Sayyad, A., Ghugal, Y. (2018). Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams. Mechanics of Advanced Composite Structures, 5(1), 1324. doi: 10.22075/macs.2018.12214.1117
Atteshamuddin S. Sayyad; Yuwaraj M. Ghugal. "Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams". Mechanics of Advanced Composite Structures, 5, 1, 2018, 1324. doi: 10.22075/macs.2018.12214.1117
Sayyad, A., Ghugal, Y. (2018). 'Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams', Mechanics of Advanced Composite Structures, 5(1), pp. 1324. doi: 10.22075/macs.2018.12214.1117
Sayyad, A., Ghugal, Y. Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams. Mechanics of Advanced Composite Structures, 2018; 5(1): 1324. doi: 10.22075/macs.2018.12214.1117
Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams
^{1}Department of Civil Engineering, SRES&#039;s College of Engineering, Savitribai Phule Pune University, Kopargaon,423601
^{2}Department of Applied Mechanics, Government College of Engineering, Karad415124, Maharashtra, India
Abstract
This study investigated bending, buckling, and free vibration responses of hyperbolic shear deformable functionally graded (FG) higher order beams. The material properties of FG beams are varied through thickness according to power law distribution; here, the FG beam was made of aluminium/alumina, and the hyperbolic shear deformation theory was used to evaluate the effect of shear deformation in the beam. The theory explains the hyperbolic cosine distribution of transverse shear stress through the thickness of a beam and satisfies zero traction boundary conditions on the top and bottom surfaces without requiring a shear correction factor. Hamilton’s principle was employed to derive the equations of motion, and analytical solutions for simply supported boundary conditions were obtained using Navier’s solution technique. The nondimensional displacements, stress, natural frequencies, and critical buckling loads of FG beams were obtained for various values of the power law exponent. The numerical results were compared to previously published results and found to be in excellent agreement with these.
Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams
A.S. Sayyad^{a}^{*}, Y.M. Ghugal^{b}
^{a }Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423601, Maharashtra, India
^{b }Department of Applied Mechanics, Government College of Engineering, Karad415124, Maharashtra, India
Paper INFO
ABSTRACT
Paper history:
Received 2017‐08‐10
Revised 2017‐12‐25
Accepted 2018‐01‐16
This study investigated bending, buckling, and free vibration responses of hyperbolic shear deformable functionally graded (FG) higher order beams. The material properties of FG beams are varied through thickness according to power law distribution; here, the FG beam was made of aluminium/alumina, and the hyperbolic shear deformation theory was used to evaluate the effect of shear deformation in the beam. The theory explains the hyperbolic cosine distribution of transverse shear stress through the thickness of a beam and satisfies zero traction boundary conditions on the top and bottom surfaces without requiring a shear correction factor. Hamilton’s principle was employed to derive the equations of motion, and analytical solutions for simply supported boundary conditions were obtained using Navier’s solution technique. The nondimensional displacements, stress, natural frequencies, and critical buckling loads of FG beams were obtained for various values of the power law exponent. The numerical results were compared to previously published results and found to be in excellent agreement with these.
Fibrous composite laminated beams are often subjected to delamination and stress concentration problems. This leads to the development of beams made of functionally graded materials (FGMs). A FGM is formed by varying the microstructure from one material to another in a specific gradient. The most commonly used FGMs are ceramic and metal for application in many engineering industries. Some typical practical applications for FGMs are given below.
1) Nuclear Projects: Fuel pellets and plasma walls of fusion reactors
2) Aerospace and Aeronautics: Stealth aircraft, rocket components, space plane frames, and space vehicles
3) Civil Engineering: Building materials, structural elements, and window glass
4) Defense: Armor plates and bulletproof vests
5) Manufacturing: Machine tools, forming and cutting tools, metal casting, and forging processes
6) Energy Sector: Thermoelectric generators, solar cells and sensors
Detailed descriptions of applications of FGMs in various fields were presented by Koizumi [1, 2], Muller et al. [3], Pompe et al. [4], and Schulz et al. [5]. Increased use of beams, plates, and shells made of FGMs has led to the development of various analytical and numerical models for predicting accurate static bending, elastic buckling, and free vibration responses in these beams. Few studies have addressed the development of elasticity solutions for the analysis of FG beams, but those that have include Sankar [6], Zhong and Yu [7], Daouadji et al. [8], Ding et al. [9], Huang et al. [10], Ying et al. [11], Chu et al. [12], and Xu et al. [13].
Twodimensional elasticity solutions are analytically difficult and computationally cumbersome. Therefore, various approximate beam theories have been developed to analyze FG beams. Recently, Sayyad and Ghugal [14, 15] presented a comprehensive literature survey on various analytical and numerical methods for the analysis of isotropic and anisotropic beams and plates using displacementbased shear deformation theories. Similarly, Carrera et al. [16] presented recent developments in refined beam theories and related applications.
