Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 3, pp. 859-870, Warsaw 2016
DOI: 10.15632/jtam-pl.54.3.859

AN APPROACH TO FREE VIBRATION ANALYSIS OF AXIALLY

GRADED BEAMS

Stanisław Kukla, Jowita Rychlewska

Czestochowa University of Technology, Institute of Mathematics, Czestochowa, Poland

e-mail: stanislaw.kukla@im.pcz.pl; jowita.rychlewska@im.pcz.pl

In this study, the solution to the free vibration problem of axially graded beams with a
non-uniform cross-section has been presented. The proposed approach relies on replacing
functions characterizing functionally graded beams by piecewise exponential functions. The
frequency equation has been derived for axially graded beams divided into an arbitrary
number of subintervals. Numerical examples show the influence of the parameters of the
functionallygradedbeamsonthe freevibration frequencies fordifferentboundaryconditions.

Keywords: axially graded beam, non-uniform beam, free vibration

1. Introduction

Functionally graded materials (FGMs) are a novel class of composites which have continuous
variation of material properties from one constituent to another. As a result, they have various
advantages over the classical composite laminates. For example, using FGMs, we avoid stress
concentrations typical for heterogeneous structureswith jumpadiscontinuity between dissimilar
materials. For this reason, FGMs are widely used inmechanical, nuclear, aerospace, biomedical
and civil engineering. Simultaneously, because ofwide applications of FGMs, it is very important
to study static and dynamic analysis of functionally graded structures, such as plates, shells and
beams. In this paper, the object of consideration is the problem of free vibration of functionally
graded (FG) beams. For FG beams, the gradient variation may be oriented in the axial and/or
in the cross-section direction.
The literature on vibration analysis of FG beams with thickness-wise gradient variation is

very extensive. For example, Anandrao et al. (2012)made free vibration analysis of functionally
graded beams using the principle of virtual work to obtain a finite element system of equations.
The variation of material properties across the thickness of the beamwas governed by a power
law distribution. The same type of variation of the beamproperties was also assumed by Sina et
al. (2009). They solved the resulting system of ordinary differential equations of free vibration
analysis by using an exactmethod.An analytical solution to study free vibration of exponential
functionally graded beams with a single delamination was developed by Liu and Shu (2014).
Pradhan and Chakraverty (2013) used the Rayleigh-Ritz model to analyse free vibration of FG
beams with material properties that continuously vary in the thickness direction according to
the power-law exponent form. This type of gradation was also assumed by Wattanasakulpong
and Ungbhakorn (2012). They applied the differential transformation method to solve the go-
verning equation of free vibration of FG beams supported by various types of general boundary
conditions. The line springmodel to solve the free vibration problem of an exponentially graded
cracked beamwas employed byMatbuly et al. (2009).
Free vibration analysis for axially graded beams has become more complicated because of

the governing equation with variable coefficients. For example, Wu et al. (2005) applied the
semi-inverse method to find solutions to the dynamic equation of axially functionally graded



860 S. Kukla, J. Rychlewska

simply supported beams. Huang and Li (2010) studied free vibration of axially functionally
graded beams by using the Fredholm integral equations. Hein and Feklistova (2011) applied
the Haar wavelet approach to analyse free vibration of axially functionally graded beams. The
differential transform element method and differential quadrature elementmethod of the lowest
order were used to solve free vibration and stability problems of FG beams by Shahba and
Rajasekaran (2012). The exact solution to free vibration of exponentially axially graded beams
was presented by Li et al. (2013). Explicit frequency equations of free vibration of exponentially
FG Timoshenko beams were derived by Tang et al. (2014). Huang et al. (2013) presented a
new approach to the investigation of free vibration of axially functionally graded Timoshenko
beams. By applying auxiliary functions, they transformed the coupled governing equations into
a single governing equation. Moreover, there are some studies related with the problem of free
vibration of FG beams where the gradation of the material is assumed to be along any of the
possible Cartesian coordinates, see Alshorbagy et al. (2011), by Shahba et al. (2013). A review
of researches on FG beam type structures can be found in Chauhan and Khan (2014).
In this contribution, we propose a new approach to free vibration analysis of FG beamswith

arbitrary axial inhomogeneity. The main idea presented in this paper is to approximate an FG
beam by an equivalent beam with piece-wise exponentially varying material and geometrical
properties. Considerations are carried out in the framework of the Euler-Bernoulli beam theory.
Taking into account various boundary conditions associated with clamped, pinned and free
ends, numerical solutions are obtained for different functions describing gradient variation of
material/geometrical properties of an FG beam. The effectiveness of the proposed approach is
confirmed by comparing the obtained numerical results with other numerical solutions available
in the existing literature for homogeneous and nonhomogeneous beams. The proposed method
is a certain generalization of the approach presented by Kukla and Rychlewska (2014).

