Vibrations Evaluation of Functionally Graded Porous Beams in Thermal Surroundings by Generalized Differential Quadrature Method

This study investigates how to obtain the natural frequency of functionally graded porous beams simply supported on an elastic substrate in thermal surroundings by the theory of third-order shear deformation. Temperature constantly changes in the beam thickness direction and step with the distribution of volume fraction power law of the ingredient has been aﬀected on the material attributes. The distribution of uniform porosity at the pass phase is examined. To achieve the equations of governing, Hamilton’s principle was carried out. To discretize these equations, the generalized diﬀerential quadrature method has been used. First, the approach’s convergence is shown. Comparison with the results of other articles was performed for validation. Here, the impacts of numerous factors like index of power law, heat ﬁeld type, temperature diﬀerence, slenderness ratio, and porosity coeﬃcient and elastic substrate factors of a functionally graded porous beam on the natural frequencies were studied for simple boundary conditions. In addition to displaying these parameters’ impact on the beam’s thermomechanical evaluation, the conclusions also conﬁrm the accuracy of the numerical technique used.


Introduction
Laminated composites are widely utilized in diverse equipment and structure, especially thermomechanical loads, due to their mechanical and thermal suitability. Due to the discontinuity of the material and many changes in the location of the layers, the stress concentration is created in this region, which will eventually lead to the separation of the layers from each other. In addition, many changes in the plastic in the boundary layer cause cracks and growth in the material [1]. Here, functionally graded (FG) materials are a suitable alternative to these materials. e development and expansion of FG materials in recent years have attracted the attention of engineers and researchers. ese materials are used due to their multiple applications in the aerospace industry, aerospace engineering, and heat shields; therefore, it is necessary to have a detailed analysis of the dynamic behavior of FG materials. is need led to research on FG graded structures and beams and their free vibrations using various beam theories.
Several researchers like Alshorbagy et al. [2], Pradhan and Chakraverty [3], and Jin and Wang [4] have utilized the classical beam theory for FG beams with a large slenderness ratio. For FG beams with a medium slenderness ratio, this theory considers the deformation to be less than the actual value and ignores the transverse shear deformation effects and estimates the natural frequencies to be larger than the actual value. Timoshenko's theory overcomes this classical theory limitation in research by researchers, for example, Li [5], Wu et al. [6], and Katili et al. [7]. A correction parameter is required in this theory because of the zero shear stress violation at the top and bottom of the beam. e high-order shear deformation theory is applied for better predicting the behavior of the beam and other structures. As mentioned before, this research was directed towards the analysis of the vibrations of FG beams using these theories. Wen and Zeng analyzed the vibrations of the beams utilizing the high-order finite element [8]. Şimşek evaluated the principle frequencies of FG beams making use of the theories of various high-order and different boundary conditions [9]. Kazemzadeh-Parsi et al. investigated free vibration of FG plates applying the theory of high-order shear deformation [10].
In engineering issues and applications, beams are usually supported along their length on a substrate and interact with that substrate. Studies about the impact of elastic substrate parameters on the free vibrations of isotropic beams were carried out by Chen [11] and Malekzadeh and Karami [12]. Additionally, several studies have been conducted by Aghazadeh et al. [13], Akbaş [14], and Mohseni and Shakouri [15] on the bending and free vibration of FG beams depending on one-and two-parametric elastic substrates and based on different theories and solution methods.
As we can see, there are few studies on the FG beams in thermal surroundings. Several studies were done with the thermal behavior approach for showing the advantages of FG materials. Vibrations of an FG beam located in a thermal surrounding were considered by Mahi et al. utilizing highorder shear deformation theory and analytical method [16]. An analytical technique was used by Trinh et al. for investigating the behaviors of buckling and vibration FG beams under thermal loads [17]. om and Kien studied the free vibrations of two-directional FG beams in thermal surroundings [18].
