Shinde, B., Sayyad, A. (2017). A Quasi3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams. Mechanics of Advanced Composite Structures, 4(2), 139152. doi: 10.22075/macs.2017.10806.1105
Bharti M. Shinde; Atteshamuddin S. Sayyad. "A Quasi3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams". Mechanics of Advanced Composite Structures, 4, 2, 2017, 139152. doi: 10.22075/macs.2017.10806.1105
Shinde, B., Sayyad, A. (2017). 'A Quasi3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams', Mechanics of Advanced Composite Structures, 4(2), pp. 139152. doi: 10.22075/macs.2017.10806.1105
Shinde, B., Sayyad, A. A Quasi3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams. Mechanics of Advanced Composite Structures, 2017; 4(2): 139152. doi: 10.22075/macs.2017.10806.1105
A Quasi3D Polynomial Shear and Normal Deformation Theory for Laminated Composite, Sandwich, and Functionally Graded Beams
^{1}Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423603, Maharashtra, India
^{2}Department of Civil Engineering, SRES&#039;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 quasi3D 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 stressfree 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 closedform 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 higherorder shear deformation theories. The comparison validates the accuracy and efficiency of the theory put forward in this work.
A Quasi3D 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, Kopargaon423603, Maharashtra, India
Paper INFO
ABSTRACT
Paper history:
Received 20170309
Revised 20170428
Accepted 20170711
Bending analyses of isotropic, functionally graded, laminated composite, and sandwich beams are carried out using a quasi3D 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 stressfree 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 closedform 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 higherorder shear deformation theories. The comparison validates the accuracy and efficiency of the theory put forward in this work.
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 wellknown 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 firstorder 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 higherorder shear deformation theories (HSDTs). Reddy [3], for example, developed a widely known thirdorder 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 strengthtoweight and stiffnesstoweight 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 crossply laminated composite beams. Chen et al. [10] constructed a stress model for the FSDTbased 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 softcore sandwich beams. Vo and Thai [15] performed a bending analysis of symmetric and antisymmetric crossply laminated composite beams by adopting a twovariable shear deformation theory, which was further extended by Sayyad et al. [16] for the bending analysis of laminated composite and softcore 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 higherorder 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 quasi3D beam theories. Some of the quasi3D beam theories discussed in the literature are the nonpolynomial 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 nonpolynomial 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 quasi3D theory resulting from this extension is computationally simpler than the other quasi3D 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. Closedformed 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.
Problem Formulation
2.1 Beam under consideration: Primary characteristics
Let us consider an advanced composite beam of length L and crosssection area (b × h) in righthand Cartesian coordinate systems. The beam occupies region 0 ≤ x ≤ L in the xdirection, region b/2 ≤ y ≤ b/2 in the ydirection, and region h/2 ≤ z ≤ h/2 in the zdirection. For simplicity, the width of the beam’s crosssection 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 xdirection and w is the displacement of any point in the zdirection, the following displacement field is derived for the thirdorder shear and normal deformation theory used in this work:
(1)
where u_{0 }and w_{0} are the displacements of the neutral axis in the x and zdirections, respectively. and denote the shear slopes. The nonzero 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 E_{1} and E_{3} are the Young’s moduli; µ_{13} and µ_{31} are the Poisson’s ratios; and G_{13} 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.
ClosedForm 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)
Illustrative Cases
The developed quasi3D 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°crossply laminated composite beams
Case 3: Bending analysis of 0°/90°/0° crossply 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 nondimensional form, are presented in Tables 1–6 and Figs. 2–13. The various nondimensional 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 nondimensional 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° crossply laminated composite beams
Table 2 presents the results of the comparison of displacements and stresses in twolayer (0°/90°) antisymmetric 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 throughthickness variations of axial displacement, axial stress, and transverse shear stress in the twolayer 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 quasi3D polynomial and nonpolynomial higherorder 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 higherorder 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° crossply laminated composite beams
Table 3 illustrates the comparison of the nondimensional displacements and stresses in threelayer (0°/90°/0°) crossply 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 threelayer laminated beam is less than that of a twolayer (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 quasi3D theory put forward in this work excellently agree with those derived through the other HSDTs. FSDT and CBT provide overestimated numerical results. The throughthickness 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 throughthickness 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 strengthtoweight ratio. Because of these attractive properties, sandwich beambased 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 throughthickness 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 nondimensional 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 powerlaw 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 higherorder theories. The stresses obtained using the proposed theory are in excellent agreement with the increasing value of k. An increase in the powerlaw 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 powerlaw index). The throughthickness 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.Throughthickness variations of in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 3. Throughthickness variations of in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Table 1. Nondimensional 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. Nondimensional displacements and stresses in 0°/90°crossply 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
SemiAnalytical [23]

