Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section

The present paper investigates the transverse vibration of a non-uniform axially functionally graded Timoshenko beam with cross-sectional and material properties varying in the beam length direction. The Chebyshev collocation method is used to spatially discretize the governing partial differential equations of motion of the beam into time-dependent ordinary differential equations in terms of Chebyshev differentiation matrices. An algebraic eigenvalue equation in matrix form is then formed to study the free vibration behavior of non-uniform axially functionally graded Timoshenko beams. Several results of natural frequencies of the beams are evaluated and compared with those in the published literature to assure the accuracy of the proposed model. The effects of taper ratio, material graded index


INTRODUCTION
Functionally graded materials (FGMs) have been used increasingly in various engineering and scientific fields recently because of their promising material properties over the traditional composites.Thus, the dynamic problems of engineering structures constructed from FGMs have received considerable attention, especially for the beam members commonly used in bridges, buildings and machine components.In the past decades, most studies numerically dealt with the dynamics of FGM beams with material properties graded in the thickness direction based on various beam theories (Simsek (2010); Thai and Vo (2012); Nguyen et al. (2013); Pradhan and Chakraverty (2014); Su and Banerjee (2015); Wattanasakulpong and Mao (2015); Chen andChang (2017, 2018); Ding et al. (2018); Esen (2019)).It is noted that the strength and weight of beam structures, which affect its vibration behavior, can be optimized by changing the crosssectional and material properties along the beam length direction.Therefore, the dynamic problems of FGM beams with axially varying properties have received considerable attention in recent years.
The free vibration of non-uniform axially FGM Euler-Bernoulli beams with various end supports was presented by Huang and Li (2010) based on the integral equation method.The bending vibration of FGM beams with axial variation of material properties was investigated by Murin et al. (2010).The free vibration and stability of tapered axially FGM Timoshenko beams with various restraint conditions was dealt with by Shahba et al. (2011) by using finite element analysis.The vibration analysis of non-uniform axially FGM beams was presented by Hein and Feklistova (2011) based on the Euler-Bernoulli beam theory and Haar wavelet approach.Based on the lowest-order differential quadrature element and differential transform element methods, the vibration and stability problems of axially FGM Euler-Bernoulli beams with tapered cross-section were investigated by Shahba and Rajasekaran (2012).Exact frequency equations of the free vibration for axially exponentially FGM beams with different end conditions were presented by Li et al. (2013) by employing an analytical approach.The lowest-order differential quadrature element and differential transform element methods were used by Rajasekaran (2013aRajasekaran ( , 2013b) ) to analyze the vibration behavior of rotating non-uniform axially FGM Euler-Bernoulli and Timoshenko beams.The free vibration of axially FGM Timoshenko beams with non-uniform cross-section was studied by Huang et al. (2013) based on a unified approach.The free vibration of clamped axially FGM uniform Timoshenko beams was studied by Sarkar and Ganguli (2014).Exact equations of vibration frequencies of tapered axially FGM Timoshenko beams were derived by Tang et al. (2014).The spline finite point method was applied by Liu et al. (2016) to study the free bending vibration of axially FGM beams with tapered cross-section by using the Euler-Bernoulli beam theory.The complementary functions method was used by Calim (2016) to investigate the transient vibration of tapered axially FGM Timoshenko beams.Based on the Euler-Bernoulli and Timoshenko beam theories, the vibration behaviors of non-uniform axially FGM beams were investigated by Zhao et al. (2017) by using the Chebyshev polynomials theory.The asymptotic development method was presented by Cao et al. (2018) to analyze the free vibration of axially FGM Euler-Bernoulli beams.The vibration of tapered axially FGM cantilevered Timoshenko beam under various axial loadings was investigated by Sun and Li (2019) based on the initial value method.The bending vibration of non-uniform axially FGM Euler-Bernoulli beams was investigated by Chen (2020) based on the Chebyshev collocation method.
As reviewed above, the vibration problems of axially FGM beams (AFGM) had been effectively studied by many researchers based on numerous numerical methods.The Chebyshev collocation method has the advantages of fast convergence rate and high accuracy of predictability over other numerical methods so it has been widely used to solve various engineering and mathematical problems.However, there exists a paucity of the contribution of the published literature to the vibration of non-uniform AFGM Timoshenko beams by using the Chebyshev collocation method.Thus, an attempt has been made to apply the Chebyshev collocation method to analyze the transverse vibration of FGM Timoshenko beams with axially varying material and cross-sectional properties.Material properties are assumed to vary along the length direction described by the exponential function.The Chebyshev collocation method is applied to reduce the partial differential equations of motion into the ordinary differential equations with time as the indendent variable.Then, an eigenvalue problem is formulated to evaluate the free vibration behavior of the tapered AFGM Timoshenko beam.The natural frequencies for various AFGM beams with different taper ratios, slenderness ratios, graded indices, material constituents and boundary conditions are determined.In comparing the calculated natural frequencies with those by other investigators, the present results agree well with those obtained by other methods.Then, relevant parameter analyses are performed to demonstrate the effects of various material and geometric parameters on the free vibration characteristics of the AFGM Timoshenko beams.

