Free vibration and buckling of tapered columns made of axially functionally graded materials

Applied Mathematical Modelling 75 (2019) 73–87

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Free vibration and buckling of tapered columns made of axially functionally graded materials Joon Kyu Lee a,∗, Byoung Koo Lee b a

Department of Civil Engineering, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea Department of Civil and Environmental Engineering, Wonkwang University, 460 Iksan-daero, Iksan-si, Jeollabuk-do 54538, Republic of Korea

b

a r t i c l e

i n f o

Article history: Received 1 October 2018 Revised 19 April 2019 Accepted 8 May 2019 Available online 15 May 2019 Keywords: Free vibration Buckling Axially functionally graded material Tapered column

a b s t r a c t This paper presents a unified model to analyze the free vibration and buckling of axially functionally graded Euler-Bernoulli columns subjected to an axial compressive force. The material properties vary linearly along the longitudinal direction, and column with circular and square cross sections is linearly tapered. The governing differential equations of the problem are derived and solved using the direct integral method combined with the determinant search technique. The computed results are compared with those reported in the literature and obtained from the finite element software ADINA. Numerical examples for natural frequency, buckling load and their corresponding mode shapes are given to highlight the effects of modular ratio, taper ratio and cross sectional shape as well as the end condition. © 2019 Published by Elsevier Inc.

1. Introduction Buckling instability and free vibration analyses of columns are inherent parts of engineering design to prevent collapse or severe damage of the structures in buildings and bridges as well as in machine parts. Columns with material inhomogeneity and geometrical nonuniformity are widely used in civil, mechanical, aeronautical and biomedical engineering. Particularly, columns fabricated by functionally graded materials (FGMs) have attracted great interest in relevant fields due to their improved performances such as thermal resistance and high stiffness [1]. The gradient variation of the columns may be orientated in cross section or/and in the axial direction. Meanwhile, in order to meet the aesthetic demand and optimize the weight and strength of the structures, non-prismatic columns may be employed in the form of tapered, stepped and continuous segmented columns [2]. Such materials and geometrical variations make the buckling and vibration problems more complicated. Numerous studies have been devoted to modeling the free vibration and buckling of axially functionally graded (AFG) beam-columns with nonuniform cross section based on Euler-Bernoulli or Timoshenko beam theory. Compared to homogenous uniform system, the governing equations of AFG beam-columns involves variable coefficients. Thus, a variety of analytical and numerical approaches have been proposed to solve the problem, including using the Adomain decomposition method [3], the asymptotic development method [4,5], the Chebyshev polynomial theory [6], the combination of differential transform element method and differential quadrature method [7,8], the differential transformation based dynamic stiffness

∗

Corresponding author. E-mail addresses: [email protected] (J.K. Lee), [email protected] (B.K. Lee).

https://doi.org/10.1016/j.apm.2019.05.010 0307-904X/© 2019 Published by Elsevier Inc.

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Nomenclature a1 , a2 A, Ae b B c, c1 , c2 Ci Ct,1 , Ct,2 , Ct,3 D D1 , D2 , D3 E, Eb , Ee , Et fi FI i I, Ie l m M n p P Q r, rb , re , rt Rf t V (x, y) α ξ ,β ξ (ξ , η ) ρ, ρb, ρe, ρt

ωi

constants areas at any point and mid-span buckling load parameter buckling load constants ith frequency parameter two adjacent trial and advanced frequencies for Regula-Falsi method determinant determinants for Ct,1 , Ct,2 and Ct,3 Young’s moduli at any point, bottom, mid-span and top ith natural frequency transverse inertia force mode number second moments of plane area at any point and mid-span column length modular ratio bending moment taper ratio axial force parameter axial force shear force circumradii at any point, bottom, mid-span and top flexural rigidity time column volume Cartesian coordinates variables in nondimensional differential equations nondimensional Cartesian coordinates mass densities at any point, bottom, mid-span and top ith angular frequency

