Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties

Composite Structures 91 (2009) 38–47

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Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties P. Malekzadeh * Department of Mechanical Engineering, Persian Gulf University, Bushehr 75168, Iran Center of Excellence for Computational Mechanics, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Available online 3 May 2009 Keywords: Free vibration Functionally graded arches Two-dimensional elasticity Temperature

a b s t r a c t In this paper, the in-plane free vibration analysis of functionally graded (FG) thick circular arches subjected to initial stresses due to thermal environment is studied. The formulations are based on the two-dimensional elasticity theory. The material properties are assumed to be temperature-dependent and graded in the thickness direction. Considering the thermal environment effects, the equations of motion are derived using the Hamilton’s principle. The initial thermal stresses are obtained by solving the two-dimensional thermoelastic equilibrium equations. Two types of temperature rise, uniform and variable through the thickness, is considered. Differential quadrature method (DQM) is adopted to solve the equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with those available in the literature for isotropic arches. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Circular arches made of functionally graded materials (FGMs) under high-temperature environment have found wide applications as structural members in modern industries such as mechanical, aerospace, nuclear engineering and reactors. Hence, their vibration characteristic is of great interest for engineering design and manufacture. The in-plane free vibration analyses of isotropic and laminated composite arches were extensively studied, see for example Refs. [1–14]. Also, most of the studies on FGMs were restricted to straight beams, rectangular and circular plates. But, to the best of author’s knowledge, there is no study available in the open literature investigating the behavior of FG circular arches in thermal environment and with temperature-dependent material properties. Usually, to model the in-plane free vibration governing equations of arches, in the previous works the one-dimensional theories such as the classical theory [5,7], and the first and the higher order shear deformation theories [6,8–12] were used. The classical theory neglects the shear deformation and rotary inertia effects. On the other hand, transverse shear deformation and rotary inertia have significant effects on the natural frequencies of the thick arches, especially on the higher order modes. Also, these arch theories neglect trans* Address: Department of Mechanical Engineering, Persian Gulf University, Bushehr 75168, Iran. Tel.: +98 771 4222150; fax: +98 771 4540376. E-mail addresses: [email protected], [email protected] 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.04.034

verse normal deformations, and generally assume that a plane stress state of deformation prevails in the arch. These assumptions may be appropriate for thin arches, but may not give good results for thick arches with length to thickness ratio equal to five or less. Recently, some research works based on the two-dimensional (2D) elasticity theory for the in-plane free vibration analyses of laminated composite arches have appeared in the literature [13,14]. Lim et al. [15] investigated the temperature-dependent in-plane vibration of functionally graded (FGM) simply supported circular arches based on the two-dimensional theory of elasticity. They used the state space formulation and Fourier series expansion to solve the problem. Here, the in-plane free vibration characteristics of FG thick circular arches in thermal environment, based on the two-dimensional elasticity theory, is presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction and can vary according to power law distributions in terms of the volume fractions of the constituents, exponentially or any other formulations in this direction. The initial thermal stresses are obtained by solving the two-dimensional thermoelastic equilibrium equations. The Hamilton’s principle is employed to derive the equations of motion and the related boundary conditions subjected to initial thermal stresses. The differential quadrature method as an efficient numerical tool [5,14,16,17] is employed to solve the differential equations with variable coefficients. Using DQM along the graded direction enables one to accurately and efficiently discretize the partial differential equations in this direction and implement the natural boundary conditions at the top and

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P. Malekzadeh / Composite Structures 91 (2009) 38–47

ature at which the arch is stress free. It is assumed that the arch is under the state of plane stress and hence the material elastic coefficients C ij for an isotropic arch are related to the elastic material properties as follows:

bottom surfaces of the arch. The accuracy and convergence of the present method are investigated through different examples. Then, the influence of uniform and non-uniform temperature rise, dependence of material properties on temperature, material property graded index and geometrical parameters on the vibration frequencies of FG arches is studied.

C 11 ¼ C 22 ¼

2. Governing equations

C 55

Consider a thick FG circular arch as shown in Fig. 1. A polar coordinate system (r, h) is used to label the material point of the arch in the unstressed reference configuration. The displacement components of an arbitrary material point of the arch are denoted as u and v in the r- and h-directions, respectively.

where E [=E(r, T)] and m = [m(r, T)] are the Young’s modulus and the Poisson’s ratio, respectively.

2.1. Temperature-dependent FGMs relations The material properties of the arch are assumed to vary continuously through the arch thickness, i.e. in the r-direction. In this study, without loss of generality of the formulations, the material properties are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents across the thickness. The material composition continuously varies such that the outer surface of the arch (r = Ro) is ceramic-rich whereas the inner surface of the arch (r = Ri) is metal-rich. Based on the power law distribution, a typical effective material property ‘P’ of the FG arch is obtained as

 p r  Ri Pðr; TÞ ¼ Pm ðTÞ þ ½P c ðTÞ  Pm ðTÞ h

ð1Þ

GðTÞ ¼ G0 ð1 þ G1 T þ G2 T 2 Þ

ð2Þ

The coefficients Gi (i = 0, 1, 2) are unique to the material constituents. The two-dimensional constitutive relations for a linear elastic orthotropic FG arch including the thermal effects can be written as

> :

