62.
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