Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams

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1 Shock and Vibration DOI /SAV IOS Press Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Ahmad Shahba a,b, Reza Attarnejad a,b, and Shahin Hajilar a a School of Civil Engineering, University College of Engineering, University of Tehran, Tehran, Iran b Centre of Numerical Methods in Engineering, University of Tehran, Tehran, Iran Received 6 March 21 Revised 11 May 21 Abstract. Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finite element method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated. Keywords: Axially functionally graded material, tapered beam element, free vibration, stability 1. Introduction Functionally graded FG materials are one of the most advanced materials whose mechanical properties vary gradually with respect to a desired spatial coordinate. In comparison with laminated composites, employing FG materials in structural systems leads to elimination of stress concentration and also improves the strength and toughness of the structure. Most of the literature on FG beams deals with beams whose mechanical properties vary through thickness [2,25,28,3,31]. There are relatively few works on axially FG beams whose mechanical properties vary along the axis of the beam where most of them concern the special case of uniform beams. Due to varying cross-sectional area, modulus of elasticity and mass density along the beam axis, the governing differential equations of axially FG tapered beams for transverse and longitudinal vibrations and buckling are differential equations with variable coefficients for which closed-form solutions could be hardly found or even impossible to obtain; hence application of numerical techniques is essential. Development of the technical literature on axially FG beams could be mostly tracked in the works written by Elishakoff and his co-workers [3 21,26,29,32,33] who used semi-inverse method for solution of the governing differential equations. Assuming a uniform beam with constant mass density, Aydogdu [27] used semi-inverse method to study the free transverse vibration and stability of axially FG simply supported beams. In this paper, a new element is proposed for analysis of tapered beams with an arbitrarily varying cross-section made of axially functionally graded materials. The element takes advantage of the shape functions of uniform homogeneous Euler-Bernoulli beam elements. The idea of formulating tapered beams in terms of uniform beams Corresponding author. Tel.: ; Fax: ; attarnjd@ut.ac.ir. ISSN /11/$ IOS Press and the authors. All rights reserved

2 684 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Fig. 1. Positive sign convention for nodal degrees of freedom and forces. could be observed in the works of Bazoune [1], Banerjee [23] and Singh and i [24]. In this paper, the varying cross-sectional area, moment of inertia, mass density and Young s modulus of elasticity are exactly considered in the derivation of the structural matrices while in the classical beam element method, the original non-prismatic beam is modeled as an assemblage of some uniform elements; that is, constant values are considered for the cross-sectional area and moment of inertia of the beam element and finally these constant values are considered in the evaluation of the structural matrices. Therefore, the present element presents a more realistic model of the original non-prismatic beam element. Although several works have been carried out on axially FG beams; there is still a gap in analysis of axially FG beams with arbitrarily varying cross-sectional area and mechanical properties along the beam axis. This could be explained in detail by recalling that the majority of previous works have followed a semi-inverse procedure in which the displacement field and the mass density distribution are prescribed and afterwards the distribution of modulus of elasticity is obtained by satisfying the governing differential equations. As a result, semi-inverse method encounters difficulty in calculation of displacement field of an axially FG beam whose distributions of mass density and modulus of elasticity are known. Finite element method FEM could overcome this problem. It should be also remembered that semi-inverse procedure should be formulated for each problem separately since the prescribed displacement field should satisfy the boundary conditions; while FEM does not necessitate the reformulation for each set of boundary conditions. Moreover, it is instructive to bear in mind that semi-inverse method provides exact results while FEM does not essentially predict the deformations of the structural system accurately. 2. New beam element Consider a general beam element with 3 degrees of freedom per node with varying cross-sectional dimensions along the element axis as shown in Fig. 1. Based on the concept of FEM which is mostly considered as a displacementbased structural analysis method, the displacement field could be interpolated within the element in terms of the nodal degrees of freedom using shape functions as w 1 θ wx 1 =Nw 1 w 2 ux =Nu θ 2 { u1 u 2 } in which x is the coordinate along the element; ux and wx are the axial and lateral displacements of the beam element respectively; Nu = { N u1 x N u2 x } and Nw = { N w1 x N w2 x N w3 x N w4 x } are respectively the axial and bending shape functions. The accuracy of the results predicted by FEM considerably depends on how well these shape functions are selected. In this paper the shape functions for homogeneous uniform Euler-Bernoulli beam elements [22] are used to formulate the displacement field given in Eqs