BENDING VIBRATION ANALYSIS
The AFGM beam with tapered rectangular cross-section as shown in Figure 1 is considered.The beam has a length L with taper ratios in the width and thickness directions.The axes x, y and z represent the respective beam length, width and thickness direction.The cross-sectional properties, area A(x) and area moment of inertia I(x), of the tapered beam are assumed to vary along the axial direction as follows.Here A o and I o are the area and area moment of inertia at the left end x = 0; C b and C h are the width and height taper ratio, respectively.The material properties, Young's modulus E(x) and shear modulus G(x) and mass density ρ(x), of the AFGM beam are represented by the effective material property P(x), which varies continuously across the beam length according to the following exponential function.

𝑃𝑃(𝑥𝑥)
where P o and P l are the material properties at the left and right surface of the beam, respectively; The exponent α is the material graded index which describe the material variation profile of the volume fraction through the beam length; the length variable x ranges from 0 to L. Figure 2 depicts the variation of property P/P o in the length direction for P o = 3P l .As can be seen, a smaller value of volume fraction index represents a more sudden increase in the property P/P o near the left surface and the material at the right surface is the dominant constituent.In contrast, the property P/P o changes abruptly near the right surface for a larger value of volume fraction index and the dominant constituent is the material at the left surface.By applying Timoshenko beam theory and Hamilton's principle to the non-uniform axially FGM beam, the following partial differential equations (Shahba et al. (2011)) with variable coefficients and boundary conditions governing the lateral free vibration behaviors of AFGM Timoshenko beams can be obtained.Clamped end: w = 0, φ = 0 (5a) Pinned end: w = 0, Free end: where w is the transverse displacement along the z direction and  is the rotation about the y axis; κ is the shear correction factor and taken to be 5/6 throughout this paper.The present study intends to use the Chebyshev collocation method to solve the free vibration equations (4) and (5).Because the space variable range of this collocation method is [-1, 1], the space variable must be transformed before analysis.Let ξ = 2   − 1 ⁄ , Eqs. ( 4) and ( 5) can be rewritten as the partial differential equations with ξ as the spatial independent variable as follows.
Latin American Journal of Solids and Structures, 2021, 18(07), e397 5/18 with Here ξ  are the Gauss-Chebyshev-Lobatto collocation points (Trefethen, 2000) within [-1, 1].Then, the matrix vector multiplication is used to obtain the first derivatives of the displacement functions in Eq. ( 8) at the collocation points as where D N denotes the (N+1)×(N+1) Chebyshev differentiation matrix; (D N ) ij is its entry at row i and column j, which is given as (Trefethen, 2000) Eq. ( 11) represents each element of the first derivative D 1 of the Chebyshev differential matrix, and the kth derivative can be obtained by D k =(D 1 ) k .It is worth pointing out that although the Chebyshev collocation method has the advantage of providing accurate and fast convergent solutions, it tends to cause physical false eigenvalues, and usually produces more roundoff errors when calculating derivatives.A detailed discussion of spurious unstable modes and roundoff errors in derivative calculations can be found in the books of Gottlieb and Orszag [1977] and Boyd [2000].Using the above-mentioned Chebyshev collocation method, the partial differential equation ( 6) can be rewritten into a timedependent ordinary differential equation represented by the Chebyshev differential matrix. where Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section Wei-Ren Chen Latin American Journal of Solids and Structures, 2021, 18(07), e397 6/18 Here () and  ̈() are the transpose displacement and acceleration vectors, respectively.Matrices  � ,  ̅ , K s1 , K s2 , K b1 and K b2 are all diagonal matrices with diagonal elements formed by evaluating m(ξ), J(ξ), Q(ξ), ′(ξ), S(ξ) and ′(ξ) at the collocation points, respectively.Similarly, the boundary conditions of the beam as described in Eq. ( 7) can be expressed in terms of Chebyshev differentiation matrices as shown in Table 1.By imposing the homogeneous boundary conditions at the supporting ends on the governing equation ( 12) and rearranging the displacement and acceleration vector () and  ̈(), Eq. ( 12) is reformulated as The subscripts 'I' and 'B' are the collocation points related to the respective governing equation and boundary condition.Considering the harmonic vibration, the displacement functions in Eq. ( 13) are assumed to be Latin American Journal of Solids and Structures, 2021, 18(07), e397 7/18 where ω is the natural frequency.Substituting Eq. ( 14) into Eq.( 13), the following algebraic eigenvalue equation is obtained Thus, Eq. ( 15) will be used to calculate the natural frequencies of the free bending vibration of various AFGM Timoshenko beams with non-uniform cross-sections in the next section.