Since the effect of shear deformation is more pronounced in thick beams made of advanced composite materials, such as FGMs, classical beam theory (CBT) [17, 18] and firstorder shear deformation theory [19] are not suitable for the analysis of thick FG beams. Therefore, higher order shear deformation theories are preferable for accurate analysis of FG thick beams. Various higher order shear deformation theories have been developed by various researchers for the analysis of isotropic and anisotropic beams, such as those by Reddy [20], Soldatos [21], Touratier [22], Karama et al. [23], Mantari et al. [24, 25], Neves et al. [26, 27], Sayyad and Ghugal [28, 29], Sayyad et al. [30, 31], Carrera et al. [32], Zenkour [33], Carrera and Ginuta [34], Carrera et al. [35], and Giunta et al. [36]. Such theories address bending, buckling, and free vibration analyses of beams using Carrera’s unified formulation.
Thai and Vo [37] obtained Naviertype analytical solutions for the bending and vibration of FG beams using various higher order shear deformation theories. Li and Batra [38] obtained the critical buckling load of FG beams in various boundary conditions using the firstorder shear deformation theory and CBT. Simsek [39] presented free vibration analysis of FG beams with various boundary conditions based on higher order shear deformation theories. Nguyen et al. [40] presented a Naviertype closed form solution for the static deformation and free vibration of FG beams using the firstorder shear deformation theory. Hadji et al. [41, 42] developed new firstorder and higher order shear deformation models for static and free vibration analysis of simply supported FG beams. Bourada et al. [43] presented a trigonometric shear and normal deformation theory that considers the effects of transverse shear and normal deformations for the analysis of FG higher order beams. The theory included three unknowns, one of which was the effect of the transverse normal. Vo et al. [44, 45] presented static bending and free vibration analysis based on higher order shear deformation theories using the finite element method. Recently, Sayyad and Ghugal [46] developed a unified shear deformation theory for the bending of FG beams and plates. Hebali et al. [47] developed five variable quasithreedimensional hyperbolic shear deformation theories for static and free vibration behavior of FG plates. Bennoun et al. [48] also developed five new variable shear and normal deformation plate theories for free vibration analysis of FG sandwich plates. Beldjelili et al. [49] investigated hygrothermomechanical bending behavior of sigmoid FG plates resting on elastic foundations using four variable trigonometric shear deformation theories. Bouderba et al. [50] developed a simple firstorder shear deformation theory for thermal buckling responses of FG sandwich plates to various boundary conditions. Bousahla et al. [51] also developed a fourvariable refined plate theory for buckling analysis of FG plates subjected to uniform, linear, and nonlinear temperature increases for various thicknesses. Boukhari et al. [52] developed a fourvariable refined plate theory for wave propagation analysis of an infinite FG plate in thermal environments. Rahmani and Pedram [53] applied Timoshenko beam theory for free vibration analysis of FG nanobeams. Akgoz and Civalek [54] studied the static bending response of singlewalled carbon nanotubes embedded in an elastic medium using higherorder shear deformation microbeams and a modified strain gradient theory. Ebrahimi and Barati [55] obtained a Naviertype solution for free vibration characteristics of FG nanobeams based on the thirdorder shear deformation beam theory.
In this paper, bending, buckling, and free vibration responses of hyperbolic shear deformable FG higher order beams were studied using the hyperbolic shear deformation theory of Soldatos [21]. Soldatos suggested using the hyperbolic function in the modeling and analysis of composite beams and plates in 1992, recommending that the hyperbolic function yields more accurate predictions of stress, frequencies, and the buckling loads of composite beams and plates. Since then, many researchers have used this function for the analysis of isotropic, laminated, and sandwich beams and plates. However, most research has not focused on the application of this function to evaluate the response of FG beams. Instead, researchers have applied hyperbolic shear deformation theory to bending, buckling, and free vibration analysis of FG beams.
The material properties of FG beams are varied through the thickness of the beam according to power law distribution. Here, the FG beam was made of aluminum (Al)/alumina (Al_{2}O_{3)}, and the hyperbolic cosine distribution of transverse shear stress through the thickness of the beam satisfied the zero traction boundary conditions on the top and bottom surfaces without using the shear correction factor. The variationally consistent governing differential equations and boundary conditions of the theory were obtained using Hamilton’s principle, and an analytical solution for simply supported boundary conditions was obtained using Navier’s solution. The nondimensional displacements, stress, natural frequencies, and critical buckling loads of FG beams were obtained for various values of the power law exponent. The numerical results were then compared to previously published results and were in excellent agreement with these.
2. Variational formulation
2.1 Kinematics
Consider a FG beam with length L, width b, and thickness h made of Al/Al_{2}O_{3} as shown in Figure 1. The bottom surface of the FG beam was ceramicrich and top surface was metalrich. The beam occupied the region 0≤ x ≤ L; b/2≤ y ≤ b/2; h/2≤ z ≤ h/2 in the Cartesian coordinate systems. The xaxis was coincident with the beam neutral axis. The zaxis was assumed to be downward positive, and the beam was assumed to be deformed in the xz plane only.
The mathematical formulation of the FG beam was based on the following kinematical assumptions.
1) The axial displacement u consists of the extension, bending, and shear components as
, (1)
where
. (2)
2) There is no relative motion in the ydirection at any point in the cross section of the beam.
3) Transverse displacement is assumed to be a function of the xcoordinate only.
(3)
4) The theory applies to the hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam.
5) The axial displacement, u, is such that the resultant axial stress , acting over the crosssection, should result only in a bending moment and should not result in force in the xdirection.
6) Displacements are small, compared to beam thickness.