3 Nu = { 1 x l { Nw = A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 685 } x l 1 3 x 2 l 2 x 3 [ l x 1 x ] 2 l 3 x 2 l 2 x [ 3 l x x l where l is the element length. ]} 2 x l Structural matrices In order to carry out any structural analyses, the elemental structural matrices should be evaluated. The strain energy of the beam element could be written as [22] U e = 1 2 l ExAx 2 u dx 1 l ExIx x w x 2 dx 1 l Nx 2 2 w dx 5 x Here Nx is the tensile axial load; Ax and Ix are respectively the cross-sectional area and moment of inertia and Ex is the modulus of elasticity. The first and second terms in Eq. 5 refer respectively to the internal energy due to stretching and bending. The third term appears in Eq. 5 due to considering the case of large deflections but small strains. Using Eqs 1 2, Eq. 5 could be rewritten in matrix form as U e = 1 2 dt Kd where d = { u 1 w 1 θ 1 u 2 w 2 θ 2 } T. K is the general stiffness matrix given by K = Ks K b Kg in which Stretching stiffness matrix: 6 7 Ks = l Bending stiffness matrix: K b = l N T u ExAxN udx Geometric stiffness matrix: Kg = l N T w ExIxN wdx 9 N T w Nx N wdx The primes designate differentiation with respect to x. The kinetic energy of the beam element reads as T e = 1 2 l ρxax 2 u dx 1 l 2 w ρxax dx 11 t 2 t in which t is time and ρx is the mass density. Similarly, kinetic energy could also be rewritten in matrix from as 8 1 T e = 1 2 ḋt Mḋ 12 Here the dot. designates differentiation with respect to t and

4 686 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams M = Ms M b where Stretching mass matrix: Ms = l Bending mass matrix: M b = l N T uρxaxnudx N T w ρxaxn wdx 15 As observed, the effects of varying cross-sectional area and being axially functionally graded are considered in the evaluationof structuralmatrices throughconsideringax, Ix, ρx and Ex. Considering a linear distribution of modulus and elasticity and a cubic distribution of mass density, the structural matrices are derived explicitly and given in Appendix A for a non-prismatic beam whose breadth and height taper linearly. In order to determine the natural frequencies and critical load of the system, the following eigenvalue problems need to be solved Free longitudinal vibration: K g s φ = ω2 Mg s φ Free transverse vibration: K g b φ = ω2 T Mg b φ Stability analysis: K g b ηkg g φ = where ω and ω T are respectively the longitudinal and transverse natural frequencies and φ is the mode shape. In stability analysis, η is the eigenvalue and the critical load is determined as P cr = ηp where P is an assumed constant compressive load. The superscript g designates the global structural matrix which is obtained through assembling the elemental matrices and imposing the boundary conditions Numerical results and discussion In this section, the present element is employed in free vibration and stability analyses of axially FG beams with different boundary conditions. Numerical results are carried out for an axially FG beam with rectangular cross-section whose height and breadth both vary linearly and distributions of modulus of elasticity and mass density are assumed to polynomially vary as E = m r b i i and ρ = a i i, respectively. Therefore, i= i= cross-sectional area and moment of inertia could be expressed respectively as A =A 1 cb 1 ch and I = I 1 cb 1 3 ch where the breadth and height taper ratios could vary in the range of c b,c h 1. Ifc b = c h =, the beam would become a uniform one and if c b = c h =1, the beam would taper to a point at x = which is merely a theoretical limit. In order to facilitate the presentation of results, the boundary conditions including clamped, simple, free and guided are abbreviated respectively as C, S, F and G. Where possible, the computed results are compared with the previous published results in the literature Stability analysis Critical load P cr is calculated for a non-prismatic beam which is subjected to a point compressive load at its both ends. Firstly, the critical load is calculated for a unit length uniform beam, i.e. c b = c h =. The results are compared with those of Elishakoff [7] in Table 1 for different boundary conditions. It is observed that the results are in good agreement with those of Elishakoff [7]. Dividing the beam into 2 equal-length elements, critical load is calculated for an axially FG tapered beam for different combinations of taper ratios in Table 2. As expected, the critical load decreases as the height and/or breadth taper ratios increases due to the softening effect caused by the decrease in moment of inertia. It is observed that the effect of height taper ratio is more pronounced than that of breadth taper ratio since the power of height in moment of inertia is 3; while it is 1 for the breadth.