RESULTS AND DISCUSSIONS
In this section, the effects of material graded indices, height and width taper ratios, slenderness ratios, material compositions and restraint types on the free vibration characteristics of the AFGM Timoshenko beams are examined.Four types of AFGM beam constructed from Alumina and Stainless steel (A/S), Zirconia and Stainless steel (Z/S), Alumina and Aluminum (A/A), and Zirconia and Aluminum (Z/A), respectively, are considered.The material properties of the typical metals and ceramics are given in Table 2 (Shahba et al. (2011); Natarajon et al. (2011)).For simplicity, the Poisson's ratio is taken to be 0.3.It is assumed that the left side of the beam is ceramic-rich and the right side is metalrich.The following frequency parameter  = ω 2 �        ⁄ is used to evaluate the dimensionless natural frequencies, in which E al and ρ al denote the Young's modulus and density of aluminum, respectively.The rotary inertial parameter r and slenderness ratio s are defined as I o /A o L 2 and (1/r) 0.5 , respectively.

Model verification
To validate the accuracy of the proposed model, the free vibration of various exponential AFGM beams composed of zirconia and aluminum with L = 1m, b o = 0.01m and h o = 0.03m (Liu et al. (2016)) is studied.The A/Z AFGM beam is aluminum rich near x = 0 and zirconia rich near x = L, whereas the Z/A AFGM beam is vice versa.Because there is a lack of data of exponential AFGM Timoshenko beams in the published literature, the presented results are compared with those obtained by the Euler-Bernoulli beam theory.Table 3 present the first four dimensionless natural frequencies of uniform A/Z AFGM beams of α = 3 under various boundary conditions.The present results match well with those obtained by Huang and Li (2010), Liu et al. (2016) and Cao et al. (2018), especially the lower frequencies.
Table 4 gives the first four dimensionless natural frequencies of uniform pinned-pinned and clamped-clamped Z/A AFGM beams with various values of α.Likewise, the presented lower mode frequencies are in good agreement with those given by Huang and Li (2010) and Liu et al. (2016).The first four dimensionless natural frequencies of non-uniform clampedfree and clamped-pinned Z/A AFGM beams of α = -10 with various values of taper ratios are presented in Table 5. Good agreement is also observed between the present results and those by Liu et al. (2016).It is noted in Tables 3-5 that the discrepancy between the higher mode frequencies is greater than that of lower ones because the Euler-Bernoulli beam theory always overestimates the frequencies, especially for the higher modes.Hence, to provide more accurate results for exponential AFGM beams, the Timoshenko beam theory is also needed.In this study, only the bending vibration about the y-axis is considered.If the 3D finite element FGM beam is modeled according to the introduction of Murin et al. [2014Murin et al. [ , 2016]], in addition to obtaining the same natural frequency of the bending mode about the y-axis as the proposed method, the natural frequencies of other bending modes around the z-axis, torsional modes and axial modes will also be determined.6 and 7.