7) Onedimensional constitutive law is used to obtain stress values
Based on these assumptions, the displacement field of the hyperbolic shear deformation theory is given by:
, (4)
where u_{0} is the axial displacement of a point on the neutral axis of the beam, w_{0} is the transverse displacement of a point on the neutral axis of the beam, and the hyperbolic function is assumed according to the transverse shearing strain distribution across the thickness of the beam (see Figures 2 and 3). The nonzero normal and shear strains at any point of the beam are
(5)
Figure 1. FG beam under bending conditions in the xz plane
Figure 2. Through thickness distribution of the transverse shearing strain function
Figure 3. Through thickness distribution of the derivative of transverse shearing strain function
2.2 Constitutive relations
The FG beam was made of Al/Al_{2}O_{3}, and the properties of the material varied continuously throughout beam thickness, according to the power law distribution given by Equation (6).
(6)
where E represents the Young’s modulus, G represents the shear modulus, represents the Poisson’s ratio, andrepresents mass density. Subscripts m and c represent the metallic and ceramic constituents, respectively, and p is the power law exponent. The variation of the Young’s modulusE(z) through the thickness z/h of the beam for various values of the power law exponent is shown in Figure 4. The stress–strain relationship at any point of the beam is given by onedimensional Hooke’s law as follows.
(7)
3. Equations of motion
Equations of motion of hyperbolic shear deformable FG beam are derived using Hamilton’s principle,
, (8)
where denotes variations in total strain energy, potential energy, and kinetic energy respectively, and t_{1} and t_{2} are the lower and upper limits of desired time period, respectively.
Figure 4. Variation in Young’s modulus E(z) through the thickness of the FG beam for various values of the power law exponent (p)
The variation of the strain energy can be stated as:
, (9)
where are the axial force, bending moment, higher order moment, and shear force resultants, respectively. Additionally,
, (10)
where
. (11)
The variation of the potential energy due to transverse and axial loads can be written as
. (12)
The variation of kinetic energy can be written in following form
, (13)
where is the mass density, and are the inertia coefficients.
(14)
Substituting Equations (9), (12), and (13) into Equation (8), doing the integrations and setting the coefficients of , , and to equal zero, the following equations of motion are obtained.
(15)
By substituting the stress resultants from Equation (10) into Equation (15), the following equations of motion can be obtained for unknown displacement variables.
(16)
4. Analytical solution
Consider a simply supported FG beam with length ‘L’ and rectangular crosssection ‘b×h’. For simply supported boundary conditions, according to Navier’s solution, the unknown displacement variables are expanded in a Fourier series as given below:
(17)
where , , and are the unknown coefficients, and is the natural frequency. The uniform transverse load (q) acting on the top surface of the beam was also expanded in the Fourier series as
, (18)
where q_{0} is the maximum intensity of the load at the center of the beam. By substituting Equations (17) and (18) into Equation (16), the analytical solution can be obtained from the following equations.
For bending, ignore time derivatives and axial force. . (19)
For buckling, ignore time derivatives and transverse load.
, (20)
For free vibrations, ignore transverse load and axial force.
. (21)
5. Numerical results and discussion
In this section, the accuracy of hyperbolic shear deformation theory for predicting bending, buckling, and vibration responses of FG higher order beams was investigated. The numerical results were obtained using Navier’s solution for simply supported boundary conditions. The beam was made of Al_{2}O_{3} for ceramic ( = 380 GPa, =3960 kg/m^{3}, = 0.3) and Al for metal ( = 70 GPa, =2702 kg/m^{3}, = 0.3). The material properties of the beam were varied across beam thickness according to power law distribution. The bottom surface of the FG beam was ceramicrich, and the top surface was metalrich.
5.1 Bending the FG beam
The bending response of the FG beam under a uniform transverse load was investigated. The displacements and stress are presented in the following nondimensional form.
Axial displacement (u) at x = 0 and z = h/2: .
Transverse displacement (w) at x = L/2 and z = 0: .
Axial stress ( ) at x = L/2 and z = h/2:
.
Transverse shear stress ( ) at x = 0 and z = 0: .
The numerical results obtained using the present theory were compared to those of other theories, which is shown in Table 1. Comparisons of numerical results are presented in Table 2. Through thickness distribution of displacements and stress are shown in Figure 5 (a–c). The displacements and stress are presented for various values of the power law exponent (p). The transverse shear stress was evaluated directly from constitutive relations. The present results were compared to higher order shear deformation theories of Reddy [20], Touratier [22], and Hadji et al. [42] and the CBT. The present results were in good agreement with those obtained using various shear deformation theories for all values of the power law exponent. Because the effect of transverse shear deformation is not included in the CBT, this theory underestimated displacement and stress. The stress presented by Hadji et al. [42] was higher compared to that obtained using shear deformation theories. The present theory gives a linear variation of axial stress through the thickness for p = 0 and p = ∞; however, for other values of the power law exponent, this is nonlinear through the thickness (see Figure 5b). Displacements and stress are increased as the power law exponent increases, creating more flexibility in FG beams.