With rapid progression in structure technology, the latest achievements in the field of FG materials can be the porosity of these materials. erefore, the effect of porosity is of special importance. ese porous structures have interesting mechanical attributes like excessive rigidity related to very low determined weight. So far, some studies have been done about the vibrational behavior of this porous structure. Linear and nonlinear studies on vibrations of FG permeable beams depending on an elastic substrate were investigated by Wattanasakulpong and Ungbhakorn [19]. Ebrahimi and Jafari studied the temperature impact on vibrations of FG beams with two porosity types and Timoshenko's theory [20]. Ait Atmane et al. checked out the thickness and permeable effect on FG beam's mechanical responses on elastic substrates [21]. Heshmati and Daneshmand investigated the vibration analysis of nonuniform porous beams with porosity distribution in FG state [22]. Akbaş et al. presented a dynamic analysis of thick beams with porous layers in FG and viscoelastic support [23].
Often, numerical approximation methods have to be sought to solve vibration problems due to the complexity of the problems. Classical techniques such as finite element and finite difference methods are well developed and well known. ese methods can provide very accurate results by using a large number of grid points; thus, they are computationally expensive. In a large number of cases, only a limited number of frequencies and mode shapes or a dynamic response at only a limited number of points needs to be found. Also, due to the difficulties with the shear deformation theory of third order, to solve the equations of governing, powerful techniques are required; therefore, the differential quadrature method (DQM) was used in this research. e DQM discretizes any derivative at a point by a weighted linear sum of functional values at its neighboring points. In the generalized differential quadrature method (GDQM), by considering the cosine division function, the number of nodes at the border increases, the calculations decrease, and the speed of convergence to the answers increases. For the free vibration analysis of FG panels, shells, plates, and beams, this method was applied by Bert and Malik [24], Farid et al. [25], Malekzadeh and Heydarpour [26], Ersoy et al. [27], Wang et al. [28], Wang et al. [29], Wang et al. [30], Al-Furjan et al. [31], Al-Furjan et al. [32], Shariati et al. [33], Huang et al. [34], Al-Furjan et al. [35], Huang et al. [34], and Al-Furjan et al. [36]. Zahedinejad investigated the free vibrations of FG beams depending on a two-parameter elastic foundation in a temperature-rise environment with DQM [37]. e novelty of this paper is the simultaneous FG porous beam analysis using the theory of third-order shear deformation on the elastic substrate in the temperature-dependent environment by GDQM. e motivation of this research is a thermomechanical analysis of thick FG porous structures under different thermal loads and also to evaluate the accuracy of GDQM. For example, hot section blades of gas turbine and plasma-facing components in divertor of a nuclear reactor are exposed to thermal loads. In this research, analysis of the free vibration FG porous beams with uniform porosity distribution is studied according to the theory of third-order shear deformation. e porosity distribution is considered through the beam thickness direction. Additionally, the effect of thermal loads and elastic substrate impact was considered on the frequency parameters. Hamilton's principle is used to obtain boundary conditions and equations of motion, and the GDQM was used to solve them. e materials attributes are a function of temperature and differ over the beam thickness direction with the distribution of power law. e behavioral convergence method was applied and the accuracy of the consequences was investigated with other solutions due to the articles. is research considered the geometric parameters impact, various temperature distributions and elastic substrate stiffness coefficients, and porosity volume fraction, according to FG porous beam natural frequencies. According to these assumptions [38], the field of the displacement is

Kinematics
where u i (i � 1, 2, 3), w s , and w b represent the axial displacement, shear, and bending components of horizontal displacement on the beam midpoint. e shear stress and the horizontal shear strain distribution at the beam depth satisfy the boundary conditions of stress-free up and low beam planes represented by the shape function f(z). According to Reddy's shear deformation theory [39], this function can be shown as (2) e present field of displacement and Reddy's theory are different from each other. Two components of the horizontal displacement are the displacements due to bending and shear. Strains can be defined as

