A Quasi-3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams

Document Type: Research Paper

Authors

1 Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423603, Maharashtra, India

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

Abstract

Bending analyses of isotropic, functionally graded, laminated composite, and sandwich beams are carried out using a quasi-3D polynomial shear and normal deformation theory. The most important feature of the proposed theory is that it considers the effects of transverse shear and transverse normal deformations. It accounts for parabolic variations in the strain/stress produced by transverse shear and satisfies the transverse shear stress-free conditions on the top and bottom surfaces of a beam without the use of a shear correction factor. Variationally consistent governing differential equations and associated boundary conditions are obtained by using the principle of virtual work. Navier closed-form solutions are employed to obtain displacements and stresses for the simply supported beams, which are subjected to sinusoidal and uniformly distributed loads. Results are compared with those derived using other higher-order shear deformation theories. The comparison validates the accuracy and efficiency of the theory put forward in this work.

Keywords


 

Mechanics of Advanced Composite Structures 4 (2017) 139-152

 

 

 

 

 

Semnan University

Mechanics of Advanced Composite Structures

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

 

A Quasi-3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams

 

B.M. Shinde, A.S. Sayyad*

 

Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon-423603, Maharashtra, India

 

Paper INFO

 

ABSTRACT

Paper history:

Received 2017-03-09

Revised 2017-04-28

Accepted 2017-07-11

Bending analyses of isotropic, functionally graded, laminated composite, and sandwich beams are carried out using a quasi-3D polynomial shear and normal deformation theory. The most important feature of the proposed theory is that it considers the effects of transverse shear and transverse normal deformations. It accounts for parabolic variations in the strain/stress produced by transverse shear and satisfies the transverse shear stress-free conditions on the top and bottom surfaces of a beam without the use of a shear correction factor. Variationally consistent governing differential equations and associated boundary conditions are obtained by using the principle of virtual work. Navier closed-form solutions are employed to obtain displacements and stresses for the simply supported beams, which are subjected to sinusoidal and uniformly distributed loads. Results are compared with those derived using other higher-order shear deformation theories. The comparison validates the accuracy and efficiency of the theory put forward in this work.

 

 

Keywords:

Laminate

Sandwich

Functionally graded

Shear and normal deformation

DOI: 10.22075/MACS.2017.10806.1105

© 2017 Published by Semnan University Press. All rights reserved.

 

 

  1. Introduction   

        In the last few decades many numerical and classical approaches based on approximate beam theories have been developed by various researchers for the analysis of isotropic and anisotropic beams. The well-known classical beam theory (CBT) developed by Euler and Bernoulli [1] is the simplest theory for the examination of beams, but its application is constrained by its failure to account for the effects of shear and normal deformations. The first-order shear deformation theory (FSDT) of Timoshenko [2] is regarded as an improvement over CBT, but it does not satisfy shear stress conditions on the top and bottom surfaces of a beam and requires a shear correction factor for appropriate explanations of strain energy due to shear deformation. To eliminate the limitations of CBT and FSDT, researchers developed higher-order shear deformation theories (HSDTs). Reddy [3], for example, developed a widely known third-order shear deformation theory for the bending analysis of isotropic and anisotropic beams. Sayyad and Ghugal [4] established a hyperbolic shear deformation theory for the examination of isotropic beams, with consideration for the combined effects of bending rotation and shear rotation. Ghugal and Sharma [5] applied a hyperbolic shear deformation theory, and Ghugal and Waghe [6] used a trigonometric shear deformation theory (TSDT) for the analysis of isotropic beams at various boundary conditions. Sayyad [7] compared various shear deformation theories for investigations into the bending and free vibration of isotropic beams.

        Two or more inherently and chemically distinct components—that is, fibers and matrices—form a material called composite material. Composite materials are characterized by improved strength-to-weight and stiffness-to-weight ratios. Nowadays, the use of beams made of composite materials is increasing in fields such as aerospace and aeronautical engineering, navigation, and construction. Accordingly, many researchers have carried out studies on the bending behavior of such beams. Carrera [8] developed a unified formulation for the analysis of laminated composite beams, and Catapano et al. [9] extended this formulation to probe into cross-ply laminated composite beams. Chen et al. [10] constructed a stress model for the FSDT-based analysis of laminated composite beams. Gherlone [11] conducted a comparative study of laminated composite and sandwich beams by using the zigzag function in an equivalent single layer theory. Sayyad et al. [12] carried out a flexural analysis of fibrous composite beams by using different refined shear deformation theories based on displacement. Nanda et al. [13] proposed a spectral finite element model by using zigzag theory, and Sayyad et al. [14] presented a simple TSDT for the bending analysis of laminated composite and soft-core sandwich beams. Vo and Thai [15] performed a bending analysis of symmetric and anti-symmetric cross-ply laminated composite beams by adopting a two-variable shear deformation theory, which was further extended by Sayyad et al. [16] for the bending analysis of laminated composite and soft-core sandwich beams. Chakraborti et al. [17] put forward a finite element model grounded in zigzag theory to examine laminated sandwich beams with a soft core. Tonelli et al. [18] carried out a bending analysis of sandwich beams by using an HSDT. Ghugal and Shikhare [19] obtained a general solution for the deflections and stresses of sandwich beams by using a TSDT, and Pawar et al. [20] analyzed the bending of sandwich and laminated composite beams by using a higher-order shear and normal deformation theory.