Effect of taper ratio
Latin American Journal of Solids and Structures, 2021, 18(07), e397 8/18 As can be seen, the increase in height and width taper ratios may decrease or increase the natural frequencies depending on the taper ratios and boundary conditions.For CF beams shown in Figure 3, when C h increases, the first frequency increases, while the other three frequencies show a downward trend.Basically, as the value of C h is higher, the rate of change of frequency is more significant.As shown in Figure 4, with the increase in C b , the first four frequencies rise.Similarly, as the value of C b is larger, the rate of change of frequency is more obvious.For the first and second frequencies, the effect of C b on the frequency is more significant than that of C h , while for the third and fourth frequencies, the influence of C h is greater.For PP beams in Figure 5, the first four frequencies are basically reduced with the increasing C h , except that the fourth frequency of the beam with C b ≤ 0.7 increases slightly from C h = 0.1 to 0.2.Its frequency is significantly affected by C h , especially the first frequency.It can be seen from Figure 6 that as C b increases, except for some cases, most of the frequencies of the PP beams decrease.The exception examples are as follows.The fourth frequency of the beam with C h = 0.1 increases as C b increases; for the beam with C h = 0.7, the second and third frequencies first increase with C b until C b = 0.8 and then start to decrease, while the fourth frequency first decreases, then increases and then reduces; the second to fourth frequencies of the beam with C h = 0.9 increase with the increasing C b .Unlike CF beams, C h has a more significant effect on the four frequencies of PP beams than C b , but the effect of C b is minor except for the first frequency.As for the CP beams in Table 6, when C h increases, except that the fourth frequency increases slightly from C h = 0.1 to 0.2, all other frequencies decrease accordingly.With the increasing C b , basically the first frequency increases.Except for beams with C h = 0.1 and 0.2, their first frequency decreases slightly when C b increases from 0.8 to 0.9.For beams with C h ≤ 0.6, the second to fourth frequencies first increase and then decrease with the increase of C b , and for beams with C h ≧0.7, they increase accordingly.Like the PP beams, the influence of C h on the frequency is significant, while the effect of C b is minimal.For CC beams in Table 7, when C h enlarges, all frequencies decrease.As C b increases, all frequencies first increase and then decrease for beams with C h ≤ 0.8.In addition, when the beam has a higher C h value, the C b value at which the frequency changes from rising to falling is greater.When the beam has a taper of C h = 0.9, all frequencies rise with the increasing C b .Like PP and CP beams, C h has a significant impact on the frequency, while C b has a slight impact.
As discussed above, the natural frequencies for the beams with the same width ratio C b reduce with the increasing height taper ratio C h except for the first frequencies of CF beams and fourth frequencies of PP and CP beams.As for the beams with the same height taper ratio C h , the first four frequencies of CF beams increase with the increasing width taper ratio C b but those of PP, CP and CC beams vary differently with C b depending on the value of C h .It is important to note that the height taper ratio has a more profound impact on the natural frequencies of all beams than width taper ratio while it shows an opposite trend for the first and second frequencies of CF beams.

