Document Type: Research Paper
Authors
^{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
Keywords

Mechanics of Advanced Composite Structures 4 (2017) 139152 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
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.




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. 
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.
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.
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)
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.
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.
References
[1] Bernaulli J. Curvatura laminae elasticae, Acta, Eruditorum Liipsiae. 1964; 262276.
[2] Timoshenko SP. on the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philosophical Magazine Series 6. 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.