method [9], the finite element method [1,10], the Harr wavelet technique [11], the direct integral method [12–15], the localized differential quadrature method [16], Newton’s eigenvalue iteration method [17], the semi-inverse method [18,19], the spline finite point method [20], and the symbolic-numeric method of initial parameters [2]. On the other hand, the closed form solutions were established for the buckling and free vibration of beam-columns [21–25]. This paper presents a mathematical model for predicting the free vibration and buckling of axially loaded AFG tapered columns. The governing differential equations of the motions are derived based on the Euler–Bernoulli beam theory. Differing from the aforementioned methods, the direct integral method in conjunction with determinant search method is suggested. Also, consideration is given to AFG tapered columns with different cross sectional shapes: circular and square cross sections. To validate the accuracy of the theories including the solution methods developed herein, the natural frequencies and buckling loads obtained from this analysis are compared with those available from literature and computed via finite element software ADINA. Column responses are observed to be functions of material and geometrical parameters as well as end condition. 2. Mathematical model Shown in Fig. 1(a) is a tapered column with span length l, made of axially functionally graded materials (AFGMs). The cross sectional shape of the columns is assumed to be circular and square. The column end is supported by free or hinged or clamped ends, allowing a total of nine end conditions. The column is externally subjected to an axial compressive load P less than the buckling load B at the top end. At any coordinate x originating from the bottom end (x = 0), the flexural rigidity, mass density and circumradius of the cross section are depicted by Rf , ρ and r, respectively. Here, the circumradius r is defined as a radius of the circular cross section and a half-diagonal length of the square cross section. It is well known that the flexural rigidity Rf is expressed as Rf = EI where E and I are the Young’s modulus of AFGM and second moment of plane area of the cross section, respectively. When the column vibrates, the column elastically deforms the mode shape, defined in the Cartesian coordinates (x, y) shown by the solid curve in Fig. 1(a). And the column is subjected to the dynamic shear force Q and bending moment M as well as the static axial load P, and the transverse inertia force FI is produced in a small element shown in Fig. 1(b) since

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75

Fig. 1. Schematics of AFG tapered column: (a) geometry and coordinate system, (b) deflection and forces of column element, and (c) linear variations of Young’s modulus, mass density and circumradii of cross sections along axial direction.

the column has mass. In this study, the free vibration is harmonic motion in which each dynamic coordinate is proportional to sin(ωi t). As an example, yx,t = yx sin (ωi t) where yx ( = y) is the transverse deflection, ω is the angular frequency, i is the mode number and t is the time. From the free body diagram based on the Euler–Bernoulli beam theory, illustrated in Fig. 1(b), the equations of equilibrium are obtained:

 

Fy = (Q + dQ ) − Q − FI dx = 0 :

dQ − FI = 0, dx

M(x,y) = (Q + dQ )dx + (M + dM ) − M + P dy = 0 :

(1) dM dy +Q +P = 0. dx dx

(2)

From Eq. (2), the first derivative dM/dx is rearranged as

dM dy = −Q − P . dx dx

(3)

Combining the second derivative d2 M/dx2 obtained from Eq. (3) and dQ/dx = FI in Eq. (1) yields

dQ d2 M d2 y d2 y =− − P 2 = −P 2 − FI . 2 dx dx dx dx

(4)

The bending moment M and transverse inertia force FI are expressed as [26,27]

M = EI

d2 y d2 y = Rf 2 , d x2 dx

(5)

FI = −ρ Aωi2 y,

(6)

where A is the area of the cross section at x. Substituting Eq. (6) into Eq. (4) yields Eq. (7) and the second derivative of Eq. (5) gives Eq. (8):

d2 M d2 y = −P 2 + ρ Aωi2 y, d x2 dx

(7)

d2 R f d2 y dR f d3 y d2 M d4 y = +2 + Rf 4 . 2 2 2 3 dx dx dx dx dx dx

(8)

Using Eqs. (7) and (8), the fourth order ordinary differential equation that governs the free vibration of the AFG column is obtained as

