Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams

Document Type: Research Paper

Authors

1 Department of Civil Engineering, SRES's College of Engineering, Savitribai Phule Pune University, Kopargaon,-423601

2 Department of Applied Mechanics, Government College of Engineering, Karad-415124, 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 non-dimensional 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.

Keywords



 

 

Mechanics of Advanced Composite Structures 5 (2018) 13–24

 

 

 

 

 

 

Semnan University

Mechanics of Advanced Composite Structures

Journal homepage: http://MACS.journals.semnan.ac.ir

 

Bending, buckling and free vibration responses of hyperbolic shear deformable FGM beams

A.S. Sayyada*, Y.M. Ghugalb

a Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423601, Maharashtra, India

b Department of Applied Mechanics, Government College of Engineering, Karad-415124, 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 non-dimensional 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.

 

Keywords:

Hyperbolic shear deformation theory

FGM beam

Bending

Buckling

Vibration

 

© 2018 Published by Semnan University Press. All rights reserved.

 

 

  1. Introduction   

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 bullet-proof 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].

Two-dimensional 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 displacement-based 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 first-order 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 Navier-type 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 first-order 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 Navier-type closed form solution for the static deformation and free vibration of FG beams using the first-order shear deformation theory. Hadji et al. [41, 42] developed new first-order 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 quasi-three-dimensional 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 hygro-thermo-mechanical bending behavior of sigmoid FG plates resting on elastic foundations using four variable trigonometric shear deformation theories. Bouderba et al. [50] developed a simple first-order shear deformation theory for thermal buckling responses of FG sandwich plates to various boundary conditions. Bousahla et al. [51] also developed a four-variable refined plate theory for buckling analysis of FG plates subjected to uniform, linear, and non-linear temperature increases for various thicknesses. Boukhari et al. [52] developed a four-variable 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 single-walled carbon nanotubes embedded in an elastic medium using higher-order shear deformation microbeams and a modified strain gradient theory. Ebrahimi and Barati [55] obtained a Navier-type solution for free vibration characteristics of FG nanobeams based on the third-order 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 (Al2O3), 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 non-dimensional 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.

  1. 2.      Variational formulation

2.1 Kinematics

Consider a FG beam with length L, width b, and thickness h made of Al/Al2O3 as shown in Figure 1. The bottom surface of the FG beam was ceramic-rich and top surface was metal-rich. The beam occupied the region 0≤ x L; -b/2≤ y b/2; -h/2≤ z h/2 in the Cartesian coordinate systems. The x-axis was coincident with the beam neutral axis. The z-axis was assumed to be downward positive, and the beam was assumed to be deformed in the x-z 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 y-direction at any point in the cross section of the beam.

3)      Transverse displacement is assumed to be a function of the x-coordinate 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 cross-section, should result only in a bending moment and should not result in force in the x-direction.

6)      Displacements are small, compared to beam thickness.

7)      One-dimensional 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 u0 is the axial displacement of a point on the neutral axis of the beam, w0 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 x-z 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/Al2O3, 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 modulus E(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 one-dimensional Hooke’s law as follows.

                              (7)

  1. 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 t1 and t2 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)

  1. 4.      Analytical solution

Consider a simply supported FG beam with length ‘L’ and rectangular cross-section ‘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 q0 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)

  1. 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 Al2O3 for ceramic ( = 380 GPa, =3960 kg/m3, = 0.3) and Al for metal ( = 70 GPa, =2702 kg/m3, = 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 ceramic-rich, and the top surface was metal-rich.

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 non-dimensional 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 non-linear 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 (N0) was investigated. A non-dimensional critical buckling load is presented in Table 3. The non-dimensional 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 length-to-thickness 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 non-dimensional form; non-dimensional quantities are reciprocal of dimensional quantities. According to Euler’s buckling theory, critical buckling loads are directly proportional to cross-sections 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 non-dimensional 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 non-dimensional form:

.

Table 4 shows the non-dimensional 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 non-dimensional (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 non-dimensional (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 non-dimensional 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 non-dimensional 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.

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 semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading. Science in China Series G, 2009; 52(8): 1244-1256.

[11]   Ying J, Lu CF, Chen WQ. Two-dimensional 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. Two-dimensional 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. Two-dimensional 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): 1-30.

[17]   Bernoulli J. Curvatura laminae elasticae. Acta Eruditorum Lipsiae, 1694; 262 – 276.

[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): 742-746.

[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 multi-layered 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 quasi-3D 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 quasi-3D 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 cross-ply 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 Non-Linear Mechanics, 1997; 32: 989-1001.

[34]   Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Applied Mechanics, 2010; 2: 117-143.

[35]   Carrera E, Giunta G, Nali P, Petrolo M. Refined beam elements with arbitrary cross-section 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: 1-10.

[37]   Thai HT, Vo TP. Bending and free vibration of functionally graded beams using various higher-order 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 higher-order 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 first-order 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 first-order shear deformation theory. Structural Engineering and Mechanics, 2016; 57: 315-325.

[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): 1835-1841.

[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): 409-423.

[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: 1-36.

[47]   Hebali H, Houari MSA, Tounsi A, Bessaim A, Bedia EEA. A new quasi-3D 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. Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory Smart Structures and Systems, 2016; 18(4): 755-786.

[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):397-422.

[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):313-335.

[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):837-859.

[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: 55-70.

[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 higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arabian Journal for Science and Engineering, 2016; 41(5): 1679-1690

 

 

 

[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 semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading. Science in China Series G, 2009; 52(8): 1244-1256.

[11]   Ying J, Lu CF, Chen WQ. Two-dimensional 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. Two-dimensional 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. Two-dimensional 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): 1-30.

[17]   Bernoulli J. Curvatura laminae elasticae. Acta Eruditorum Lipsiae, 1694; 262 – 276.

[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): 742-746.

[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 multi-layered 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 quasi-3D 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 quasi-3D 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 cross-ply 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 Non-Linear Mechanics, 1997; 32: 989-1001.

[34]   Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Applied Mechanics, 2010; 2: 117-143.

[35]   Carrera E, Giunta G, Nali P, Petrolo M. Refined beam elements with arbitrary cross-section 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: 1-10.

[37]   Thai HT, Vo TP. Bending and free vibration of functionally graded beams using various higher-order 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 higher-order 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 first-order 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 first-order shear deformation theory. Structural Engineering and Mechanics, 2016; 57: 315-325.

[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): 1835-1841.

[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): 409-423.

[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: 1-36.

[47]   Hebali H, Houari MSA, Tounsi A, Bessaim A, Bedia EEA. A new quasi-3D 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. Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory Smart Structures and Systems, 2016; 18(4): 755-786.

[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):397-422.

[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):313-335.

[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):837-859.

[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: 55-70.

[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 higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arabian Journal for Science and Engineering, 2016; 41(5): 1679-1690