2. Equations of motion

An axially graded and non-uniform beam of length L is considered. In this contribution, the
material properties and/or cross-section of the beam are assumed to vary continuously along
the axial direction. Based on the Euler-Bernoulli beam theory, Lebed and Karnovsky (2000),
the governing differential equation is given by

∂2

∂x2

[

E(x)I(x)
∂2w

∂x2

]

+ρ(x)A(x)
∂2w

∂t2
=0 0<x<L (2.1)

where x is the axial coordinate, A(x) is the cross-section area, I(x) is the moment of inertia,
E(x) denotes themodulus of elasticity, ρ(x) is thematerial density andw(x,t) is the transverse
deflection at the position x and time t.
In order to investigate free vibration of the beam, we assume that

w(x,t) =W(x)sinωt (2.2)

whereW(x) is the amplitude of vibration andω is the circular frequency of vibration. Substitu-
ting (2.2) into (2.1) and introducing the non-dimensional coordinate ξ=x/L, we can transform
governing equation (2.1) into

d2

dξ2

[

E(ξ)I(ξ)
d2W

dξ2

]

−L4ω2ρ(ξ)A(ξ)W =0 0<ξ< 1 (2.3)

In the subsequent analysis, it is assumed that

E(ξ)I(ξ) = d0g(ξ) ρ(ξ)A(ξ) =m0h(ξ) 0<ξ< 1 (2.4)



An approach to free vibration analysis of axially graded beams 861

where d0 = E(0)I(0) and m0 = ρ(0)A(0). Subsequently, we shall approximate the FG beam
under consideration by an equivalent beam with piecewise exponentially varying geometrical
andmaterial properties, setting

g(ξ)∼= die
2βiξ h(ξ)∼=mie

2βiξ ξi−1 <ξ<ξi i=1, . . . ,n (2.5)

where ξ0 = 0 and ξn = 1. The coefficients di, mi, βi, i = 1, . . . ,n we determine by using the
following relationships (i=1, . . . ,n)

g(ξi−1)= die
2βiξi−1 g(ξi)= die

2βiξi g(ξ0)= 1 (2.6)

and (i=1, . . . ,n)

h
(ξi+ ξi−1

2

)

=mie
βi(ξi+ξi−1) h(ξ0)= 1 (2.7)

Hence (i=1, . . . ,n)

βi =
1

2(ξi− ξi−1)
ln
g(ξi)

g(ξi−1)
di = g(ξi)e

−2βiξi

mi =h
(ξi+ ξi−1
2

)

e−βi(ξi+ξi−1)
(2.8)

We shall also assume that the transverse deflection of the beam has the form

W(ξ)=Wi(ξ) ξi−1 <ξ<ξi i=1, . . . ,n (2.9)

Hence, the governing system of equations for such a piecewise beam can be expressed by

d2

dξ2

[

d0die
2βiξ
d2Wi
∂ξ2

]

−L4ω2m0mie
2βiξWi =0

ξi−1 <ξ<ξi
i=1, . . . ,n

(2.10)

Introducing denotations

Ω2 =
m0m1
d0d1

L4ω2 µ2i =
mid1
m1di

(2.11)

equations (2.10) can be rewritten as

d2

dξ2

[

e2βiξ
d2Wi
dξ2

]

−Ω2µ2ie
2βiξWi =0 ξi−1 <ξ<ξi i=1, . . . ,n (2.12)

After somemanipulations. equations (2.12) reduce to the form

d4Wi
dξ4
+4βi

d3Wi
dξ3
+4β2i

d2Wi
dξ2
−Ω2µ2iWi =0

ξi−1 <ξ<ξi
i=1, . . . ,n

(2.13)