2 dR f d3 y d4 y =− − R f dx dx3 d x4



d2 R f P+ d x2



1 d2 y A + ρωi2 y. R f d x2 Rf

(9)

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J.K. Lee and B.K. Lee / Applied Mathematical Modelling 75 (2019) 73–87

It is a well-known fact that the natural frequency vanishes when the axial compressive load P is identical to the buckling load B. That is, ωi is set to zero in buckling analysis. Accordingly, the following equation is derived directly from Eq. (9) for determining the buckling load:

2 dR f d3 y d4 y =− − R f dx dx3 d x4



d2 R f B+ d x2



1 d2 y . R f d x2

(10)

Let us consider the boundary conditions of the column ends (x = 0 and x = l). At the free end, the bending moment M in Eq. (5) and shear force Q in Eq. (3) are zero, that is

d2 y d3 y P dy = 0, + = 0. 2 R f dx dx d x3

(11)

At the hinged end, the deflection y and bending moment M are zero:

y = 0,

d2 y = 0. d x2

(12)

At the clamped end, the deflection y and rotation dy/dx are zero:

y = 0,

dy = 0. dx

(13)

Now define the variable functions of E, ρ and r for material and geometrical properties of the AFG columns. In the governing equations of Eqs. (9) and (10), arbitrary variable functions of the material properties of E and ρ are available. There are various kinds of functions in the literature: linear [5,7,15]; polynomial [2,5,12]; exponent [14,16]; periodic [12]; trigonometric [4,19] and piece-wised [17], etc. In this study, the function of E is selected to linear one, as shown Fig. 1(c), where Young’s moduli at the bottom end (x = 0) and top end (x = l) are denoted by Eb and Et , respectively. The modular ratio m defined as

m=

Et . Eb

(14)

The value of modular ratio m practically ranges between 0.05 and 10, which corresponds to the typical range of 27.9 and 380 GPa for AFG materials made of ceramics and metals [28]. By assuming linear function of the Young modulus E at x is given by





2Ee x 1 + (m − 1 ) , m+1 l

E=

(15)

where Ee is the Young’s modulus at the mid-span (x = l/2) of the column expressed as

Ee = Ex=l/2 =

1 (E + Et ). 2 b

(16)

As pointed out by Huang and Li [12], the mass density can be parallel to Young’s modulus and hence the density ratio is defined as same as the modular ratio m (see Fig. 1(b)), or

m=

ρt , ρb

(17)

where ρ t and ρ b are the mass densities at the top end and bottom end, respectively. Similar to Eqs. (15) and (16), the mass densities ρ at x and ρ e at the mid-span are given by

ρ=





2 ρe x 1 + (m − 1 ) , m+1 l 1 2

ρe = (ρb + ρt ).

(18) (19)

As shown in Fig. 1(c), a linear distribution of the circumradius r is considered for the circular and square cross sections of the column. The taper ratio n is defined as

n=

rt . rb

(20)

The circumradius r at x and re at the mid-span are expressed as

r=





2re x 1 + (n − 1 ) , n+1 l

re = rx=l/2 =

1 (r + rt ). 2 b

(21) (22)

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77

Variable functions of the area A and second moment of plane area I in terms of the circumradius r at x are derived as

A = c1 r 2 = I = c2 r 4 =

4c1 re2



(n + 1 )2  16c r 4 2 e

(n + 1 )

4

1 + (n − 1 )

1 + (n − 1 )

x l

x l

2

4

,

(23)

,

(24)

where c1 and c2 are constants, i.e., c1 = π and c2 = π /4 for circular cross section and c1 = 2 and c2 = 1/3 for square cross section. Using Eqs. (15) and (24), the variable function of the flexural rigidity Rf ( = EI) at x is established:

Rf =



32c2 Ee re4

(m + 1 ) (n + 1 )

4

1 + (m − 1 )

x l



1 + (n − 1 )

x l

4

.

