Small amplitude vibrations of a functionally graded material beam under in-plane thermal loading in the prebuckling and postbuckling regimes is studied in this paper. The material properties of the FGM media are considered as function of both position and temperature. A three parameters elastic foundation including the linear and nonlinear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The solution is sought in two regimes. The first one, a static phase with large amplitude response, and the second one, a dynamic regime near the static one with small amplitude. In both regimes, nonlinear governing equations are discretized using the generalized differential quadrature (GDQ) method and solved iteratively via the Newton–Raphson method. It is concluded that depending on the type of boundary condition and loading type, free vibration of a beam under in-plane thermal loading may reach zero at a certain temperature which indicates the existence of bifurcation type of instability.

References

1.
Li
,
S. R.
,
Zhou
,
Y. H.
, and
Zheng
,
X.
,
2002
, “
Thermal Post-Buckling of a Heated Elastic Rod With Pinned-Fixed Ends
,”
J. Thermal Stresses
,
25
(
1
), pp.
45
56
.10.1080/014957302753305862
2.
Li
,
S. R.
,
Cheng
,
C. J.
, and
Zhou
,
Y. H.
,
2003
, “
Thermal Post-Buckling of an Elastic Beam Subjected to a Transversely Non-Uniform Temperature Rising
,”
Appl. Math. Mech.
,
24
(
5
), pp.
514
520
, English edition.10.1007/BF02435863
3.
Li
,
S. R.
,
Teng
,
Z. C.
, and
Zhou
,
Y. H.
,
2004
, “
Free Vibration of Heated, Euler-Bernoulli Beams With Thermal Post-Buckling Deformations
,”
J. Thermal Stresses
,
27
(
9
), pp.
843
856
.10.1080/01495730490486352
4.
Song
,
X.
, and
Li
,
S. R.
,
2007
, “
Thermal Buckling and Post-Buckling of Pinned-Fixed Euler-Bernoulli Beams on an Elastic Foundation
,”
Mech. Res. Commun.
,
34
(
2
), pp.
164
171
.10.1016/j.mechrescom.2006.06.006
5.
Li
,
S. R.
, and
Batra
,
R. C.
,
2007
, “
Thermal Buckling and Postbuckling of Euler-Bernoulli Beams Supported on Nonlinear Elastic Foundations
,”
AIAA J.
,
45
(
3
), pp.
712
720
.10.2514/1.24720
6.
Bhangale
,
R. K.
, and
Ganesan
,
N.
,
2006
, “
Thermoelastic Buckling and Vibration Behavior of a Functionally Graded Sandwich Beam With Constrained Viscoelastic Core
,”
J. Sound Vib.
,
295
(
1–2
), pp.
294
316
.10.1016/j.jsv.2006.01.026
7.
Ramkumar
,
K.
, and
Ganesan
,
N.
,
2008
, “
Finite-Element Buckling and Vibration Analysis of Functionally Graded Box Columns in Thermal Environments
,”
Proc. Inst. Mech. Eng. Part L
,
222
(
1
), pp.
53
64
.10.1243/14644207JMDA129
8.
Zhao
,
F. Q.
,
Wang
,
Z. M.
, and
Liu
,
H. Z.
,
2007
, “
Thermal Post-Buckling Analyses of Functionally Graded Material Rod
,”
Appl. Math. Mech.
,
28
(
1
), pp.
59
67
, English edition.10.1007/s10483-007-0107-z
9.
Ke
,
L. L.
,
Yang
,
J.
, and
Kitipornchai
,
S.
,
2009
, “
Postbuckling Analysis of Edge Cracked Functionally Graded Timoshenko Beams Under End Shortening
,”
Composite Struct.
,
90
(
2
), pp.
152
160
.10.1016/j.compstruct.2009.03.003
10.
Anandrao
,
K. S.
,
Gupta
,
R. K.
,
Ramchandran
,
P.
, and
Rao
,
G. V.
,
2010
, “
Thermal Post-Buckling Analysis of Uniform Slender Functionally Graded Material Beams
,”
Struct. Eng. Mech.