4.7080
30.019
2.7192

5.900
36.6784
3.8488
HSDTN1 [37]
1.7066
4.4411
33.5966
2.4774
2.2613
5.5824
40.1544
3.5557
HSDTN2 [37]
1.7068
4.4378
33.6027
2.4794
2.2620
5.5789
40.1618
3.5522
HSDTN3 [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
SemiAnalytical [23]

2.9611
176.53
7.2550

3.744
217.330
10.738
HSDTN1 [37]
23.1462
2.9495
181.5245
6.2994
30.0738
3.7312
222.6837
9.5100
HSDTN2 [37]
23.1429
2.9489
181.6364
6.3082
30.0701
3.7304
222.8253
9.5148
HSDTN3 [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. Nondimensional displacements and stresses in 0°/90°/0° crossply 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
SemiAnalytical [23]

2.890
18.819
1.5776

3.605
21.761
2.4880
HSDTN1 [37]





3.3496
19.6712

HSDTN2 [37]





3.3496
19.6784

HSDTN3 [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
SemiAnalytical [23]

0.933
73.610
4.4390

1.170
89.030
6.1500
HSDTN1 [37]





1.0966
85.0144

HSDTN2 [37]





1.0970
85.0504

HSDTN3 [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. Nondimensional 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
SemiAnalytical [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
SemiAnalytical [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. Nondimensional 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. Nondimensional 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.Throughthickness variations of in 0°/90° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 5.Throughthickness variations of in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 6.Throughthickness variations of in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 7.Throughthickness variations of in 0°/90°/0° laminated beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 8.Throughthickness variations of in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 9.Throughthickness variations of in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 10. Throughthickness variations of in sandwich beams subjected to single sinusoidal and uniformly distributed loading at L/h = 4.
Figure 11.Throughthickness variations of in functionally graded beams subjected to single sinusoidal loading at L/h = 4.
Figure 12. Throughthickness variations of in functionally graded beams subjected to single sinusoidal loading at L/h = 4.
Figure 13.Throughthickness variations of in functionally graded beams subjected to single sinusoidal loading at L/h = 4.
Concluding Remarks
In this research, a quasi3D 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 tractionfree 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 powerlaw index given the fact that an increase in the index improves the flexibility of functionally graded beams.
[2] Timoshenko SP. on the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philosophical Magazine Series6. 1921; 41: 744746.
[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: 113122.
[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): 585604.
[6] GhugalYM, Waghe UP. Flexural analysis of deep beams using trigonometric shear deformation theory. IEI (India) J 2011; 92: 39.
[7] Sayyad A.S. Static flexure and free vibration analysis of thick isotropic beams using different higher order shear deformation theories. Int J Appl Math Mech 2012; 8(14): 7187.
[8] Carrera E. Giunta G. Refined beam theories based on a unified formulation. Int J Appl Mech 2010; 2(1): 117–143.
[9] Catapano A. Giunta G. Belouettar S. Carrera E. Static analysis of laminated beams via a unified formulation. Compos Struct 2011;94:75–83.
[10] Chen W. Li L. Xua M. A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos Struct 2011; 93: 2723–2732.
[11] Gherlone M, Tessler A, Sciuva MD. C^{0} beam elements based on the refined zigzag theory for multilayered composite and sandwich laminates. Compos Struct 2011;93:2882–2894.
[12] Sayyad AS, Ghugal YM, Borkar RR. Flexural analysis of fibrous composite beams under various mechanical loadings using refined shear deformation theories. Compos: Mech Comput Appl An Int J 2014; 5(1): 1–19.
[13] Nanda N, Kapuria S, Gopalakrishnan S. Spectral finite element based on an efficient layerwise theory for wave propagation analysis of composite and sandwich beams. J Sound Vib 2014; 333: 3120–3137.
[14] Sayyad AS, Ghugal YM, Naik NS. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved Layered Struct 2015; 2: 279–289.
[15] Vo TP, Thai HT. Static behavior of composite beams using various refined shear deformation theories. Compos Struct 2012; 94: 2513–2522.
