Comparison of Deflection in Two-Directional Functionally Graded Tapered Beam

1. C. M. Reddy ^a* , P. Bridjesh ^a, B. S. Reddy ^b, K. V. K. Reddy ^c

^a Department of Mechanical Engineering, MLR Institute of Technology, Hyderabad, India

^b Department of Mechanical Engineering, Rajeev Gandhi Memorial College of Engineering &Technology, Nandyal, India.

^c Department of Mechanical Engineering, JNTU Hyderabad, Hyderabad, India.

1.     Introduction

Functionally Graded Material (FGM) [1] was developed in Japan in the mid-1980s for structural and functional applications requiring high-temperature resistance and strength. FGM is an advanced composite material with dimension-dependent properties. High-temperature FGMs are inhomogeneous composites made of ceramic and metal phases. FGM effortlessly changes mechanical and physical properties by mixing metal and ceramic at suitable volumes. Ceramic resists high temperatures due to low thermal conductivity [2]. Ductile metal can withstand high-temperature gradient stresses for a short time [3]. Thus, low and high-temperature FGM structural components can use metals and ceramics. FGM is mostly used in structural applications that integrate refractoriness and toughness. Rocket engine combustion chambers, high-performance tools, etc. Heat shields in spacecraft, tubes in a heat exchanger, implants in biomedical applications, and fusion reactor plasma facings can use FGMs [4, 5]. Chemical facilities, nuclear reactors, aircraft, and other incompatible applications use FGM [6]. High-temperature thermal barrier coatings attach ceramic layers to metallic structures.

In high-temperature applications, a beam's end surfaces often have different temperatures (top and bottom). Consequently, the member is subjected to thermal loading because steady-state conditions facilitate heat conduction across the beam thickness [7].

Beam theories describe beam-type structures. Euler–Bernoulli beam theory is the oldest and most famous beam theory is Classical Beam Theory (CBT). Top and bottom shear stress violates free boundary conditions. First-order shear deformation theory's shear force discrepancy requires a shear correction factor (k) [8]. Researchers developed Higher Order Shear Deformation Theory (HSDT) to overcome this problem. Shear deformation is widely noticeable in the thick beam. Beam theory ignores these effects. HSDT must accurately describe thick beam bending behavior, including cross-sectional warping. Appropriate kinematics and constitutive models enable this. The higher-order theory is accounted for transverse shear deformation by incorporating the function f(x) into the displacement field [9-10].

Ebrahimi et al. [11] using a Navier-type solution, this work investigates the free vibration characteristics of Functionally Graded (FG) nanobeams based on the third-order shear deformation beam theory. Material properties of FG nanobeam should vary continuously along the thickness according to the power-law model. The nonlocal governing equations are developed using Hamilton's principle and third-order shear deformation beam theory, then solved using an analytic solution. Cukanovi et al. [12] describe the bending analysis of thick and moderately thick FG square and rectangular plates on a Winkler-Pasternak elastic basis. The plates are assumed to have an isotropic, two-component material distribution throughout their thickness, with the modulus of elasticity increasing in terms of the volume percentages of the constituents according to a power-law distribution. Sayyad [13] the purpose of their work was to create a unified theory of shear deformation based on displacements to study advanced composite beams and plates that are susceptible to shear deformation. In order to model transverse shear deformation, a variety of coordinate shape functions based on trigonometry, hyperbola, parabola, and exponential were created. Bending and shear rotations affected in-plane displacements. This theory's parabolic shape function produces Reddy's theory.   Prashik Malhari Ramteke et al. [14] in this study, the nonlinear Eigen frequency responses of an FGM panel in a temperature environment are computed using finite elements. Consideration is given to the elements of the FGM as a function of temperature. The material properties of the FGM panel are evaluated using Voigt's micromechanical model in conjunction with three unique material distribution patterns. Thom et al. [15] Two-Dimensional Functionally Graded Plates (2D-FGP) were numerically analyzed for buckling and bending using a finite element model described plate kinematics without a shear correction factor or shear locking treatment.

