Hygro thermal effects on vibration and thermal buckling behaviours of functionally graded beams

Voc,d, Huu-Tai Thaie a Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam b Faculty

Trang 1

Hygro-thermal effects on vibration and thermal buckling behaviours of

functionally graded beams

Trung-Kien Nguyena,⇑, Ba-Duy Nguyena,b, Thuc P Voc,d, Huu-Tai Thaie

a

Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam

b

Faculty of Civil Engineering, Thu Dau Mot University, 6 Tran Van On Street, Phu Hoa District, Thu Dau Mot City, Binh Duong Province, Viet Nam

c

Duy Tan University, Da Nang, Viet Nam

d

Faculty of Engineering and Environment, Northumbria University, Ellison Place, Newcastle upon Tyne NE1 8ST, UK

e School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, VIC 3086, Australia

a r t i c l e i n f o

Article history:

Received 21 April 2017

Revised 8 June 2017

Accepted 9 June 2017

Available online 15 June 2017

Keywords:

Advanced composite beams

Hygro-thermal loadings

Buckling

Vibration

a b s t r a c t The hygro-thermal effects on vibration and buckling analysis of functionally graded beams are presented

in this paper The present work is based on a higher-order shear deformation theory which accounts for a hyperbolic distribution of transverse shear stress and higher-order variation of in-plane and out-of-plane displacements Equations of motion are obtained from Lagrange’s equations Ritz solution method is used

to solve problems with different boundary conditions Numerical results for natural frequencies and crit-ical buckling temperatures of functionally graded beams are compared with those obtained from previ-ous works Effects of power-law index, span-to-depth ratio, transverse normal strain, temperature and moisture changes on the results are discussed

1 Introduction

Hygro-thermal stresses arising from a variation of temperature

and moisture content can affect structural responses of

engineer-ing structures Therefore, an accurate evaluation of environmental

exposure is important to investigate hygro-thermal effects on their

behaviours Owing to the low density and high stiffness and

strength, composite structures become popular in several

applica-tions of aerospace, automotive engineering, construction, etc They

became more attractive due to an introduction of functionally

graded (FG) materials The general benefit of these structures

com-pared to conventional ones is a continuous variation of

hygro-thermo-elastic properties in a required direction so that interfacial

issues found in laminated composite structures could be neglected

In order to accurately predict hygro-thermo-mechanical

beha-viours of FG nanobeams and FG plates/beams, several models

and approaches have been developed in recent years Ebrahimi

and Salari[1,2]investigated nonlocal thermo-mechanical buckling

and free vibration of FG nanobeams in thermal environments

dynamic analysis of nonlocal heterogeneous nanobeams in

responses of FG plates under hygro-thermo-mechanical loading using a four variable refined plate theory Zenkour et al.[5,6] inves-tigated hygro-thermo-mechanical effects on behaviours of FG

thermal stability of FG sandwich plates under various through-the-thickness temperature distributions Vibration and buckling analysis of FG beams under mechanical loads have been investi-gated by many authors based on classical beam theory (CBT) ([8,9]), first-order shear deformation beam theory (FSBT) ([10–14]),

For thermal environments, the thermal stability and vibration analysis of FG beams have studied by many authors with different methods Esfahani et al.[28]studied nonlinear thermal buckling of

FG beams The nonlinear thermal dynamic buckling of FG beams is also investigated by Ghiasian et al.[29] Ma and Lee[30]proposed exact solutions for nonlinear bending behaviour of FG beams under

investigated the dynamic thermal response of FG beams under a

HSBT to study the buckling and vibration of FG beams under the uniform thermal loading Sun et al.[34]investigated thermal buck-ling and post-buckbuck-ling of FG beams on nonlinear elastic

thermo-mechanical responses of FG beams Bhangale and Ganesan

http://dx.doi.org/10.1016/j.compstruct.2017.06.036

⇑Corresponding author.

E-mail address: kiennt@hcmute.edu.vn (T.-K Nguyen).