The use of beams and plates made of functionally graded materials (FGMs) in different engineering fields has recently increased. In a functionally graded beam, material properties gradually change along the spatial direction, thus generating a higher resistance against temperature than that achieved with conventional materials. Giunta et al. [21] analyzed functionally graded beams by using classical and advanced shear deformation theories. Li et al. [22] formulated a general solution for the static and dynamic analysis of functionally graded Timoshenko and Euler beams by extending Levinson’s beam theory. Pendhari et al. [23] applied a mixed semi- analytical model for the bending analysis of FGM narrow beams under plane stress conditions. With consideration for warping and shear deformation effects, Benatta et al. [24] inquired into the static analysis of functionally graded beams. Kadoli et al. [25] and Kapuria et al. [26] developed a new HSDT for the bending analysis of FGM beams. A static and dynamic analysis of functionally graded Timoshenko and Euler–Bernoulli beams was carried out by Li [27], with the author considering rotary inertia and shear deformation effects. Ying et al. [28] developed exact solutions for the bending analysis of functionally graded beams resting on an elastic foundation. Sayyad and Ghugal [29] recently developed a unified shear deformation theory for the analysis of functionally graded beams.   

 

1.1    Contributions of the current work

        Transverse shear and normal deformations play an important role in the accurate prediction of the structural behavior of beams and plates made of advanced composite materials. Therefore, any refinements to CBTs are generally meaningless unless the effects of transverse shear and normal strains are taken into account. Such effects are neglected in Euler and Bernoulli’s CBT [1], FSDT [2], Reddy’s parabolic shear deformation theory (PSDT) [3], Touratier’s TSDT [30], Soldatos’ HSDT [31], Karama et al.’s exponential shear deformation theory (ESDT) [32], and Thai and Vo’s theory [33].

    Theories that consider the effects of transverse shear and normal deformations are called quasi-3D beam theories. Some of the quasi-3D beam theories discussed in the literature are the non-polynomial shear deformation theories of Sayyad and Ghugal [34], Nguyen et al. [35], Yarasca [36], Mantari and Canales [37], and Osofero et al. [38] and the polynomial shear deformation theory of Vo et al. [39]. A recent initiative by Sayyad and Ghugal [40] involved a review of various beam theories available in the literature for the analysis of isotropic and anisotropic beams.

        The use of a non-polynomial shear strain function is computationally more difficult than the adoption of a polynomial shear strain function. The present study therefore extends Murphy’s [41] polynomial shear deformation theory by accounting for the effects of thickness stretching (i.e., normal deformation). The quasi-3D theory resulting from this extension is computationally simpler than the other quasi-3D theories cited above. In the theory proposed in the current work, both axial and transverse displacements are functions of x and z coordinates. The theory satisfies the transverse shear strain conditions on the top and bottom surfaces of a beam without the use of a shear correction factor. Governing equations are obtained by using the principle of virtual work and applying a fundamental lemma of calculus. Closed-formed solutions are derived using Navier’s solution for simply supported boundary conditions. The accuracy of the theory is confirmed by applying it to bending analyses of advanced composite beams made of isotropic materials, fibrous composite materials, and FGMs. Numerical results are obtained for the simply supported beams, which are subjected to sinusoidal and uniformly distributed loads. The findings are then compared with those in the literature for validation.

 

  1. Problem Formulation

2.1 Beam under consideration: Primary characteristics

        Let us consider an advanced composite beam of length L and cross-section area (b × h) in right-hand Cartesian coordinate systems. The beam occupies region 0 ≤ xL in the x-direction, region -b/2 ≤ yb/2 in the y-direction, and region -h/2 ≤ zh/2 in the z-direction. For simplicity, the width of the beam’s cross-section is assumed to be unity. The beam is made of advanced composite materials, and its top surface is subjected to transverse loading.

 

2.2 Kinematics and constitutive relations

        Assuming that u is the displacement of any point in the x-direction and w is the displacement of any point in the z-direction, the following displacement field is derived for the third-order shear and normal deformation theory used in this work:

   (1)

where u0 and w0 are the displacements of the neutral axis in the x- and z-directions, respectively.  and  denote the shear slopes. The non-zero strains associated with the theory are obtained from the linear theory of elasticity.

                  (2)

where ‘,x’ indicates the derivative with respect to x. The constitutive relations for advanced composite beams are also obtained from the linear theory of elasticity.

                            (3)

where are the reduced stiffness coefficients.

