Buckling analysis of open-section beams with thin-walled functionally graded materials along the contour direction
Introduction
The main concept of Functionally graded material (FGMs) first appeared in Japan in the 1990s, report by oprKoizumi (1993); Koizumi (1997), Yamaoka et al. (1993) and Pindera et al. (1994); Pindera et al. (1997) for development and implication of thermal barrier materials. FGMs are known as special composite mixture of two or more materials in which the volume fractions are continuously varied depending on the position along a certain dimension to achieve the desired function. Interest in FGMs has recently escalated due to the ability to produce materials with tailored properties which are suitable for numerous high tech applications, such as, aerospace, bioengineering and nuclear industries. As the use of FGMs increases, it is necessary to develop new methodologies for their characterization, as well as for the development and processing of these materials. As such, the number of publications in this area of research has been increasing exponentially in the last decades, also become gradually attractive in many engineering fabrication areas. With the potential benefits of using FGMs, many researchers have been investigating various aspects of static, vibration, buckling (Abrate, 2006; Neves et al., 2013; Eltaher et al., 2014; Ziane et al., 2015, 2017; Mantari and Monge, 2016; Şimşek, 2019; Nguyen et al., 2019) and the microstructure of the FGMs (Ebrahimi et al., 2019; Uzun et al., 2020).
Many researches on buckling analysis of FGMs structures have been carried out in the recent years. Sofiyev (2010) proposed the buckling analysis of FGMs shells under the axial extension loads and hydrostatic pressure. Dong and Li (2017) presented a unified nonlinear analytical solution of bending, buckling and vibration for the temperature-dependent FG rectangular plates subjected to thermal load. The higher order shear and normal deform-able plate theory for buckling and free vibration analysis of in-compressible FG rectangular thick plates is introduced by Mohammadi et al. (2019). Nateghi et al. (2012) analyzed the buckling analysis of FG micro beams based on modified couple stress theory with three different beam theories. Şimşek and Reddy (2013) proposed a unified higher order beam theory to proposed buckling of a FG microbeams. It is found that the elastic medium constants have an increasing effect on the critical buckling load. Dabbagh et al. (2020) studied the analysis of the post-buckling behaviors of multi-scale hybrid nanocomposite beam-type structures and the impact of the existence of agglomeration phenomenon in the multi-scale hybrid nanocomposite materials.
According to those papers, the authors had mainly focused on the overall stability of the above-mentioned FG structures in which the flexural, torsional and flexural-torsional buckling configurations were commonly investigated. Furthermore, with the help of extensive mathematical theories, e.g., first-order shear deformation, inverse trigonometric deformation theory,… FGMs can be designed for specific function and applications. They showed that FGMs core may mitigate or even prevent impact damage on structures while also yields a significant weight reduction.
Up to now, the attention has rarely been concentrated on lateral buckling behavior which is an important information for revealing the response of the structural failure under various loading conditions. Andrade et al. (2007) employed the so-called 3-factor formula to estimate the elastic critical moment of cantilever I-beams prone to lateral-torsional buckling. Mohri et al. (2010) developed the 3-factor formula for the lateral buckling stability of thin-walled beams with consideration of pre-buckling deflection and load height effects. Dourakopoulos and Sapountzakis (2010) investigated the post-buckling analysis of beams of arbitrary cross section with large displacements and large angles of twist. The post-buckling behavior of multi-directional FGMs place are investigated through the combination of NURBS-based IsoGeonetric Analysis in Yang et al., (2020).
For thin-walled laminated composite beam, buckling by bending and torsion separately occurring under axial load were developed in (Lee and Kim, (2001); Zhang and Tong, (2004); Vo and Lee, (2007); Asadi et al., (2019)), the lateral buckling analysis under various configurations were also parametrically studied in (Lee and Kim, (2002); Machado and Cortínez, (2005); Sapkás and Kollár, (2002); Lee, (2006); Asgarian et al., (2013)) by many researchers. In many years, the separate modeling of in-plane and through-the-thickness properties is developed to better tackle the numerical complexity of multi-layer composites, as well as yield accurate prediction of the buckling capacity of structures.
Regarding to thin-walled FGMs, the locations of center of gravity and shear center are directly depend on its geometries and material properties, they therefore have strong influences over buckling characteristics. It is important to remind that the behavior and the stability of thin-walled beams are more complex in the presence of cross-sections. In such situations, all cross-section constants and positions of shear and centroid positions are not evident. Buckling thin-walled FGMs, therefore, needs to be investigated to predict accurately the critical load for various material distributions and properties throughout the structures.