2

C 11

C 12

rhh ¼ 6 4 C 12 C 22 > rrh ; 0 0

9 38 > = < err  a1 ðr; TÞDT > 7 0 5 ehh  a2 ðr; TÞDT > > ; : crh C 55 0

ð3Þ

where rij (i, j = r, h) are the stress tensor components; eii (i = r, h) and crh are the normal and shearing components of strain tensor, respectively; C ij ½¼ C ij ðr; TÞ; i; j ¼ 1; 2; 5 are the material elastic coefficients; ai(r, T) (i = 1, 2) are the thermal expansion coefficients; DTð¼ T  T 0 Þ is the temperature rise and T0 is the reference temper-

θ

r θ0

mðr; TÞEðr; TÞ ; 1  m2 ðr; TÞ ð4Þ

It is assumed that the arch is stress free at the temperature T0. If the arch operates in a thermal environment, due to non-uniform temperature rise or mechanical constraints at its edges, some stresses are produced in it. These stresses affect the vibration characteristic of the arch. In order to evaluate these thermal stresses, the temperature distribution in the arch should be first obtained. In this study, it is assumed that the temperature rise is uniform or varies across the section of arch and no heat generation source exists within the arch. Hence, for the second case, the temperature distribution along the thickness direction can be obtained by solving the following steady state one-dimensional heat transfer equation through the thickness of arch 2

d T dr

2

þ

dKðrÞ dT ¼0 dr dr

ð5Þ

where K is the thermal conductivity of the arch. Different thermal boundary conditions can be considered at the inner and the outer surfaces of the arch. One of the usual thermal boundary conditions that were considered in the literature for beams and plates is the prescribed temperature at the upper and lower surfaces of these structural elements. Hence, for brevity purpose and without loss of generality, here these boundary conditions are considered, which for the arch problem become

T ¼ T m at r ¼ Ri

and T ¼ T c at r ¼ Ro

ð6Þ

The solution of Eq. (5) subjected to the boundary conditions (6) can be obtained by means of polynomial series solutions. The result is given as

"   pþ1 ðT c  T m Þ r  Ri K cm r  Ri TðrÞ ¼T m þ  C h ðp þ 1ÞK m h    3pþ1 2pþ1 K 2cm r  Ri K 3cm r  Ri þ  h h ð2p þ 1ÞK 2m ð3p þ 1ÞK 3m  4pþ1  5pþ1 # 4 5 K cm r  Ri K cm r  Ri þ  h h ð5p þ 1ÞK 5m ð4p þ 1ÞK 4m

ð7Þ

where

C ¼1  þ

h

C 12 ¼

2.2. Initial thermal stress evaluation

KðrÞ

where subscripts m and c refer to the metal and ceramic constituents, respectively; p is the power law index or the material property graded index; h the arch thickness; and T[=T(r)] is the temperature at an arbitrary material point of the arch. For FG arch constituents, i.e. ceramic and metal, the material properties are temperature-dependent and a typical material property ‘G’ can be expressed as a function of temperature as [18,19]

9 8 > = < rrr >

Eðr; TÞ ; 1  m2 ðr; TÞ Eðr; TÞ ¼ 2½1 þ mðr; TÞ

K cm K 2cm K 3cm þ  2 ðp þ 1ÞK m ð2p þ 1ÞK m ð3p þ 1ÞK 3m K 4cm

ð4p þ 1ÞK 4m



K 5cm ð5p þ 1ÞK 5m

;

K cm ¼ K c  K m

Hereafter, a subscript ‘0’ is used to represent the deformation field variables and stress components in the equilibrium state of the arch in thermal environment. The linear strain–displacement relations can be expressed as

Ri

Rm

e0rr ¼

@u0 ; @r

e0hh ¼

  1 @v 0 þ u0 ; r @h

c0rh ¼

  1 @u0 @v 0  v0 þ r @h @r ð8Þ

Fig. 1. Geometry and coordinate system of the FGM circular arch.

where u0 = u0(r, h) and

v0 = v0(r, h).

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P. Malekzadeh / Composite Structures 91 (2009) 38–47

Using the two-dimensional constitutive relations as stated in Eqs. (3) and (4), the linear strain–displacement relations (8) and elasticity equilibrium equations in polar coordinate system, the equilibrium equations of FG arch in terms of displacement components can be written as

    @ 2 u0 C 11 @u0 C 55 @ 2 u0 C 22 u0 0 0 þ þ C þ þ C  11 12 2 @r 2 r @r r 2 @h r r  0  2 ðC 12 þ C 55 Þ @ v 0 rC 12  C 22  C 55 @ v 0 þ þ r @r@h r2 @h

C 11

¼ ½ðC 011 þ C 012 Þ a þ ðC 11 þ C 12 Þa0  DT þ ðC 11 þ C 12 ÞaðDTÞ0  0  ðC 12 þ C 55 Þ @ 2 u0 rC 55 þ C 22 þ C 55 @u0 @2v 0 þ þ C 55 2 r @r@h r @h @r 2   0 2 C 55 @ v 0 C 22 @ v 0 ðrC 55 þ C 55 Þ þ C 055 þ  v0 ¼ 0 þ 2 r2 r @r r @h2

ð9Þ

ð10Þ

where ðÞ0 ¼ dðÞ . Eqs. (9) and (10) represent the in-plane equations of dr motion along the r- and h-axes, respectively. The related boundary conditions are as follows: At r = Ri and r = Ro:



@u0 C 12 @ v 0 þ þ u0  ðC 11 þ C 12 ÞaDT ¼ 0; @r r @h   @ v 0 v 0 1 @u0 ¼ C 55  þ ¼0 r @h @r r

r0rr ¼ C 11 r0rh



ð11Þ

At the edges h = 0 and h0:   @ v 0 v 0 1 @u0 Either u0 ¼ 0 or r0rh ¼ C 55  þ ¼ 0; @r r r @h   @u0 C 22 @ v 0 þ þ u0  ðC 12 þ C 22 ÞaDT ¼ 0 either v 0 ¼ 0 or r0hh ¼ C 12 @r r @h

ð12Þ Different types of classical boundary conditions at the edges h = 0 and h = h0 of the arch can be obtained by combining the conditions stated in Eq. (12) as

Simply supported ðSÞ : u0 ¼ 0; Clamped ðCÞ : u0 ¼ 0;

v 0 ¼ 0;

r0hh ¼ 0; FreeðFÞ : r0rh ¼ 0;

r0hh ¼ 0

ð13Þ

Although the system given by Eqs. (9)–(13) is not impossible to solve analytically, it certainly is quite difficult to obtain such a solution. Hence, here the differential quadrature method as an efficient and accurate numerical tool [5,14,16,17] is employed to solve these system of equations and consequently, to obtain the initial stress components. A brief review of DQM is presented in Appendix A. Based on the DQM, the arch is discretized by Nr and Nh grid points along the r- and h-directions, respectively. Then, at each domain grid point (ri, hj) with i ¼ 2; 3; . . . ; N r  1 and j ¼ 2; 3; . . . ; N h  1, the equilibrium Eqs. (9) and (10) can be discretized as Eq. (9):

ðC 11 Þi

  Nr Nh ðC 11 Þi X ðC 55 Þ X Brim u0mj þ ðC 011 Þi þ Arim u0mj þ 2 i Bh u0in ri ri m¼1 jn m¼1 m¼1 Nr X

  Nh Nr X ðC 22 Þi u0ij ðC 12 þ C 55 Þi X þ ðC 012 Þi  þ Arim Ahjn v 0mn ri ri ri m¼1 n¼1  þ

 Nh ri ðC 012 Þi  ðC 22 Þi  ðC 55 Þi X Ahjn v 0in r 2i n¼1

Eq. (10):

  Nh Nh Nr X ðC 12 þ C 55 Þi X r i ðC 055 Þi þ ðC 22 þ C 55 Þi X Arim Ahjn u0mn þ Ahjn u0in ri r 2i m¼1 n¼1 n¼1   Nr Nr X ðC 55 Þi X þ ðC 55 Þi Brim u0mj þ ðC 055 Þi þ Arim v 0mj ri m¼1 m¼1   Nh ðC 22 Þ X r i ðC 055 Þi þ ðC 55 Þi þ 2 i Bhjn v 0in  v 0ij ¼ 0 ð15Þ r i n¼1 r2i where Anij and Bnij (n = r, h) are the first and the second order DQ weighting coefficients in the n-direction, respectively. The discretized form of the boundary conditions at the inner and the outer surfaces of the arch, i.e. Eq. (11), becomes

ðr0rr Þij ¼ ðC 11 Þi

Nr X

Arim u0mj þ

m¼1

Nh ðC 12 Þi X Ahjn v 0in þ u0ij ri n¼1

!

 ðC 11 þ C 12 Þi ai ðDTÞi ¼ 0; " !# Nh Nr X 1 X r h ¼0 ðr0rh Þij ¼ ðC 55 Þi Aim v 0mj þ A u0in  v 0ij r i n¼1 jn m¼1 for i ¼ 1; N r and j ¼ 1; 2; . . . ; N h

ð16Þ

where i = 1 on the inner surface and i = Nr on the outer surface. Similarly, the discretized form of the boundary conditions at the two edges of the arch can be obtained. In the matrix form, the discretized equilibrium equations and the related boundary conditions become

½SfU 0 g ¼ ff g

ð17Þ

where [S], {U0} and {f} are the stiffness matrix, the vector of unknown degrees of freedom and the load vector, respectively. After solving this system of algebraic equations, the displacement components at the DQ grid points are obtained. Then, the stress components at each DQ grid points ðr i ; hj Þ are obtained from the constitutive relations (3) as

9 2 8 3 ðC 11 Þi ðC 12 Þi 0 > = < ðr0rr Þij > 6 7 ðr0hh Þij ¼ 4 ðC 11 Þi ðC 22 Þi 0 5 > ; : ðr Þ > 0 0 ðC 55 Þi 0rh ij 8 Nr P > r > Aim u0mj  ai ðDTÞi > > > > m¼1 > > N  < Pr h 1   ai ðDTÞi A v 0in þ u0ij jn r i > n¼1 > > > N  > Nr > Pr h P > > A u0  v 0 þ Ar v 0 :1 ri

n¼1

jn

in

ij

m¼1

im

9 > > > > > > > > =

> > > > > > > > ; mj

ð18Þ

3. Free vibration analysis To study the free vibration characteristic of the arch in thermal environment, the displacement components of an arbitrary material point (r, h) are perturbed around its equilibrium position in thermal environment, and hence the total displacement components measured from the arch undeformed configurations become u0(r, h) + u(r, h, t) and v0(r, h) + v(r, h, t) along the r- and h-directions, respectively. The two-dimensional equations of motion and the related boundary conditions can be obtained in a systematic manner by using the Hamilton’s principle, which has the following form for the free vibration analysis:

Z

t2

ðdK  dUÞdt ¼ 0

ð19Þ

t1

¼ ½ðC 011 þ C 012 Þi ai þ ðC 11 þ C 12 Þi a0i ðDTÞi þ ðC 11 þ C 12 Þi ai ðDTÞ0i ð14Þ

where K and U are the kinetic and the potential energy of the arch; t1 and t2 are the beginning and end of motion time, respectively.