5 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 687 Table 1 Critical load for an axially FG uniform beam Boundary Condition Coefficients of E Present NE =1 Elishakoff [7] S-S b =1, b 1 =1, b 2 = C-C b = 1 6, b 1 =1, b 2 = S-C b = 9 16, b 1 = 3 4, b 2 = NE stands for number of elements Free longitudinal vibration Table 2 Non-dimensional critical load λ = P cr 2 /E I for an axially FG tapered beam with E =E 1 c b c h C-C C-F S-S In order to investigate the convergence of the method in free longitudinal vibration, the fundamental longitudinal frequency ω 1 for two types axially FG uniform beams is determined for different numbers of elements and compared with exact results from Candan and Elishakoff [29] in Fig. 2. It is well-observed that the computed results converge rapidly to exact ones as the number of elements increases. It is also instructive to know that FEM is a type of stiffness method which is formulated on the basis of variational calculus; therefore it is expected that the results provided by FEM show an upper bound of the exact ones. This fact could be clearly verified in Fig. 2. Dividing the beam into 4 elements, the first three non-dimensional longitudinal frequencies of an axially FG tapered beam for two different boundary conditions are reported in Table 3. It is worthy to note that cross-sectional area plays a key role in free longitudinal vibration of rods and it is obvious that the order of contribution of c h and c b in cross-sectional area is equal; thus the natural longitudinal frequencies of a rod with c h = c 1 and c b = c 2 are not different from those of a bar with c h = c 2 and c b = c 1. This point is considered in tabulating the results in Table 3. It is observed that unlike lower modes, the higher modes are not considerably affected by taper such that higher modes of an axially FG tapered rod is very close to those of uniform rod. Moreover, it is observed that natural frequencies of a C-F rod increase with taper ratio; while they show a decreasing trend for a clamped-clamped rod. The first three mode shapes of both C-F and C-C axially FG tapered rods with c h =and c b =.8 are plotted in Fig Free transverse vibration Firstly the fundamental transverse frequency ω T 1 of a unit-length axially FG uniform beam is obtained for different boundary conditions and distributions of mass density and modulus of elasticity. The results are compared with those of Elishakoff and his co-workers [14 16] in Table 4 where it is observed that the results are comparable with exact ones.

6 688 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Table 3 Non-dimensional longitudinal frequencies μ = ω ρ 2 /E for an axially FG tapered beam with ρ = ρ 1 2 and E = E 1 C-F C-C c h c b μ 1 μ 2 μ 3 μ 1 μ 2 μ Fig. 2. Convergece of the present element in determination of the fundamental longitudinal frequency of a unit length C-F inhomogeneous rod. Case 1: ρ =1, E =22 2 ;Case2:ρ =1, E = Results in squares are exact ones reported in Candan and Elishakoff [29]. Dividing the beam into 2 elements, the first three non-dimensional transverse frequencies of an axially FG tapered beam are given in Tables 5 7 for different combinations of height and breadth taper ratios with different boundary conditions. As observed in Eq. 4, the shape functions do not reflect the effects of varying moment of inertia; thus it could be expected that the error introduced by this element is greater for those problems where the rate of change of moment of inertia is high i.e. height and/or breadth taper ratios increase towards unity. It is also obvious that the effect of height tapering on moment of inertia is greater than that of breadth tapering; therefore the effects of breadth tapering is negligibly small on the convergence rate of the element. Seeking brevity, the convergence rate of the method is higher for those beam configurations where the height taper ratio is small. This point could be observed in Fig. 4 which depicts the convergence of the method in determination of the first three natural frequencies of a clamped-clamped axially FG tapered beam with ρ = ρ 1 2 and E =E 1 with respect

7 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 689 Table 4 Fundamental transverse frequency of an axially FG uniform beam Boundary Coefficients Coefficients of E Present Present Exact condition of ρ NE = 6 NE = 1 G-S a =1 b =61,b 1 =,b 2 = 14 b 3 =,b 4 = C-G a =1 C-F a =1 C-F a =1,a 1 =1 C-F a =1,a 1 =1 a 2 =1 b = 88 9,b 1 = 16 3,b 2 = 4 3 b 3 = 4, b 4 =1 b =26,b 1 =16,b 2 =6 b 3 = 4, b 4 =1 b = 324 5,b 1 = 214 5,b 2 = 14 5 b 3 = 6 5,b 4 = 11 5,b 5 =1 b = 578 5,b 1 = 46 5,b 2 =45 b 3 = 44 5,b 4 = 3 5,b 5 = 2 b 6 =1 Results from Ref. [14], Results from Ref. [15], Results from Ref. [16] Table 5 Non-dimensional transverse frequencies μ T = ω T ρ A 4 /E I for an axially FG tapered beam with ρ = ρ 1 2 and E = E 1 ; Boundary Conditions: C-C c b c h First Mode Second Mode Third Mode Fig. 3. The first three mode shapes of an axially FG tapered rod with c h =and c b =.8 for a C-F and b C-C boundary conditions.