,
36
(
5
), pp.
545
560
.
11.
Ma
,
L. S.
, and
Lee
,
D. W.
,
2011
, “
A Further Discussion of Nonlinear Mechanical Behavior for FGM Beams Under In-Plane Thermal Loading
,”
Composite Struct.
,
93
(
2
), pp.
831
842
.10.1016/j.compstruct.2010.07.011
12.
Ma
,
L. S.
, and
Lee
,
D. W.
,
2012
, “
Exact Solutions for Nonlinear Static Responses of a Shear Deformable FGM Beam Under an In-Plane Thermal Loading
,”
Eur. J. Mech. A
,
31
(
1
), pp.
13
20
.10.1016/j.euromechsol.2011.06.016
13.
Li
,
S. R.
,
Su
,
H. D.
, and
Cheng
,
C. J.
,
2009
, “
Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment
,”
Appl. Math. Mech.
,
30
(
8
), pp.
969
982
, English edition.10.1007/s10483-009-0803-7
14.
Fu
,
Y.
,
Wang
,
J.
, and
Mao
,
Y.
,
2012
, “
Nonlinear Analysis of Buckling, Free Vibration and Dynamic Stability for the Piezoelectric Functionally Graded Beams in Thermal Environment
,”
Appl. Math. Model.
,
36
(
9
), pp.
4323
4340
.10.1016/j.apm.2011.11.059
15.
Pradhan
,
S. C.
, and
Murmu
,
T.
,
2009
, “
Thermo-Mechanical Vibration of FGM Sandwich Beam Under Variable Elastic Foundations Using Differential Quadrature
,”
J. Sound Vib.
,
321
(
1–2
), pp.
342
362
.10.1016/j.jsv.2008.09.018
16.
Xiang
,
H. J.
, and
Yang
,
J.
,
2008
, “
Free and Forced Vibration of a Laminated FGM Timoshenko Beam of Variable Thickness Under Heat Conduction
,”
Composites Part B
,
39
(
2
), pp.
292
303
.10.1016/j.compositesb.2007.01.005
17.
Komijani
,
M.
,
Kiani
,
Y.
, and
Eslami
,
M. R.
,
2013
, “
Non-Linear Thermoelectrical Stability Analysis of Functionally Graded Piezoelectric Material Beams
,”
J. Intell. Mater. Syst. Struct.
,
24
(
4
), pp.
399
410
.10.1177/1045389X12461079
18.
Komijani
,
M.
,
Kiani
,
Y.
,
Esfahani
,
S. E.
, and
Eslami
,
M. R.
,
2013
, “
Vibration of Thermo-Electrically Post-Buckled Rectangular Functionally Graded Piezoelectric Beams
,”
Composite Struct.
,
98
(
1
), pp.
143
152
.10.1016/j.compstruct.2012.10.047
19.
Sahraee
,
S.
, and
Saidi
,
A. R.
,
2009
, “
Free Vibration and Buckling Analysis of Functionally Graded Deep Beam-Columns on Two-Parameter Elastic Foundations Using the Differential Quadrature Method
,”
Proc. Inst. Mech. Eng. Part C
,
223
(
6
), pp.
1273
1284
.10.1243/09544062JMES1349
20.
Fallah
,
A.
, and
Aghdam
,
M. M.
,
2012
, “
Thermo-Mechanical Buckling and Nonlinear Free Vibration Analysis of Functionally Graded Beams on Nonlinear Elastic Foundation
,”
Composites Part B
,
43
(
3
), pp.
1523
1530
.10.1016/j.compositesb.2011.08.041
21.
Fallah
,
A.
, and
Aghdam
,
M. M.
,
2011
, “
Nonlinear Free Vibration and Post-Buckling Analysis of Functionally Graded Beams on Nonlinear Elastic Foundation
,”
Eur. J. Mech. A
,
30
(
4
), pp.
571
583
.10.1016/j.euromechsol.2011.01.005
22.
Hetenyi
,
M.
,
1948
,
Beams on Elastic Foundation
,
University of Michigan Press
,
Ann Arbor, MI
.