Table 1. Displacement fields of the present and referred theories
Reference
Displacement field
Present
Hadji et al. [42]
Reddy [20]
Touratier [22]
TBT
CBT
Figure 5(b) shows that an increase in the power law exponent increased the compression zone in the beam, while Figure 5(c) shows the hyperbolic cosine variation of transverse shear stress that was across the thickness of the beam and that satisfied the traction free conditions at the top and bottom surfaces of the beam. Figure 5(c) also shows an increase in the power law exponent neutral axis that shifted toward the bottom. This was due to ceramic, with which metal has a low elastic modulus.
5.2 Buckling an FG beam
In this section, the buckling response of an FG beam subjected to axial force (N_{0}) was investigated. A nondimensional critical buckling load is presented in Table 3. The nondimensional form of the buckling load was as follows:
.
The critical buckling load was obtained for various values regarding the power law exponent (p) and a lengthtothickness ratio (L/h). Results were compared with those presented by Li and Batra [38], Nguyen et al. [40], and Vo et al. [45]. Table 3 reveals that this study's results agreed with those available in the literature. Specifically, the critical buckling load was higher for a thin, slender beam and lower for a thick beam. However, the critical buckling load was in a nondimensional form; nondimensional quantities are reciprocal of dimensional quantities. According to Euler’s buckling theory, critical buckling loads are directly proportional to crosssections of beams (i.e., moments of inertia). Therefore, it can be noted that the dimensional critical buckling load for the slender beam was actually smaller than the load for the thicker beam.
Table 2. A comparison of the nondimensional displacements and stress of the FG beams subjected to uniform loads with various power law exponent values
L/h = 5
L/h = 20
p
Theory
0
Present
0.9274
3.1224
3.7529
0.7259
0.2275
2.8585
14.8179
0.7259
Hadji et al. [42]
0.9233
3.1673
3.9129
0.7883
0.2290
2.8807
15.4891
0.7890
Reddy [20]
0.9397
3.1654
3.8019
0.7330
0.2306
2.8962
15.0129
0.7437
Touratier [22]
0.9409
3.1649
3.8053
0.7549
0.2306
2.8962
15.0138
0.7686
CBT
0.9211
2.8783
3.7500

0.2303
2.8783
15.0000

1
Present
2.2735
6.2586
5.8077
0.7187
0.5611
5.7292
22.9038
0.7259
Hadji et al. [42]
2.2115
6.1805
6.0709
0.7883
0.5498
5.6965
24.0095
0.7890
Reddy [20]
2.3036
6.2594
5.8836
0.7330
0.5686
5.5685
23.2051
0.7432
Touratier [22]
2.3058
6.2586
5.8892
0.7549
0.5686
5.8049
23.2067
0.7686
CBT
2.2722
5.7746
5.7959

0.5680
5.7746
23.1834

2
Present
3.0720
7.9627
6.7938
0.6573
0.7591
7.3450
26.7470
0.6648
Hadji et al. [42]
2.9629
7.9106
7.0925
0.7274
0.7366
7.2458
27.9844
0.728
Reddy [20]
3.1127
8.0677
6.8824
0.6704
0.7691
7.4421
27.0989
0.6812
Touratier [22]
3.1153
8.0683
6.8901
0.6933
0.7692
7.4421
27.1010
0.7069
CBT
3.0740
7.4003
6.7676

0.7685
7.4003
27.0704

5
Present
3.6612
9.6986
8.0059
0.5786
0.9014
8.7031
31.3997
0.5863
Hadji et al. [42]
3.5429
9.6933
8.3581
0.6523
0.8775
8.6182
32.8183
0.6540
Reddy [20]
3.7097
9.8281
8.1104
0.5904
0.9134
8.8182
31.8127
0.6013
Touratier [22]
3.7140
9.8367
8.1222
0.6155
0.9134
8.8188
31.8159
0.6292
CBT
3.6496
8.7508
7.9428

0.9124
8.7508
31.7711

10
Present
3.8351
10.7949
9.5870
0.6412
0.9412
9.5641
37.6432
0.6426
Hadji et al. [42]
3.7462
10.8680
9.9878
0.7064
0.9262
9.5513
39.2717
0.7091
Reddy [20]
3.8859
10.9381
9.7119
0.6465
0.9536
9.6905
38.1382
0.6586
Touratier [22]
3.8913
10.9420
9.7238
0.6708
0.9537
9.6908
38.1414
0.6858
CBT
3.8097
9.6072
9.5228

0.9524
9.6072
38.0913

5.3 The free vibrations of FG beams
The free vibration responses of FG beams were investigated. Fundamental frequencies were obtained for various power law exponent values and L/h ratios. The results were compared to those presented by Reddy [20], Simsek [39], Thai and Vo [37], Vo et al. [45], and Timoshenko [19] and those obtained with the CBT. Fundamental frequencies were presented in the following nondimensional form:
.
Table 4 shows the nondimensional fundamental frequencies ( ) of simply supported FG beams. The natural frequencies of first three bending modes are presented. Table 4 reveals that the fundamental frequencies obtained using the theory presented in this research were in excellent agreement with those obtained by other researchers. The numerical results showed that all shear deformation theories predicted more or less the same frequencies, whereas the CBT overestimated all frequencies due to a neglect of shear deformation. The effects of a power law exponent, p, on the frequencies of FG beams are shown in Figure 6(b). It was observed that increases in power law exponent values led to reductions of fundamental frequencies. This was because the increases in power law exponent values resulted in decreases in elasticity modulus values. It should be noted that the fundamental frequencies were higher when there were higher modes of vibration.