(25)

To facilitate the numerical analysis, following dimensionless parameters are introduced:

ξ

x = , l

η

y Pl 2 Bl 2 = , p= , bi = , Ci = ωi l 4 l Ee re Ee re4



ρe l 2 Ee re2

,

(26)–(30)

where (ξ , η) are the normalized Cartesian coordinates, (p, bi ) are the dimensionless axial and buckling load parameters and Ci is the dimensionless frequency parameter. Substituting the functions of ρ in Eq. (18), A in Eq. (23) and Rf in Eq. (25) into Eq. (9) and using the dimensionless parameters, Eqs. (26)–(30), yields

2 dαξ d3 η 1 d4 η =− − αξ dξ dξ 3 αξ dξ 4



d2 αξ a1 p+ c2 dξ 2



d2 η c1 a2 2 + C β η, c2 i ξ dξ 2

(31)

where,

a1 =

( m + 1 ) ( n + 1 )4 32

; a2 =

(n + 1 )2 4

;

αξ = (1 + m1 ξ )(1 + n1 ξ )4 ;

dαξ 4 3 = m1 ( 1 + n1 ξ ) + 4n1 ( 1 + m1 ξ ) ( 1 + n1 ξ ) ; dξ d2 αξ dξ 2

βξ =

= 8m1 n1 (1 + n1 ξ ) + 12n21 (1 + m1 ξ )(1 + n1 ξ ) ; 3

1

( 1 + n 1 ξ )2

2

with m1 = m − 1 and n1 = n − 1.

It is noteworthy that the governing equation for the buckling of AFG columns is obtained by eliminating the last term including Ci in Eq. (31) and replacing the buckling load parameter bi to the load parameter p as previously discussed in Eq. (10), or

2 dαξ d3 η 1 d4 η = − − αξ dξ dξ 3 αξ dξ 4



d2 αξ a1 bi + c2 dξ 2



d2 η . dξ 2

(32)

Using the dimensionless parameters Eqs. (26)–(30), the boundary conditions of Eqs. (11)–(13) in the dimensional forms are transformed into the nondimensional ones, or

Free end (ξ = 0 ) :

d2 η d3 η a1 d η = 0 , + p ; c2 d ξ dξ 2 dξ 3

(33.1)

Free end (ξ = 1 ) :

d2 η d3 η a1 p d η = 0, + ; 2 c2 mn 4 d ξ dξ dξ 3

(33.2)

Hinged end (ξ = 0 and

ξ = 1 ) : η = 0,

Clamped end (ξ = 0 and

d2 η = 0; dξ 2

(34)

dη = 0, dξ

(35)

ξ = 1 ) : η = 0,

where the load parameter p in Eq. (33) is replaced to the buckling load parameter bi for the buckling problem. The above fourth order ordinary differential equation of Eqs. (31) and (32) with the boundary conditions, Eqs. (33)–(35), govern the free vibration and buckling, respectively, of axially loaded AFG columns. The end condition, cross sectional shape, modular ratio m, taper ratio n and load parameter p (except in the buckling analysis) are the input parameters, while Ci and bi are the eigenvalues which will be calculated with their mode shapes (ξ i ,ηi ).

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J.K. Lee and B.K. Lee / Applied Mathematical Modelling 75 (2019) 73–87 Table 1 Assumed boundary conditions at bottom (ξ = 0) and calculated trial results at top (ξ = 1) for hinged–clamped ends. Hinged end (ξ = 0)

η

Set no 1 2

0 0

η 1 3

η  0 0

Clamped end (ξ = 1)

η   2 4

Execution of Runge–Kutta method

η

η

η1,T η2,T

η

1,T

η2 ,T

η  

η1,T  η2,T

η   

η1,T  η2,T

Note: Figures ‘1,2,3,4’ in Set 1 and 2 are assumed values, and the subscript ‘T’ at the clamped end stands for ‘at the top end’.