Coskun et al. [16] by applying general third-order plate theory studied the bending, free vibration, and critical buckling behavior of FG porous microplates. Analytical results were obtained using the Navier method. In the context of third-order plate theory, analyzed the effects of three different porosity distributions on the resulting porosity variations. Kim et al. [17] examined the FG porous microplates for bending, free vibration, as well as buckling responses with classical and first-order shear deformation plate theories. Analysis using Navier’s solution of supported rectangular plates was executed. Two material constituents were modeled using power-law distribution across plate thickness. Singh et al. [18] an analytical model for FG thin-walled beam flexural response was developed using first-order shear deformation theory and Vlasov's theory. The power law distribution of mild steel and alumina volume fractions varied material properties along the depth. FG thin beams under uniform vertical loading were numerically analyzed. Mergen et al. [19] established the dynamic and static responses in a nonlinearly forced axially FG tapered beam. Hareram et al. [20], axial FG thin taper beams with different taper profiles and material gradations were studied for free vibration. The mathematical formulation uses energy and that could be resolved in two halves, static as well as dynamic.

Prashik Malhari Ramteke et al. [21] in this study, nonlinear thermal frequency characteristics of multidirectional, double-curved, FG panels are investigated numerically using finite elements. The material properties of the FG panel are estimated utilizing the temperature-dependent individual properties of the FG constituents, Voigt's micromechanical model, and various forms of material distribution patterns. Rajasekarana et al. [22] discussed the mechanical behavior of non-uniform nano-beams. Small-scale effects are described according to the nonlocal strain gradient theory, the beam is represented according to the Euler-Bernoulli beam theory, and non-uniformity with a general change in cross section is postulated. Nguyen et al. [23] beams that are either transversely or axially tapered in FG are analyzed statically. The numerical results show that a T-FG beam behaves differently than an isotropic beam. On the contrary, an A-FG beam behaves exactly like an isotropic beam in terms of axial displacement and stress. Ketabdari et al. [24] using their elastic basis, Winkler and Pasternak analyzed the free vibration in homogenous and FG plates. The elastic foundation was an electric support by Pasternak and Winkler, whose stiffness coefficients varied both parabolically and linearly in each direction. Khoram et al. [25] there were tests conducted on the magnetic and mechanical deflection of FG nanobeams. The assumption was made that the nanobeam would be supported by the Winkler-Pasternak basis. The mechanical behavior of nanobeams was described by the Timoshenko beam model and the nonlocal elasticity theory of Eringen.

Ammar Melaibari et al. [26] present the first mathematical definition of the dynamical behavior of Bi-Directional Functionally Graded Porous Plates (BDFGPP) resting on a Winkler–Pasternak foundation based on unified higher-order plate theories. Using the kinematic displacement fields, the null shear strain/stress at the bottom and top surfaces of the plate is satisfied without the requirement for a shear factor correction. Mohammadi et al. [27] FG was used to create a core for a composite nanoplate with lipid face sheets to study nonlinear vibrations. There are three distinct types of FG core materials. The viscoelasticity of lipid membranes was studied using the Kelvin-Voigt model. Adapting the Von-Karman hypotheses, a nonlinear differential equation was derived for analyzing vibration in composite nanoplates. Mohammadi et al. [28] used FG material with a lipid core and two layers for vibration analysis. Nonlocal elasticity theory yields nonlinear differential governing equations. Kelvin-Voigt equation modeling lipid layer viscoelasticity. Nejad et al.  [29] The phenomenon of buckling in a nanoplate constructed from a BDFG material of arbitrary composition was investigated. Eringen studied micro buckling loads with his nonlocal theory. Using minimum potential energy, the governing equations were derived. Trung-Kien Nguyen et al. [30] new HSDT was proposed for buckling, and free vibration analysis of isotropic and FG sandwich beams. The transverse shear stress distribution was hyperbolic and boundary conditions were satisfied.

Rahmani et al. [31] Timoshenko's beam theory examined FG material size dependence. FG nanobeam material properties varied power law-like along the thickness. Ammar Melaibari et al. [32] offer a novel mathematical framework to represent the static deflection of a Bi-Directionally Functionally Graded (BDFG) porous plate lying on an elastic base. Detailing the validity of the static response produced by middle surface versus neutral surface formulations and the location of the boundary conditions. Şimşek [33] the effects of a shifting load on a Timoshenko beam which is actually made of bi-directional FGM were investigated. Both the width and length of the beam expanded and contracted at an exponential rate. Slimane et al. [34] using HSDT, performed a vibration behavior of a porous, simply supported FG plate. The thickness of the FG porous plate had an effect on the material's behavior.