Figure 5. Through thickness distribution of the nondimensional (a) axial displacement ( ), (b) the axial stress ( ), and (c) transverse shear stress ( ) simply supported the FG beam under a uniform load throughout various power law exponent values (L/h = 5)
Figure 6. The variations in nondimensional (a) critical buckling loads and (b) natural frequencies with respect to the power law exponents of simply supported FG beams.
Table 3. A comparison of the nondimensional critical buckling loads ( ) of the FG beams subjected to axial forces in regards to various power law exponent values
L/h
Theory
p
0
1
2
5
10
5
Present
48.596
24.584
19.071
15.645
14.052
Li and Batra [38]
48.835
24.687
19.245
16.024
14.427
Nguyen et al. [40]
48.835
24.687
19.245
16.024
14.427
Vo et al. [45]
48.837
24.689
19.247
16.026
14.428
Vo et al. [45]
48.840
24.691
19.160
16.740
14.146
10
Present
52.238
26.141
20.366
17.082
15.500
Li and Batra [38]
52.309
26.171
20.416
17.192
15.612
Nguyen et al. [40]
52.309
26.171
20.416
17.194
15.612
Vo et al. [45]
52.308
26.172
20.418
17.195
15.613
Vo et al. [45]
52.308
26.172
20.393
17.111
15.529
Table 4. A comparison of the first three nondimensional fundamental frequencies of the FG beams in regards to various power law exponent values
p
L/h
Mode
Theory
0
1
2
5
10
5
1
Present
5.1527
3.9904
3.6264
3.4014
3.2816
Reddy [20]
5.1527
3.9904
3.6264
3.4012
3.2816
Simsek [39]
5.1527
3.9904
3.6264
3.4012
3.2816
Thai and Vo [37]
5.1527
3.9904
3.6264
3.4012
3.2816
Vo et al. [45]
5.1527
3.9716
3.5979
3.3742
3.2653
Timoshenko [19]
5.1524
3.9902
3.6343
3.4311
3.3134
CBT
5.3953
4.1484
3.7793
3.5949
3.4921
2
Present
17.881
14.010
12.640
11.544
11.024
Thai and Vo [37]
17.881
14.009
12.640
11.544
11.024
CBT
20.618
15.798
14.326
13.587
13.237
3
Present
34.202
27.098
24.316
21.720
20.556
Thai and Vo [37]
34.208
27.097
24.315
21.718
20.556
CBT
43.348
33.027
29.745
28.085
27.475
20
1
Present
5.4603
4.2050
3.8361
3.6485
3.5390
Reddy [20]
5.4603
4.2050
3.8361
3.6485
3.5389
Simsek [39]
5.4603
4.2050
3.8361
3.6485
3.5389
Thai and Vo [37]
5.4603
4.2050
3.8361
3.6484
3.5389
Vo et al. [45]
5.4603
4.2038
3.8342
3.6466
3.5378
Timoshenko [19]
5.4603
4.2050
3.8367
3.6508
3.5415
CBT
5.4777
4.2163
3.8472
3.6628
3.5547
2
Present
21.573
16.634
15.161
14.374
13.926
Thai and Vo [37]
21.573
16.634
15.161
14.374
13.926
CBT
21.843
16.810
15.333
14.595
14.167
3
Present
47.593
36.768
33.469
31.579
30.095
Thai and Vo [37]
47.593
36.767
33.469
31.5789
30.537
CBT
48.899
37.617
34.295
32.6357
31.688
6. Conclusions
A hyperbolic shear deformation theory developed by Soldatos [21] was extended in this paper to conduct bending, buckling, and free vibration analyses of FG beams. With the theory, hyperbolic cosine variations of transverse shear stress were found at the top and bottom surfaces of the beams. Subsequently, Hamilton’s principle was employed to derive equations of motion. The equations of motion, with the theory, were variationally consistent and allowed the avoidance of a shear correction factor. Then, an analytical solution for a simply supported boundary condition was obtained using Navier’s solution procedure.
The numerical results were compared to those obtained by other researchers to determine the accuracy of the theory. Based on the comparisons and a discussion, it was concluded that the displacements, stress, critical buckling loads, and natural frequencies obtained using the theory were accurate and in agreement with those obtained using other refined shear deformation theories. It was seen that varying material properties had significant effects on the dimensionless stress, frequencies, and buckling loads of the FG beams. Increasing power law exponent values reduced the stiffnesses of the FG beams and consequently led to increases in displacements and reductions of frequencies and buckling loads. Overall, the investigation of the bending, buckling, and free vibration responses of the FG beams confirmed the effects and credibility of the hyperbolic shear deformation theory.
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[2] Koizumi M. FGM activities in Japan. Composites Part B,1997; 28: 1–4.
[3] Muller E, Drasar C, Schilz J, Kaysser WA. Functionally graded materials for sensor and energy applications. Mater. Sci. Eng. A,2003; 362: 17–39.
[4] Pompe W, Worch H, Epple M, Friess W, Gelinsky M, Greil P, Hempele U, Scharnweber D, Schulte K. Functionally graded materials for biomedical applications. Mater. Sci. Eng.A,2003; 362: 40–60.