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P. Malekzadeh / Composite Structures 91 (2009) 38–47

Either ‘u’ is prescribed or

The variational form of the arch potential energy can be stated as

dU ¼

Z

h0

0

Z

ðC 11 þ r0rr Þ

Ro

½ðr0rr þ rrr Þdðe0rr þ err Þ þ ðr0hh þ rhh Þdðe0hh þ ehh Þ

Ri

Either ‘v’ is prescribed or

þðr0rh þ rrh Þdðc0rh þ crh Þ rdrdh

ð20Þ

To include the effects of the initial stresses due to thermal environment in the equations of motion, the nonlinear terms of the strain components should be considered in the strain–displacement relations

"   2 # 2 @u 1 @u @v ; err ¼ þ þ @r 2 @r @r    2  2 1 @v 1 @u 1 @v ehh ¼ v þ 2 þu þ 2 þu ; r @h 2r @h 2r @h       1 @u @ v 1 @u @u 1 @ v @v v þ v þ þ þu crh ¼ r @h @r r @h @r r @h @r

Z

h0

0

Z

Ro



q Ri

 @u @du @ v @dv rdrdh þ @t @t @t @t

ð21Þ

ð22Þ

where q ½¼ qðr; TÞ is the mass density of the arch. Inserting Eqs. (20)–(22) into Eq. (19) and performing the integration by parts with respect to spatial coordinate variables (r, h) and time t together with the use of constitutive relations and the equilibrium equations of initial stresses, the linearized form of the equations of motion and the related boundary conditions become, Governing equations:

  @2u C 11 þ r0hh @u ðC 55 þ r0hh Þ @ 2 u du : ðC 11 þ r0rr Þ 2 þ C 011 þ þ @r @r r2 r @h2 0 2 2r0rh @ u 2r0rh @u ðrC 12  C 22  r0hh Þ 2r0rh @ v þ u  2 þ @h r2 r @r@h r r @r   ðC 12 þ C 55 Þ @ 2 v ðrC 012  C 22  C 55  2r0hh Þ @ v 2r0rh þ þ þ r r2 @r@h @h r2

v ¼q

@2u @t2

ð23Þ

ðC 12 þ C 55 Þ @ 2 u 2r0rh @u ðrC 055 þ C 22 þ C 55 þ 2r0hh Þ @u þ þ r2 @h r @r  r  @r@h 2 2 2r0rh @ v 2r0rh @ v u þ ðC 55 þ r0rr Þ 2 þ  r @r r @r@h   2 C þ r @ v ðC þ r 2r0rh @ v 55 0hh 22 0hh Þ @ v þ  þ C 055 þ 2 r2 r @r r @h @h  0  rC 55 þ C 55 þ r0hh @2v v¼q 2 ð24Þ  r2 @t

dv :

Boundary conditions: At the edges h = 0 and h0: Either ‘u’ is prescribed or

  ðC 55 þ r0hh Þ @u @v @u  v þ C 55 ¼ 0; þ r0rh r @h @r @r Either ‘v’ is prescribed or

  ðC 22 þ r0hh Þ @ v @u @v þ r0rh þ u þ C 12 ¼0 r @r @h @r On the surfaces r = Ri and Ro:

C 55

    1 @u v @ v @ v r0rh @ v þ r0rr  þ þ þu ¼0 r @h r @r @r r @h

ð25Þ

ð26Þ

Since the free vibration is a harmonic motion, the displacement components can be formulated as

uðr; h; tÞ ¼ Uðr; hÞeIxt ;

Also, one should note that since u0 and v0 were obtained previously, we obtain de0rr ¼ de0hh ¼ dc0rh ¼ 0. The variational form of the kinetic energy of the FG arch is obtained from the following equation:

dK ¼

    @u C 12 @ v r0rh @u þ  v ¼ 0; þu þ @r @h r @h r

v ðr; h; tÞ ¼ Vðr; hÞeIxt

ð27Þ

pffiffiffiffiffiffiffi where x is the natural frequency and Ið¼ 1Þ is the imaginary number. Substituting for the displacement components from Eq. (27) and using the DQ rules for the spatial derivatives, the discretized form of the equations of motion (23) and (24) at each domain grid point (ri, hj) with i = 2, . . . , Nr  1 and j = 2, . . . , Nh  1, becomes Eq. (23):

  Nr ðC 11 Þi þ ðr0hhÞij X Brim U mj þ ðC 011 Þi þ Arim U mj ri m¼1 m¼1 !   Nh Nh Nr X ðC 55 Þi þ ðr0hh Þij X 2ðr0rh Þij X U in h r h þ B U þ A A U  mn in jn im jn ri ri r2i m¼1 n¼1 m¼1   Nr X ðC 22 Þi þ ðr0hh Þij U ij 2ðr0rh Þij  Arim V mj þ ðC 012 Þi  ri ri ri m¼1

½ðC 11 Þi þ ðr0rr Þij 

Nr X

Nr X h ðC 12 þ C 55 Þi X Arim Ahjn V mn ri m¼1 n¼1 " # Nh r i ðC 012 Þi  ðC 22 Þi  ðC 55 Þi  2ðr0hh Þij X N