23.
Emam
,
S. A.
, and
Nayfeh
,
A. H.
,
2009
, “
Postbuckling and Free Vibrations of Composite Beams
,”
Composites Struct.
,
88
(
4
), pp.
636
642
.10.1016/j.compstruct.2008.06.006
24.
Kiani
,
Y.
,
Taheri
,
S.
, and
Eslami
,
M. R.
,
2011
, “
Thermal Buckling of Piezoelectric Functionally Graded Material Beams
,”
J. Thermal Stresses
,
34
(
8
), pp.
835
850
.10.1080/01495739.2011.586272
25.
Kiani
Y.
,
Rezaei
,
M.
,
Taheri
,
S.
, and
Eslami
,
M. R.
,
2011
, “
Thermo-Electrical Buckling of Piezoelectric Functionally Graded Material Timoshenko Beams
,”
Int. J. Mech. Mater. Design
,
7
(
3
), pp.
185
197
.10.1007/s10999-011-9158-2
26.
Reddy
,
J. N.
,
2003
,
Mechanics of Laminated Composite Plates and Shells, Theory and Application
, 2nd ed.,
CRC
,
Boca Raton, FL
.
27.
Hetnarski
,
R. B.
, and
Eslami
,
M. R.
,
2009
,
Thermal Stresses, Advanced Theory and Applications
, 1st ed.,
Springer
,
Berlin
.
28.
Vosoughi
,
A. R.
,
Malekzadeh
,
P.
,
Banan
,
Ma. R.
, and
Banan
,
Mo. R.
,
2012
, “
Thermal Buckling and Postbuckling of Laminated Composite Beams With Temperature-Dependent Properties
,”
Int. J. Non-Linear Mech.
,
47
(
3
), pp.
96
102
.10.1016/j.ijnonlinmec.2011.11.009
29.
Liew
,
K. M.
,
Yang
,
J.
, and
Kitipornchai
,
S.
,
2004
, “
Thermal Post-Buckling of Laminated Plates Comprising Functionally Graded Materials With Temperature-Dependent Properties
,”
ASME J. Appl. Mech.
,
71
(
6
), pp.
839
850
.10.1115/1.1795220
30.
Reddy
,
J. N.
, and
Chin
,
C. D.
,
1998
, “
Thermomechanical Analysis of Functionally Graded Cylinders and Plates
,”
J. Thermal Stresses
,
21
(
6
), pp.
593
626
.10.1080/01495739808956165
31.
Ying
,
J.
,
Lu
,
C. F.
, and
Chen
,
W. Q.
,
2008
, “
Two-Dimensional Elasticity Solutions for Functionally Graded Beams Resting on Elastic Foundations
,”
Composite Struct.
,
84
(
3
), pp.
209
219
.10.1016/j.compstruct.2007.07.004
32.
Shen
,
H. S.
,
2007
, “
Thermal Postbuckling of Shear Deformable FGM Cylindrical Shells With Temperature-Dependent Properties
,”
Mech. Adv. Mater. Struct.
,
14
(
6
), pp.
439
452
.10.1080/15376490701298942
33.
Bellman
,
R. E.
,
Kashef
,
B. G.
, and
Casti
,
J.
,
1972
, “
Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equations
,”
J. Comput. Phys.
,
10
(
1
), pp.
40
52
.10.1016/0021-9991(72)90089-7
34.
Quan
,
J. R.
, and
Chang
,
C. T.
,
1989
, “
New Insights in Solving Distributed System Equations by the Quadrature Methods
,”
Comput. Chem. Eng.
,
13
(
7
), pp.
779
788
.10.1016/0098-1354(89)85051-3
35.
Wu
,
T. Y.
, and
Liu
,
G. R.
,
1999
, “
A Differential Quadrature as a Numerical Method to Solve Differential Equations
,”
Comput. Mech.
,
24
(
3
), pp.
197
205
.10.1007/s004660050452
36.
Shu
C.
,
2000
,
Differential Quadrature and Its Application in Engineering
,
Springer
,
London
.
You do not currently have access to this content.