[5] Schulz U, Peters M, Bach FW, Tegeder G. Graded coatings for thermal, wear and corrosion barriers. Mater. Sci. Eng.A, 2003; 362: 61–80.
[6] Sankar BV. An elasticity solution for functionally graded beams. Compos Sci Technol., 2001; 61(5): 689–696.
[7] Zhong Z, Yu, T. Analytical solution of a cantilever functionally graded beam. Compos Sci Technol., 2007; 67:481–488.
[8] Daouadji TH, Henni AH, Tounsi A, Bedia EAA. Elasticity solution of a cantilever functionally graded beam. Appl Compos Mater, 2013; 20:1–15.
[9] Ding JH, Huang DJ, Chen WQ. Elasticity solutions for plane anisotropic functionally graded beams. Int J of Solids and Structures, 2007; 44(1): 176–196.
[10] Huang DJ, Ding JH, Chen WQ. Analytical solution and semianalytical solution for anisotropic functionally graded beam subject to arbitrary loading. Science in China SeriesG, 2009; 52(8): 12441256.
[11] Ying J, Lu CF, Chen WQ. Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 2008; 84:209–219.
[12] Chu P, Li XF, Wu JX, Lee KY. Twodimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending. Acta Mechanica, 2015; 226: 2235–2253.
[13] Xu Y, Yu T, Zhou D. Twodimensional elasticity solution for bending of functionally graded beams with variable thickness. Meccanica, 2014;49: 2479–2489.
[14] Sayyad AS, Ghugal YM. On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Composite Structures, 2015; 129: 177–201.
[15] Sayyad AS, Ghugal YM. Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Composite Structures, 2017; 171: 486–504.
[16] Carrera E, Pagani A, Petrolo M, Zappino E. Recent developments on refined theories for beams with applications. Mechanical Engineering Reviews, 2015; 2(2): 130.
[18] Euler L. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne and Geneva, 1744.
[19] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 1921; 41(6): 742746.
[20] Reddy JN. A simple higher order theory for laminated composite plates. J of Applied Mechanics, 1984; 51: 745–752.
[21] Soldatos KP. A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica, 1992; 94: 195–200.
[22] Touratier M. An efficient standard plate theory. Int J Engineering Science, 1991; 29:901–916.
[23] Karama M, Afaq KS, Mistou S. Mechanical behavior of laminated composite beam by new multilayered laminated composite structures model with transverse shear stress continuity. Int J Solids and Structures, 2003; 40: 1525–1546.
[24] Mantari JL, Oktem AS, Soares CG. A new higher order shear deformation theory for sandwich and composite laminated plates. Composites Part B, 2012; 43: 1489–1499.
[25] Mantari JL, Oktem AS, Soares CG. A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids and Structures, 2012; 49: 43–53.
[26] Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, Soares CMM. A quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Composite Structures, 2012; 94: 1814–1825.
[27] Neves AMA, Ferreira AJM, Carrera E. A quasi3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Composites Part B, 2012; 43: 711–725.
[28] Sayyad AS, Ghugal YM. Effect of transverse shear and transverse normal strain on the bending analysis of crossply laminated beams. Int J of Applied Mathematics and Mechanics, 2011; 7: 85–118.
[29] Sayyad AS, Ghugal YM. Flexure of thick beams using new hyperbolic shear deformation theory. Int J of Mechanics, 2011;5: 113–122.
[30] Sayyad AS, Ghugal YM, Naik NS. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved and Layered Structures, 2015; 2: 279–289.
[31] Sayyad AS, Ghugal YM, Shinde PN. Stress analysis of laminated composite and soft core sandwich beams using a simple higher order shear deformation theory. J Serbian Society of Compututational Mechanics, 2015; 9: 15–35.
[32] Carrera E, Giunta G, Petrolo M. Beam structures, John Wiley & Sons, 2011.
[33] Zenkour AM. Maupertuis  Lagrange mixed variational formula for laminated composite structure with a refined higher order beam theory. Int J NonLinear Mechanics, 1997; 32: 9891001.
[34] Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Applied Mechanics, 2010; 2: 117143.
[35] Carrera E, Giunta G, Nali P, Petrolo M. Refined beam elements with arbitrary crosssection geometries. Computers and Structures, 2010; 88: 283–93.
[36] Giunta G, Belouettar S, Carrera E. Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Materials Science and Engineering,2010; 10: 110.
[37] Thai HT, Vo TP. Bending and free vibration of functionally graded beams using various higherorder shear deformation beam theories. Int J of Mechanical Sciences, 2012; 62: 57–66.
[38] Li SR, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Composite Structures, 2013; 95: 5–9.
[39] Simsek M. Fundamental frequency analysis of functionally graded beams by using different higherorder beam theories. Nuclear Engineering and Design, 2010; 240: 697–705.
[40] Nguyen TK, Vo TP, Thai HT. Static and free vibration of axially loaded functionally graded beams based on the firstorder shear deformation theory. Composites Part B, 2013; 55: 147–157.
[41] Hadji L, Daouadji TH, Meziane MAA, Tlidji Y, Bedia EAA. Analysis of functionally graded beam using a new firstorder shear deformation theory. Structural Engineering and Mechanics, 2016; 57: 315325.