The parameters βi in equations (2.8) have been determined from the function g(·) corre-
sponding to the stiffness of the beam. These parameters can be determined also by using the
function h(·) corresponding to mass of the beam. In this case, we assume that

h(ξi−1)=mie
2βiξi−1 h(ξi)=mie

2βiξi h(ξ0)= 1 i=1, . . . ,n (2.14)

and

g
(ξi+ ξi−1

2

)

= die
βi(ξi+ξi−1) g(ξ0)= 1 i=1, . . . ,n (2.15)

Then we have

βi =
1

2(ξi− ξi−1)
ln
h(ξi)

h(ξi−1)
mi =h(ξi)e

−2βiξi

di = g
(ξi+ξi−1

2

)

e−βi(ξi+ξi−1) i=1, . . . ,n

(2.16)

Differential equation (2.13) is valid also for di,mi, βi, given by formulae (2.16).



862 S. Kukla, J. Rychlewska

3. Solution to the free vibration problem

On the assumption β2i <µiΩ, the general solution to equations (2.13) has the form

Wi(ξ)= e
−βiξ(Aicosδiξ+Bi sinδiξ+Cicoshδiξ+Di sinhδiξ) ξi−1 <ξ<ξi (3.1)

where δi =
√

µiΩ−β
2
i , δi =

√

µiΩ+β
2
i ,Ai,Bi,Ci,Di ∈R, i=1, . . . ,n.

In order to analyse the free vibration of functionally graded beams, solution (3.1) has to be
applied to certain boundary conditions. In this paper we shall consider the following types of
boundary conditions:

— clamped-clamped beam (C-C)

W1(0)= 0
dW1
dξ
(0)=0 Wn(1)= 0

dWn
dξ
(1)= 0 (3.2)

— pinned-pinned beam (P-P)

W1(0)= 0
d2W1
dξ2
(0)= 0 Wn(1)= 0

d2Wn
dξ2
(1)= 0 (3.3)

— clamped-pinned beam (C-P)

W1(0)= 0
dW1
dξ
(0)=0 Wn(1)= 0

d2Wn
dξ2
(1)= 0 (3.4)

— pinned-clamped beam (P-C)

W1(0)= 0
d2W1
dξ2
(0)= 0 Wn(1)= 0

dWn
dξ
(1)= 0 (3.5)

— clamped-free beam (C-F)

W1(0)= 0
dW1
dξ
(0)=0

d2Wn
dξ2
(1)= 0

d

dξ

(

e2βξ
d2Wn
dξ2

)

(1)= 0 (3.6)

— free-clamped beam (F-C)

d2W1
dξ2
(0)= 0

d

dξ

(

e2βξ
d2W1
dξ2

)

(0)= 0 Wn(1)= 0
dWn
dξ
(1)= 0 (3.7)

The matching conditions between two connecting elements of the piecewise beams satisfy
the following continuity conditions

Wi(ξi)=Wi+1(ξi)
dWi
dξ
(ξi)=

dWi+1
dξ
(ξi)

d2Wi
dξ2
(ξi)=

d2Wi+1
dξ2

(ξi)
d3Wi
dξ3
(ξi)=

d3Wi+1
dξ3

(ξi) i=1, . . . ,n−1

(3.8)

Substituting functions (3.1) into one of the set of boundary conditions (3.2)-(3.7) and continuity
conditions given by equations (3.8), we obtain a system of 4n equations which can bewritten in
the matrix form

A(ω)X=0 (3.9)



An approach to free vibration analysis of axially graded beams 863

where X= [A1,B1,C1,D1, . . . ,An,Bn,Cn,Dn]
T and A(ω) = [akj]4n×4n. The matrix A can be

expressed as

A(ω)=





















B1
C1
C2
...
Cn−1
Bn





















4n×4n

(3.10)

where the matrices B1,Bn of size (2×4n) represent the boundary conditions andmatricesCi,
i = 1, . . . ,n− 1 of size (4× 4n) represent the continuity conditions. The matrices associated
with the boundary conditions corresponding to the four kinds of end supports can bewritten as
follows:

— clamped-clamped beams

B1 =

[

1 0 1 0 0 · · · 0

−β1 δ1 −β1 δ1 0 · · · 0

]

2×4n

Bn =

[

0 · · · 0 cosδn sinδn coshδn sinhδn
0 · · · 0 a4n,4n−3 a4n,4n−2 a4n,4n−1 a4n,4n

]