Ahmad Reshad Noori et al. [35] an effective numerical method for solving the dynamic response of functionally graded porous (FGP) beams is introduced. Along the thickness direction, the elastic modulus and mass density of porous materials are believed to have non-uniform distributions. It is assumed that the standard open-cell metal foam governs the material constitutive law. Thai et al. [36] vibration analysis in an FG beam was determined by HSD theories. Adjustments to the shear strain boundary produce parallel beam surfaces on top and bottom. Vo, et al. [37] FGB static bending and vibration were examined using an enhanced shear deformation theory.  Simsek [38] investigated the behavior of the Timoshenko beam made of bidirectional FGM while it was subjected to a varying load. Both the width and length of the beam expanded and contracted at an exponential rate. Chen et al. [39] Timoshenko's beam theory analyzed porous FG beam buckling and statics. Porous composite materials had different thickness-dependent elasticity modules and mass densities.

A displacement-based unified shear deformation theory was developed by Sayyad et al. [40] for advanced composite plates and beams. Transverse shear deformation affected shape function in this theory. Zenkour et al. [41] temperature and humidity impact FG porous beam bending. Normal and shear strains affect this investigation.

Menasria et al. using revised shear deformation theory, the current work present a dynamic analysis of an FG-sandwich plate sat on an elastic foundation with various types of support. The current analytical model is simplified by reducing the number of unknowns. Without inserting correction factors, the free surfaces of the FG-sandwich plate are guaranteed to have zero shear stresses. The study of Prashik Malhari Ramteke et al. [42] describes the massive geometric deformation-induced deformation and stress analysis of a bidirectional (2D) functionally graded material (FGM) shell panel subjected to changing mechanical loads (static and dynamic). The reactions of graded structures are computed numerically using an in-house finite element model with displacement functions of higher order.

Non-classical Timoshenko beam model [43] is created using a variational formulation based on the modified couple stress theory and Gurtin-Murdoch surface elasticity theory to investigate the mechanics of 2D-FG nonuniform micro/nanobeams. The Hamilton principle is used to simultaneously derive the equations of motion and boundary conditions that account for bending and axial deformations, Poisson's effect, and the physical neutral axis concept.

Nguyen et al. [44] analytical solutions for the static analysis of functionally graded transversely or axially tapered cross-section beams are given. Depending on the power form, the beam's elastic modulus changes continuously in the thickness or axial direction.

Samaniego et al. [45] Using DNNs to tackle boundary value problems has been investigated. Many extremely relevant computational mechanics cases have been solved using DNNs to construct the approximation space, demonstrating that it is possible to approach the solution of extremely relevant BVPs using concepts and methods derived from deep learning.

Hongwei et al. [46] Deep Collocation Method (DCM) for thin plate bending problems is proposed in this work. Deep learning employs computational graphs and backpropagation algorithms, which are utilized in this method. In addition, the planned DCM is based on a feed-forward deep neural network (DNN) and varies from the majority of previous deep learning applications for mechanical challenges.

The majority of the research is concentrated on static, dynamic, and buckling studies of conventional, FG one-dimensional, and two-dimensional beams. As a result of this, we are motivated to apply higher-order theory in this research on cross-section (tapered) in FGM. The author is only vaguely aware of a small handful of studies that have been conducted and presented on the subject of the deflection of two-directional tapered FGBs that are predicated on a third-order shear deformation theory

In this study, a comparison is made between the analytical solutions of beam deflection under evenly distributed loads in 2D-FGTB. It is presumable that the material properties in the beam will vary according to a power law distribution in thickness as well as length directions. Simply supported (SS) and Clamped-clamped (CC) beams are subjected to homogenous load with varying length-to-thickness ratios (L/h) and gradient exponents so that the deflection of each type of beam can be compared. Hamilton's principle and RHSDT are used to generate governing equations of motion.

2.     Material Properties of FG Tapered Beam

2D-FGTBs are favored in this study. The composition of a material can vary along the length or thickness axes, as well as simultaneously along the length (x-axis) and thickness (z-axis) axes. Material parameters such as Young's modulus ' ,' Poisson's ratio ' μ,' and mass density ( ) vary (z-axis). The thicknesses at the left and right ends of the non-uniform beams are h₂ and h₁ (<h₂), respectively. At any axial location, the thickness h(x) in the tapered beam is given by h(x) =h₂ [1 - n(x/L)]. Where n is the taperness parameter given by n= [1-h₁/h₂]. In view of this, the present theory is having following important assumptions.

