Bending, buckling and free vibration behaviors of thin-walled functionally graded sandwich and composite channel-section beams

Abstract This article proposes static, free vibration, and buckling analysis of thin-walled functionally graded (FG) sandwich and composite channel-section beams. It is based on the first-order shear deformable beam theory, which can recover to classical one by ignoring the shear effect. Ritz’s approximation functions are developed to solve the characteristic problems. Both results from classical and the first-order shear deformable theories are given in a unified fashion. Ritz solutions are applied for thin-walled FG sandwich channel-section beams for the first time. Numerical examples are presented in relation to many important effects such as span-to-height ratio, material parameter, lay-ups, fiber orientation and boundary conditions on the beams’ deflections, natural frequencies, and critical buckling loads. New results presented in this study can be of interests to the scientific and engineering community in the future.


Introduction
Functionally graded (FG) and composite materials are widely used in many engineering fields owing to their high strength-to-weight and stiffness-to-weight ratio, long-term durability, and noncorrosive nature. Most recent applications in civil, transportation, and mechanical industries show the effectiveness of thin-walled FG and composite structures (Dechao, Zhongmin, and Xingwei 2001;Librescu and Song 2005;Pawar and Ganguli 2006;Moon et al. 2010;Arnaud et al. 2011;Alizada, Sofiyev, and Kuruoglu 2012;Harursampath, Harish, and Hodges 2017b;Xu, Zhang, and Zhang 2018;Sofiyev 2019). They also attracted a large number of researchers to study the structural responses, in which, vibration, bending, and buckling behaviors are the importance and interest to the performance.
Thin-walled beam theory was first introduced for isotropic material by Vlasov, which was also called Vlasov's classical beam theory (Vlasov 1961). It was then extended to composite material by many authors and some of them were mentioned here. Bauld and Lih-Shyng (1984) developed Vlasov's theory for bending and buckling analysis of thin-walled composite beams. Kim, Shin, and Kim (2006) analyzed thin-walled composite channel and I-beams under torsion. Bending behaviors of composite I-beams were presented by Shin, Kim, and Kim (2007). Lee (2016a, 2016b) predicted the deflections and vibration of thin-walled FG sandwich beams with channel and I-sections. Vo and Lee (2013) proposed a finite element method (FEM) for vibration and buckling analysis of thin-walled composite I-beams with arbitrary lay-ups under axial loads and end moments. Kim and Lee (2015) developed refined theory for analyzing thinwalled composite beams on elastic foundations. Recently, Kim and Lee (2017a) analyzed bending responses of thin-walled FG sandwich beams under torsion and vertical load. Zhu et al. (2016), Malekshahi, Hosseini, and Ansari (2020) proposed theoretical estimation for vibration and postbuckling structure with hollow sections. It can be seen that although the classical beam theory is simply but it ignores the shear effect, which becomes significant for the thick beams. Therefore, to account this effect, a large number of studies are developed to predict behaviors of thin-walled composite beams. Mao and Lin (1996) studied buckling and post-buckling behaviors of thinwalled composite beams with simply supported boundary conditions using trigonometric series solutions and perturbation method. Pagani, Carrera, and Ferreira (2016) investigated free vibrations of thin-walled beams using higher-order shear deformation theories and radial shape functions. Ascione, Feo, and Mancusi (2000) proposed a shear deformable model to determine the deflections of composite channel beams. Lee (2005) analyzed bending behaviors of thin-walled composite I-beams. Sheikh and Thomsen (2008) presented a new beam element to analyze thin-walled composite beams with open or closed section. Cort ınez and Piovan (2006) analyzed nonlinear buckling thin-walled composite beams including shear effects. Back and Will (2008) predicted the buckling loads and deflections of thin-walled composite I-beams. Kim and Lee (2014) proposed exact solution for vibration and buckling of thin-walled composite beams with open section. Maceri and Vairo (2009) proposed a new model for thin-walled anisotropic beams. The shear deformable theory was also used by Kim and Lee (2017b) to analyze thin-walled FG sandwich I-beams. Pavazza, Matokovi c, and Vukasovi c (2020) proposed a torsion theory for isotropic thin-walled beams considering shear effect. Sofiyev (2014), Sofiyev and Osmancelebioglu (2017), and Sofiyev et al. (2016aand Sofiyev et al. ( , 2016b analyzed buckling and vibration of FG cylindrical shells based on a first-order shear deformation theory. In these studies, authors investigated the effects of shear stress, FG core, sandwich shell geometry on critical loads, and frequencies of shell. For computational approach, the FEM is increasingly used for bending and buckling analysis of thinwalled composite beams. Koll ar and Pluzsik (2012) analyzed bending and torsion behaviors of thin-walled composite beams. Aguiar, Moleiro, and Soares (2012) developed FEM to analyze bending of composite beams with various cross-sections. G€ unay and Timarci (2017) presented static behaviors of thin-walled composite beams with closed-section by the classical beam theory. Kim and Lee (2018) proposed a nonlinear model of thin-walled FG I-beams.  analyzed static and dynamic behaviors of thin-walled composite channel beams by a shear deformable theory. FEM is also used to analyze bending, buckling, and vibration of FG beam with honeycomb core (Li, Shen, and Wang 2019a, 2019b, 2019c, 2019d. Later, by stiffness matrix method and FEM, Kim, Jeon, and Lee (2013), Kim and Shin (2009), and Kim (2011, 2012 determined the deflections of thin-walled composite beams with mono-symmetric I-, L-, and channel section. Lanc et al. (2016) used FEM to analyze buckling behaviors of thin-walled FG beams. Isogeometric analysis method was used to deal with thin-walled composite curved beams (C ardenas et al. 2018). Harursampath, Harish, and Hodges (2017a) proposed the variational asymptotic method and used Monte-Carlo-type stochastic approach for behavior analysis of thinwalled composite beams. Ritz method is simple and effective to analyze bending, buckling, and free vibration of composite beams with rectangular cross-section (Aydogdu 2006a(Aydogdu , 2006bŞimşek 2009;Pradhan and Chakraverty 2013;Mantari and Canales 2016;Nguyen et al. 2017), however, it is rarely used for thin-walled composite beams. Qiao and Zou (2002) presented free vibration of fiber-reinforced plastic composite cantilever I-beams using the Ritz method with transcendental and polynomial shape functions satisfying the boundary conditions. Nguyen et al. (2019) developed the Ritz method for vibration and buckling of thin-walled composite I-beams. From literature review, Ritz method has not been previously used to analyze vibration, bending, and buckling behaviors of thin-walled composite channel beams. Due to asymmetric geometric and material anisotropic of composite channel section, shear center, and centroid are not coincided. This causes coupling responses from axial, bending, lateral, torsional, and warping behaviors, thus, their structural responses are is very complex. Besides, it can be seen that the effect of shear deformation on the structural responses of thin-walled FG sandwich channel-section beams has not been available yet. Therefore, there is a need for further studies related to these complicated problems.
This article, which is extended from previous study (Nguyen et al. 2019), focuses on bending, vibration, and buckling analysis of thin-walled FG sandwich and composite channel beams. It is based on the first-order shear deformation theory, which can recover to classical one by ignoring the shear effect. Lagrange's equations are employed to formulate the governing equations to describe the structural responses of beams and Ritz method is used to solve the problems. Numerical examples are performed to verify accuracy and efficiency of the present solutions. Many significant effects such as span-to-height ratio, material parameter, lay-ups, fiber orientation, boundary conditions on the beams' deflection, frequency, and critical buckling load are investigated.