2×4n

(3.11)

where

a4n,4n−3 =−βncosδn−δn sinδn

a4n,4n−2 =−βn sinδn+ δncosδn

a4n,4n−1 =−βncoshδn+δn sinhδn

a4n,4n =−βn sinhδn+ δncoshδn

(3.12)

— pinned-pinned beams

B1 =

[

1 0 1 0 0 · · · 0

β21 − δ
2
1 −2β1δ1 β

2
1 + δ

2
1 −2β1δ1 0 · · · 0

]

2×4n

Bn =

[

0 · · · 0 cosδn sinδn coshδn sinhδn
0 · · · 0 a4n,4n−3 a4n,4n−2 a4n,4n−1 a4n,4n

]

2×4n

(3.13)

where

a4n,4n−3 =2βnδn sinδn+(β
2
n− δ

2
n)cosδn

a4n,4n−2 =−2βnδncosδn+(β
2
n− δ

2
n)sinδn

a4n,4n−1 =−2βnδn sinhδn+(β
2
n+δ

2
n)coshδn

a4n,4n =−2βnδncoshδn+(β
2
n+ δ

2
n)sinhδn

(3.14)

— free-clamped beams

B1 =

[

β21 − δ
2
1 −2β1δ1 β

2
1 + δ

2
1 −2β1δ1 0 · · · 0

β31 +β1δ
2
1 −β

2
1δ1− δ

3
1 β

3
1 −β1δ

2
1 −β

2
1δ1+ δ

3
1 0 · · · 0

]

2×4n

(3.15)

the matrixBn is given by (3.11)2



864 S. Kukla, J. Rychlewska

—clamped-free beams
the matrixB1 is given by (3.11)1

Bn =

[

0 · · · 0 a4n−1,4n−3 a4n−1,4n−2 a4n−1,4n−1 a4n−1,4n
0 · · · 0 a4n,4n−3 a4n,4n−2 a4n,4n−1 a4n,4n

]

2×4n

(3.16)

where

a4n−1,4n−3 =2βnδn sinδn+(β
2
n− δ

2
n)cosδn

a4n−1,4n−2 =−2βnδncosδn+(β
2
n− δ

2
n)sinδn

a4n−1,4n−1 =−2βnδn sinhδn+(β
2
n+ δ

2
n)coshδn

a4n−1,4n =−2βnδncoshδn+(β
2
n+ δ

2
n)sinhδn

a4n,4n−3 =(β
2
nδn+ δ

3
n)sinδn+(β

3
n+βnδ

2
n)cosδn

a4n,4n−2 =−(β
2
nδn+ δ

3
n)cosδn+(β

3
n+βnδ

2
n)sinδn

a4n,4n−1 =(−β
2
nδn+ δ

3
n)sinhδn+(β

3
n−βnδ

2
n)coshδn

a4n,4n =(−β
2
nδn+ δ

3
n)coshδn+(β

3
n−βnδ

2
n)sinhδn

(3.17)

For clamped-pinned and pinned-clamped beams, the matrices B1, Bn are given by equations
(3.11)1-(3.13)2 and (3.13)1-(3.11)2, respectively. The matrices associated with the continuity
conditions are represented by

Ci =











0 · · · 0 a4i−1,4i−3 · · · a4i−1,4i+4 0 · · · 0
0 · · · 0 a4i,4i−3 · · · a4i,4i+4 0 · · · 0
0 · · · 0 a4i+1,4i−3 · · · a4i+1,4i+4 0 · · · 0
0 · · · 0 a4i+2,4i−3 · · · a4i+2,4i+4 0 · · · 0











4×4n

i=1, . . . ,n−1 (3.18)

The non-zero elements of these matrices are given in Appendix.

The determinant of the matrix A has to vanish for a non-trivial solution of equation (3.9)
to exist. The frequency equation

detA(ω)= 0 (3.19)

is then solved numerically using an approximate method.

4. Numerical results

The numerical computations have been carried out for an FG beam which was divided into n
segments of the same length.The functions g(·),h(·) introduced into equations (2.4) are assumed
in the form g(ξ) = (1+γξ)α,h(ξ)= 1+γξ. In the computations, the formulae given by equation
(2.8) have been used. The first three non-dimensional free vibration frequencies obtained in the
present study for n = 100 are listed in Tables 1 and 2 in comparison with those presented by
Huang and Li (2010) and calculated by using a power series expansion. FromTables 1-2, it can
be seen that the present results are in good agreement with the existing results. For γ =0, we
have the case of a homogeneous beam. It is seen inTables 1 and2 that in this case the agreement
is excellent.