1. The origin of the Cartesian coordinate system is taken at the neutral surface of the FG beam.
2. The transverse normal stress is negligible in comparison with in-plane stress
3. This theory assumes a constant transverse shear stress and requires a shear correction factor to satisfy the lower and upper beam boundary requirements.

Fig. 1. Geometry of FG Tapered Beam

Due to variations in the volume fraction of constituent materials, the FG beam's properties vary continuously. Assume a functional relationship between thickness coordinate and material property.  The volume fraction of metal (V_m) could be expressed based on power law as [43],

Material properties of TDFG tapered beam can be defined as [43]:

Metal phases are denoted by “m” and ceramic phases by “c”. 'P_x' and 'P_z' stands for power law exponents that indicate the material variation profile through the axial and transverse directions of the beam.  The power law exponents are (p_x = p_z = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1) as shown in Figure 2.

Fig. 2. Assessing metal volume fractions
 in the z/h and x/L directions.

2.1.   2D-FGTB Formula

Modulus of elasticity "E" Poisson's ratio "μ” and density "ρ" of FG tapered beam are listed below [43].

2.2.  Field of displacement and constitutive equations

For the purpose of presenting the displacement equations, Reddy's HSDT with a shear strain function is taken into account [30].

                                                                                        (5)

(6)

The axial displacement, transverse displacement shear slope at a given point on the neutral axis is represented by u₀, w₀,  Determining transverse shear deformation distribution using the inverse elastoplastic function (shape function) i.e. f (z).

where

=

=

The beam in functionally graded materials obeys Hooke’s law, so the behavioral relations can be given as follows:

2.3.  Laws of Motion Equations

Governing equations derived from the principle of virtual displacements yields, [30].

where U is variation strain energy, k is variation in kinetic energy, V is variation in work done and t is the time.

The bending stress of the tapered beam in terms of virtual strain energy and work done can be shown by:

where,  is axial resultant force,  is the resultant bending moment,  is resultant moment owing to shear deformation and  are resultant shear forces. By integrating displacement gradations, governing equations of equilibrium can be derived.

The motion equations admit Navier’s solutions applied to get analytical results for the various boundary conditions. The coefficients  can be written assuming the following variations:

where ,  and  are unknown coefficients, and complex number  and  .

The transverse load q is also expressed by double series of Fourier sine as follows:

Substitute the expressions  of Eq. (33) in the equations of motion (24), (25), and (26). The analytical solutions are given in the following form:

3.     Results and Discussion

To foretell the static analysis of the 2D-FGTB subject to several boundary conditions, such as for SS and CC, the HSDT-based numerical studies presented in Table 1 are carried out.

Deflection analysis is presented and validates the accuracy of the current theory. 2D-FGTB is considered for numerical results made of Alumina and Aluminum with material properties as follows.

Table 1. Kinematic boundary conditions can be
 applied to numerical calculations.

Transverse deflection ( )

 = transverse deflection

 = transverse deflection at any point in the beam

= Young's modulus of the metal

 = Height of taper beam

 = load

 = Length of beam

3.1.  Validation

A methodical and exhaustive validation of the 2D-FGTB with higher-order shear deformation theory was conducted. Eq. (42) represents the nondimensional deflection. The results presented in Table 2 are based on idealized beams with different boundary conditions. The results from the current code differ marginally from those reported by Rabab et al. [43]. The deflection results of Table 3 are also identical for the CC beam.

Table 2. Comparison of dimensionless transverse deflection ( ) values of SS 2D-FGTB with different aspect ratios (L/h=5, 20), taper ratio (n=0.1), and
 gradation exponents(p).

Table 3. Comparison of dimensionless transverse deflection ( ) values of CC 2D-FGTB with different aspect ratios (L/h=5, 20), taper ratio (n=0.1), and
gradation exponents(p).

Table 4. Dimensionless transverse deflection ( ) values of 2D-FGTB with different aspect ratio (L/h=5, 20), taper
 ratio (n=0.1), and gradation exponents (p_x, p_z).

Fig. 3. Dimensionless transverse deflection ( ) values of CC TDFG tapered beam at aspect ratios (L/h=5),
 taper ratio (n=0.1).

Fig. 4. Dimensionless transverse deflection ( ) values of CC 2D-FGTB at aspect ratios (L/h=5), taper ratio (n=0.5).

Fig. 5. Dimensionless transverse deflection ( ) values of CC TDFG tapered beam at aspect ratios (L/h=20),
 taper ratio (n=0.1).