3. Methods of solution Solution methods are developed to calculate the frequency parameter Ci and the buckling load parameter bi with their mode shapes (ξ i ,ηi ). For free vibration problem, to obtain the mode shape, the governing differential equation Eq. (31) is integrated numerically by the fourth order Runge–Kutta method [29] as one of the direct integral method, which is known to be algebraically stable [30]. Then, to compute eigenvalues Ci in Eq. (31) the determinant search technique in conjunction with the Regula–Falsi method [29] is used. For the sake of clarity, the algorithm of solution methods is summarized: 1) Define m, n and p with the cross sectional shape and end condition. 2) Set a trial frequency Ct in Eq. (31) as a trial eigenvalue Ci . The first trial Ct is zero. 3) According to the input end condition, subject initial boundary conditions of Eqs. (33)–(35) to Eq. (31) at ξ = 0 and assume two sets of unknown initial conditions as an initial value problem. For example, exemplary schemes of Steps (3) and (4) for the hinged–clamped end condition are shown in Table 1. 4) Integrate Eq. (31) numerically from ξ = 0 to ξ = 1 using Runge–Kutta method. This result gives trial coordinates of η, η , η , η at ξ , where the prime denotes differentiation with respect to ξ . 5) Using results of two trial coordinates separately calculated by Set 1 and Set 2, the following linear combinations are established:

η = η1 + cη2 , η = η1 + cη2 , η = η1 + cη2 , η = η1 + cη2 ,

(36)

where c is a constant. If the trial frequency Ct assumed in Step (2) is a characteristic eigenvalue Ci , then the boundary conditions of Eq. (35) at the clamped end (ξ = 1) are satisfied for the linear combination equations, that is

ηT = η1,T + cη2,T = 0, ηT = η1 ,T + cη2 ,T = 0,

(37)

where subscript T represents the top end of column. The above equations can be written in the matrix form:

   η1,T η2,T 1 ηT = η1 ,T η2 ,T c = 0. ηT

(38)

Since c = 0, to satisfy the above matrix equation, the following determinant D must be zero:



η1,T η2,T D= = 0. η 1,T η 2,T

(39)

where the first convergence criterion is

|D| ≤ 10−8 .

(40)

If this criterion is met, the trial Ct is just the characteristic eigenvalue Ci and go to Step (8). Example of D versus Ct curve is shown in Fig. 2 where the computed frequencies (lowest three) of C1 , C2 and C3 marked by  are presented. 6) If not, increase the trial frequency Ct = Ct + Ct and perform Steps (2)–(5). During executions, note the sign of D1 × D2 where D1 is D of previous execution and D2 is D of present execution. When the sign changes, the characteristic eigenvalue of Ci lies between Ct,1 and Ct,2 where Ct,1 is the trial frequency corresponding to D1 and Ct,2 corresponding to D2 , respectively. An advanced trial frequency Ct,3 more approaching to the eigenvalue Ci can be obtained by Regula-Falsi method, a solution method of nonlinear equation:

Ct,3 =

Ct,1 |D2 | + Ct,2 |D1 | . |D1 | + |D2 |

(41)

Here, the relationship between Ct,1 , Ct,2 and Ct,3 marked by  are shown in Fig. 2 where the input parameters are presented. See coordinates of (Ct,1 , D1 ) and (Ct,2 , D2 ) marked by ●, from which Ct,3 = 22.08 is computed by Eq. (39), much more approached to the eigenvalue C2 = 23.37 comparing its corresponding Ct,1 = 20 and Ct,2 = 30. The residual errors for Ct,1 and Ct,2 are − 14.4% and + 28.4%, respectively, and the residual error for Ct,3 is − 5.51%. The trial Ct,3 after applying Regula–Falsi scheme is much closer to C2 , compared to Ct,1 and Ct,2 trials before the application of the Regula–Falsi method. 7) Second convergence criterion is

Ct,2 − Ct,1 ≤ 10−5 . Ct,2

(42)

J.K. Lee and B.K. Lee / Applied Mathematical Modelling 75 (2019) 73–87

79

Fig. 2. D versus Ct curve.