An approach to free vibration analysis of axially graded beams 865

Table 1. The first three non-dimensional free vibration frequencies for g(ξ) = (1 + γξ)3,
h(ξ)= 1+γξ, clamped-clamped beam

γ Power series method Huang and Li (2010) Present study

−0.1 21.2409777868 21.24097778688 21.242905

58.5500545739 58.55005461550 58.567526

114.780241659 114.78027750905 114.824704

0 22.3732854478 22.37328544806 22.373285

61.6728228676 61.67282294761 61.672823

120.903391727 120.90340027002 120.903392

0.1 23.4796072481 23.47960724845 23.460013

64.7210676329 64.72106768601 64.678046

126.878016311 126.87805071630 126.802905

0.2 24.5634175322 24.5634175326 24.508817

67.7047553171 67.7047553184 67.596273

132.723976757 132.7240684027 132.546612

Table 2. The first three non-dimensional free vibration frequencies for g(ξ) = (1 + γξ)3,
h(ξ)= 1+γξ, clamped-pinned beam

γ Power series method Huang and Li (2010) Present study

−0.1 14.8488960557 14.84889605539 14.844562

47.6370371901 47.63703719174 47.647237

99.171635183 99.17165323722 99.206918

0 15.4182057169 15.41820571698 15.418206

49.964862032 49.96486203816 49.964862

104.247696458 104.24770194514 104.247696

0.1 15.968709884 15.96870988416 15.950015

52.2372268871 52.23722689317 52.198883

109.202352455 109.20235370558 109.134912

0.2 16.5028988943 16.50289889399 16.445277

54.4614625302 54.46146253076 54.360368

114.051623344 114.05163085534 113.888586

The effects of parameters α, γ and the number of segments n on the first three non-
dimensional frequencies for different boundary conditions are presented in Tables 3-5. It can
be observed that an increase in the value of the parameterα causes an increase in the difference
between the results obtained for n=5, n=10 and n=20, respectively.

Figure 1 presents the first free vibration frequencies calculated for the functions
g(ξ) = (1+ γξ)α and h(ξ) = 1+ γξ for α = 1, α = 2 and α = 3. The calculations have
been performed for six types of boundary conditions. It can be noticed that variation of the
parameter γ has a significant effect on the free vibration frequency. For the clamped-free beams,
the greatest impact of the parameter γ occurs for α=1, and for all the other boundary condi-
tions under considerations it is for α=3. For the clamped-clamped and pinned-pinned beams,
the differences between the values of free vibration frequencies for α=1 are negligible.



866 S. Kukla, J. Rychlewska

Table 3. The first non-dimensional free vibration frequency for different boundary conditions,
g(ξ) = (1+γξ)α, h(ξ)= 1+γξ

BC n
α=1 α=2 α=3

γ =−0.5 γ =0.5 γ=−0.5 γ=0.5 γ=−0.5 γ=0.5

C-C 5 22.03982 22.264535 19.265636 24.165700 16.814251 26.214186

10 22.051059 22.262618 19.048822 24.459508 16.441212 26.863566

20 22.053796 22.261942 18.935593 24.609820 16.247100 27.197036

P-P 5 9.583045 9.773619 8.177437 10.524969 6.896407 11.294071

10 9.588353 9.772740 8.086959 10.653015 6.746380 11.574244

20 9.589777 9.772464 8.039484 10.718554 6.667729 11.718081

C-P 5 15.838604 14.907051 14.080008 15.909067 12.483974 16.939622

10 15.845921 14.905829 13.922110 16.102144 12.209467 17.358429

20 15.847897 14.905405 13.839671 16.201108 12.066042 17.573873

P-C 5 14.448440 15.749301 12.216596 17.289925 10.244436 18.957895

10 14.456396 15.747708 12.079634 17.500534 10.017769 19.429126

20 14.458368 15.747212 12.008080 17.608138 9.899795 19.670529

C-F 5 4.379586 3.121248 4.321434 3.188805 4.234717 3.251220

10 4.380502 3.120991 4.252879 3.232930 4.103112 3.342365

20 4.380745 3.120903 4.222757 3.254157 4.045646 3.386566

F-C 5 2.887779 3.995266 2.290903 4.643804 1.807925 5.384977

10 2.889157 3.994728 2.270108 4.689088 1.774671 5.492166

20 2.889508 3.994563 2.257891 4.715110 1.755475 5.553767

Table 4.The secondnon-dimensional freevibration frequency for differentboundaryconditions,
g(ξ) = (1+γξ)α, h(ξ)= 1+γξ