Fig. 6. Dimensionless transverse deflection ( ) values of CC 2D-FGTB at aspect ratios (L/h=20), taper ratio (n=0.5).

Table 5. Dimensionless transverse deflection ( ) values of SS 2D-FGTB with different aspect ratio (L/h=5, 20),
taper ratio (n=0.5).

Fig. 7. Dimensionless transverse deflection ( ) values of SS 2D-FGTB at aspect ratios (L/h=5), taper ratio (n=0.1).

Fig. 8. Dimensionless transverse deflection ( ) values of SS 2D-FGTB at aspect ratios (L/h=5), taper ratio (n=0.5).

Fig. 9. Dimensionless transverse deflection ( ) values of SS 2D-FGTB at aspect ratios (L/h= 20), taper ratio (n=0.1).

Fig. 10. Dimensionless transverse deflection ( ) values of SS 2D-FGTB at aspect ratios (L/h= 20), taper ratio (n=0.5).

The results of transverse deflections of the present theory are compared with the results of a taper nano beam [43] for SS boundary conditions. It is found that the results are agreeable. The remaining values are also acceptable for CC as well.

The effects of non-uniform parameters on thickness variation, h(x), and on the maximum dimensionless deflection at different power-law exponents and supporting types are presented ahead.

In this scenario, the width of the beam is constant; therefore, the applied distribution load is constant along the beam, but the moment of inertia and elastic modulus change along the length of the beam. The variation of the moment of inertia is due to the thickness variation along the length of the beam. It is obvious that, for all supported types, the maximum dimensionless deflection decreases as shown in Tables 4, and 5.

while the non-uniformity parameter decreases ‘n’ and the rate of increase depends on the supporting types as shown in Figures 3-6 for the CC beam and Figures 7-10 for the SS beam.

 The maximum rate of dimensionless deflection appears in the SS beam as shown in Table 5, in comparison with other supports like CC as shown in Table 4. The main factors that affected the maximum dimensionless deflection and its rate of change are the distribution of the young modulus, the position of the clamped end, and the change in moment of inertia.

Generally, the width variation effects are not significant on the deflection in the beam in comparison with the thickness variation effects and the combination of both width and thickness variations, where it has the greater effects. In other words, the effects of variations mean that the dimensionless deflection increases as the non-uniformity parameters are reduced. For SS beam the effects of width variation, thickness variation, and both width and thickness variation cause to increase in the dimensionless deflection, in addition to changing the position of maximum dimensionless deflection. Generally, the mid-span is the position of the maximum deflection in a SS beam (i.e., x = l/2). It seems that the position of dimensionless deflection is changed due to the variation in the dimensions of the beam and the distribution of elastic modulus.

The effects of aspect ratio (L/h) on the maximum dimensionless deflection, when the aspect ratio increases the maximum dimensionless deflection decrease as shown in Figure 5, Figure 6 as compared with Figure 3, Figure 4 for CC beam, and Figure 9, Figure 10 as compared with Figure 7, Figure 8 for SS beam due to the structure becomes relatively narrower or shorter in one direction compared to the other, which reduces its bending stiffness and makes it more susceptible to deflection under load.

4.     Conclusions

Using a variational formulation based on Reddy's higher-order shear deformation theory, a new 2D-FGTB model is developed in this article. In order to derive the equation of motion and corresponding boundary conditions that account for deflection and the neutral axis concept, the Hamilton principle is utilized simultaneously. In addition to the material properties, the model includes a material length scale parameter and surface elasticity constants that vary along the axial and transverse directions of the beam according to a power law.

Navier's method is applied to develop a 2D-FGTB model to derive an analytical solution for the deflection responses of simply supported and clamped-clamped 2D-FGTB. A comprehensive parametric study is performed to investigate the effect of various material and geometrical parameters on the deflection responses. The principal findings can be summed up as follows:

1. The structure effect and surface residual stress increase the beam's stiffness, thereby decreasing the predicted deflection.
2. Increasing the gradient exponents in the beam's thickness pz and/or length px increases the beam's stiffness-hardening, thereby reducing its static deflection.
3. The stiffness-hardening effect of the taper beam decreases its static deflection proportionally to the aspect ratio of the material.
4. The influence of the taperness parameters is reduced when the structure and surface effects are incorporated.

Boundary conditions: Reddy's HSDT assumes that the plate's edges are simply supported or clamped. However, in practice, plates may have more complex boundary conditions, such as free or partially clamped edges, which can affect the analysis accuracy.

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