If the criterion is met, the trial Ct,3 is the characteristic eigenvalue Ci . 8) From Eq. (37), the constant c of the linear combination relationship is computed as

c=−

η1,T η or c = −  1.T , η2,T η 2,T

(43)

where two c values in Eq. (43) are the same. All characteristic coordinates of η, η , η , η at ξ are computed using Eq. (36). 9) In order to compute higher frequencies, set a new Ct = Ci + Ct and repeat all above Steps until the number of frequency reaches the desired number being computed. 10) Output the calculation results of Ci along with the corresponding coordinates of η, η , η , η and stop calculation. For buckling load problem, the numerical procedure is straightforward the same as the free vibration problem except that the load parameter p is not an input parameter and the eigenvalue is bi instead of Ci . Based on the proposed formulations, two FORTRAN computer programs were coded for computing Ci and bi with their corresponding mode shapes (ξ i ,ηi ). 4. Numerical results and discussion In this section, the effectiveness of the proposed methods is confirmed, and then the effects of the end condition, cross sectional shape, modular ratio, taper ratio and load parameter on the free vibration and buckling behavior of AFG columns are explored. For verification, the natural frequencies and buckling loads obtained from this study are compared with the existing solutions. Fig. 3 shows the comparison of Ci and b( = b1 ) for homogeneous material with circular cross section: (a) for the frequency parameters C1,2,3 of axially loaded uniform columns, the obtained results are very close to the closed form solutions by Karnovsky and Lebed [31], regardless of end conditions; (b) for the critical buckling load parameters b of tapered columns, the obtained results agree well with the results by Riley [32] using Newmark method. Table 2 compares the predicted values of natural frequencies fi ( = ωi /2π ) and critical buckling loads B of AFG tapered circular columns with those determined via the finite element software ADINA. Here, the graded material is composited with pure aluminum (Al) at the bottom end and pure zirconia (ZrO2 ) at the top end. Here, the material and geometrical properties of column in both the free vibration and buckling problems are: l = 1 m, Eb = 70 GPa, ρ b = 2700 kg/m3 , rb = 0.1 m, Et = 140 GPa, ρ t = 5400 kg/m3 , rt = 0.05 m and P = 0. Using these column properties, the distributions of E, ρ and r can be computed at x. The results are in good agreement within 3% error, verifying the theories including the numerical methods developed herein. Table 3 shows the frequency parameters Ci for AFG tapered circular columns with different material and geometrical gradients. The values of Ci vary greatly, depending on end condition. For a given example, the largest and smallest values

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Fig. 3. Comparison of results of this study with those available from literature: (a) Ci and (b) b. Table 2 Comparisonsa of natural frequencies and critical buckling loads of this study with those obtained from finite element ADINA for AFG columns. (a) Natural frequency fi in Hz End condition

Data source

Natural frequency fi in Hz i= 1

i= 2

i=3

Free–clamped

ADINA This study ADINA This study ADINA This study ADINA This study ADINA This study ADINA This study

65.05 65.96 281.2 288.3 403.8 408.1 151.8 155.4 478.2 490.6 632.2 647.0

551.4 571.8 1134. 1173. 1387. 1431. 715.6 742.4 1462. 1509. 1735. 1802.

1675. 1753. 2513. 2632. 2887. 3033. 1820. 1913. 2956. 3111. 3376. 3547.

Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped (b) Buckling load B in MN End condition

Critical buckling load B in MN ADINA

This study

Free–clamped Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped

3.365 21.32 43.67 7.793 42.79 82.93

3.375 22.00 44.48 8.032 43.61 84.64

a

See text for column parameters.

of Ci are attained at the clamped-clamped and clamped-free ends, respectively. Interestingly, even though the values of m and n are exchanged each other, their corresponding Ci values do not change. For the first frequency parameter C1 (called fundamental frequency), C1 value of free-free end with m = 2 and n = 0.5 equals that of free-free ends with m = 0.5 and n = 2. The effect of modular ratio m on C1 of AFG tapered circular columns is highlighted in Fig. 4, where the first frequency parameter C1 is only presented because the higher natural frequencies are often of no practical interest [33]. As m increases within 0 < m ≤ 5, C1 gradually increases for the hinged–clamped, hinged–hinged and free–clamped ends. In contrast, the