 

 

Figure 1.Beam under consideration.

 

 

 

 

These can be expressed for different materials as follows:

(a)    Isotropic material   

(4)

 

where E denotes the Young’s modulus, G represents the shear modulus, and is the Poisson’s ratio.

(b)   Fibrous composite material  

                      (5)

where E1 and E3 are the Young’s moduli; µ13 and µ31 are the Poisson’s ratios; and G13 is the shear modulus.    

(c)    FGM

                                (6)

where,

                             (7)

where Em and Ec are the Young’s moduli of metal and ceramic, respectively, and k is the volume fraction exponent, whose value varies from zero to infinity. The beam is fully ceramic when k is equal to zero and fully metallic when k is infinity.   

 

2.3 Governing differential equations of equilibrium

        The governing differential equations of equilibrium can be derived by using the principle of virtual displacements thus:

 

                  (8)

Substituting the values of stresses and strains from Eqs. (2) and (3) into Eq. (8) and integrating these by parts yield the following governing differential equations:

(9)

 

                (10)

 

          (11)

 

          (12)

where the stiffness coefficients are as follows:

(13)

In this manner, the variationally constant governing differential equations that underlie the theory developed in this study are obtained.

 

  1. Closed-Form Solution

        Following Navier’s solution procedure, the following solution form is assumed for unknown variables that satisfy simply supported boundary conditions:

                 (14)

where  are the arbitrary parameters to be determined subject to the condition that the solution in (13) satisfies differential equations (9)–(12). Transverse load q is also expanded in the Fourier sine series as

          (15)

Substituting the solution form from Eqs. (14) and (15) into governing equations (9)–(12) derives

  

                                                                                             (16)

where [K] is the stiffness matrix, is the vector of unknowns, and is the force vector.

(17)

 

                                (18)

                                                                                                                                              

  1. Illustrative Cases

        The developed quasi-3D polynomial shear and normal deformation theory is applied in the bending analyses of advanced composite beams subjected to single sinusoidal and uniformly distributed loads. To confirm the accuracy and validity of the theory, the following cases are solved:

Case 1: Bending analysis of isotropic beams

Case 2: Bending analysis of 0°/90°cross-ply laminated composite beams

Case 3: Bending analysis of 0°/90°/0° cross-ply laminated composite beams

Case 4: Bending analysis of 0°/core/0° sandwich beams

Case 5: Bending analysis of FGMs

        The following material properties are used for the detailed numerical study:

        MAT 1:

        MAT 2:

        MAT 3:

        MAT 4:

        The numerical results, which are expressed in non-dimensional form, are presented in Tables 1–6 and Figs. 2–13. The various non-dimensional parameters used are as follows:

(a)    Isotropic, laminated composite, and sandwich beams

              (19)

 

 

 

 

 

(b)    FGMs

 

        (20) 

Case 1: Bending analysis of isotropic beams

        In this case, the displacements and stresses of isotropic beams subjected to single sinusoidal and uniformly distributed loads are obtained for aspect ratios (L/h) of 4 and 10. The non-dimensional results are presented in Table 1. The beams are made of an isotropic material MAT 1 (i.e., steel). The findings are compared with the numerical results derived with HSDT [4], PSDT [3], FSDT [2], and CBT [1]. Table 1 shows that the transverse displacement obtained using the proposed theory is of a higher value for an aspect ratio of 4 and produces the exact result for an aspect ratio of 10 compared with the values obtained with PSDT [3]. The stresses obtained for aspect ratios 4 and 10 are in excellent agreement with those derived with other theories for single sinusoidal loads. In the case of isotropic materials, the axial stress is zero at the neutral axis and reaches its maximum at the top and bottom surfaces of the beams. By contrast, the transverse shear stress is at its maximum at the neutral axis and zero at the top and bottom surfaces of the beams. CBT [1] underestimates the deflections and stresses because of this theory’s disregard of transverse shear and normal deformations. The same pattern of results is observed for the beam subjected to a uniformly distributed load. Overall, the proposed theory generates excellent results for isotropic beams because of its inclusion of the effects of transverse normal deformations.

 

Case 2: Bending analysis of 0°/90° cross-ply laminated composite beams

        Table 2 presents the results of the comparison of displacements and stresses in two-layer (0°/90°) anti-symmetric laminated composite beams subjected to single sinusoidal and uniformly distributed loads. The layers are of equal thickness, expressed as h/2, where h is the overall thickness. The beams are made of fibrous composite materials (MAT 2). The through-thickness variations of axial displacement, axial stress, and transverse shear stress in the two-layer beams are shown in Figs. 2–4. The numerical results are compared with those presented by Reddy [3], Soldatos [31], Karama et al. [32], and Mantari and Canales [37] and those derived using FSDT [2] and CBT [1]. Table 2 indicates that the transverse displacements obtained using the proposed theory are in excellent agreement with those derived with the other quasi-3D polynomial and non-polynomial higher-order theories. FSDT and CBT respectively overestimates and underestimates the transverse displacements because of their neglect of transverse shear and normal deformations. Compared with the values derived with the other higher-order theories, FSDT and CBT generate identical underestimated axial stresses. Transverse shear stresses are obtained using equations of equilibrium to ascertain stress continuity at the layer interface. Figs. 3 and 4 show that the stresses are at their maximum level at the 0° layer—a result attributed to the high elastic modulus along the direction of the fiber in the materials. The stresses are at their minimum at the 90°  layer.