Contents lists available atScienceDirect

Composite Structures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c t

Trang 2

[36]analyzed thermoelastic buckling and vibration behaviours of

FG sandwich beam with constrained viscoelastic core By using

thermo-mechanical vibration of FG sandwich beams However, a

limited number of researches has been considered to investigate

studied nonlinear analysis of composite laminated beams in

hygro-thermal environments Moreover, it is known that Ritz

method is efficient to deal with composite and FG beams with

arbi-trary boundary conditions The accuracy and efficiency of this

43,24,26,44]

The objective of this paper is to present hygro-thermal

responses of FG beams using a higher-order shear deformation

the-ory in which a higher-order variation of both in-plane and

out-of-plane displacement is taken into account FG beams are composed

of ceramic and metal mixtures, and the material properties are

var-ied according to power-law form Ritz solution is developed for

dif-ferent boundary conditions to verify the accuracy of the present

theory and to investigate the effects of power-law index,

span-to-depth ratio, temperature and moisture content on the vibration

and buckling responses of FG beams under hygro-thermal

loadings

2 Theoretical formulation

2.1 Material properties

A FG beam made of a mixture of ceramic and metal isotropic

materials, which is embedded in a moisture and temperature

envi-ronment, with length L and uniform section b h is considered as

shown inFig 1 The material properties are varied according to

power-law form:

PðzÞ ¼ ðPc PmÞ 2zþ h

2h

p

where p is the power-law index and Pcand Pmare Young’s modulus

E, mass densityq, coefficient of thermal expansiona, coefficient of

moisture expansion b, thermal conductivity coefficient k of ceramic

and metal materials, respectively

Moreover, the thermo-elastic material properties of FG beams

are also expressed in terms of temperature T(K) ([31]):

PðT; zÞ ¼ H0ðH1T1þ 1 þ H1Tþ H2T2þ H3T3Þ ð2Þ

where H0; H1; H2; H3are temperature dependent coefficients for var-ious types of materials (Table 1) It should be noted that both tem-perature dependency (TD) and temtem-perature independency (TID) are considered in this paper

2.2 Moisture and temperature distribution Three different moisture and temperature distributions through the beam depth are considered: uniform moisture and tempera-ture rise, linear moistempera-ture and temperatempera-ture rise and nonlinear mois-ture and temperamois-ture rise

Uniform moisture and temperature rise: the temperature and moisture are supposed to be uniform in the beam and increased from a reference T0and C0, thus their current values of temper-ature and moisture are:

respectively, which are supposed to be at the bottom surface

of the beam

Linear moisture and temperature rise: the temperature and moisture are linearly increased as follows:

TðzÞ ¼ðTt TbÞ 2zþ h

2h

CðzÞ ¼ðCt CbÞ 2zþ h

2h

where Ttand Tbare temperatures as well as Ctand Cbare mois-ture content at the top and bottom surfaces of the beam

Nonlinear moisture and temperature rise: the temperature and moisture are varied nonlinearly according to a sinusoidal law ([7]) as follows:

TðzÞ ¼ðTt TbÞ 1 cosp

2

2zþ h 2h

CðzÞ ¼ðCt CbÞ 1 cosp

2

2zþ h 2h

In addition, the temperature distribution obtained from Fourier equation of steady-state one-dimensional heat conduction is also considered:

TðzÞ ¼ Tbþ Tt Tb

Rh =2

h=2 kðzÞ1 dz

Z z

h=2

1

2.3 Kinematics The displacement field is chosen from previous study[25]:

u1ðx; z; tÞ ¼uðx; tÞ zw;xþ sinh1 z

h

8z3

3 ffiffiffi 5

p

h3

hðx; tÞ

¼uðx; tÞ zw;xþ f1ðzÞhðx; tÞ ð7aÞ

u3ðx; z; tÞ ¼wðx; tÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

h2þ z2

p 8zffiffiffi2

5

p

h3

!

wzðx; tÞ

¼wðx; tÞ þ f2ðzÞwzðx; tÞ ð7bÞ

where the comma indicates partial differentiation with respect to the coordinate subscript that follows; f2¼ f1;z; u and h are the axial

dis-placements, respectively

The nonzero strains are given by:

Trang 3

xxðx; z; tÞ ¼u;x zw;xxþ f1h;x ð8aÞ

cxzðx; z; tÞ ¼f2ðh þ wz ;xÞ ð8cÞ

The elastic constitutive equations are given by:

rxx

rzz

rxz

8

>

9

>

>¼

Q11 Q13 0

Q13 Q11 0

0 0 Q55

2

64

3

75 xxzz

cxz

8

>

9

>

where

Q11¼ EðzÞ

1m2; Q13¼ EðzÞm

1m2; Q55¼ EðzÞ

2ð1 þmÞ ð10Þ

If the transverse normal strain effect is omitted (zz¼ 0), the

components of Qijin Eq.(9)are reduced as:

Q11¼ EðzÞ

1m2; Q13¼ 0; Q55¼ EðzÞ

through the beam thickness and its value is evaluated as the

aver-age of ceramic and metal ones

2.4 Lagrange’s equations

The strain energyU of system is expressed by:

U ¼12

Z

VðrxxxxþrzzzzþrxzcxzÞdV

¼1

2

Z L

0

Au2;x2Bu;xw;xxþDw2

;xxþ2Bs

u;xh;x2Ds

w;xxh;xþHs

h2;x h

þ2ðXu;xwzYw;xxwzþYs

h;xwzÞþZw2

zþAs

ðh2

þ2hwz ;xþw2

z;xÞidx ð12Þ

where

ðA; B; D; Bs

; Ds

; Hs

; ZÞ ¼

Zh =2

h=2Q11ðzÞ 1; z; z2; f1; zf1; f2

1; f2

2 ;z

bdz ð13aÞ ðX; Y; Ys

Þ ¼

Zh =2

h=2Q13ðzÞ 1; z; fð 1Þf2 ;zbdz ð13bÞ

As¼

Zh=2

h=2

Q55ðzÞf2

expressed by:

V ¼ 1 2

Z L 0

ðNtþ NmÞðw;xÞ2

where

Nt¼

Zh =2

h=2

Q11ðzÞaðzÞ TðzÞ Th 0i

Nm¼

Zh=2

h=2

Q11ðzÞbðzÞ CðzÞ Ch 0i

The kinetic energyK is expressed by:

K ¼1 2

Z

VqðzÞð _u2þ _u2ÞdV

¼1 2

Z L 0

I0_u2 2I1_u _w;xþ I2_w2

;xþ 2J1_h_u 2J2_h _w;xþ K2_h2þ I0_w2

h þ2L1_w _wzþ L2_w2

z

where dot-superscript denotes the differentiation with the time t; and I0; I1; I2; J1; J2; K2; L1; L2are the inertia coefficients defined by:

ðI0; I1; I2; J1; J2; K2; L1; L2Þ ¼

Z h =2

h=2qðzÞ 1; z; z2; f1; zf1; f2

1; f2; f2 2

bdz ð17Þ

Lagrangian functional is used to derive the governing equations

of motion:

P¼12

Z L 0

Au2;x 2Bu;xw;xxþ Dw2

;xxþ 2Bs

u;xh;x 2Ds

w;xxh;xþ Hs

h2;x h

þ2ðXu;xwz Yw;xxwzþ Ys

h;xwzÞ þ Zw2

zþ Asðh2þ 2hwz;xþ w2

z ;xÞidx

1 2

Z L 0

ðNtþ NmÞðw;xÞ2dx

12

Z L 0

I0_u2 2I1_u _w;xþ I2_w2

;xþ 2J1_h_u 2J2_h _w;xþ K2_h2

h

þ I0_w2þ 2L1_w _wzþ L2_w2

z

The displacement field is expressed by the approximation func-tions according to the Ritz method as follows:

Table 1

Temperature dependent coefficients for ceramic and metal materials.

