Flexural analysis of I-section beams functionally graded materials

. This paper aims to present the flexural behavior of thin functionally graded (FG) I-beams. The characteristics of metal-ceramic materials are defined using a power-law function dependent on volume fraction. The bending-torsion equations for this problem are derived based on Vlasov's theory for thin-walled beams and the principle of minimum total potential energy. All geometric properties are expressed according to the functional graded power index law. A nonlinear algebraic system is obtained, and the deflection of the structure is numerically derived. To confirm the precision and effectiveness of the suggested method, a standard benchmark test case is implemented, that concerns the flexural analysis of an FGM I-section beam according to power index and aspect ratio parameters.


Introduction
Thin-walled open section beams have garnered considerable interest across diverse engineering applications, spanning from automotive design to the development of aeronautical components.These beams possess several advantageous characteristics: enhancing buckling structural capacity, improving mechanical strength and stiffness distribution, optimizing material utilization, and meeting architectural requirements [1,2].
In these structural components may experience flexural-torsional instability because of external transverse loading, eccentric compressive axial force, low cross-sectional torsional stiffness, and the symmetry or asymmetry of the member's cross-section.Therefore, advancements in manufacturing processes and the need for optimal weight distribution with superior stability have enabled the production of beams with varying cross-sections using different structural materials like wood, steel, and composites.
A Functionally graded material (FGM) is a sophisticated multi-phase composite that exhibits a gradual variation in the volume fraction of components in one or more privileged directions.Initially used as thermal barrier materials in aerospace structures and fusion reactors due to their ceramic properties, FGMs offer distinct advantages over traditional laminated composites.They eliminate stress concentration, the risks associated with cracks and delamination.Featuring remarkable mechanical properties such as excellent thermal resistance, minimal stress concentrations, and high strength, FGMs find suitable applications in automotive, aerospace, military, and biomedical fields [3,5].
With the growing use of Functionally Graded Materials (FGMs), there is a pressing need to advance methodologies for their characterization, development and processing.The concept of Functionally Graded Materials (FGMs) first emerged in Japan during the 1990s [6].
Consequently, the volume of research publications in this field has experienced exponential growth over the past decades, making FGMs progressively appealing in various engineering fabrication domains.Bourihane et al. [7,8] proposed novel finite elements to explore the buckling and vibration behavior of rectangular plates made of Functionally Graded Materials (FGM) using an eigenvalues approach.Sitli and co-workers [9] analyzed the post-buckling behavior of rectangular plates by means of the ANM method.Abrat [10] explored the issues related to natural vibrations, buckling, and static deflections in functionally graded plates with varying material properties across their thickness.The Spectral high-order continuation approach proposed in [11] was used to study the bending and buckling of FG and sandwich beams in a geometrically nonlinear context by Mesmoudi and co-workers.Koutoati et al. [12] introduced a finite element model for beams based on the zigzag theory.This model was utilized for analyzing the static and free vibration behavior of sandwich beams made of Functionally Graded Materials (FGM) with viscoelastic nonlinear material properties.Ziane et al [13] studied the free vibration of FGM box girders by formulating an accurate dynamic stiffness matrix utilizing the first-order shear deformation theory (FSDT).Lanc et and co-workers [14] conducted a buckling analysis of thin-walled functionally graded (FG) sandwich box beams using the Euler-Bernoulli beam bending theory and Vlasov torsion theory.The study explored various boundary conditions, including simply supported, freely constrained, and constrained configurations.Phi et al. [15] presented a free vibration study of a thin-walled beams using functionally graded materials along the contour direction.Recently, Soltani et al. [16] conducted an evaluation to examine the influence of axial preload on the lateral stability capacity of I-beam-column elements, taking into account size-dependent characteristics and variable material properties along the axial direction under lateral loads.Kim and Lee [17] investigated the bending-torsional behavior of shear-flexible thin-walled sandwich I-beams constructed from functionally graded materials.Their study took into account warping shear deformations and transverse shear effects.Nguyen et al. [18] investigated the flexural, torsional, and flexural-torsional buckling responses of thin-walled functionally graded beams subjected to axial loading.The analysis included beams with open cross-sections and diverse material distributions.An analysis was conducted using a non-linear model to study the significant torsional behavior of thin-walled tapered beams with open cross sections.byMohri and co-workers.[19].
Based on these papers, the authors primarily emphasized the overall stability of the mentioned FG structures, with particular attention to the investigation of flexural, torsional, and flexural-torsional buckling configurations.The researchers demonstrated that FGMs can be tailored for specific functions and applications.They also highlighted that the FGM core has the potential to mitigate or even prevent impact damage in structures, leading to a substantial reduction in weight.
The objective of this research is to introduce a comprehensive analytical model for investigating the flexural behavior of I-section beams made from functionally graded (FG) E3S Web of Conferences 469, 00043 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900043materials.This model is constructed based on Vlasov's thin-walled beam theory, and the governing equations are derived using the principle of stationary conditions of total potential energy.The analysis specifically focuses on mono-symmetric I-sections, the resultant equilibrium equations are transformed into a linear algebraic system.Consequently, the deflection is expressed based on the maximum bending moment.