 

Case 3: Bending analysis of 0°/90°/0° cross-ply laminated composite beams

        Table 3 illustrates the comparison of the non-dimensional displacements and stresses in three-layer (0°/90°/0°) cross-ply laminated composite beams subjected to single sinusoidal and uniformly distributed loads. The overall thickness (i.e., h/3) is equally distributed among all the layers of the beams, which are made of fibrous composite materials (MAT 2). The numerical results are compared with those presented in the literature [1–3, 31, 32, 37]. Table 3 reveals that the transverse deflection of a three-layer laminated beam is less than that of a two-layer (0°/90°) laminated beam. This finding is ascribed to the increase in stiffness along the length of the beams. The displacements and stresses obtained using the quasi-3D theory put forward in this work excellently agree with those derived through the other HSDTs. FSDT and CBT provide overestimated numerical results. The through-thickness variations of axial displacement and stress are shown in Figs. 5 and 6. The figures indicate that because the laminated beams are symmetric, the axial displacement and stress are zero at the neutral axis (i.e., 90°layer) and at their maximum at the top and bottom surfaces of the beam (i.e., 0° layer). The through-thickness variations of transverse shear stress obtained using the equations of equilibrium is shown in Fig. 7.

 

Case 4: Bending analysis of 0°/core/0° sandwich beams

        Sandwich composite beams are constituted by hard face sheets and soft cores. The modulus of the core materials is significantly lower than that of the face sheets. The main benefit of using a sandwich beam lies in its high bending stiffness and high strength-to-weight ratio. Because of these attractive properties, sandwich beam-based structures have been widely used in many industries.  

        The proposed theory is also validated on the basis of a bending analysis of sandwich beams. The comparison of the numerical results for displacement and stresses in 0°/core/0° sandwich beams subjected to single sinusoidal and uniformly distributed loads is shown in Table 4. Values are obtained for aspect ratios of 4, 10, and 100. The thickness of the face sheets is 0.1 h, whereas that of the core is 0.8 h. The face sheets are made of MAT 2, whereas the core is composed of MAT 3. The numerical results are compared with those presented by Reddy [3], Soldatos [31], and Karama et al. [32] and those obtained by FSDT [2] and CBT [1]. Table 4 indicates that the central deflection and stresses obtained in the central core are less than those at the top and bottom face sheets. This finding is attributed to the fact that the core is made up of soft transversely isotropic material. The through-thickness variations of axial displacement and stress are shown in Figs. 8–10. As seen in Fig. 9, minimal axial stress is experienced by the core material, thus reflecting that the soft core is resistant only to transverse shear stress.

 

Case 5: Bending analysis of functionally graded beams

        Tables 5 and 6 show the comparison of non-dimensional displacements and stresses in functionally graded beams subjected to single sinusoidal and uniformly distributed loads, respectively. The results on displacements and stresses are obtained for various values of the power-law index (i.e., k = 0, 1, 2, 5, and 10). When k = 0, a beam is fully ceramic. The deflection obtained using the proposed theory is in good agreement with that derived with other higher-order theories. The stresses obtained using the proposed theory are in excellent agreement with the increasing value of k. An increase in the power-law index reduces the stiffness of the functionally graded beams, thereby elevating the displacements and axial stresses. Transverse shear stress decreases with decreasing stiffness of a beam (i.e., increased power-law index). The through-thickness variations of axial displacement and stress are shown in Figs. 11–13. The proposed theory yields a parabolic distribution of transverse shear stress across the depth of the beams and satisfies the zero shear stress conditions on the top and bottom surfaces of the beams (Fig. 12). The axial stress is not zero at the neutral axis, and the transverse shear stress is not at its maximum at such axis. This result is due to the fact that the material properties continuously vary throughout the thickness of the beams.