Al 2 O 3

q(kg/m 3

Si 3 N 4

SUS304

q(kg/m 3

Trang 4

uðx; tÞ ¼X

N

j ¼1

wðx; tÞ ¼X

N

j ¼1

hðx; tÞ ¼XN

j¼1

wzðx; tÞ ¼X

N

j ¼1

wherex is the natural frequency, andðuj; wj; hj; yj) are unknown

values The approximation functions wjðxÞ andujðxÞ are chosen as

follows:

The Lagrange multipliers (di) are used to impose the boundary

conditions, that leads to a new Lagrangian functional:

whereuiðxÞ denote prescribed displacement at two ends (x ¼ 0 and

L) Substituting Eq (20) into Eq.(19), and using Lagrange’s equations:

@P

@qj

d

dt

@P

where qj representing the values of ðuj; wj; hj; yj), a characteristic

problem for hygro-thermal vibration and buckling response is

obtained through the stiffness matrix K and mass matrix M:

K11 K12 K13 K14 K15

TK13 TK23 K33 K34 K35

TK14 TK24 TK34 K44 K45

TK15 TK25 TK35 TK45 0

2

66

4

3 77 77 77 5

0

B

@

x2

M11 M12 M13 0 0

0 TM24 0 M44 0

2

66

4

3 77 77 77 5

1 C C C A

u w h y d

8

>

<

>

:

9

>

=

>

;

¼

0 0 0 0 0

8

>

<

>

:

9

>

=

>

; ð24Þ

where

K11ij ¼A

Z L

0 wi ;xwj ;xdx; K12

ij ¼ B

Z L

0 wi ;xuj;xxdx; K13

ij ¼ BsZ L

0 wi ;xwj ;xdx

K14ij ¼X

Z L

0

wi;xujdx; K22

ij ¼ D

Z L

0 ui ;xxuj ;xxdx NtZ L

0 ui ;xuj ;xdx

Nm

Z L

0 ui ;xuj ;xdx

K23

ij ¼ Ds

Z L

0 ui ;xxwj ;xdx; K24

ij ¼ Y

Z L

0 ui ;xxujdx

K33ij ¼HsZ L

0 wi ;xwj ;xdxþ AsZ L

0 wiwjdx; K34

ij ¼ YsZ L

0 wi ;xujdx

þ AsZ L

0

wiuj ;xdx

K44

ij ¼Z

Z L

0 uiujdxþ AsZ L

0 ui ;xuj ;;xdx

M11

ij ¼I0

Z L

0

wiwjdx; M12

ij ¼ I1

Z L 0

wiuj ;xdx; M13

ij ¼ J1

Z L 0

wiwjdx

M22ij ¼I0

Z L

0 uiujdxþ I2

Z L

0 ui;xuj;xdx; M23

ij ¼ J2

Z L

0 ui;xwjdx

M24ij ¼L1

Z L

0 uiujdx; M33

ij ¼ K2

Z L 0

wiwjdx; M44

ij ¼ L2

ZL 0

wiwjdx ð25Þ

The components of matrix K15; K25

, K35and K45, which depend

on boundary conditions (Table 2), are list below:

For hinged-hinged (H-H) beams:

K15 i1 ¼wið0Þ; K15

i2 ¼ wiðLÞ; K15

ij ¼ 0 with j ¼ 3; 4; ; 6

K25i3 ¼uið0Þ; K25

i4 ¼uiðLÞ; K25

ij ¼ 0 with j ¼ 1; 2; 5; 6

K35ij ¼0 with j ¼ 1; 2; ; 6

K45 i5 ¼uið0Þ; K45

i6 ¼uiðLÞ; K45

ij ¼ 0 with j ¼ 1; 2; 3; 4 ð26Þ

For clamped-hinged (C–H) beams:

K15 i1 ¼wið0Þ; K15

i2 ¼ wiðLÞ; K15

ij ¼ 0 with j ¼ 3; 4; ; 8

i4 ¼uiðLÞ; K25

i5 ¼ui;xð0Þ; K25

ij ¼ 0 with j ¼ 1; 2; 6; 7; 8

K35i6 ¼wið0Þ; K35

ij ¼ 0 with j ¼ 1; 2; 3; 4; 5; 7; 8

K45 i7 ¼uið0Þ; K45

i8 ¼uiðLÞ; K45

ij ¼ 0 with j ¼ 1; 2; ; 6 ð27Þ

For clamped–clamped (C–C) beams:

i2 ¼ wiðLÞ; K15

ij ¼ 0 with j ¼ 3; 4; ; 10

i4 ¼uiðLÞ; K25

i5 ¼ui ;xð0Þ; K25

i6 ¼ui ;xðLÞ; K25

ij ¼ 0 with j¼ 1; 2; 7; ; 10

i8 ¼ wiðLÞ; K35

ij ¼ 0 with j ¼ 1; 2; ; 6; 9; 10

i10¼uiðLÞ; K45

ij ¼ 0 with j ¼ 1; 2; ; 8 ð28Þ

3 Numerical results and discussion

In this section, a number of numerical examples are analyzed to verify the accuracy of present theory and investigate the effects of power-law index, span-to-depth ratio, transverse normal strain, temperature and moisture content on buckling and vibration responses of FG beams for various boundary conditions (H–H, C–