FG Porous Materials Properties
e beam material composition changes from the metal-rich lowest surface (z � − h/2) to the ceramic-rich upper surface (z � h/2), following a simple power law relating to the volume fractions of the constituents; that is, where − h/2 ≤ z ≤ h/2 and p is the material index of power law, which is greater than or equal to zero. e case where p is equal to zero shows a completely ceramic beam, while the approaching infinity shows an almost fully metallic beam. Parameters V m and V c denote the fractions of the metal and ceramic volume. e main attributes P eff , such as Poisson's ratio υ, Young's modulus E, and thermal expansion coefficient, are specified as Shock and Vibration 3 where a is the porosity coefficient. FG materials are often applied in high-temperature surroundings with unavoidable changes in material attributes. Here position and temperature are important factors in the FG beam material attributes, so accurately predicting the structural response should be considered in accordance with the temperature. Material properties are stated as environmental nonlinear function at temperature T(K) as whereas T(z) � T 0 + ΔT(z), T 0 � 300K, and P 0 , P − 1 , P 1 , P 2 , and P 3 are unique temperature-dependent coefficients for the materials of constituent. It is assumed that, in the thickness direction, the temperature varies with upper and lower surfaces temperatures being known. Due to this case, to calculate the temperature function along the thickness direction, the following steady-state heat transfer equation can be solved: Imposing the boundary conditions solved this equation: e equation solution in [40] is as follows: where k(z,T) is thermal conductivity.

Equations of Governing
Hamilton's principle is determined for the FG beam equations of the motion.
where t, t 1 , and t 2 , respectively, are the time and the initial and end times; δK, δU, and δV ef are the kinetic and the total strain energy variations and the potential energy of the elastic substrate variations. e total beam strain energy can be shown as where U d and U T are, respectively, the strain energy because of mechanical stresses and the initial stresses of the strain energy as a result of temperature rise. ese strains are represented by the following equations [41]: e thermal stress in (12) is presented by e potential energy and the kinetic energy in the elastic substrate can be given by where the Winkler and shearing-layer elastic coefficients of the substrate are k w and k p depending on the soil and underlayer properties, like elastic modulus, Poisson's soil ratio, and the soil length. By replacing equations (10)-(16) in (9) and integrating by components according to space and time, the motion equations of FG beam are considered as Where N, N T ،M b ،M T b ،M s ،M T s , and Q parameters are defined as 4 Shock and Vibration e estimations of coefficients are as follows: e motion equations of the substation parts are estimated by replacing relations (20)-(26) with (17)- (19) as follows: Shock and Vibration 5

Conditions of Boundary
ese parameters are determined due to the FG beam conditions of the boundary: x � 0 and x � L are different conditions which can be obtained by combining the conditions in (30)- (35). e simple soft support conditions of boundary in this research are examined as follows: 6 Shock and Vibration For substitution components, these answers apply to analyze free vibrations: and ω is natural frequency.

Differential Quadrature Separation
e DQ separation rules based on Gauss-Lobatto-Chebyshev sampling points known as Generalized Differential Quadrature (GDQ) are attained to transfer the constitutive equations and related boundary conditions into algebraic equations [24].
e nth-order partial derivative of f(x) with respect to x at the discrete point x i is where N x is the number of the differential quadrature separation items in the direction of length. C (n) ij are weighting coefficients associated with the nth-order partial derivative of f(x) with respect to x at the discrete point x i . e first one is where e weighting coefficients of higher-order derivatives can be obtained through the following recurrence relation: Based on this method, the partial derivatives of a function can be given as where A ij , B ij , C ij , and D ij are the first-, second-, third-, and fourth-order weighting coefficients of the DQM, respectively. e conditions of boundary and the motion equations are converted to DQM as the equations related algebra. e Shock and Vibration differential equations of governing and conditions of boundary are considered as separation use for derivatives (24) and relations (28)- (30) and (36) and DQM as follows: Boundary conditions based on this method are written in relation (45). e boundary degrees and domain of freedom separate parts of the freedom degrees for obtaining the eigenvalue system of equations as In the matrix form, using (46), the motion equations separated form can be arranged differently as where To find the natural frequencies, the former eigenvalue equation can be solved.