BC n
α=1 α=2 α=3

γ =−0.5 γ =0.5 γ=−0.5 γ=0.5 γ=−0.5 γ=0.5

C-C 5 61.187699 61.527290 53.563835 66.810777 46.705060 72.453327

10 61.220229 61.522622 52.964685 67.624175 45.679608 74.250130

20 61.227661 61.520754 52.649784 68.039773 45.140471 75.171899

P-P 5 39.248490 39.415763 34.281704 42.767233 29.772507 46.312345

10 39.265508 39.411680 33.896833 43.288075 29.121704 47.465112

20 39.270192 39.410436 33.695664 43.554164 28.779150 48.054717

C-P 5 50.300388 49.461958 44.282315 53.482420 38.806190 57.735593

10 50.321872 49.457835 43.792788 54.127769 37.971984 59.156518

20 50.327702 49.456336 43.532945 54.460171 37.525512 59.890410

P-C 5 48.939014 50.267141 42.517999 54.777959 36.759687 59.599505

10 48.964462 50.262091 42.035739 55.451783 35.941623 61.097469

20 48.970497 50.260490 41.785685 55.792915 35.517228 61.857034

C-F 5 23.381744 21.158807 21.233450 22.452525 19.191540 23.771357

10 23.393474 21.157385 20.890872 22.761849 18.565383 24.434588

20 23.396111 21.156800 20.740129 22.911071 18.295994 24.757241

F-C 5 20.476134 22.847483 17.187639 25.342189 14.317750 28.060961

10 20.488032 22.845673 17.036003 25.587931 14.064074 28.609593

20 20.490837 22.844949 16.945280 25.728924 13.914191 28.926634



An approach to free vibration analysis of axially graded beams 867

Table 5.The third non-dimensional free vibration frequency for different boundary conditions,
g(ξ) = (1+γξ)α, h(ξ)= 1+γξ

BC n
α=1 α=2 α=3

γ=−0.5 γ=0.5 γ =−0.5 γ=0.5 γ=−0.5 γ =0.5

C-C 5 120.319511 120.746274 105.344423 131.127496 91.681928 142.145173

10 120.392769 120.740300 104.218260 132.739432 89.853340 145.728137

20 120.407372 120.736681 103.598604 133.555211 88.792805 147.537078

P-P 5 88.594258 88.779880 77.531290 96.389774 67.426814 104.429449

10 88.629715 88.770369 76.684167 97.575681 66.023866 107.072172

20 88.639792 88.767515 76.229088 98.175746 65.248530 108.403315

C-P 5 104.554645 103.740552 91.766692 112.472339 80.084560 121.703952

10 104.597719 103.733256 90.815730 113.815304 78.503258 124.685866

20 104.609566 103.730098 90.278366 114.513546 77.584066 126.231330

P-C 5 103.144606 104.556740 90.048791 113.706807 78.140870 123.417530

10 103.202283 104.546600 89.029672 115.147269 76.462403 126.629660

20 103.214761 104.543267 88.499082 115.856814 75.558342 128.206906

C-F 5 62.982515 60.852831 56.140476 65.522940 49.819399 70.432539

10 63.020704 60.849312 55.256707 66.417417 48.227924 72.379515

20 63.028371 60.847560 54.856960 66.851872 47.524147 73.333012

F-C 5 60.136355 62.488331 51.918494 68.523131 44.593816 75.023941

10 60.171203 62.485112 51.449765 69.200653 43.789492 76.516820

20 60.178811 62.483270 51.173617 69.581994 43.319244 77.363991

The presented numerical results have been obtained by using coefficients di, mi, βi,
i = 1, . . . ,n, given by equations (2.8). Numerical computations show that the application of
equations (2.16) leads to results which are in good agreement with the obtained by using equ-
ations (2.8).