[16] 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 Serb Soc Comput Mech 2015; 9(1): 1535.
[17] Chakrabarti A, Chalak HD, Iqbal MA, Sheikh AH. A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core. Compos Struct 2011; 93: 271–279.
[18] Tonelli D, Bardella L, Minelli M. A critical evaluation of mechanical models for sandwich beams. J Sand Struct Mater 2012; 14(6): 629–654.
[19] Ghugal YM, Shikhare GU, Bending analysis of sandwich beams according to refined trigonometric beam theory. J Aerospace Eng Technol 2015; 5(3): 2737.
[20] Pawar EG, Banerjee S, Desai YM. Stress analysis of laminated composite and sandwich beams using a novel shear and normal deformation theory. Lat Am J Solids Struct 2015; 12:134161.
[21] Giunta G, Crisafulli D, Belouettar S, Carrera E. A thermomechanical analysis of functionally graded beams via hierarchical modeling. Compos Struct 2013; 95: 676–690.
[22] Li XF, Wang BL, Han JC. A higherorder theory for static and dynamic analyses of functionally graded beams. Arch Appl Mech 2010; 80: 1197–1212.
[23] Pendhari SS, Kant T, Desai YM, Subbaiah CV. On deformation of functionally graded narrow beams under transverse loads. Int J Mech Mater Des 2010; 6: 269–282.
[24] Benatta MA, Mechab I, Tounsi A, Bedia EAA. Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci 2008; 44: 765–773.
[25] Kadoli R, Akhtar K, Kadoli NG. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model 2008; 32: 2509–2525.
[26] Kapuria S, Bhattacharyya M, Kumar AN. Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation. Compos Struct 2008; 82: 390–402.
[27] Li XF. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams J Sound Vib 2008; 318: 1210–1229.
[28] Ying J, Lu CF, Chen WQ. TwoDimensional elasticity solutions for functionally graded beams resting on elastic foundations. Compos Struct 2008; 84: 209–219.
[29] Sayyad AS, GhugalYM. A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. Int J Appl Mech 2017; 9(1): 136.
[30] Touratier M. An efficient standard plate theory. Int J Eng Sci 1991; 29(8): 901–916.
[31] Soladatos KP. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech 1992;94: 195200.
[32] 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 Struct 2003; 40: 15251546.
[33] Thai HT, Vo TP. Bending and free vibration of functionally graded beams by using various higher order shear deformation beam theories. Int J Mechanical Science. 2012; 62: 5766.
[34] Sayyad AS, GhugalYM. Effect of transverse shear and transverse normal strain on bending analysis of crossply laminated beams. Int J Appl Math Mech 2011; 7(12): 85118.
[35] Nguyen TK, Vo TP, Nguyen BD, Lee J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi3D shear deformation theory. Compos Struct 2016; 156: 238–252.
[36] Yarasca J, Mantari JL, Arciniega RA. Hermite–Lagrangian finite element formulation to study functionally graded sandwich beams. Compos Struct 2016; 140: 567–581.
[37] Mantari JL, Canales FG. Finite element formulation of laminated beams with capability to model the thickness expansion. Compos Part B 2016; 101:107115.
[38] Osofero AI, Vo TP, Nguyen TK, Lee J. Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi3D theories. J Sand Struct Mater 2015; 1–27 (In press).
[39] Vo TP, Thai HT, Nguyen TK, Inam F, Lee J. Static behaviour of functionally graded sandwich beams using a quasi3D theory. Compos Part B 2015; 68: 59–74.
[40] Sayyad AS, GhugalYM. Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Compos Struct 2017; 171: 486–504
[41] Murty K, Vellaichamy S. Higherorder theory of homogeneous plate flexure. AIAA J 1988; 26: 719725.
[2] Timoshenko SP. on the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philosophical Magazine Series6. 1921; 41: 744746.
[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: 113122.
[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): 585604.
[6] GhugalYM, Waghe UP. Flexural analysis of deep beams using trigonometric shear deformation theory. IEI (India) J 2011; 92: 39.
[7] Sayyad A.S. Static flexure and free vibration analysis of thick isotropic beams using different higher order shear deformation theories. Int J Appl Math Mech 2012; 8(14): 7187.
[8] Carrera E. Giunta G. Refined beam theories based on a unified formulation. Int J Appl Mech 2010; 2(1): 117–143.
[9] Catapano A. Giunta G. Belouettar S. Carrera E. Static analysis of laminated beams via a unified formulation. Compos Struct 2011;94:75–83.
[10] Chen W. Li L. Xua M. A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Compos Struct 2011; 93: 2723–2732.