 

 

Figure 2.Through-thickness variations of  in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

Figure 3. Through-thickness variations of  in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 


 

Table 1. Non-dimensional displacements and stresses in isotropic beams (MAT 1)

 

Theory

SSL

UDL

h/L

               

0.25

Proposed

12.248

1.445

9.960

1.897

15.830

1.816

12.135

2.893

 

HSDT [4]

12.704

1.427

9.977

1.896

16.486

1.804

12.254

2.882

 

PSDT [3]

12.715

1.429

9.986

1.895

16.504

1.806

12.263

2.908

 

FSDT [2]

12.385

1.430

9.727

1.910

16.000

1.806

12.000

1.969

 

CBT [1]

12.297

1.232

9.727

1.900

16.000

1.563

12.000

-

0.1

Proposed

193.20

1.261

60.98

4.769

249.51

1.599

75.078

7.353

 

HSDT [4]

194.31

1.263

61.04

4.769

251.23

1.601

75.259

7.312

 

PSDT [3]

194.34

1.264

61.05

4.769

251.27

1.602

75.268

7.361

 

FSDT [2]

193.51

1.264

60.79

4.769

250.00

1.602

75.000

4.922

 

CBT [1]

192.95

1.232

60.91

4.769

250.00

1.563

75.000

-


 

 

Table 2. Non-dimensional displacements and stresses in 0°/90°cross-ply laminated composite beams (MAT 2)

 

h/L

Theory

SSL

UDL

               

0.25

Proposed

1.7059

4.4409

33.608

2.9796

2.2524

5.5768

40.2535

5.0407

 

PSDT [3]

1.7100

4.4511

33.592

2.9768

2.2580

5.590

40.2390

5.0236

 

HSDT [31]

1.6930

4.4039

33.253

2.9513

2.2299

5.533

39.9207

4.8144

 

ESDT [32]

1.7450

4.2305

34.264

2.8484

2.3085

5.316

40.9211

5.8468

 

Semi-Analytical [23]

---

4.7080

30.019

2.7192

---

5.900

36.6784

3.8488

 

HSDT-N1 [37]

1.7066

4.4411

33.5966

2.4774

2.2613

5.5824

40.1544

3.5557

 

HSDT-N2 [37]

1.7068

4.4378

33.6027

2.4794

2.2620

5.5789

40.1618

3.5522

 

HSDT-N3 [37]

1.7179

4.3931

33.8186

2.5192

2.2745

5.5245

40.3980

3.5999

 

FSDT [2]

1.4210

4.7966

27.904

2.9468

1.8360

6.008

34.4272

4.5567

 

CBT [1]

1.4210

2.6254

27.904

2.9468

1.8360

3.329

34.4272

4.5567

0.1

Proposed

22.889

2.9158

180.38

7.3604

29.735

3.688

221.260

11.548

 

PSDT [3]

22.942

2.9225

180.18

7.3780

29.840

3.696

221.017

11.544

 

HSDT [31]

22.901

2.9161

179.86

7.3679

29.7390

3.688

220.692

11.421

 

ESDT [32]

23.028

2.8864

180.86

7.3247

29.9363

3.652

221.704

10.698

 

Semi-Analytical [23]

----

2.9611

176.53

7.2550

---

3.744

217.330

10.738

 

HSDT-N1 [37]

23.1462

2.9495

181.5245

6.2994

30.0738

3.7312

222.6837

9.5100

 

HSDT-N2 [37]

23.1429

2.9489

181.6364

6.3082

30.0701

3.7304

222.8253

9.5148

 

HSDT-N3 [37]

23.1769

2.9427

181.7649

6.4236

30.1162

3.7229

222.9276

9.6752

 

FSDT [2]

22.206

2.9728

174.40

7.3670

28.6882

3.758

215.170

11.391

 

CBT [1]

22.206

2.6254

174.40

7.3670

28.6883

3.329

215.170

11.391

0.01

Present

22166

2.6229

17468

73.433

28638.4

3.326

21549.7

113.57

 

PSDT [3]

22214

2.6285

17447

73.675

28701.2

3.333

21524.1

113.94

 

HSDT [31]

22213

2.6283

17446

73.670

28699.0

3.333

21522.6

113.92

 

ESDT [32]

22214

2.6281

17447

73.668

28701.6

3.333

21524.1

113.85

 

FSDT [2]

22207

2.6290

17441

73.674

28689.6

3.334

21518.1

113.91

 

CBT [1]

22206

2.6254

17440

73.670

28688.2

3.329

21517.0

113.91

 

 

 


 

Table 3. Non-dimensional displacements and stresses in 0°/90°/0° cross-ply laminated composite beams (MAT 2)


h/L

Theory

SSL

UDL

               

0.25

Proposed

0.8624

2.700

16.986

1.5561

1.1590

3.367

19.646

1.8346

 

PSDT [3]

0.8653

2.700

16.989

1.5570

1.1617

3.368

19.670

1.8310

 

HSDT [31]

0.8630

2.698

16.944

1.5594

1.1590

3.365

19.615

1.8312

 

ESDT [32]

0.9678

2.687

19.003

1.3320

1.2895

3.366

22.139

1.7557

 

Semi-Analytical [23]

---

2.890

18.819

1.5776

---

3.605

21.761

2.4880

 

HSDT-N1 [37]

---

---

---

---

---

3.3496

19.6712

---

 

HSDT-N2 [37]

---

---

---

---

---

3.3496

19.6784

---

 

HSDT-N3 [37]