Theoretical formulation
In this section, a displacement field and constitutive equations of thin-wall FG sandwich and composite beams are established. Next, their strain energy, work done by external forces and total potential energy are determined, and finally, the Ritz method is proposed to solve structural responses of such beams with various boundary conditions.
To develop displacement field of thin-walled beams, local coordinate system (n, s, z), Cartesian coordinate system (x, y, z), and contour coordinate S along the profile of the section are used as shown in Fig. 1 (Lee 2001). The P is called the shear center and the axis, which is through P and parallel to the axis z, is called the pole axis. h is an angle of orientation between (n, s, z) and (x, y, z) coordinate systems. r and q are coordinates of any point on the contour measured from P in (n, s, z) coordinate system. The basic assumptions are (a) the contour of section does not deform in its own plane; (b) shear strains c 0 xz , c 0 yz and warping shear c 0 -are uniform over the section; (c) Poisson's coefficient is constant.

Kinematics
The displacements (u, v, w) at any point in the contour are expressed through the mid-surface displacements ( u, v, w) and the rotations of transverse normal about s and z ð w s , w z Þ as followings (Lee 2005;Vo and Lee 2009;Nguyen et al. 2019): The mid-surface displacements ( u, v, w) and rotations of transverse normal about s and z ð w s , w z Þ are related to displacements of P in x-, y-, and z-directions ðU, V, WÞ and rotations of the cross-section with respect to x, y,and pole axis ðw x , w y , w -, /Þ as (Lee 2005;Vo and Lee 2009;Kim and Lee 2017b;Nguyen et al. 2019): w -¼ c 0 -À / 0 (3c) and the prime designates the derivative with respect to z;is warping function given by: The nonzero strains of thin-walled beams are defined as (Lee 2005): It can be seen that e 0 z , j x , j y , j -, and j sz are axial strain, biaxial curvatures in the x-and y-directions, warping cuvature with respect to the shear center and twisting cuvature in the beam, respectively.

Constitutive equations
2.2.1. Thin-walled FG sandwich beams Young's modulus (E) and mass density ðqÞ of thin-walled FG beams is expressed through the volume fraction of ceramic ðV c Þ, Young's modulus and mass density of ceramic and metal ðE c , E m , q c , q m Þ: Three types of material distributions are considered as follows ( Fig. 2): Type A: where h ðh 1 , h 2 , h 3 Þ is the thickness of the flanges or web and p is material parameter. Type B: , À 0:5h n À0:5ah or 0:5ah n 0:5h (9a) where a ða 1 , a 2 , a 3 Þ is thickness ratio of ceramic material of the flanges or web. Type C: The stress and strain relations can be written as: where and is Poisson's coefficient.

Thin-walled composite beams
The stress and strain relations at the kth-layer in (n, s, z) coordinate systems can be determined as: where: In Eq. (14), Q ij are the transformed reduced stiffnesses (Reddy 2003).

Variational formulation
The strain energy P E of the system is defined by: where X is volume and k s is shear correction factor, which is assumed to be a unity in previous study (Nguyen et al. 2019). Substituting Eqs. (5a)-(5c), (11), and (13) into Eq. (15) leads to: where L is length of beam and E ij are the stiffness coefficients of thin-walled FG and composite beam, which depend on the geometry and material distributions in cross-section (see (Lee 2005) for more details). The work done P W of the system by uniform load q y and concentrated load P y applied at z L and axial load N 0 can be expressed as (Lee 2005;Back and Will 2008): The kinetic energy P K of the system is given by: where dot-superscript denotes the differentiation with respect to the time t, and the inertia coefficients are defined in Vo and Lee (2009).
The total potential energy of the system is obtained by:

Ritz solutions
The displacement fields of the thin-walled composite beams are approximated by using Ritz's approximation functions: where i 2 ¼ À1 is the imaginary unit; x is the frequency; W j , U j , V j , / j , w yj , w xj , and w -j are Ritz's parameters, which need to be determined and u j ðzÞ are Ritz's approximation functions which depend on boundary conditions (BCs) as seen in Table 1. Four typical BCs as simply supported (S-S), clamped-free (C-F), clamped-simply supported (C-S), and clamped-clamped (C-C) are considered. By substituting Eq. (20) into Eq. (19), Lagrange's equations are used to formulate the governing equations: with p j representing the values of ðW j , U j , V j , / j , w yj , w xj , w -j Þ: Bending, vibration, and buckling behaviors of the thin-walled beams can be obtained by solving the following equation, which presents relations of stiffness matrix K, mass matrix M, displacement, and force vetor F: The explicit forms of stiffness matrix K, mass matrix M, and force vector F are given in Appendix A.
In case of ignoring the shear effect as the classical beam theory, Eqs. (3a)-(3c) degenerate to w y ¼ ÀU 0 , w x ¼ ÀV 0 , w -¼ À/ 0 and only four unknown variables ðW, U, V, /Þ are available. Thus, the bending, vibration, and buckling behaviors of the thin-walled beams in this case can be reduced: Figure 7. E 33 =E 77 ratio of FG sandwich beams for C2-and C3-section with respect to material parameter.
The coefficients of the stiffness matrix NS K, mass matrix NS M, and force vector NS F are given in Appendix B. Figure 8. Shear effect on the deflections of FG sandwich C3-beams (L=b 3 ¼ 10) with respect to ceramic's thickness ratio of top and bottom flanges ða 3 ¼ 0:1, a 1 ¼ a 2 Þ and ceramic's thickness ratio of web ða 1 ¼ a 2 ¼ 0:9, a 3 Þ:

Numerical results
In this section, numerical examples are carried out to show the accuracy of the present solutions, and then, investigate bending, vibration, and buckling behaviors of thin-walled FG sandwich and composite channel beams. The shear effect is defined as R S À R NS j j =R S Â 100%; where R NS and R S denote the results from classical and shear deformable theory, respectively. Unless other states, the material and geometry properties used in this section are given as follows: For FG sandwich channel beams (Fig. 2): Bending and buckling analysis: . Shear effect on the critical buckling load and fundamental frequency of FG sandwich beams (C2-section, a 1 ¼ a 2 ¼ a 3 ¼ 0:4 and p ¼ 2) with respect to L/b 3 for various BCs.

Convergence study
To study convergence of the present solutions, FG sandwich C1-beams (p ¼ 10 and L=b 3 ¼ 20) and composite channel C4-beams, whose lay-ups in the flanges and web are 30= À 30 ½ 4s and L=b 3 ¼ 20) subject to a vertical concentrated load (P y ¼ 1kN) acting at mid-span with the various Figure 10. Shear effect on the critical buckling load and fundamental frequency of FG sandwich C-C C2-beams (L=b 3 ¼ 5) with respect to ceramic's thickness ratio or material parameter.
BCs are considered. Their mid-span deflections, critical buckling loads, and fundamental frequencies are shown in Tables 2 and 3 with various series number m: For all BCs, it can be found that the present solutions converge at m ¼ 12 for deflections, and m ¼ 10 for critical buckling loads and frequencies. These numbers of series terms will be used in the next sections.

Verification
For bending problem, FG sandwich channel cantilever beams ðL=b 3 ¼ 50Þ with C1-, C2-, and C3-sections are considered. The thickness ratio of ceramic material is taken as ða 1 ¼ a 2 ¼ a 3 ¼ 0:4Þ for C2-section, and ða 1 ¼ a 2 ¼ 0:9, a 3 ¼ 0:1Þ for C3-section. To compare with the results of Nguyen, Kim, and Lee (2016a), which used classical beam theory, nondimensional vertical Figure 11. Shear effect on first three buckling load and natural frequencies of FG sandwich C-C C2-beams (a 1 ¼ a 2 ¼ 0:4, p ¼ 2, and L=b 3 ¼ 5) with respect to ceramic's thickness ratio of web.    Fig. 4. The present results are an excellent agreement with those of Nguyen, Kim, and Lee (2016a) for all sections. It is noted that there is not much discrepancy between results of shear and no shear models due to their slenderness ðL=b 3 ¼ 50Þ: To verify further, FG sandwich C1-beams ( with L=b 3 ¼ 12:5 for buckling and L=b 3 ¼ 40 for free vibration are considered. Nondimensional frequency is defined as : Their results are given in Tables 4 and 5, and compared with those from Lanc et al. (2016) and Nguyen, Kim, and Lee (2016b), which based on classical beam theory. It is seen that the present results are good agreement with those of previous studies.

Parameter study
3.2.2.1 Bending analysis. FG sandwich channel beams under a uniform load (q y ¼ 0.5 kN/m) for various BCs, L=b 3 , and p are considered to study span-to-height ratio ðL=b 3 Þ and material parameter ðpÞ: Tables 6-8 show their mid-span deflections with C1-, C2-sections ða 1 ¼ a 2 ¼ a 3 ¼ 0:4Þ, and C3-section ða 1 ¼ a 2 ¼ 0:9anda 3 ¼ 0:1Þ: It can be seen that they are the largest for C-F and smallest for C-C beams, as expected. Besides, they increase as p increases for all configurations.
The shear effect on the deflections with respect to L=b 3 ðp ¼ 10Þ, and with respect to pðL=b 3 ¼ 10Þ are shown Figs. 5 and 6 for C2-and C3-sections. As L=b 3 increases, this effect decreases and has the highest value for C-C beams and the lowest one for C-F beams. For C2-section, it does not depend on material parameter for all BCs (Fig. 5b), which can be explained partly by the constant ratio E 33 =E 77 in Fig. 7. However, for C3-section, it strongly increases from 0 p 5 after that, increases slowly from 5 p 40 as shown in Fig. 6(b) and again can be justified by ratio ðE 33 =E 77 Þ: Next, FG sandwich channel C3-beams under uniform load (L/b 3 ¼ 10, p ¼ 2, and q y ¼ 10 kN/ m) are used to study the shear effect with respect to variation of ceramic's thickness ratio in the flanges and web in Fig. 8. This effect increases as ceramic's thickness ratio in the top and bottom flanges increases, whilst it decreases as ceramic's thickness ratio in the web increases.