þ þ þ

r 2i 2ðr0rh Þij r 2i

Ahjn V in

n¼1

2

V ij þ qi x U ij ¼ 0

ð28Þ

Eq. (24): Nh Nr X Nr 2ðr0rh Þij X ðC 12 þ C 55 Þi X Arim Ahjn U mn þ Arim U mj ri r i m¼1 n¼1 m¼1 " # Nh X r i ðC 055 Þi þ ðC 22 þ C 55 Þi þ 2ðr0hh Þij 2ðr0rh Þij  þ Ahjn U in  U ij 2 ri ri n¼1 Nh Nr Nr X h iX 2ðr0rh Þij X þ ðC 55 Þi þ ðr0rr Þij Brim V mj þ Arim Ahjn V mn r i m¼1 m¼1 n¼1   Nr ðC 55 Þi þ ðr0hh Þij X r 0 þ ðC 55 Þi þ Aim V mj ri m¼1   Nh Nr ðC 22 Þi þ ðr0hh Þij X 2ðr0rh Þij X þ Bhjn V in  Ahjn V in 2 2 ri r i n¼1 m¼1 " # r i ðC 055 Þi þ ðC 55 Þi þ ðr0hh Þij 2 V ij þ qi x V ij ¼ 0  r 2i

ð29Þ

In a similar manner, the geometrical and natural boundary conditions stated in Eqs. (25) and (26) can also be discretized, however, for brevity purpose they are not presented here. In order to carry out the eigenvalue analysis, the domain and boundary degrees of freedom should be separated; hence, in vector forms they are denoted as {d} and {b}, respectively. Based on these definitions, the discretized form of the equations of motion and the related boundary conditions can be rearranged in the matrix form as Equations of motion:

½K db fbg þ ½K dd fdg  x2 ½Mfdg ¼ f0g

ð30Þ

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P. Malekzadeh / Composite Structures 91 (2009) 38–47

Boundary conditions:

½K bd fdg þ ½K bb fbg ¼ f0g

ð31Þ

The elements of the stiffness matrices [Kdi] (i = b, d) and the mass matrix [M] are obtained from equations of motion and those of the stiffness matrices [Kbi] (i = b, d) are obtained from the boundary conditions. Eliminating the boundary degrees of freedom in Eq. (30) using Eq. (31), this equation becomes 2

ð½K  x ½MÞfdg ¼ f0g

ð32Þ 1

where [K] = [Kdd]  [Kdb][Kbb] [Kbd]. Solving the above eigenvalue system of equations, the natural frequencies and mode shapes of the arches will be obtained. 4. Numerical results In this section, firstly, the convergence and accuracy of the method is investigated and then the effects of the geometrical

and the material parameters on the in-plane free vibration characteristics of FG circular arches under uniform and non-uniform temperature rise is presented. In the solved examples, the following non-dimensional frequency parameters are used:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki ¼ xi ðR2m =hÞ q0c =E0c ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki ¼ xi ðL2 =hÞ q0c =E0c

ð33Þ

o where Rm ð¼ Ri þR Þ is the mean radius of the arch and L = Rmh0. The 2 material properties of Ti–6Al–4V and ZrO2, as given in Table 1, are used in the numerical computations. They are valid for the temperature range of 300 K 6 T 6 1100 K [18,19]. Due to lack of appropriate results of functionally graded circular arches for direct comparison, validation of the presented formulation is conducted in two ways. Firstly, the results are compared with those of isotropic arches, and then, the results of presented formulations are given in the form of convergence studies with respect to Nr and Nh, the number of discrete points distributed along the r- and h-directions.

Table 1 Temperature-dependent coefficients of material properties for ceramic (ZrO2) and metals (Ti–6Al–4V) (Refs. [18,19]). Material

G0

G1

G2

E (GPa)

Ti–6Al–4V ZrO2

122.7 132.2

4.605  104 3.805  104

0 6.127  108

m

Ti–6Al–4V ZrO2

0.2888 0.3330

1.108  104 0

0 0

q (kg/m3)

Ti–6Al–4V ZrO2

4420 3657

0 0

0 0

a (1/K)

Ti–6Al–4V ZrO2

7.43  106 13.3  106

7.483  104 1.421  103

3.621  107 9.549  107

K (W/mK)

Ti–6Al–4V ZrO2

6.10 1.78

0 0

0 0

Table 2 Convergence and accuracy of the first three non-dimensional natural frequency parameters of the clamped isotropic arch ðh0 ¼ 60 ; m ¼ 0:3Þ. Nh

Nr

h/L = 0.1

h/L = 0.3 k2

k3

9

7 9 13

10.439 10.439 10.439

14.821 14.822 14.822

28.221 28.222 28.222

5.255 5.258 5.258

8.539 8.549 8.550

11.159 11.161 11.161

13

7 9 13

10.436 10.436 10.436

14.803 14.805 14.805

28.194 28.200 28.200

5.250 5.255 5.255

8.528 8.542 8.542

11.154 11.158 11.158

17

7 9 13

10.435 10.435 10.435

14.797 14.801 14.802

28.183 28.192 28.194

5.248 5.254 5.256

8.524 8.540 8.544

11.152 11.156 11.158

10.435

14.802

28.194

5.256

8.545

11.158

2D-LW-DQ [14]

k1

k2

k3

k1

Table 3 Convergence and accuracy of the first three non-dimensional natural frequency parameters of the clamped isotropic arches ðh0 ¼ 120 ; m ¼ 0:3Þ: Nh