Results and Discussion
First, the method convergence is considered for simply supported conditions of boundary. e conclusion accuracy is confirmed by comparing other available solutions. A beam containing aluminum (Al) and alumina (Al 2 O 3 ) is considered for temperature-independent states. Table 1 shows the FG material characteristics. A beam concluding stainless steel (SUS304) and silicon nitride (Si 3 N 4 ) is estimated for the temperature-dependent cases. In Table 2, the temperature-dependent material attributes for some metals and ceramics are listed, according to the data given in [40]. e natural frequency of nondimensional (λ n ) are explained as Parameter I is defined as Here, the number of differential quadrature grid points along the length was investigated for frequency convergence. However, the elastic substrate and porosity effects were ignored. Table 3 presents the natural frequency converging for simply supported conditions. In this table, it is clear that the results converged with N x � 29.
In Table 4 the parameters of frequency were estimated and compared with those by Ebrahimi and Jafari [20] and Şimşek [9]. e good agreement was observed.
Further confirming the results obtained from the GDQM, Zahedinejad, in an article [37], investigated the isotropic beam frequencies depending on an elastic layer and in comparison with Matsunga [34]. e impacts of elastic substrate and slenderness ratio parameters are considered. e method accuracy was observed between the presented results in close agreement.
In Table 5, the first nondimensional frequency computed under the temperature-dependent environments with conditions of the simply supported boundary is compared with those by Ebrahimi and Jafari [20]. Here, three thermal states are considered: (1) uniform temperature rise (UTR), (2) linear temperature rise (LTR), and (3) nonlinear temperature rise (NLTR). ere is a good agreement between the two solutions. e validity of using the GDQM is confirmed because of the above consequences comparison and the good agreement of these results and other articles. In the following, the various factors' impacts on the first frequencies are investigated.
For considering the elastic substrates impact on FG beam nondimensional natural frequencies, studies were performed on both parameters of the elastic substrates. A beam with uniform porosity distribution applies a uniform thermal load. By canceling Pasternak's parameter, for porosity coefficient values, the dimensionless frequencies variation versus the Winkler's parameter can be figured in Figure 2. Also, by canceling Winkler's parameter for various porosity coefficients, these variations are plotted versus Pasternak's parameter in Figure 3. From these figures, for the porosity coefficient value, enhancing Winkler's and Pasternak's parameters causes the natural frequencies to increase. Because of enhancing both elastic substrate factors, the complete system stiffness increases, and therefore the dimensionless frequencies increase. e higher frequencies can be obtained by the larger porosity coefficient values. As  Material shown in Figure 4, natural frequencies versus different temperature differences are in the range of 4 to 7 for different values of Winkler's parameter, and those are in the range of 25 to 40 for different values of Pasternak's parameter.
Pasternak's parameter has a greater effect on natural frequencies than Winkler's parameter; also, combining both elastic substrate factors caused higher frequencies. In other words, Pasternak's parameter and combining both elastic  substrate parameters increase the stiffness and thus increase the natural frequency. Figure 5 shows the temperature impact on frequencies, the variation of nondimensional natural frequency versus temperature for different power law indexes, and a constant slenderness ratio. A uniform temperature rise is subjected by the beam with a uniform porosity distribution. For power law indexes smaller than one, first, the natural frequency decreases with increasing temperature and then further natural frequency increases with rising temperature. But, always for indexes of power law greater than one, decreasing the natural frequencies of the beam concurred with increasing temperature. In the case of porous beams, given the modulus of elasticity, density, coefficients of inertia, and other mechanical properties in addition to the observance of temperature and porosity, it can also be affected by the power law index. e natural frequency with increasing temperature depending on the value of the power law index and porosity coefficient has been down to a point, and, from there, with the predominance of the effects of the power law index and the porosity coefficient, this trend will increase. In this case, by approaching the ceramic material, this issue will be more evident. Also, with enhancing the power law index, the frequency decreases for a certain temperature variation. Figure 6 shows the frequency consequences contrary to the different porosity parameters temperature at fixed values of the slenderness ratio and index of power law to display the porosity coefficient impact with a uniform porosity distribution on the natural frequency. Higher natural frequencies are required for a higher value of the porosity coefficient. As shown in Figure 6, for the nonporosity mode, with increasing temperature variation for a thick beam, the natural frequency decreases because of the change in beam stiffness and greater flexibility with increasing temperature. In the case of porous beams, given that porosity can affect all the elasticity modulus, coefficients of inertia density, and other mechanical attributes with observing temperature and  power law index, the natural frequency with increasing temperature depending on the power law index value and coefficient of porosity to a point and, from there, with the predominance of the index of power law impact and the coefficient of porosity, can cause this trend increase. For a larger porosity coefficient, this will occur at lower temperature differences. In Figure 7, the thermal load's effect on frequencies is presented. e uniform porosity distribution of the beam is displayed in three uniform, linear, and nonlinear thermal loads for fixed values coefficient of porosity, slenderness ratio, and index of power law. e frequencies are higher than the other two states for the uniform temperature distribution with various temperatures differences, and the slope of the frequency decreasing variations is smaller. is is due to the greater flexibility of the beam due to the higher temperature of different points in the beam thickness direction.
For different slenderness ratio values and constant values coefficient of the porosity and the index of power law, the differences of the dimensionless natural frequencies of FG beam with the temperature versus a uniform porosity distribution are presented in Figure 8. e beam is exposed in a uniform temperature rise. As shown in the figure, at a high slenderness ratio, thermal buckling occurs at a lower temperature difference. erefore, the comparison of frequencies should be made below the critical temperature. It is clear, in the figure, that the natural frequencies were higher at a given temperature difference for a smaller slenderness ratio, and this is due to the greater geometric stiffness of the beam with a smaller slenderness ratio.