Most of the works concerning the buckling stability of thin-walled composite beams concentrated on material properties graded through the wall thickness (Oh et al., 2005; Librescu et al., 2005; Sahraei et al., 2015; Lanc et al., 2016). Lanc et al. (2015) performed the global buckling behavior of thin-walled FG sandwich box beams with various boundary conditions. Nguyen et al., 2017a, 2017b also dealt with similar problem in which effects of gradual law, ceramic core and skin, span-to-height on the buckling parameters of an axially loaded thin-walled FG open-section beam.
In furtherance of thin-walled FG analysis (Phi et al., 2020), the study aims to present a general analytical model in examining the global buckling of thin-walled FG channel-section beam, as well as the lateral buckling of thin-walled FG mono-symmetric I-section beam. The model is based on Vlasov's theory and accounts for the material anisotropy, warping of cross-section and various boundary conditions. A finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled beam, also that preserving the performance of the overall analysis. The principle of the stationary value of total potential energy is used to solve governing buckling equations. With the assumption the material properties are graded through the contour direction by the power law distribution of volume fraction. To access the theory, various types of material distributions are evaluated. The critical load is obtained for thin-walled FG beams under uniformly distributed load and moment gradient. In order to verify the results of this investigation, several numerical examples are presented and compared with those obtained in the literature.
The paper is structured as follows. The coordinate systems, assumptions and strain of thin-walled open-section are given in Section 2. The variational formulations are shown in Section 3, several types of material distributions and governing buckling equations in Section 4 Material distribution, 5 Governing equations for buckling. In addition, a finite element method using 2-node 14-degree-of-freedom element is also provided in Section 6. Numerical verification and examples for flexural-torsional buckling with channel-section and lateral buckling with mono-symmetric I-section are discussed in Section 7. Finally, the paper ends up with several remarked statements in Section 8.
Section snippets
Displacement fields
In order to build a general model for thin-walled beam, three different coordinate systems are considered: an orthogonal Cartesian coordinate global system , an orthogonal local coordinate system and a contour coordinate axis as shown in Fig. 1. The n-axis is normal to the middle surface of a plate element, the s-axis is tangent to the middle surface and is directed along the contour line of the cross-section. The and coordinate systems are related through an
Variational formulation
The strain energy in a system domain is defined by
The variation of strain energy is calculated by substituting Eq. (11) into Eq. 13where L is the length of beam, is the internal force vector, defined bywhere and are axial force, bending moments in the x- and y-directions, warping and torsional moments, respectively. The explicit form can be expressed as
Material distribution
Using a simple rule of mixture of constituent material properties where represent the characteristics of ceramic and metal, respectively, same for Young's modulus, shear modulus, Poisson's ratio and density.
Assume that volume fraction distribution of the ceramic and metal can be written as
The transformation of Young's modulus, for example, for rule of mixture (Reddy, 2000) with respect to various power-law index p from ceramic to metal (,
Governing equations for buckling
The stress-strain relation for the FG beam are given bywhere and are normal axial and shearing stress, respectively, and is the transformed reduced stiffness defined by
The axial force can be obtained with respect to the generalized strains as
Similarly, the other stress resultants can also be written in terms of the generalized strains . Consequently, the constitutive equations for a thin-walled FG
Finite element formulation
The present theory for thin-walled FG beams described in the previous section was implemented via a displacement based finite element method. Each note of beam element have seven nodal degrees of freedom including the warping. The generalized displacements are obtained over each element as a linear combination of the Lagrange interpolation function and Hermite-cubic interpolation function associated with node j and the nodal values
The finite element model of a
Numerical examples
In the following section, the performance of flexural-torsional and lateral buckling for a thin-walled FG open section beam has been conducted. Firstly, the special case via benchmark problems from the literature of isotropic material which provided a reference solution are considered. Several examples including various materials distributions of FGMs are studied. In addition, effects of gradual law, load height, also span-to-height on the buckling behavior of a thin-walled FG beam have been
Concluding remarks
Buckling analysis with an axially loading and lateral buckling analysis with different kinds of loading of thin-walled open-section beam with functionally graded materials have been studied. The material properties are considered to vary in the contour directions by the power gradation laws, and they are estimated by the conventional Lagrange and Hermite interpolations. Several examples for a variety of shell geometries have been performed such as mono-symmetric I- and channel-sections.
Authorship statement
Linh T. M. Phi: Methodology, Visualization, Software, Investigation, Data curation, Writing – original draft. Tan- Tien Nguyen: Software, Investigation. Jaehong Lee: Conceptualization, Writing – review & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was funded by a grant (2020R1A4A2002855) from NRF (National Research Foundation of Korea) funded by MEST (Ministry of Education and Science Technology) of Korean government.
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