Effect of material distribution
Table 8 gives the effects of material graded index α on the first four frequencies of Z/A AFGM beams of r = 0.01 and C b /C h = 0.3/0.5 under various boundary conditions.As can be seen, the first and second frequencies of the CF beam reduce as the value of || approaches to zero, but its third and fourth frequencies enlarge with the increasing α.For the PP beam, all the first four frequencies increase with the increase in α.For the CP and CC beams, the first and second modes varies irregularly with α, but the third and fourth modes increase as α increases.

Effect of slenderness ratio
Table 9 shows the effects of the slenderness ratio s on the first four frequencies of Z/A AFGM beams of α = 3 and C b /C h = 0.5/0.5 under different boundary conditions.As expected, all frequencies decrease with the increasing s.The influence is becoming more significant for higher mode frequencies and restraint.

Effect of material composition
Tables 10-13 give the variations of the natural frequencies against different material constituents of uniform and non-uniform AFGM beams of r = 0.01 with CF, PP, CP and CC boundary conditions, respectively.As far as CF beams are concerned, except for some exceptions, basically A/A beam has the largest frequency, followed by A/S, Z/A and Z/S beams.However, when α = 0 and -3, the order of the first mode frequency is A/A > Z/A > A/S > Z/S, and Z/A is larger than A/S.Similarly, the first four frequencies of the Z/A beam are larger than those of the A/S beam as α = -10.Furthermore, it can be seen from Table 10 that different material compositions have a significant effect on the frequency of the CF beam, and the influence is different under different values of α.
For PP beams, in most cases, the order of frequencies of beams composed of different materials is A/A, A/S, Z/A and Z/S beams from high to low.The exceptions are as follows.When α = -10, the first frequency of uniform Z/A beam is larger than that of A/S beam; when α = -3, 0 and 3, the first frequency of non-uniform Z/S beam is greater than that of Z/A beam.For uniform beam with α = 10, the order of the first three frequencies from high to low is the A/A, A/S, Z/S and Z/A beams.For nonuniform beam with α = 10, the order of the first frequency is the A/S, A/A, Z/S and Z/A beams, and that of second to fourth frequencies is A/A, A/S, Z/S and Z/A beams.In addition, it can be found from Table 11 that the difference between the frequencies of A/A and A/S beams is small, especially when α = 10, the discrepancy is even smaller.The same phenomenon also occurs between Z/A and Z/S beams.When α = -10, the difference of frequencies among the four beams A/A, A/S, Z/A and Z/S is very small.
Like the CF and PP beams, the order of frequencies for the four types of CP beams is A/A, A/S, Z/A and Z/S beams except for some typical cases.For example, the order of frequencies becomes A/A, Z/A, A/S and Z/S for all four frequencies of beams with α = -10, and for the first frequency of nonuniform beams with α = -3.In addition, as α = 10, the first three frequencies of the uniform Z/S beam and the first four frequencies of the nonuniform Z/S beam are larger than those of the corresponding Z/A beam.As can be seen in Table 12, when α = 10, the difference between the frequencies of A/A and A/S beams is small except for the third and fourth frequencies of the beams with uniform crosssection.The same phenomenon also occurs between Z/A and Z/S beams.
Latin American Journal of Solids and Structures, 2021, 18(07), e397 11/18   For CC beams, basically A/A beam has the largest frequency, followed by A/S, Z/A and Z/S beams except for some exceptions.Like CF and CP beams, the order of frequencies is A/A, Z/A, A/S and Z/S for the beams with α = -10; however, the difference between the frequencies of Z/A and A/S beams is slight.When α = 3, the first and second frequencies of uniform Z/A beam and the first frequency of non-uniform Z/A beam are larger than those of the corresponding Z/A beam.When α = 10, the order of frequencies is A/S, A/A, Z/S and Z/A for the first and second frequencies, and is A/A, A/S, Z/S and Z/A for the third and fourth frequencies.