H and C–C) FG beams are made of ceramic (Si3N4, Al2O3) and metal (SUS304) with material properties inTable 1 Three types of tem-perature and moisture distribution through the beam depth are considered: uniform moisture and temperature rise (UMR, UTR), linear moisture and temperature rise (LMR, LTR), nonlinear mois-ture and temperamois-ture rise (NLMR, NLTR) The following non-dimensional parameters are used:

x¼xL

2

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I0

Rh =2

h=2EðzÞdz

v u

; ^x¼xL

2

h

ffiffiffiffiffiffiffiffiffiffiffi

12qc

Ec

s

; k ¼DTcr

L2

h2am ð29Þ

whereamis thermal expansion coefficient of metal at T0(K) Notic-ing that the followNotic-ing relations are used in this paper: T0= 300 (K),

C0= 0%, Tb T0= 5 (K)

For convergence test,Table 3reports the first natural frequency with respect to the number of series N of Si3N4/SUS304 beams with

p = 1, L=h = 5 andDT = 20,DC = 0 The results are calculated with different boundary conditions and Fourier-law NLTR In order to obtain good solution, the number of series N are chosen 8, 12,

Table 2 Kinematic boundary conditions.

h ¼ 0, w z ¼ 0

u ¼ 0; w ¼ 0; w z ¼ 0,

h ¼ 0, w z ¼ 0

u ¼ 0; w ¼ 0; w ;x ¼ 0,

h ¼ 0, w z ¼ 0

Trang 5

and 14 for H–H, C–H and C–C beams, respectively For this reason,

these numbers are used in the following examples

As the first example, FG beams under uniform temperature rise

(UTR) are considered.Table 4presents the normalized critical

depen-dency (TD) and temperature independepen-dency (TID) solutions with

different values of power-law index p It is noted that the results

reported in this example are based on the assumption that the

Q11¼ EðzÞ=ð1 mÞ The results are compared with those of

Wat-tanasakulpong et al.[33]and Trinh et al.[35]using HSBT The

pre-sent results without normal strain (zz¼ 0) are in good agreement

with earlier works Fig 2a presents the effect of the power-law

index p on the normalized critical temperatures of Si3N4/SUS304

beams with L=h = 20 It is plotted with both TD and TID solutions

as well as with and without normal strain It can be seen that

the normalized critical temperatures decrease with the increase

zz¼ 0 This can be explained by the fact that the effect of

trans-verse normal strain made beams softer This figure also shows that

the TD solutions give lower values than the TID ones, which

emphasizes the importance of temperature dependency in the FG

beams Similarly, the accuracy of present theory in predicting the

vibration response of Al203/SUS304 FG beams is studied inTable 5

The results are calculated with p = 0.2, 2 andDT = 0, 50 and 100 It

is seen that good agreements between HSBTs are again found for

all cases.Fig 2b displays the effects of UTR on the normalized

p = 2) Obviously, the result decreases with the increase ofDT up

to critical temperatures at which the fundamental frequencies van-ish In this case, the critical temperatures of H–H, C–H and C–C beams are 52.6580 (K), 103.5923 (K) and 192.1833 (K), respectively