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81

Table 3 Frequency parameters Ci of AFG tapered circular columns with different end conditions. (a) For m = 2, n = 0.5 and p = 0 End condition

Frequency parameter Ci

Free–free Free–hinged Free–clamped Hinged–free Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped

i=1

i=2

i=3

11.41 6.930 1.085 8.466 4.743 6.715 2.557 8.071 10.64

30.68 24.00 9.407 25.38 19.30 23.55 12.22 24.83 29.65

59.51 50.43 28.83 51.78 43.31 49.90 31.49 51.19 58.36

11.41 6.930 1.085 8.466 4.743 6.715 2.557 8.071 10.64

30.68 24.00 9.407 25.38 19.30 23.55 12.22 24.83 29.65

59.51 50.43 28.83 51.78 43.31 49.90 31.49 51.19 58.36

(b) For m = 0.5, n = 2 and p = 0 Free–free Free–hinged Free–clamped Hinged–free Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped

Fig. 4. C1 versus m curves.

value of C1 monotonically decreases for the clamped–free and clamped–hinged ends. For the clamped–clamped ends, C1 increases first, reaches a peak value and then decreases with an increase of m. The peak frequency parameter marked by  is 11.16, when the value of m is 0.391. This finding indicates that there exists a specified material property such that an AFG tapered column is capable of carrying a maximum natural frequency, which is regarded as dynamically strongest column [34].

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Fig. 5. C1 versus n curves. Table 4 Critical buckling load parameters b by end condition for circular cross section. End condition

Critical buckling load parameter b m = 2; n = 0.5

m = 0.5; n = 2

Free–clamped Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped

1.016 6.622 13.29 2.418 13.13 25.48

2.418 6.622 13.13 1.016 13.29 25.48

Fig. 5 demonstrates the effect of taper ratio n on C1 of AFG tapered circular column. When n increases, C1 increases for the clamped–clamped, hinged–clamped and free–clamped ends, and decreases for the clamped–free end. Also, the convex shape and prominent peak (denoted by ) are observed in C1 curves with the clamped–hinged and hinged–hinged ends, suggesting that there is the dynamical optimal taper ratio at which the column has the maximum natural frequency. Fig. 6 explains the effect of load parameter p on C1 of AFG tapered circular column. With an increase of p, the C1 value continuously decreases and the rate of decrease in C1 increases, irrespective of end condition. When p = b, the value of C1 becomes 0, and the column transversely buckles. For an instance, the AFG tapered column with clamped–clamped ends exhibits the critical buckling parameter of 25.48 (see symbol ). Table 4 presents the critical buckling load parameter b of AFG tapered circular columns with different material and geometrical gradients. The b is sensitive to the end condition and the largest and smallest values of b are attained at the clamped–clamped and free–clamped ends. Interestingly, the b values are the same as those for the reverse end supports (e.g., hinged–clamped versus clamped–hinged). In addition, b values of free–free, free–hinged and hinged–free ends are zero, not given in this table, indicating that the columns with these end conditions are unstable in the structural system. The effect of modular ratio m on b of AFG tapered circular columns is displayed in Fig. 7, where m < 1 corresponds to the case of softer column toward the top while m > 1 corresponds to the case of stiffer column toward the top. In general, b increases as m increases and the dependence of m on b is significant for smaller m. However, for the clamped–clamped end, a peak b marked by  is observed. This means that in a similar manner, as that of the dynamically strongest column, the proposed model can estimate a statically strongest column that is capable of bearing applied compressive load [35]. It is also noticed in this figure that the highest value of b is obtained at the clamped–clamped ends, followed by the hinged–

J.K. Lee and B.K. Lee / Applied Mathematical Modelling 75 (2019) 73–87

Fig. 6. C1 versus p curves.

Fig. 7. b versus m curves.