CONCLUSION
The eigenvalue problem for the free vibration of non-uniform AFGM Timoshenko beams with various boundary conditions is established based on the Chebyshev collocation method.The axially graded material properties of the beam are assumed to vary exponentially.Four types of AFGM Timoshenko beams constructed from two metals and two ceramics are examined to demonstrate the effect of material compositions on the vibration behavior.Based on the results discussed previously, some major conclusions are highlighted as follows.
1. Depending on the values of taper ratios and boundary conditions, the natural frequencies of AFGM beams may decrease or increase with the increasing height and width taper ratios.The height taper ratio has a more considerable effect on the natural frequencies than the width taper ratio except for the fundamental frequencies of clamped-free beams.
2. The variation of natural frequencies against the values of material graded index varies differently depending on the restraint types of the AFGM beams.
3. The natural frequencies of AFGM beams always reduce as the slenderness ratio increases.
4. Except for certain cases, basically the AFGM beam constructed from Alumina/Aluminum has the highest frequency, followed by the beams made from Alumina/Stainless steel, Zirconia/Aluminum and Zirconia/Stainless steel.

Figure 1 :
Figure 1: Axially FGM beam configuration and coordinate systems.

Figure 2 :
Figure 2: Variation of effective property P(x)/P o versus beam length for exponentially AFGM beam with various values of material graded index α.

Figures
Figures 3-6 present the effects of various values of C h and C b on the varying trend of the first four dimensionless frequencies for the CF and PP Z/A AFGM beams with r = 0.01 and α = -10.The variations of the first four natural frequencies with respect to the taper ratios C h and C b for the CP and CC Z/A AFGM beams are presented in Tables6 and 7.

Figure 3 :
Figure 3: Effect of C h on dimensionless natural frequencies for clamped-free Z/A AFGM beams of r = 0.01 and α = -10 with various values of C b .(a) First mode (b) Second mode (c) Third mode (d) Fourth mode

Figure 4 :
Figure 4: Effect of C b on dimensionless natural frequencies for clamped-free Z/A AFGM beams of r = 0.01 and α = -10 with various values of C h .(a) First mode (b) Second mode (c) Third mode (d) Fourth mode.

Figure 5 :
Figure 5: Effect of C h on dimensionless natural frequencies for pinned-pinned Z/A AFGM beams of r = 0.01 and α = -10 with various values of C b .(a) First mode (b) Second mode (c) Third mode (d) Fourth mode.

Figure 6 :
Figure 6: Effect of C b on dimensionless natural frequencies for pinned-pinned Z/A AFGM beams of r = 0.01 and α = -10 with various values of C h .(a) First mode (b) Second mode (c) Third mode (d) Fourth mode.

Table 1
Boundary condition equations in terms of Chebyshev differentiation matrices.

Table 2
Properties of materials.

Table 3
Comparison of frequencies of uniform A/Z AFGM beams with different boundary conditions.

Table 4
Comparison of frequencies of uniform PP and CC Z/A AFGM beams with different α.

Table 5
Comparison of frequencies of CF and CP Z/A AFGM beams with different C b and C h

Table 6
Effects of height and width taper ratios on first four dimensionless natural frequencies of clamped-pinned Z/A AFGM beams with r = 0.01 and α = -10.

Table 7
Effects of height and width taper ratios on first four dimensionless natural frequencies of clamped-clamped Z/A AFGM beams with r = 0.01 and α = -10.

Table 8
Effects of material graded index on frequencies of various Z/A AFGM beams.

Table 10
Effects of material constituents on frequencies of CF AFGM beams with various α.

Table 11
Effects of material constituents on frequencies of PP AFGM beams with various α.Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section Wei-Ren ChenLatin American Journal of Solids and Structures, 2021, 18(07), e39716/18

Table 12
Effects of material constituents on frequencies of CP AFGM beams with various α.

Table 13
Effects of material constituents on frequencies of CC AFGM beams with various α.