[11] Gherlone M, Tessler A, Sciuva MD. C^{0} beam elements based on the refined zigzag theory for multilayered composite and sandwich laminates. Compos Struct 2011;93:2882–2894.
[12] Sayyad AS, Ghugal YM, Borkar RR. Flexural analysis of fibrous composite beams under various mechanical loadings using refined shear deformation theories. Compos: Mech Comput Appl An Int J 2014; 5(1): 1–19.
[13] Nanda N, Kapuria S, Gopalakrishnan S. Spectral finite element based on an efficient layerwise theory for wave propagation analysis of composite and sandwich beams. J Sound Vib 2014; 333: 3120–3137.
[14] Sayyad AS, Ghugal YM, Naik NS. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved Layered Struct 2015; 2: 279–289.
[15] Vo TP, Thai HT. Static behavior of composite beams using various refined shear deformation theories. Compos Struct 2012; 94: 2513–2522.
[16] 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 Serb Soc Comput Mech 2015; 9(1): 1535.
[17] Chakrabarti A, Chalak HD, Iqbal MA, Sheikh AH. A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core. Compos Struct 2011; 93: 271–279.
[18] Tonelli D, Bardella L, Minelli M. A critical evaluation of mechanical models for sandwich beams. J Sand Struct Mater 2012; 14(6): 629–654.
[19] Ghugal YM, Shikhare GU, Bending analysis of sandwich beams according to refined trigonometric beam theory. J Aerospace Eng Technol 2015; 5(3): 2737.
[20] Pawar EG, Banerjee S, Desai YM. Stress analysis of laminated composite and sandwich beams using a novel shear and normal deformation theory. Lat Am J Solids Struct 2015; 12:134161.
[21] Giunta G, Crisafulli D, Belouettar S, Carrera E. A thermomechanical analysis of functionally graded beams via hierarchical modeling. Compos Struct 2013; 95: 676–690.
[22] Li XF, Wang BL, Han JC. A higherorder theory for static and dynamic analyses of functionally graded beams. Arch Appl Mech 2010; 80: 1197–1212.
[23] Pendhari SS, Kant T, Desai YM, Subbaiah CV. On deformation of functionally graded narrow beams under transverse loads. Int J Mech Mater Des 2010; 6: 269–282.
[24] Benatta MA, Mechab I, Tounsi A, Bedia EAA. Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci 2008; 44: 765–773.
[25] Kadoli R, Akhtar K, Kadoli NG. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model 2008; 32: 2509–2525.
[26] Kapuria S, Bhattacharyya M, Kumar AN. Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation. Compos Struct 2008; 82: 390–402.
[27] Li XF. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams J Sound Vib 2008; 318: 1210–1229.
[28] Ying J, Lu CF, Chen WQ. TwoDimensional elasticity solutions for functionally graded beams resting on elastic foundations. Compos Struct 2008; 84: 209–219.
[29] Sayyad AS, GhugalYM. A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. Int J Appl Mech 2017; 9(1): 136.
[30] Touratier M. An efficient standard plate theory. Int J Eng Sci 1991; 29(8): 901–916.
[31] Soladatos KP. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech 1992;94: 195200.
[32] 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 Struct 2003; 40: 15251546.
[33] Thai HT, Vo TP. Bending and free vibration of functionally graded beams by using various higher order shear deformation beam theories. Int J Mechanical Science. 2012; 62: 5766.
[34] Sayyad AS, GhugalYM. Effect of transverse shear and transverse normal strain on bending analysis of crossply laminated beams. Int J Appl Math Mech 2011; 7(12): 85118.
[35] Nguyen TK, Vo TP, Nguyen BD, Lee J. An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi3D shear deformation theory. Compos Struct 2016; 156: 238–252.
[36] Yarasca J, Mantari JL, Arciniega RA. Hermite–Lagrangian finite element formulation to study functionally graded sandwich beams. Compos Struct 2016; 140: 567–581.
[37] Mantari JL, Canales FG. Finite element formulation of laminated beams with capability to model the thickness expansion. Compos Part B 2016; 101:107115.
[38] Osofero AI, Vo TP, Nguyen TK, Lee J. Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi3D theories. J Sand Struct Mater 2015; 1–27 (In press).
[39] Vo TP, Thai HT, Nguyen TK, Inam F, Lee J. Static behaviour of functionally graded sandwich beams using a quasi3D theory. Compos Part B 2015; 68: 59–74.
[40] Sayyad AS, GhugalYM. Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature. Compos Struct 2017; 171: 486–504
[41] Murty K, Vellaichamy S. Higherorder theory of homogeneous plate flexure. AIAA J 1988; 26: 719725.