Equilibrium equations
Consider a thin-walled beam with an open section of length .Here,  represents the initial axis along the beam's length, while  and  denote the two principal axes within the section.The origin of the coordinate system is denoted as , and (  ,   ) indicates the shear center.Let's examine a point  within the cross-section of the beam, specified by coordinates (, , ), where  represents the angular coordinate characterizing the warping of the section due to non-uniform torsion.The parameter  serves as the sectorial coordinate used in Vlasov's model for non-uniform torsion [20,21].The linear Vlasov beam model, presented in the context of these structures, describes the torsion and warping of the cross-section while accounting for all structural couplings resulting from material anisotropy and cross-sectional warping, based on two fundamental assumptions, including: 1. Rigidity in Plane: The contour of the cross-section remains rigid within its plane, implying no distortion deformation of the section.2. Shear Deformation Absence: There is no shear deformation in the mean surface of the section.In the framework of small deformations and small rotations which leads to the approximation of trigonometric functions  ≃ 1 and  ≃ , the three displacement components of a point  are typically expressed linearly according to Vlasov's theory [11]: where  represents the longitudinal displacement and is considered to be zero. and  are the transverse displacements of the shear center , and  represents the torsion angle.The components of the Green's strain tensor of thin-walled beams are obtained by applying equation (1) as described in reference [22] and are given by: In this model, it is assumed that the beam demonstrates elastic behavior.The equilibrium equations are derived based on the stationary conditions of the total potential energy, which is calculated from the following formulation: with  denotes the virtual variation,  is the strain energy and  is the external load work.The variation of the strain energy is given by: The variation of the strain tensor components is determined using Equation ( 2), where   represents the Piola-Kirchhoff stress tensor.
The variation in strain energy can be represented as a function of the stress resultants by substituting Equation (6) into Expression (5) in the following manner: +    ′′ −    ′′ −    ′′ +    ′ ) where  is the normal stress,   and   denote the bending moments,   signifies the bimoment and   stands for the Saint-Venant torsion moment given by (see Fig. Examining the initial bending response of the Functionally Graded Material (FGM) beam around its principal axis, the applied loads are simplified to a lateral distributed load   , acting on point  located along the section contour.The external work  is given as follows: where   is the eccentricity of the applied loads from the shear center .
Utilizing Equations ( 7) and ( 9), the variation in the total potential (4) can be articulated as a function of virtual displacements and their derivatives.Employing integration by parts on Equation (7) results in an expression dependent solely on virtual displacements , , , and .The equilibrium equations, derived from the stationary conditions, can thus be represented as follows: The stress resultants are derived from Equations ( 8), representing functions of strain components and the stiffness properties of thin-walled functionally graded (FG) beams.Consequently, the constitutive equations for a thin-walled FG beam can be formulated in a matrix representation as described in reference [18]: where {S} and {γ} are the vector of generalized stresses and the vector of generalized strains.Note that matrix [] is diagonal in the context of homogenous open section beams.However, all coupled terms   ( ≠ ,  ≠    ≠ ) are present due to the material gradation.These coefficients   are the different stiffnesses of thin-walled FG beam, defined as follows [15]: Using Equations ( 10) and ( 11), the equilibrium equations can be developed as: − 100  ′′ +  200  (4) +  110  (4) −  101  (4) = 0 − 010  ′′ +  110  (4) +  020  (4) −  011  (4) =    001  ′′ −  101  (4) −  011  (4) −  002  (4) −  ′′ =     (13) In these equations, (. ) (4) represents the fourth-order derivative with respect to .The bending and torsion displacement modes for simply supported beams with free warping are estimated through an approximation employing sinusoidal functions, and  0 ,  0 and  0 are the associated displacement amplitudes: The equilibrium equations can be presented as:  (15) In order to get an expression of the deflection  0 as function of  0 , the maximal bending moment of the beam, this expression  0 =    2 8 will be insert in Equations system [23] and the linear system (15) will be solved.

Applications and numerical results
In this research, the deflection analysis of an I-section beam made from functionally graded (FG) material, represented here as a standard IPE300, is calculated under a distributed load   (see Fig. 2).The specific section type considered in this example is Pure Functionally Graded Material or Type A (see Fig. 2), This type comprises ceramic (c) and metal (m) components and is characterized using the P-FGM law.The volume fraction in this law is expressed in Equation ( 16), with   = 210,   = 70 and υ = 0.
Fig. 2 Section dimensions and   load In this study, various values of the power index were considered.The deflection values, corresponding to maximum moment  0 , were calculated for two different beam lengths 6  and 3 , and the results are presented in Table 1.
For  = 0, the deflection values are 0.016 when the applied moment  0 is 284.3  for  = 3, and 0.022  when  0 equals 98.3 .The numerically obtained deflection values are approximately consistent with those obtained in the referenced study [23] when the power index  is equal to 0.  1, the present example shows that for the two different values of  0 , the beam deflection  0 increases as the value of  rises and the material distribution has an impact on the beam's bending behavior.

Conclusion
In conclusion, this study conducted a flexural analysis of an I-section beam made from functionally graded material (FGM).The derivation of equilibrium equations was a central focus, particularly within the context of a simply supported bisymmetric I-section configuration.This project resulted in the acquisition of an algebraic system, and subsequently, an analytical expression for deflection was established in terms of the maximal bending moment.It's worth noting that the insights gained from this investigation lay a foundation for potential extensions, particularly towards addressing lateral buckling challenges within the realm of FGM beams.

Table 1 .
Numerical results