Vibration and buckling analysis.
To investigate shear effect on the critical buckling load and natural frequencies, FG sandwich channel C2-beams (h Variations of shear effect with respect to L=b 3 , ceramic's thickness ratio in flanges ða 1 , a 2 Þ, web ða 3 Þ, and material parameter (p) are showed in Figs. 9 and 10. It can be observed that this effect decreases as ceramic's thickness ratio in flanges increases; increase as ceramic's thickness ratio in web increase, and hardly depend on p. It is significant for higher buckling and vibration modes as shown in Fig. 11.

Verification
For bending problem, a cantilever composite channel C4-beam (L=b 3 ¼ 20Þ under a vertical load ðP y ¼ 1kNÞ acting at free end is analyzed. This beam is made by 16 layers of symmetric angle-ply lay-ups in the flanges and web. Maximum deflections are given in Table 9 for both shear and no shear case, and compared with results of Kim, Jeon, and Lee (2013). Again, the current results are coincided with those from previous research.
For vibration and buckling problems, composite cantilever C4-beams  Tables 10 and 11, and compared with Kim and Lee (2014). It can be seen that there are absolutely coincided between the present results and those of Kim and Lee (2014). 3.3.2. Parameter study 3.3.2.1. Bending analysis. Table 12 presents the mid-span deflections of C4-section beams subjected to a uniform load ðq y ¼ 0:1kN=mÞ for various ðL=b 3 Þ: Figure 12 shows the shear effect on the deflection of beams with lay-up ½45= À 45 4S for various BCs. It can be found that the deflections increase as ðL=b 3 Þ and fiber orientation increases.
To further examine the shear effect with respect to the fiber orientation, Fig. 13 illustrates the results for C-C composite channel beams ðL=b 3 ¼ 10, q y ¼ 0:1kN=mÞ with C5-and C6-sections. The C5-section has the web considered as unidirectional, and the top and bottom flanges assumed to be angle-ply laminates ½h= À h, while the C6-section has the top and bottom flanges considered as unidirectional, and the web assumed to be angle-ply laminates ½h= À h as shown in Fig. 3. It is clear that the shear effect depends on fiber orientation, and it is stronger for C6-section than C5-section. It is interesting that for C5-section, this effect is minimum at fiber angle h ¼ 60 whereas for C6-section, it is minimum at h ¼ 40: This phenomenon is explained in Fig.  14, where ðE 33 =E 77 Þ ratio is plotted with respect to fiber orientation.

Vibration and buckling analysis.
To investigate shear effect on the critical buckling loads and natural frequencies, composite channel C4-beams Variation of shear effect on critical buckling load and frequency of beams (½45= À 45 4S in flanges and web) respect to L=b 3 is plotted in Fig. 15. This effect also depends on fiber angle as shown in Fig. 16. It is more pronounced for higher buckling and vibration modes (Fig. 17).

Conclusions
Based on the first-order shear deformation theory, bending, vibration, and buckling analysis of thinwalled FG sandwich and composite channel beams is studied. Lagrange's equations are used to formulate the governing equations. Ritz method is developed to obtain the deflections, natural frequencies, and buckling loads of thin-walled FG sandwich and composite channel beams. Both results from classical and first-order shear deformation beam theories are derived in a unified fashion. Numerical results are obtained and compared with those available in the literature. Some new results are displayed as benchmark values in the future. The results indicate that the present study is efficient and accurate to analyze the structural responses of FG sandwich and composite channel beams.