[42] Hadji L, Khelifa Z, Bedia EAA. A new higher order shear deformation model for functionally graded beams. KSCE J of Civil Engineering, 2016; 20(5): 18351841.
[43] Bourada M, Kaci A, Houari MSA, Tounsi A. A new simple shear and normal deformations theory for functionally graded beams. Steel and Composite Structures, 2015; 18(2): 409423.
[44] Vo TP, Thai HT, Nguyen TK, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica, 2014;49:155–168.
[45] Vo TP, Thai HT, Nguyen TK, Maheri A, Lee J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Engineering Structures, 2014;64:12–22.
[46] Sayyad AS, Ghugal YM. A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. Int J Applied Mechanics, 2017; 9: 136.
[47] Hebali H, Houari MSA, Tounsi A, Bessaim A, Bedia EEA. A new quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. ASCE Journal of Engineering Mechanics, 2014;140:374 – 383.
[48] Bennoun M, Houari MSA, Tounsi A. A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mechanics of Advanced Materials and Structures, 2016;23(4):423 – 431.
[49] Beldjelili Y, Tounsi A, Hassanet S. Hygrothermomechanical bending of SFGM plates resting on variable elastic foundations using a fourvariable trigonometric plate theory Smart Structures and Systems, 2016; 18(4): 755786.
[50] Bouderba B, Houari MSA, Tounsi A, Mahmoud SR. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Structural Engineering and Mechanics, 2016;58(3):397422.
[51] Bousahla AA, Benyoucef S, Tounsi A, Mahmoud SR. On thermal stability of plates with functionally graded coefficient of thermal expansion. Structural Engineering and Mechanics, 2016;60(2):313335.
[52] Boukhari A, Atmane HA, Tounsi A, Bedia EEA, Mahmoud SR. An efficient shear deformation theory for wave propagation of functionally graded material plates. Structural Engineering and Mechanics, 2016;57(5):837859.
[53] Rahmani O, Pedram O. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 2014; 77: 5570.
[54] Akgoz B, Civalek O. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory, Acta Astronautica, 2016;119:1–12.
[55] Ebrahimi F, Barati MR. A nonlocal higherorder shear deformation beam theory for vibration analysis of sizedependent functionally graded nanobeams. Arabian Journal for Science and Engineering, 2016; 41(5): 16791690
References
[1] Koizumi M. The concept of FGM, Functionally Gradient Material, 1993; 34:3–10.
[2] Koizumi M. FGM activities in Japan. Composites Part B,1997; 28: 1–4.
[3] Muller E, Drasar C, Schilz J, Kaysser WA. Functionally graded materials for sensor and energy applications. Mater. Sci. Eng. A,2003; 362: 17–39.
[4] Pompe W, Worch H, Epple M, Friess W, Gelinsky M, Greil P, Hempele U, Scharnweber D, Schulte K. Functionally graded materials for biomedical applications. Mater. Sci. Eng.A,2003; 362: 40–60.
[5] Schulz U, Peters M, Bach FW, Tegeder G. Graded coatings for thermal, wear and corrosion barriers. Mater. Sci. Eng.A, 2003; 362: 61–80.
[6] Sankar BV. An elasticity solution for functionally graded beams. Compos Sci Technol., 2001; 61(5): 689–696.
[7] Zhong Z, Yu, T. Analytical solution of a cantilever functionally graded beam. Compos Sci Technol., 2007; 67:481–488.
[8] Daouadji TH, Henni AH, Tounsi A, Bedia EAA. Elasticity solution of a cantilever functionally graded beam. Appl Compos Mater, 2013; 20:1–15.
[9] Ding JH, Huang DJ, Chen WQ. Elasticity solutions for plane anisotropic functionally graded beams. Int J of Solids and Structures, 2007; 44(1): 176–196.
[10] Huang DJ, Ding JH, Chen WQ. Analytical solution and semianalytical solution for anisotropic functionally graded beam subject to arbitrary loading. Science in China SeriesG, 2009; 52(8): 12441256.
[11] Ying J, Lu CF, Chen WQ. Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 2008; 84:209–219.
[12] Chu P, Li XF, Wu JX, Lee KY. Twodimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending. Acta Mechanica, 2015; 226: 2235–2253.
[13] Xu Y, Yu T, Zhou D. Twodimensional elasticity solution for bending of functionally graded beams with variable thickness. Meccanica, 2014;49: 2479–2489.
[14] Sayyad AS, Ghugal YM. On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Composite Structures, 2015; 129: 177–201.
[15] Sayyad AS, Ghugal YM. Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Composite Structures, 2017; 171: 486–504.
[16] Carrera E, Pagani A, Petrolo M, Zappino E. Recent developments on refined theories for beams with applications. Mechanical Engineering Reviews, 2015; 2(2): 130.
[18] Euler L. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne and Geneva, 1744.
[19] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 1921; 41(6): 742746.
[20] Reddy JN. A simple higher order theory for laminated composite plates. J of Applied Mechanics, 1984; 51: 745–752.
[21] Soldatos KP. A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica, 1992; 94: 195–200.
[22] Touratier M. An efficient standard plate theory. Int J Engineering Science, 1991; 29:901–916.