The next example aims to investigate the effects of linear and nonlinear temperature rise (LTR, NLTR) on the thermal buckling and vibration of FG beams For verification purpose, the critical temperatures of Si3N4/SUS304 beams with L=h = 40 are reported

in Table 6 These results are compared with those of Esfahani

et al.[28], Ebrahimi and Salari[2]based on FSDT It is observed that the present solutions are in good agreement with those of[28]for C–C beams under the Fourier-law NLTR while there is slight devi-ations for several values of p between the present solutions and those of[2]for H-H beams under LTR It is noted that the super-script ‘‘a” is used to indicate that Poisson’s ratio effect is not included in the constitutive equation and thermal stress resultant (Q11¼ EðzÞ) and this index will be used in the next examples for

critical temperatures from the present solutions and those from

[35] It shows that there are small differences between the HSBT models The effect of normal strain is again found in which the HSBTs over-predict critical temperatures in comparison with the quasi-3D theory.Fig 3a displays the variation of fundamental fre-quency for UTR, LTR and Fourier-law NLTR It can be seen that the results decrease with the increase ofDT and vanish at the critical temperatures.Table 9andFig 3b consider the effects of tempera-ture distribution under Fourier- and sinusoidal-law through the beam depth for different boundary conditions For comparison, the critical temperature with Fourier law is smaller than that with

Table 3

Convergence test for the nondimensional fundamental frequency (^x) of Si 3 N 4 /SUS304 beams under Fourier-law NLTR with p ¼ 1; L=h ¼ 20;DT = 20 (K), TD, DC = 0%.

Table 4

Normalized critical temperatures (k) of Si 3 N 4 /SUS304 beams under UTR (L=h = 20).

Trang 6

sinusoidal one Moreover,Table 10presents the normalized

funda-mental frequency of Si3N4/SUS304 beams with L=h = 20, p = 0.1, 0.5

and 1,DT = 20 and 80, subjected to the LTR and Fourier-law NLTR

The results are compared to those of[33,35]for different boundary conditions and good agreements between the HSBT models are again found

Table 5

Fundamental frequency (x ) of Al 2 O 3 /SUS304 beams under UTR (L=h = 30).

Table 6

Critical temperature of Si 3 N 4 /SUS304 FG beams under LTR and Fourier-law NLTR (L=h = 40, TD).

a Q11¼ EðzÞ.

Fig 2 Variation of normalized critical temperature and fundamental frequency of FG beams with respect to the power-law index p and uniform temperature rise DT.

Trang 7

Table 8

Critical temperatures of Si 3 N 4 /SUS304 beams under Fourier-law NLTR (L=h = 20, TD).

Present a

a

Q 11 ¼ EðzÞ.

Fig 3 Variation of normalized fundamental frequency of Si 3 N 4 /SUS304 beams with respect to the power-law index p and temperature rise (TD).

Table 7

Critical temperature of Si 3 N 4 /SUS304 FG beams under LTR for different boundary conditions (L=h = 20, TD).

a

Q 11 ¼ EðzÞ.

Trang 8

The final example is to analyse the effects of moisture content

on the thermal vibration behaviour of FG beams.Tables 11–13

pre-sent the normalized fundamental frequencies of Si3N4/SUS304 FG

beams under the uniform, linear and nonlinear moisture (UMR,

LMR, NLMR) and temperature rises It is noted that the

sinusoidal-law NLMR is used in this example The results are

calcu-lated for the power-law indices p = 0.2, 1 and 5,DT = 0, 20 and 40,

DC = 0%,1% and 2% The present solutions are compared with those

beam The present solutions based on HSBT without Poisson’s ratio

are in good agreement with those of[3]for all moisture and

tem-perature changes The effect of normal strain is clearly observed in

presents the effect of the power-law index p on the normalized

the fundamental frequency decreases with the increase of p and the moisture content rise makes the beams softer This

fundamental frequency with respect to the UTR It can be seen from this figure that the frequency of the FG beams with

content rise, and that the critical temperatures decrease with the increase ofDC

Table 10

Fundamental frequency (x^ ) of Si 3 N 4 /SUS304 beams under LTR and Fourier-law NLTR (L=h = 20, TD).

Temperature

distribution

Present a

a Q11¼ EðzÞ.

Table 9

Critical temperatures of Si 3 N 4 /SUS304 beams under Fourier- and sinusoidal-law NLTR (L=h = 30, TD).

Trang 9

Table 12

Fundamental frequency (x^ ) of Si 3 N 4 /SUS304 beams under linear moisture and temperature rise (L=h = 20, TD).