8 69 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Table 6 Non-dimensional transverse frequencies μ T = ω T ρ A 4 /E I for an axially FG tapered beam with ρ = ρ 1 2 and E = E 1 ; Boundary Conditions: C-F c b c h First Mode Second Mode Third Mode Table 7 Non-dimensional transverse frequencies μ T = ω T ρ A 4 /E I for an axially FG tapered beam with ρ = ρ 1 2 and E =E 1 ; Boundary Conditions: S-S c b c h First Mode Second Mode Third Mode to different number of elements. It is observed that the convergence rate for the case of c h =.4 and c b =.8 is higher than the other two cases. 5. Concluding remarks A new element is proposed for matrix analysis of axially FG tapered beams by using the shape functions of homogeneous uniform beams. The effects of varying cross-sectional area and mechanical properties of the beam

9 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 691 Fig. 4. Cnvergence of the element in free transverse vibration of a C-C axially FG tapered beam. element are taken into account in the evaluation of the structural matrices. Carrying out several numerical examples, it was shown that the element could provide acceptable results in free vibration and stability analyses. Since limited studies have been carried out on axially FG tapered Euler-Bernoulli beams, the results given in this paper could be used as a means of comparison for any future studies. Moreover, this idea could be extended to derivation of a new element for shear deformable beams. Acknowledgment The financial support from University College of Engineering, University of Tehran Grant No /1/2 is deeply appreciated. Appendix A. Consider a non-prismatic beam whose cross-sectional area and moment of inertia vary respectively as 3 A =A 1 c b 1 c h and I =I 1 c b 1 c h where is the coordinate along the beam; A and I are respectively the cross-sectional area and moment of inertia at =; c h and c b are respectively the height and breadth taper ratios and is the beam length. It is assumed that ρ =a a 1 a 2 2 and E =b b 1. The beam is divided into several elements of length l. The structural matrices are evaluated for the n th element.

10 692 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams A.1. Stretching stiffness matrix [ ] The stretching stiffness matrix is obtained as Ks = A l k s 2 i,j,i,j=1, 2. k1,1 s = α 1 n 1 n 2 n 1 α 2 n 2 n α 3 n 1 2 b 2 where k s 2,2 = ks 2,1 = ks 1,1 α 1 = c h c b b 1 l 3, α 2 = l 2 [b 1 c h c b b c b c h ], α 3 = l [b 1 b c h c b ] A.2. Bending stiffness matrix [ ] The bending stiffness matrix is obtained as K b = I l 3 k b 4 i,j,i,j=1,..., 4. k1,1 b =12 [β 1 n 4 2n 3 3n 2 2n 4 n 1 β 2 n 4 2n n2 7 5 n β 3 n 2 n 7 n 1 β 4 n 2 n 2 β 5 n 1 ] b [ k2,1 b =6l β 1 n n n3 5n n β 3 n 3 2n n 1 2 k b 3,1 = k b 1,1 = k b 3,3 k b 4,1 =6l [ β 1 n n4 7 3 n3 2n 2 19 β 4 n n n 1 6 β 3 n 3 n n 1 β 4 n n 7 3 [ k2,2 b =4l 2 β 1 n n n n n k b 3,2 = kb 2,1 β 3 n n n 11 2 [ k4,2 b =2l2 β 1 n 4 2n n2 7 3 n β 3 n 2 n 4 n β 4 n n 19 3 β 4 n 2 n 13 3 β 2 n n3 17 β 5 n n2 2n ] b 4 β 2 n n3 7 5 n2 4 5 n β 5 n 1 ] b 4 3 β 2 n 4 3n 3 19 β 5 n 3 4 n 1 β 2 n 4 2n ] b 4 β 5 n n n ] b 4 5 n2 8 5 n k b 4,3 = k b 4,1