Nr

h/L = 0.1

h/L = 0.3 k2

k3

9

7 9 13

12.952 12.954 12.954

16.775 16.775 16.775

28.924 28.926 28.926

7.174 7.177 7.178

7.269 7.284 7.286

12.008 12.014 12.015

13

7 9 13

12.935 12.938 12.938

16.765 16.766 16.766

28.841 28.846 28.846

7.173 7.176 7.177

7.258 7.277 7.281

12.002 12.010 12.011

17

7 9 13

12.930 12.934 12.935

16.764 16.765 16.765

28.831 28.839 28.840

7.172 7.176 7.176

7.254 7.275 7.280

12.001 12.009 12.011

12.935

16.765

28.841

7.177

7.281

12.011

2D-LW-DQ [14]

k1

k2

k3

k1

43

P. Malekzadeh / Composite Structures 91 (2009) 38–47 Table 4 Convergence of the first three non-dimensional natural frequency parameters of the clamped FG arches under uniform temperature rise (p = 2, DT = 800 K, h0 = 120°). Nh

Nr

h/L = 0.1 k1

k2

k3

k1

k2

k3

9

7 9 13a

8.312 8.313 8.313

10.769 10.769 10.769

18.551 18.553 18.553

6.182 6.188 6.188

6.246 6.246 6.246

11.656 11.657 11.657

13

7 9 13a

8.291 8.293 8.293

10.768 10.769 10.769

18.502 18.506 18.506

6.172 6.182 6.182

6.245 6.245 6.245

11.646 11.648 11.648

17

7 9 13a

8.287 8.290 8.291

10.768 10.768 10.768

18.496 18.502 18.503

6.167 6.178 6.180

6.244 6.245 6.245

11.645 11.647 11.647

a

h/L = 0.2

After this value the results have no change.

Table 5 Convergence of the first three (p = 2, DTm = 0, DTc = 800 K, h0 = 120°). Nh

Nr

non-dimensional

natural

frequency

parameters

of

the

clamped

h/L = 0.1

FG

arches

under

non-uniform

temperature

rise

h/L = 0.2 k2

k3

9

7 9 13a

10.013 9.978 9.976

13.038 13.022 13.021

22.387 22.306 22.304

7.527 7.528 7.528

7.612 7.609 7.609

13.892 13.887 13.886

13

7 9 13a

9.994 9.978 9.976

13.034 13.022 13.021

22.325 22.306 22.304

7.517 7.521 7.522

7.611 7.608 7.608

13.882 13.877 13.876

17

7 9 13a

9.990 9.975 9.974

13.033 13.021 13.020

22.318 22.302 22.301

7.512 7.518 7.520

7.610 7.608 7.608

13.880 13.876 13.876

a

k1

k2

k3

k1

After this value the results have no change.

Table 6 Convergence of the first three non-dimensional natural frequency parameters of the clamped-simply supported FG arches under uniform temperature rise (h/ L = 0.3, DT = 800 K, h0 = 120°). Nh

Nr

p=1

p=5

k1

k2

k3

k1

k2

k3

9

7 9 13

1.149 1.153 1.154

3.944 3.949 3.950

5.976 5.981 5.981

1.091 1.096 1.096

3.752 3.758 3.758

5.688 5.694 5.694

13

7 9 13

1.147 1.153 1.155

3.940 3.947 3.948

5.971 5.976 5.977

1.090 1.096 1.097

3.748 3.755 3.756

5.684 5.689 5.690

17

7 9 13

1.146 1.153 1.155

3.939 3.946 3.948

5.971 5.976 5.977

1.088 1.095 1.097

3.747 3.754 3.756

5.684 5.689 5.689

Table 7 The first three non-dimensional natural frequency parameters of the FG arches under uniform temperature rise (h/L = 0.2, p = 2). Initial thermal stress evaluation

h0

60°

k2

k3

k1

k2

k3

Eq. (18) Eq. (34)

5.069 4.966 2.04

8.401 8.261 1.66

12.606 12.548 0.46

4.281 4.082 4.66

7.094 6.822 3.83

10.712 10.599 1.06

Eq. (18) Eq. (34)

7.279 7.156 1.70

7.344 7.242 1.38

13.694 13.570 0.91

6.180 5.941 3.87

6.245 6.054 3.06

11.647 11.403 2.09

Eq. (18) Eq. (34)

6.088 5.981 1.75

9.180 9.069 1.20

14.586 14.419 1.15

5.177 4.967 4.06

7.815 7.607 2.66

12.413 12.085 2.64

% Relative error 180° % Relative error a

DTm = DTc = 800 K

k1

% Relative errora 120°

DTm = DTc = 400 K

% Relative error = 100  [(ki)Eq.

(18)

 (ki)Eq.

(34)]/[(ki)Eq. (18)].