[23] Karama M, Afaq KS, Mistou S. Mechanical behavior of laminated composite beam by new multilayered laminated composite structures model with transverse shear stress continuity. Int J Solids and Structures, 2003; 40: 1525–1546.
[24] Mantari JL, Oktem AS, Soares CG. A new higher order shear deformation theory for sandwich and composite laminated plates. Composites Part B, 2012; 43: 1489–1499.
[25] Mantari JL, Oktem AS, Soares CG. A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids and Structures, 2012; 49: 43–53.
[26] Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, Soares CMM. A quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Composite Structures, 2012; 94: 1814–1825.
[27] Neves AMA, Ferreira AJM, Carrera E. A quasi3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Composites Part B, 2012; 43: 711–725.
[28] Sayyad AS, Ghugal YM. Effect of transverse shear and transverse normal strain on the bending analysis of crossply laminated beams. Int J of Applied Mathematics and Mechanics, 2011; 7: 85–118.
[29] Sayyad AS, Ghugal YM. Flexure of thick beams using new hyperbolic shear deformation theory. Int J of Mechanics, 2011;5: 113–122.
[30] Sayyad AS, Ghugal YM, Naik NS. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved and Layered Structures, 2015; 2: 279–289.
[31] Sayyad AS, Ghugal YM, Shinde PN. Stress analysis of laminated composite and soft core sandwich beams using a simple higher order shear deformation theory. J Serbian Society of Compututational Mechanics, 2015; 9: 15–35.
[32] Carrera E, Giunta G, Petrolo M. Beam structures, John Wiley & Sons, 2011.
[33] Zenkour AM. Maupertuis  Lagrange mixed variational formula for laminated composite structure with a refined higher order beam theory. Int J NonLinear Mechanics, 1997; 32: 9891001.
[34] Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Applied Mechanics, 2010; 2: 117143.
[35] Carrera E, Giunta G, Nali P, Petrolo M. Refined beam elements with arbitrary crosssection geometries. Computers and Structures, 2010; 88: 283–93.
[36] Giunta G, Belouettar S, Carrera E. Analysis of FGM beams by means of a unified formulation. IOP Conference Series: Materials Science and Engineering,2010; 10: 110.
[37] Thai HT, Vo TP. Bending and free vibration of functionally graded beams using various higherorder shear deformation beam theories. Int J of Mechanical Sciences, 2012; 62: 57–66.
[38] Li SR, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Composite Structures, 2013; 95: 5–9.
[39] Simsek M. Fundamental frequency analysis of functionally graded beams by using different higherorder beam theories. Nuclear Engineering and Design, 2010; 240: 697–705.
[40] Nguyen TK, Vo TP, Thai HT. Static and free vibration of axially loaded functionally graded beams based on the firstorder shear deformation theory. Composites Part B, 2013; 55: 147–157.
[41] Hadji L, Daouadji TH, Meziane MAA, Tlidji Y, Bedia EAA. Analysis of functionally graded beam using a new firstorder shear deformation theory. Structural Engineering and Mechanics, 2016; 57: 315325.
[42] Hadji L, Khelifa Z, Bedia EAA. A new higher order shear deformation model for functionally graded beams. KSCE J of Civil Engineering, 2016; 20(5): 18351841.
[43] Bourada M, Kaci A, Houari MSA, Tounsi A. A new simple shear and normal deformations theory for functionally graded beams. Steel and Composite Structures, 2015; 18(2): 409423.
[44] Vo TP, Thai HT, Nguyen TK, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica, 2014;49:155–168.
[45] Vo TP, Thai HT, Nguyen TK, Maheri A, Lee J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Engineering Structures, 2014;64:12–22.
[46] Sayyad AS, Ghugal YM. A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. Int J Applied Mechanics, 2017; 9: 136.
[47] Hebali H, Houari MSA, Tounsi A, Bessaim A, Bedia EEA. A new quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. ASCE Journal of Engineering Mechanics, 2014;140:374 – 383.
[48] Bennoun M, Houari MSA, Tounsi A. A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mechanics of Advanced Materials and Structures, 2016;23(4):423 – 431.
[49] Beldjelili Y, Tounsi A, Hassanet S. Hygrothermomechanical bending of SFGM plates resting on variable elastic foundations using a fourvariable trigonometric plate theory Smart Structures and Systems, 2016; 18(4): 755786.
[50] Bouderba B, Houari MSA, Tounsi A, Mahmoud SR. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Structural Engineering and Mechanics, 2016;58(3):397422.
[51] Bousahla AA, Benyoucef S, Tounsi A, Mahmoud SR. On thermal stability of plates with functionally graded coefficient of thermal expansion. Structural Engineering and Mechanics, 2016;60(2):313335.
[52] Boukhari A, Atmane HA, Tounsi A, Bedia EEA, Mahmoud SR. An efficient shear deformation theory for wave propagation of functionally graded material plates. Structural Engineering and Mechanics, 2016;57(5):837859.
[53] Rahmani O, Pedram O. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 2014; 77: 5570.
[54] Akgoz B, Civalek O. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory, Acta Astronautica, 2016;119:1–12.
[55] Ebrahimi F, Barati MR. A nonlocal higherorder shear deformation beam theory for vibration analysis of sizedependent functionally graded nanobeams. Arabian Journal for Science and Engineering, 2016; 41(5): 16791690