---

---

---

---

---

3.3852

20.2936

---

 

FSDT [2]

0.5136

2.410

10.085

1.7690

0.6636

2.991

12.442

2.7355

 

CBT [1]

0.5136

0.510

10.085

1.7690

0.6636

0.648

12.442

2.7355

0.1

Proposed

8.9160

0.873

70.264

4.3342

11.703

1.095

85.098

6.0721

 

PSDT [3]

8.9398

0.875

70.212

4.3344

11.733

1.098

85.029

6.0900

 

HSDT [31]

8.9329

0.874

70.158

4.3355

11.724

1.097

84.973

6.0922

 

ESDT [32]

9.2585

0.889

72.716

4.2051

12.714

1.115

87.629

5.9196

 

Semi-Analytical [23]

--

0.933

73.610

4.4390

---

1.170

89.030

6.1500

 

HSDT-N1 [37]

---

---

---

---

---

1.0966

85.0144

---

 

HSDT-N2 [37]

---

---

---

---

---

1.0970

85.0504

---

 

HSDT-N3 [37]

---

---

---

---

---

1.1062

85.6388

---

 

FSDT [2]

8.0257

0.814

63.033

4.4226

10.368

1.023

77.767

6.8388

 

CBT [1]

8.0257

0.510

63.033

4.4226

10.368

0.648

77.767

6.8388

0.01

Proposed

8018.81

0.513

6319.2

43.999

10361.9

0.651

7794.8

68.243

 

PSDT [3]

8034.9

0.514

6310.6

44.217

10382.8

0.652

7784.1

68.243

 

HSDT [31]

8034.8

0.514

6310.5

44.217

10382.6

0.652

7784.0

68.244

 

ESDT [32]

8038.3

0.514

6313.3

44.204

10388.0

0.653

7786.8

68.046

 

FSDT [2]

8025.7

0.514

6303.4

44.226

10368.5

0.651

7776.7

68.387

 

CBT [1]

8025.7

0.510

6303.4

44.226

10368.5

0.648

7776.7

68.687

 

 

 

Table 4. Non-dimensional displacements and stresses in 0°/core/0° sandwich beams (Face sheet: MAT 2, Core: MAT 3)



h/L

Theory

SSL

UDL

               

0.25

Proposed

1.7471

10.052

34.435

1.377

2.3770

12.455

39.429

2.583

 

PSDT [3]

1.7393

10.034

34.181

1.372

2.3653

12.494

39.161

2.662

 

HSDT [31]

1.7368

10.027

34.132

1.372

2.3616

12.447

39.110

2.655

 

ESDT [32]

1.7618

10.045

34.622

1.371

2.3940

12.473

39.647

2.672

 

FSDT [2]

1.0120

5.2798

19.898

1.410

1.3080

6.5480

24.549

2.181

 

CBT [1]

1.0120

1.0070

19.898

1.410

1.3080

1.2770

24.549

2.181

 

Semi-Analytical [23]

---

11.060

37.552

1.356

---

13.750

43.488

2.280

0.1

Proposed

17.706

2.4807

139.55

3.508

23.291

3.0966

168.89

5.305

 

PSDT [3]

17.670

2.4772

138.41

3.509

23.24

3.0923

168.13

5.287

 

HSDT [31]

17.664

2.4763

138.85

3.509

23.231

3.0911

168.08

5.288

 

ESDT [32]

17.731

2.4824

139.38

3.508

23.328

3.0988

168.61

5.286

 

FSDT [2]

15.821

1.6910

124.36

3.526

20.439

2.1210

153.43

5.452

 

CBT [1]

15.821

1.0070

124.36

3.526

20.439

1.2770

153.43

5.452

 

Semi-Analytical [23]

---

2.6680

143.14

3.504

---

3.3300

172.60

5.240

0.01

Proposed

15860

1.0233

12498.6

35.20

20494

1.2973

15416.9

54.42

 

PSDT [3]

15839

1.0220

12451.1

35.26

20468

1.2957

15358.4

54.50

 

HSDT [31]

15839

1.0219

12451.1

35.26

20468

1.2957

15358.4

54.35

 

ESDT [32]

15840

1.0220

12451.7

35.26

20469

1.2958

15358.9

54.49

 

FSDT [2]

15820

1.0140

12436.5

35.26

20439

1.2829

15343.3

54.52

 

CBT [1]

15821

1.0072

12436.6

35.26

20439

1.2775

15343.5

54.52



 

Table 5. Non-dimensional displacements and stresses in functionally graded beams under single sinusoidal loading (MAT 4)

 

k

Theory

L/h = 5

L/h = 20

               

0

Proposed

0.9150

3.1397

3.8341

0.7230

0.2302

2.8947

15.0719

0.7376

Li et. al [22]

0.9402

3.1657

3.8020

0.7500

0.2306

2.8962

15.0130

0.7500

TBT [33]