5. Conclusions

In the paper, a solution to the free vibration problem of axially functionally graded beams is
presented. An exact solution is derived for axially piece-wise exponential graded beams. The
frequency equation for beams with various combinations of clamped, pinned and free ends has
been obtained. In this approach, the distributed parameters which describe continuous axial
changes of thematerial properties of the beamare approximated by piecewise exponential func-
tions. The non-dimensional free vibration frequencies for a chosen function characterizing the
functionally graded beams have been numerically computed. An improvement of the accuracy
of the numerical results for a larger number of beam subsections applied in the method has
been demonstrated. A high agreement of the numerical results obtained by using the presented
method with the results obtained by using the power series method as well as with results gi-
ven by other authors has also been observed. The numerical investigation shows that the beam
stiffness distribution in the axial direction significantly effects free vibration frequencies of the
system.



868 S. Kukla, J. Rychlewska

Fig. 1. The first non-dimensional free vibration frequency as a function of γ for α=1 (solid line),
α=2 (dashed line), α=3 (dotted line) for different boundary conditions

Appendix

Letusdenoteexi=eξi(βi+1−βi), ci=cos(δiξi),si=sin(δiξi), chi=cosh(δiξi), shi=sinh(δiξi),
ci1= cos(δi+1ξi), si1= sin(δi+1ξi), chi1= cosh(δi+1ξi), shi1= sinh(δi+1ξi).

The non-zero elements of thematrixCi, i=1, . . . ,n−1, which occur in equation (3.18) are
given by



An approach to free vibration analysis of axially graded beams 869

a4i−1,4i−3 = exi · ci a4i−1,4i−2 = exi ·si a4i−1,4i−1 = exi · chi

a4i−1,4i = exi ·shi a4i−1,4i+1 =−ci1 a4i−1,4i+2 =−si1

a4i−1,4i+3 =−chi1 a4i−1,4i+4 =−shi1

a4i,4i−3 =−exi(βici+ δisi) a4i,4i−2 =−exi(βisi− δici)

a4i,4i−1 =−exi(βichi− δishi) a4i,4i =−exi(βishi− δichi)

a4i,4i+1 =βi+1ci1+ δi+1si1 a4i,4i+2 =βi+1si1− δi+1ci1

a4i,4i+3 =βi+1chi1+ δi+1shi1 a4i,4i+4 =βi+1shi1− δi+1chi1

a4i+1,4i−3 = exi[(β
2
i −δ

2
i )ci+2βiδisi]

a4i+1,4i−2 = exi[(β
2
i −δ

2
i )si−2βiδici]

a4i+1,4i−1 = exi[(β
2
i +δ

2
i)chi−2βiδishi]

a4i+1,4i = exi[(β
2
i + δ

2
i)shi−2βiδichi]

a4i+1,4i+1 =−2βi+1δi+1si1+(−β
2
i+1+ δ

2
i+1)ci1

a4i+1,4i+2 =2βi+1δi+1ci1+(−β
2
i+1+ δ

2
i+1)si1

a4i+1,4i+3 =2βi+1δi+1shi1− (β
2
i+1+ δ

2
i+1)chi1

a4i+1,4i+4 =2βi+1δi+1chi1− (β
2
i+1+ δ

2
i+1)shi1

a4i+2,4i−3 =−exi[(β
2
i −3δ

2
i )βici+(3β

2
i − δ

2
i )δisi]

a4i+2,4i−2 =−exi[(β
2
i −3δ

2
i )βisi+(−3β

2
i + δ

2
i )δici]

a4i+2,4i−1 =−exi[(β
2
i +3δ

2
i)βichi− (3β

2
i + δ

2
i)δishi]

a4i+2,4i =−exi[(β
2
i +3δ

2
i)βishi− (3β

2
i + δ

2
i)δichi]

a4i+2,4i+1 =(β
2
i+1−3δ

2
i+1)βi+1ci1− (−3β

2
i+1+ δ

2
i+1)δi+1si1

a4i+2,4i+2 =(β
2
i+1−3δ

2
i+1)βi+1si1+(−3β

2
i+1+ δ

2
i+1)δi+1ci1

a4i+2,4i+3 =−(3β
2
i+1+ δ

2
i+1)δi+1shi1+(β

2
i+1+3δ

2
i+1)βi+1chi1

a4i+2,4i+4 =−(3β
2
i+1+ δ

2
i+1)δi+1chi1+(β

2
i+1+3δ

2
i+1)βi+1shi1

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Manuscript received July 8, 2015; accepted for print December 4, 2015