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

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

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 ≤ x ≤ L in the x-direction, region -b/2 ≤ y ≤ b/2 in the y-direction, and region -h/2 ≤ z ≤ h/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)

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

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

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

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

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