Present a

a Q11¼ EðzÞ.

Table 13

Fundamental frequency (x^ ) of Si 3 N 4 /SUS304 beams under sinusoidal moisture and temperature rise(L=h = 20, TD).

Present a

a

Table 11

Fundamental frequency (x^ ) of Si 3 N 4 /SUS304 beams under uniform moisture and temperature rise for different boundary conditions (L=h = 20, TD).

Present a

a

Q 11 ¼ EðzÞ.

Trang 10

4 Conclusions

Hygro-thermal vibration and stability analysis of FG beams is

presented It is based on a higher-order shear deformation

the-ory, which considers a higher-order distribution of transverse

shear stress and both in-plane and out-of-plane displacements

These beams are subjected to hygro-thermal loadings under

uniform, linear and nonlinear distributions through the beam

depth Lagrange’s equations are applied to derive the

character-istic dynamic equations and Ritz solution method is developed

to solve the problems for different boundary conditions The

proposed Ritz solution converges quickly and agrees well with

that from other studies The obtained numerical results showed

that:

The critical buckling temperatures and natural frequencies

derived from the quasi-3D theory, which includes normal

strain, is smaller than those from the HSBT, which neglects it

It implies that the effect of normal strain is important and needs

to be taken into account for the analysis of hygro-thermal

beha-viours of FG beams

The increase of the power-law index leads to the increase of

metal volume fraction, that makes the beams softer and

decrease of the critical temperature and natural frequency

The temperature dependency solutions give lower values than

the temperature independency ones, so the importance of

tem-perature dependency in the FG beams is confirmed

For a temperature rise, the critical temperature and

fundamen-tal frequency derived from nonlinear temperature rise are

lar-ger than those from uniform one

The critical temperature and fundamental frequency calculated

from Fourier-law nonlinear temperature distribution are

smal-ler than those from sinusoidal-law one

The thermal buckling and vibration responses of FG beams

decrease with the increase of moisture content

In conclusion, the proposed beam model and approach is found

to be simply and efficient for hygro-thermal buckling and vibration

of FG beams

Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant

No 107.02-2015.07

References [1] Ebrahimi F, Salari E Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment Acta Astronaut 2015;113:29–50

[2] Ebrahimi F, Salari E Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments Compos Struct 2015;128:363–80

[3] Ebrahimi F, Barati MR A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment Appl Phys A 2016;122(9):792

[4] Zidi M, Tounsi A, Houari MSA, Bedia EAA, Beg OA Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory Aerosp Sci Technol 2014;34:24–34

[5] Zenkour AM Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations Compos Struct 2010;93(1):234–8

[6] Zenkour AM, Allam MNM, Radwan AF Effects of transverse shear and normal strains on FG plates resting on elastic foundations under hygro-thermo-mechanical loading Int J Appl Mech 2014;06(05):1450063

[7] Fazzolari FA, Carrera E Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions J Therm Stresses 2014;37(12):1449–81

[8] Aydogdu M, Taskin V Free vibration analysis of functionally graded beams with simply supported edges Mater Des 2007;28:1651–6

[9] Pradhan K, Chakraverty S Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method Compos Part B: Eng 2013;51:175–84 [10] Chakraborty A, Gopalakrishnan S, Reddy JN A new beam finite element for the analysis of functionally graded materials Int J Mech Sci 2003;45(3):519–39 [11] Li X-F A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams J Sound Vib 2008;318(4–5):1210–29

[12] Sina S, Navazi H, Haddadpour H An analytical method for free vibration analysis of functionally graded beams Mater Des 2009;30(3):741–7 [13] Li S-R, Batra RC Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams Compos Struct 2013;95:5–9

[14] Nguyen T-K, Vo TP, Thai H-T Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory Compos Part B: Eng 2013;55:147–57

[15] Kapuria S, Bhattacharyya M, Kumar AN Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation Compos Struct 2008;82(3):390–402

Fig 4 Variation of normalized fundamental frequency of Si 3 N 4 /SUS304 beams with respect to the power-law index, moisture and temperature rise (L=h = 20, TD).

Định dạng
Số trang	11
Dung lượng	682,7 KB