11 where A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 693 [ k4,4 b =4l 2 β 1 n n4 4 3 n3 n n β 3 n n2 2 1 n 1 β 4 n n 2 15 β 1 = b 1 l 5 c 3 hc b, β 2 =3l 4 c 2 h 13 [c h b 1 13 ] c bb c b b 1, [ ] 1 β 3 =3l 3 c h 3 b c 2 h c h b 1 c b b c b b 1 β 2 n 4 n n2 2 5 n 3 35 β 5 n 3 ] b 4 4 β 4 = l 2 2 [ 3c 2 hb c h 3b 1 3c b b b 1 c b ], β 5 = l 3 b 1 c b b 3c h b A.3. Geometric stiffness matrix Assuming Nx =P, the geometric stiffness matrix is obtained as Kg = P [ k g i,j],i,j=1,..., 4. k g 1,1 = 6 5l, kg 2,1 = 1 1, kg 3,1 = kg 1,1 = kg 3,3, kg 4,1 = kg 3,2 = kg 4,3 = kg 2,1, kg 2,2 = 2 15 l, k g 4,2 = 1 3 l, kg 4,4 = kg 2,2 A.4. Stretching mass matrix The stretching mass matrix is obtained as Ma = Al [ ] 3 M a 2 i,j,i,j=1, 2. M1,1 a = χ 1 n 4 3n n2 2n 3 χ 2 n n2 9 5 n 1 2 where χ 4 n 3 a 2 4 χ 3 n n 3 5 M2,1 a = 1 [χ 1 n 4 2n n2 4 5 n 1 χ 2 n 2 n 2 n 1 χ 3 n 2 n χ 4 n 1 ] a 2 2 M a 2,2 = χ 1 n 4 n n2 1 5 n 1 35 χ 4 n 1 a 2 4 χ 1 = l 4 c b c h a 2, χ 2 = l 3 [a 2 c h c b a 1 c h c b ], χ 2 n n2 3 1 n 1 χ 3 n n 1 1 χ 3 = l 2 [ a 2 2 a 1 c h c b a c h c b ], χ4 = l [a 1 a c h c b ]

12 694 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams A.5. Bending mass matrix [ ] The bending mass matrix is given by M b = A M b 2 i,j,i,j=1,..., 4. M1,1 b = 13l [γ 1 n n n n γ 3 n n M2,1 [γ b = 11l2 1 n γ 3 n n M3,1 b = 9l [γ 1 n 4 2n M b 4,1 = 13l2 42 γ 3 n 2 n γ 4 n 1 13 a 2 ] 11 n n n γ 4 n 15 ] a n n ] a 2 γ 4 n 1 2 [γ 1 n n n n ] a 2 γ 3 n n γ 4 n 7 13 M2,2 [γ b = l3 1 n n3 5 2 n2 7 6 n 7 33 γ 3 n n 5 12 M3,2 b = [γ 13l2 1 n γ 3 n n γ 4 n 5 8 γ 2 n 3 3 γ 2 n 3 45 γ 2 n 2 n γ 2 n 3 21 γ 2 n 3 15 a 2 ] 13 n n n γ 4 n 6 ] a 2 13 M4,2 [γ b = l3 1 n 4 2n n2 2 3 n 7 66 γ 3 n 2 n 5 18 M3,3 b = 13l [γ 1 n γ 3 n n γ 4 n 1 2 a 2 ] 13 n n n γ 4 n 3 ] a 2 13 γ 2 n n n n n n n n n2 5 4 n n n γ 2 n 2 n 1 n γ 2 n n n