P. Malekzadeh / Composite Structures 91 (2009) 38–47

As a first example, the convergence behavior of the method against the number of DQ grid points along the r- and h-directions for the first three non-dimensional natural frequency parameters of the clamped isotropic arches is presented in Tables 2 and 3. The results are prepared for two different values of the opening angle (h0) and thickness-to-length ratio. In order to demonstrate the accuracy of the converged results, the same problem is solved by using the two-dimensional layerwise-differential quadrature method (2D-LW-DQ) presented by Malekzadeh et al. [14], where its high accuracy was demonstrated. Fast rate of convergence of the presented method and its excellent agreement with those of the 2D-LW-DQ in all cases are obvious. The converged results of the 2D-LW-DQ are obtained by using twelve mathematical layers in the radial direction and 70 grid points in the tangential direction.

p=0

2.4

p=0.5 p=1 p=5

2.2

p=10 λ1

44

2.0

1.8

1.6 0

4.9

100

200

300

400

500

ΔΤ (Κ)

600

700

800

(a) 4.5

2.75 p=0

λi

4.1

p=0.5 p=1

2.50

3.7

p=5 p=10 λ2

3.3 Temperature Temperature dependent independent

2.9

2.25

First mode (i=1) Second mode (i=2)

2.00

2.5 0

100

200

300

400

500

600

700

800

ΔΤ (Κ) Fig. 2. The effects of temperature-dependence of material properties on the first two frequency parameters of clamped circular arches subjected to uniform temperature rise (DTc = DTm = DT, h/Rm = 0.2, p = 2, ho = 90°).

1.75 0

100

200

300

400

500

ΔΤ (Κ)

600

700

800

(b) 5.2 p=0

75

First mode (i=1)

p=0.5

4.9

Second mode (i=2)

p=1

Third mode (i=3)

60

p=5

4.6 λ3

p=10

45

λi

4.3

4.0

30

3.7

15

3.4 0

0 5

30

55

80

105

130

155

180

Opening angle (degree) Fig. 3. The effects of opening angle on the first three frequency parameters of clamped arches subjected to non-uniform temperature rise (DTc = 800 K, DTm = 0, h/Rm = 0.3, p = 2).

100

200

300

400

500

ΔΤ (Κ)

600

700

800

(c) Fig. 4. (a)–(c) The effects of uniform temperature rise (DTc = DTm = DT) and the material graded index (p) on the first three frequency parameters of C–C arches ðh=Rm ¼ 0:3; h0 ¼ 120 Þ.

45

P. Malekzadeh / Composite Structures 91 (2009) 38–47

2.4

p=0

p=1

p=0.5

p=5

5.00 p=0 p=0.5

4.75

p=10

p=1

+ 10

4.50

2.2

p=5 p=10

4.25

λ1

λ1

2.3

4.00

2.1 3.75

2.0

3.50 3.25

1.9 0

100

200

300

400

500

600

700

0

800

100

200

300

400

500

600

700

800

ΔΤ (Κ)

ΔΤ (Κ)

(a)

(a)

2.05

2.70

p=0

p=1

p=0.5

p=5

p=0 p=0.5

p=10

1.90

p=1 p=5

2.55

p=10

λ2

λ2

1.75

2.40

1.60

2.25

1.45

1.30

2.10

0

0

100

200

300

400

500

ΔΤ (Κ)

600

700

100

200

300

400

5.0

600

700

800

(b)

(b)

5.2

500

ΔΤ (Κ)

800

p=0

p=1

p=0.5

p=5

p=0

3.5

p=0.5 p=1

3.3

p=10

p=5 p=10

3.1

λ3

λ3

4.8

2.9

4.6

2.7

4.4

2.5

4.2 2.3 0

4.0 0

100

200

300

400

500

600

700

800

ΔΤ (Κ)

(c) Fig. 5. (a)–(c) The effects of non-uniform temperature rise ðDT c ¼ DT; DT m ¼ 0Þ and the material graded index (p) on the first three frequency parameters of C–C arches ðh=Rm ¼ 0:3; h0 ¼ 120 Þ:

As another attempt to validate the presented formulations, the convergence of the method for FG arches with temperature-depen-

100

200

300

400

500

600

700

800

ΔΤ (Κ)

(c)

Fig. 6. (a)–(c) The effects of uniform temperature rise ðDT c ¼ DT m ¼ DTÞ and the material graded index (p) on the first three frequency parameters of C–S arches ðh=Rm ¼ 0:3; h0 ¼ 120 Þ.

dent material properties under high-temperature thermal environment is studied here. The results for the FG clamped arches under

46

P. Malekzadeh / Composite Structures 91 (2009) 38–47

high uniform temperature rise are presented in Table 4 and those of the FG clamped arches subjected to non-uniform temperature

4.9

p=0

p=1

p=0.5

p=5 p=10

λ 1+ 10

4.7

4.5

4.3

4.1

r0rr ¼ r0hh ¼ ðC 11 þ C 12 Þaðr; TÞDT; r0rh ¼ 0

3.9 0

100

200

300

400

500

600

700

800

ΔΤ (Κ)

(a)

2.0

p=0

p=1

p=0.5

p=5 p=10

λ2

1.9

1.8

1.7

1.6 0

100

200

300

400

500

600

700

800

ΔΤ (Κ)

(b) 3.50

p=0

p=1

p=0.5

p=5 p=10

3.35

λ3

rise are shown in Table 5. In Table 6, the convergence of the method for FG clamped-simply supported arches under high-temperature rise and for two different values material graded index (p) are presented. Again, in all cases, fast rate of convergence of the method is evident. It should be mentioned that for both uniform and non-uniform temperature rise, only seven and nine DQ grid points in the radial and the tangential direction are sufficient to obtain results with high accuracy. Also, in all cases for N r P 13 and N h P 17 the results remain the same up to four significant digits. Hence, hereafter, the results for the FG arches are prepared using Nr = 13 and N h ¼ 17. In some previous studies regarding the free vibration analysis of FG plates, the pre-stress analysis was not performed and the initial thermal stresses were usually obtained approximately from the thermal strains; see for example Refs. [19,20]. For the circular arch these approximate formulations reduce to