0.9398

3.1654

3.8020

0.7332

0.2306

2.8962

15.0129

0.7451

SBT [33]

0.9409

3.1649

3.8053

0.7549

0.2306

2.8962

15.0138

0.7686

HBT [33]

0.9397

3.1654

3.8017

0.7312

0.2306

2.8962

15.0129

0.7429

EBT [33]

0.9420

3.1635

3.8083

0.7763

0.2306

2.8961

15.0145

0.7920

Vo et al. [39]

---

3.1397

3.8005

0.7233

---

2.8947

15.0125

0.7432

HSDT2 [36]

---

3.1397

3.8028

0.7235

---

2.8947

15.0197

0.7443

HSDT3 [36]

--

3.1397

3.8021

0.7224

--

2.8947

15.0195

0.7433

CBT [1]

0.9211

2.8783

3.7500

---

0.2303

2.8783

15.0000

---

1

Proposed

2.1975

6.1338

5.7941

0.7230

0.5517

5.7201

23.2714

0.7376

Li et. al [22]

2.3045

6.2599

5.8837

0.7500

0.5686

5.8049

23.2054

0.7500

TBT [33]

2.3038

6.2594

5.8836

0.7332

0.5686

5.8049

23.2053

0.7451

SBT [33]

2.3058

6.2586

5.8892

0.7549

0.5686

5.8049

23.2067

0.7686

HBT [33]

2.3036

6.2594

5.8831

0.7312

0.5685

5.8049

23.2052

0.7429

EBT [33]

2.3075

6.2563

5.8943

0.7763

0.5686

5.8047

23.2080

0.7920

Vo et al. [39]

---

6.1338

5.8812

0.7233

---

5.7201

23.2046

0.7432

HSDT2 [36]

---

6.1334

5.8855

0.7235

---

5.7197

23.2184

0.7443

HSDT3 [36]

--

6.1334

5.8843

0.7224

--

5.7197

23.2181

0.7433

CBT [1]

2.2722

5.7746

5.7959

---

0.5680

5.7746

23.1834

---

2

Proposed

2.9460

7.8606

6.6179

0.6620

0.7397

7.2805

27.2030

0.6757

Li et. al [22]

3.1134

8.0602

6.8812

0.6787

0.7691

7.4415

27.0989

0.6787

TBT [33]

3.1130

8.0677

6.8826

0.6706

0.7691

7.4421

27.0991

0.6824

SBT [33]

3.1153

8.0683

6.8901

0.6933

0.7692

7.4421

27.1010

0.7069

HBT [33]

3.1127

8.0675

6.8819

0.6685

0.7691

7.4420

27.0989

0.6802

EBT [33]

3.1174

8.0667

6.8969

0.7157

0.7692

7.4420

27.1027

0.7315

Vo et al. [39]

---

7.8606

6.8818

0.6622

---

7.2805

27.0988

0.6809

HSDT2 [36]

---

7.8598

6.8871

0.6625

---

7.2797

27.1158

0.6800

HSDT3 [36]

--

7.8597

6.8857

0.6613

--

7.2797

27.1154

0.6790

CBT [1]

3.0740

7.4003

6.7676

---

0.7685

7.4003

27.0704

---

5

Proposed

3.5050

9.6038

7.9579

0.5838

0.8797

8.6479

31.9586

0.5966

Li et. al [22]

3.7089

9.7802

8.1030

0.5790

0.9133

8.8151

31.8112

0.5790

TBT [33]

3.7100

9.8281

8.1106

0.5905

0.9134

8.8182

31.8130

0.6023

SBT [33]

3.7140

9.8367

8.1222

0.6155

0.9134

8.8188

31.8159

0.6292

HBT [33]

3.7097

9.8271

8.1095

0.5883

0.9134

8.8181

31.8127

0.5998

EBT [33]

3.7177

9.8414

8.1329

0.6404

0.9135

8.8191

31.8185

0.6562

Vo et al. [39]

---

9.6037

8.1140

0.5840

---

8.6479

31.8137

0.6010

HSDT2 [36]

---

9.6030

8.1202

0.5843

---

8.6471

31.8341

0.6019

HSDT3 [36]

--

9.6025

8.1184

0.5829

--

8.6471

31.8337

0.6014

CBT [1]

3.6496

8.7508

7.9428

---

0.9124

8.7508

31.7711

---

10

Proposed

3.6922

10.7578

9.6903

0.6394

0.9267

9.5749

37.9164

0.6534

Li et. al [22]

3.8860

10.8979

9.7063

0.6436

0.9536

9.6879

38.1372

0.6436

TBT [33]

3.8864

10.9381

9.7122

0.6467

0.9536

9.6905

38.1385

0.6596

SBT [33]

3.8913

10.9420

9.7238

0.6708

0.9537

9.6908

38.1414

0.6858

HBT [33]

3.8859

10.9375

9.7111

0.6445

0.9536

9.6905

38.1383

0.6572

EBT [33]