13 where M b 4,3 = 11l2 21 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams 695 [γ 1 n n n n γ 3 n n γ 4 n 7 22 M4,4 [γ b = l3 1 n n3 n n 1 22 γ 3 n n 1 6 γ 4 n 3 8 a 2 ] γ 2 n 3 21 γ 2 n n2 1 2 n 1 12 a 2 ] γ 1 = l 4 a 2 c b c h, γ 2 = l 3 [a 2 c h c b a 1 c b c h ], γ 3 = l 2 [ a 2 2 a 1 c h c b a c h c b ],γ4 = l [a 1 a c h c b ] 22 n n 2 33 References [1] A. Bazoune, Effect of tapering on natural frequencies of rotating beams, Shock and Vibration [2] A. Chakraborty, S. Gopalakrishnan and J.N. Reddy, A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Sciences 45 23, [3] G. Catellani and I. Elishakoff, Apparently first closed-form solutions of semi-inverse buckling problems involving distributed and concentrated loads, Thin-Walled Structures 42 24, [4] I. Calio and I. Elishakoff, Can a trigonometric function serve both as the vibration and the buckling mode of an axially graded structure, Mechanics Based Design of Structures and Machines 32 24, [5] I. Calio and I. Elishakoff, Closed-form trigonometric solutions for inhomogeneous beam-columns on elastic foundation, International Journal Structural Stability and Dynamics 4 24, [6] I. Calio and I. Elishakoff, Closed-form solutions for axially graded beam-columns, Journal of Sound and Vibration 28 25, [7] I. Elishakoff, Inverse buckling problem for inhomogeneous columns, International Journal of Solids and Structures 38 21, [8] I. Elishakoff, Some unexpected results in vibration of non-homogeneous beams on elastic foundation, Chaos Soliton Fractal 12 21, [9] I. Elishakoff, Euler s problem revisited: 222 years later, Meccanica 36 21, [1] I. Elishakoff, Apparently first closed-form solution for frequency of beam with rotational spring, AIAA Journal 39 21, [11] I. Elishakoff and A. Perez, Design of a polynomially inhomogeneous bar with a tip mass for specified mode shape and natural frequency, Journal of Sound and Vibration , [12] I. Elishakoff and D. Pentaras, Apparently the first closed-form solution of inhomogeneous elastically restrained vibrating beams, Journal of Sound and Vibration , [13] I. Elishakoff and O. Rollot, New closed-form solutions for buckling of a variable stiffness column by Mathematica, Journal of Sound and Vibration , [14] I. Elishakoff and R. Becquet, Closed-form solutions for natural frequencies for inhomogeneous beams with one sliding support and the other pinned, Journal of Sound and Vibration 238 2, [15] I. Elishakoff and R. Becquet, Closed-form solutions for natural frequencies for inhomogeneous beams with one sliding support and the other clamped, Journal of Sound and Vibration 238 2, [16] I. Elishakoff and S. Candan, Apparently first closed-form solution for vibrating inhomogeneous beams, International Journal of Solids and Structures 38 21, [17] I. Elishakoff and V. Johnson, Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass, Journal of Sound and Vibration , [18] I. Elishakoff and Z. Guede, A remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Euler-Bernoulli beams, Chaos Soliton Fractal 12 21, [19] I. Elishakoff and Z. Guede, Novel closed-form solutions in buckling of inhomogeneous columns under distributed variable loading, Chaos Soliton Fractal 12 21, [2] I. Elishakoff and Z. Guede, Analytical polynomial solutions for vibrating axially graded beams, Mechanics of Advanced Material and Structure 11 24, [21] J. Neuringer and I. Elishakoff, Inhomogeneous beams that may possess a prescribed polynomial second mode, Chaos Soliton Fractal 12 21, [22] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd Edition, John Wiley & Sons, 22. [23] J.R. Banerjee, Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method, Journal of Sound and Vibration 233 2,

14 696 A. Shahba et al. / Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams [24] K.V. Singh and G. i, Buckling of functionally graded and elastically restrained non-uniform columns, Composites: Part B [25].. Ke, J. Yang, S. Kitipornchai and Y. iang, Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials, Mechanics of Advanced Materials and Structures 16 29, [26]. Wu, Q.S. Wang and I. Elishakoff, Semi-inverse for axially functionally graded beams with an anti-symmetric vibration mode, Journal of Sound and Vibration , [27] M. Aydogdu, Semi-inverse method for vibration and buckling of axially functionally graded beams, Journal of Reinforced Plastics and Composites 27 28, [28] M.A. Benatta, A. Tounsi, I. Mecha and M.B. Bouiadjra, Mathematical solution for bending of short hybrid composite beams with variable fibers spacing, Applied Mathematics and Computation , [29] S. Candan and I. Elishakoff, Constructing the axial stiffness of longitudinally vibrating rod from fundamental mode shape, International Journal of Solids and Structures 38 21, [3] S.A. Sina, H.M. Navazi and H. Haddadpour, An analytical method for free vibration analysis of functionally graded beams, Materials and Design 3 29, [31].F. i, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams, Journal of Sound and Vibration , [32] Z. Guede and I. Elishakoff, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos Soliton Fractal 12 21, [33] Z. Guede and I. Elishakoff, Apparently the first closed-form solution for inhomogeneous vibrating beams under axial loading, Proceedings of Royal Society A ,

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