Conclusion
is research studies functionally graded beams dynamic behavior with an elastic substrate of two parameters in a temperature-dependent environment and uniform porosity. Temperature-dependent and constant variation in the direction of the beam thickness and the distribution of power law of the material constituents' volume fraction are the material attributes. Uniform, linear, and nonlinear thermal loads are carried out in the direction of beam thickness. For considering the thick beam's vibrational behavior, the theory of third-order shear deformation is used. For the motion equations, Hamilton's principle was administered. For solving the equations of governing, generalized differential quadrature method has been used. e impacts of power law index, type of thermal load porosity coefficient, and beam slenderness ratio on natural beam frequencies were investigated. We can conclude the numerical consequences as follows: (i) the decrease in frequency, there is an increase in the material index (p). (ii) e shorter (or thicker) beam has a higher natural frequency. (iii) the increase in temperature, there is a decrease in natural frequency for nonporous beam, because high temperatures can reduce the material stiffness with fixed mass. (iv) In the case of porous beams, given the modulus of elasticity, density, coefficients of inertia, and other   mechanical properties in addition to the observance of temperature and power law index, it can also be affected by porosity. e natural frequency with increasing temperature depending on the value of the power law index and porosity coefficient has been down to a point, and, from there, with the predominance of the effects of the power law index and the porosity coefficient, this trend will increase. (v) e natural frequency under the uniform increase in temperature for thick functionally graded porous beam is greater than those under the linear and nonlinear increase in temperature. (vi) e porosity increase will enhance the natural frequency in the distribution of uniform porosity and a certain thermal variation. (vii) ere is a significant effect on the natural frequency by changes in the elastic substrate rigidity. e total system stiffness and, accordingly, the natural frequency are enhanced by increasing both elastic substrate factors. In other words, combining both elastic substrate parameters increases the stiffness and thus increases the natural frequency. (viii) e presented numerical method is in good agreement with the solution methods in other articles. In other words, generalized differential quadrature method provides very accurate results, and, with fewer divisions, the answer can be reached faster.

Data Availability
e manuscript is a mathematical method; therefore, no data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.