3.8957

10.9404

9.7341

0.6944

0.9538

9.6907

38.1440

0.7115

Vo et al. [39]

---

10.7578

9.7164

0.6396

---

9.5749

38.1395

0.6583

HSDT2 [36]

---

10.7573

9.7234

0.6399

---

9.5742

38.1624

0.6614

HSDT3 [36]

--

10.7569

9.7215

0.6386

--

9.5743

38.1636

0.6529

CBT [1]

3.8097

9.6072

9.5228

---

0.9524

9.6072

38.0913

---



 

Table 6. Non-dimensional displacements and stresses in functionally graded beams under uniformly distributed loading (MAT 4)


k

Theory

L/h = 5

       

0

Proposed

0.7086

2.5047

3.1048

0.4769

PSDT [3]

0.7251

2.5020

3.0916

0.4769

TSDT [29]

0.7259

2.5016

3.0949

0.4920

HSDT [29]

0.7247

2.5003

3.0899

0.4739

ESDT [29]

0.7280

2.4974

3.1039

0.4871

FSDT [2]

0.7129

2.5023

3.0396

0.3183

CBT [1]

0.7129

2.2693

3.0396

----

1

Proposed

1.7051

4.8435

5.0392

0.4769

PSDT [3]

1.7793

4.9458

4.7856

0.5243

TSDT [29]

1.7806

4.9451

4.7912

0.5331

HSDT [29]

1.7517

4.9257

4.7165

0.6025

ESDT [29]

1.7819

4.9432

4.7944

05430

FSDT [2]

1.7588

4.8807

4.6979

0.5376

CBT [1]

1.7588

4.5228

4.6979

----

5

Proposed

2.7143

7.5938

6.9216

0.3856

PSDT [3]

2.8644

7.7723

6.6057

0.5314

TSDT [29]

2.8671

7.7792

6.6172

0.5144

HSDT [29]

2.8641

7.7715

6.6047

0.5332

ESDT [29]

2.8697

7.7830

6.6281

0.5022

FSDT [2]

2.8250

7.5056

6.4382

0.9942

CBT [1]

2.8250

6.8994

6.4382

----

10

Proposed

2.8591

8.5088

8.2877

0.4224

PSDT [3]

2.9989

8.6530

7.9080

0.4226

TSDT [29]

3.0022

8.6561

7.9195

0.4392

HSDT [29]

2.9986

8.6527

7.9070

0.4211

ESDT [29]

3.0054

8.6547

7.9301

0.4558

FSDT [2]

2.9488

8.3259

7.7189

1.2320

CBT [1]

2.9488

7.5746

7.7189

----

 

 

 

 

Figure 4.Through-thickness variations of  in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

 

 

 

 

 

 

 

Figure 5.Through-thickness variations of  in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

Figure 6.Through-thickness variations of  in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

Figure 7.Through-thickness variations of  in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

 

Figure 8.Through-thickness variations of  in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

Figure 9.Through-thickness variations of  in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

Figure 10. Through-thickness variations of  in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.

 

 

Figure 11.Through-thickness variations of  in functionally graded beams subjected to single sinusoidal loading at L/h = 4.

 

Figure 12. Through-thickness variations of  in functionally graded beams subjected to single sinusoidal loading at L/h = 4.

 

 

 

Figure 13.Through-thickness variations of  in functionally graded beams subjected to single sinusoidal loading at L/h = 4.

 

  1. Concluding Remarks

        In this research, a quasi-3D polynomial shear and normal deformation theory is applied for the bending analyses of composite beams made of fibrous composite materials and FGMs. The proposed theory considers the effects of transverse shear and normal deformations. It also satisfies the traction-free conditions on the top and bottom surfaces of beam without the application of a shear correction factor. Governing equations are obtained using the virtual work principle, and displacements and stresses are determined using Navier’s solution. Numerical results are presented for isotropic, laminated composite, sandwich, and functionally graded beams. On the basis of the findings, we can conclude that the proposed theory derives excellent results on displacements and stresses for the examined beams. Shear stress continuity is satisfied by equations of equilibrium. The transverse displacement obtained using the proposed theory for functionally graded beams increases with increasing power-law index given the fact that an increase in the index improves the flexibility of functionally graded beams.

 

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[1]   Bernaulli J. Curvatura laminae elasticae, Acta, Eruditorum Liipsiae. 1964; 262-276.

[2]   Timoshenko SP. on the correction for shear of the differential equation for transverse   vibrations of prismatic bar.  Philosophical Magazine Series 6. 1921; 41: 744-746.

[3]   Reddy JN. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 2007; 45: 288–307.

[4]   Sayyad AS, Ghugal YM. Flexure of thick beams using new hyperbolic shear deformation theory. Int J Mech 2011; 5: 113-122.

[5]   Ghugal YM, Sharma R. A hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams.  Int J Comput Meth 2009; 6(4): 585-604.

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