83

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J.K. Lee and B.K. Lee / Applied Mathematical Modelling 75 (2019) 73–87

Fig. 8. b versus n curves. Table 5 Comparisonsa of fundamental frequency f1 with P = 0 and critical buckling load B between circular and square cross sections under same volume. End condition

Fundamental frequency f1 in Hz

Critical buckling load B in MN

Circular section

Square section

Circular section

Square section

Free–clamped Hinged–hinged Hinged–clamped Clamped–free Clamped–hinged Clamped–clamped Radius of cross section Ratio of square to circular

49.62 216.9 307.0 116.9 369.0 486.6 re = 0.0564 m f1,squ. /f1,cir. = 1.0233

50.78 221.9 314.2 119.6 377.6 498.0 re = 0.0707 m

1.081 7.045 14.14 2.573 13.97 27.11 re = 0.0564 m Bsqu. /Bcir. = 1.0472

1.132 7.377 14.81 2.694 14.63 28.39 re = 0.0707 m

a

See text for column parameters.

clamped and clamped–hinged ends. The smallest is at free–clamped one. This supports the fact that the higher degree of end condition the higher buckling performance of the column. The effect of taper ratio n on b of AFG tapered circular columns is depicted in Fig. 8, where n < 1 represents the case of tapered down toward the top whereas m > 1 represents the case of tapered up toward the top. It is found that the influence of n on b is predominant for smaller n. Smooth convex-shape variation is shown with the peak values of b for all end conditions. For an example, the clamped–clamped column has a peak b = 30.21 at n = 0.900. Table 5 presents the effect of cross sectional shape on the fundamental frequencies f1 and buckling loads B for AGF tapered columns with a given volume. The circular and square cross sections are taken into account in this study. The column parameters are: (l = 1 m, V = 0.01 m3 ), (Ee = 105 GPa, ρ e = 4050 kg/m3 ) and (m = 2, n = 0.5). The column circumradius re can be computed using V = c1 re2 l from which re is calculated as re = 0.0564 m for circular cross section and re = 0.0707 m for square one. The results indicate that irrespective of end condition, the values of f1 and B for square column are 2.33% and 4.72% larger than those for circular column, respectively. It is revealed these quantitative differences between circular and square cross sections are always constant despite material and geometrical variations. The mode shapes to vibrating the AFG tapered circular columns with different end conditions are illustrated in Fig. 9, where the effects of m, n and p are highlighted. As shown, the mode shape is independent on the modular ratio m and load

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Fig. 9. Free vibration mode shape for AFG tapered circular columns with different end conditions: effects of (a) modulus ratio m for first mode, (b) taper ratio n for second mode, and (c) load parameter p for third mode.

Fig. 10. Effect of column parameters on buckling mode shape for AFG tapered circular columns with different end conditions: (a) modular ratio m for clamped–free ends and (b) taper ratio n for hinged–hinged ends.

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parameter p, but it is highly affected by the taper ratio n and end support. Fig. 10 shows the buckled shapes of AFG tapered circular columns with different end conditions. From these mode shapes, the positions of nodes and maximum deflections of the load-carrying capacity can be evaluated, which is useful for the stability analysis and design of AFG columns. 5. Concluding remarks This paper presented a unified model for analyzing the free vibration and buckling of the AFG tapered columns subjected to an axial compressive force. It is assumed that the material properties and circumradius of the cross section vary along the axial direction according to linear distributions. The ordinary differential equations of the motion are derived with their boundary conditions based on the Euler–Bernoulli beam theory. To solve the governing equations with the material and geometrical parameters, the direct integral method is applied to compute the mode shape and then the determinant search technique enhanced by the Regula–Falsi method is used. The validation of the approach is proven through the comparisons of the predicted natural frequencies and buckling loads with those reported in literature and determined from finite element software ADINA. The results reveal that the buckling load increase with the modular ratio while the fundamental frequencies exhibit different trends for different end conditions. Fundamental frequency dramatically decreases to zero provided that applied compressive load is close to the buckling load. For a fixed volume of the column, the fundamental frequency and buckling load of AFG columns with square cross section are higher than those with circular cross section. 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