3.20

ð34Þ

In Table 7, a comparison between the results based on using Eqs. (18) and (34) for the initial thermal stresses are made. The first three frequency parameters of FG circular arches under uniform temperature rise and for different opening angles are presented. It is interesting to note that the effect of approximate evaluation of the initial stresses on the fundamental frequency parameters is higher than other frequency parameters. Also, the difference between the results of the two approaches increases as the temperature rise increases. In order to show the importance of considering the variation of material properties with temperature, the first two frequency parameters of FG clamped circular arches subjected to uniform temperature rise with and without temperature-dependence of material properties are compared in Fig. 2. The frequency parameters are greatly overestimated when the temperature-dependence of material parameters is not taken into account. The discrepancy between temperature-dependent and temperature-independent solutions increases dramatically as the temperature rise increases. The difference reaches as high as approximately 41.7% and 42.0% for the first and the second frequency parameters, respectively. The influence of the opening angle (h0) on the first three frequency parameters of FG clamped circular arches under non-uniform temperature rise are presented in Fig. 3. It can be seen that when the opening angle increases, the frequency parameters decrease dramatically. The effects of the material index (p) on the first three frequency parameters of FG clamped arches subjected to uniform and nonuniform temperature rise are presented in Figs. 4 and 5, respectively. Also, the results for the same FG arches but with clamped-simply supported edges under uniform and non-uniform temperature rise are shown in Figs. 6 and 7, respectively. For all cases, the frequency parameters decrease monotonically when the temperature rise increases. Also, when the material index (p) increases, the frequency parameters decrease.

3.05

5. Conclusions 2.90

2.75 0

100

200

300

400

500

600

700

800

ΔΤ (Κ)

(c)

Fig. 7. (a)–(c) The effects of non-uniform temperature rise ðDT c ¼ DT; DT m ¼ 0Þ and the material graded index (p) on the first three frequency parameters of C–S arches ðh=Rm ¼ 0:3; h0 ¼ 120 Þ:

The two-dimensional in-plane free vibration analysis of the FG circular arches with temperature-dependent material properties subjected to thermal environment are presented. The initial thermal stresses are obtained by solving the two-dimensional thermoelastic governing equations of the arch. Using the Hamilton’s principle, the equations of motion and the related boundary conditions subjected to initial thermal stresses are derived. The differential quadrature method as an efficient and accurate numerical tool is used to solve the system of equations. The present approach has the advantage of providing solutions for arches with some different

P. Malekzadeh / Composite Structures 91 (2009) 38–47

types of boundary conditions. Since no assumptions on stresses and displacements have been made, it can be applied to the free vibration analysis of arches with arbitrary thickness. The effects of the temperature rise, the material graded index, and different geometrical parameters such as the thickness-to-length ratio and the opening angle on the frequency parameters of the FG arches are investigated. It is shown that the temperature-dependence of the material properties have significant effects on the natural frequency parameters. The solutions can be used as benchmark for other numerical methods and also the refined arch theories.

The basic idea of the differential quadrature method is that the derivative of a function, with respect to a space variable at a given sampling point, is approximated as the linear weighted sums of its values at all of the sampling points in the domain of that variable. In order to illustrate the DQ approximation, consider a function f(n, g) having its field on a rectangular domain 0 6 n 6 a and 0 6 g 6 b. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to DQ method, the rth derivative of a function f(n, g) can be approximated as

ðA:1Þ

for i = 1, 2, . . . , Nn, j = 1, 2, . . . , Ng and r = 1, 2, . . . , Nn  1. From this equation one can deduce that the important components of DQ approximations are weighting coefficients and the choice of sampling points. In order to determine the weighting coefficients a set of test functions should be used in Eq. (A.1). For polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the first-order derivatives in n-direction are thus determined as [17]

Anij

¼

8 1 > > L > < n  > > > :

Mðni Þ ðni nj ÞMðnj Þ Nn P j¼1 i–j

Anij

for i – j for i ¼ j

i; j ¼ 1; 2 ; . . . ; N n

ðA:2Þ

where Ln is the length of domain along the n-direction and

Mðni Þ ¼

Nn Y

ðni  nk Þ

k¼1;i – k

The weighting coefficients of second order derivative can be obtained as [17],

½Bnij  ¼ ½Anij ½Anij  ¼ ½Anij 2

In numerical computations, Chebyshev-Gauss-Lobatto quadrature points are used, that is [17],

   ni 1 ði  1Þp 1  cos ¼ ; a 2 ðNn  1Þ    gj 1 ðj  1Þp 1  cos ¼ b 2 ðNg  1Þ for i ¼ 1; 2; . . . ; N n and j ¼ 1; 2; . . . ; N g

ðA:4Þ

References

Appendix A. DQ weighting coefficients

 Nn Nn X X @ r f ðn; gÞ nðrÞ nðrÞ ¼ Aim f ðnm ; gj Þ ¼ Aim fmj r  @n ðn;gÞ¼ðni ;gj Þ m¼1 m¼1

47

ðA:3Þ

In a similar manner, the weighting coefficients for g-direction can be obtained.

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