288
7 Mathematical Models of Functionally Graded Beams in Temperature Field
55. Janghorban, M., Zare, A.: Free vibration analysis of functionally graded carbon nanotubes
with variable thickness by differential quadrature method. Phys. E 43, 1602–1604 (2011)
56. Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A., Ahmadian, M.T.: A strain gradient functionally graded Euler-Bernoulli beam formulation. Int. J. Eng. Sci. 52, 65–76 (2012)
57. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent
functionally graded microbeams. Int. J. Eng. Sci. 50(1), 256–267 (2012)
58. Kiani, K.: Longitudinal and transverse instabilities of moving nanoscale beam-like structures
made of functionally graded materials. Compos. Struct. 107, 610–619 (2014)
59. Nateghi, A., Salamat-Talab, M., Rezapour, J., Daneshian, B.: Size dependent buckling analysis
of functionally graded micro beams based on modified couple stress theory. Appl. Math.
Model. 36, 4971–4987 (2012)
60. Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams.
Mech. Phys. Solids 59, 2382–2399 (2011)
61. Simsek, M., Yurtcu, H.H.: Analytical solutions for bending and buckling of functionally
graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97,
378–386 (2013)
62. Simsek, M.: Nonlocal effects in the free longitudinal vibration of axially functionally graded
tapered nanorods. Comput. Mater. Sci. 61, 257–265 (2012)
63. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T.: The modified couple
stress functionally graded Timoshenko beam formulation. Mater. Des. 32, 1435–1443 (2011)
64. Arbind, A., Reddy, J.N., Srinivasa, A.R.: Modified couple stress-based third-order theory
for nonlinear analysis of functionally graded beams. Lat. Am. J. Solids Struct. 11, 459–487
(2014)
65. Ansari, R., Sahmani, S., Rouhi, H.: Rayleigh-Ritz axial buckling analysis of single-walled
carbon nanotubes with different boundary conditions. Phys. Lett. A 375, 1255–1263 (2011)
66. Anjomshoa, A.: Application of Ritz functions in buckling analysis of embedded orthotropic
circular and elliptical micro/nano-plates based on nonlocal elasticity theory. Meccanica 48,
1337–1353 (2013)
67. Rahmani, O., Jandaghian, A.A.: Buckling analysis of functionally graded nanobeams based
on a nonlocal third-order shear deformation theory. Appl. Phys. A 119, 1019–1032 (2015)
68. Ansari, R., Gholami, R., Sahmani, S.: Size-dependent vibration of functionally graded curved
microbeams based on the modified strain gradient elasticity theory. Arch. Appl. Mech. 83,
1439–1449 (2013)
69. Liu, Y.P., Reddy, J.N.: A nonlocal curved beam model based on a modified coupled stress
theory. Int. J. Struct. Stab. Dyn. 11, 495–512 (2011)
70. Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: A novel size-dependent functionally graded
curved microbeam model based on the strain gradient elasticity theory. Compos. Struct. 106,
374–392 (2013)
71. Xiang, H.J., Yang, J.: Free and forced vibration of a laminated FGM Timoshenko beam of
variable thickness under heat conduction. Compos. Part B: Eng. 39(2), 292–303 (2008)
72. Dehrouyeh-Semnani, A.M., Dehrouyeh, M., Torabi-Kafshgari, M., Nikkhah-Bahrami, M.: A
damped sandwich beam model based on symmetric-deviatoric couple stress theory. Int. J.
Eng. Sci. 92, 83–94 (2015)
73. Chen, W.J., Li, X.P.: Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Arch. Appl. Mech. 83(3), 431–444
(2013)
74. Chen, W.J., Li, X.P., Ma, X.: A modified couple stress model for bending analysis of composite
laminated beams with first order shear deformation. Compos. Struct. 93(11), 2723–2737
(2011)
75. Mohammadimehr, M., Shahedi, S., Navi, B.R.: Nonlinear vibration analysis of FG-CNTRC
sandwich Timoshenko beam based on modified couple stress theory subjected to longitudinal
magnetic field using generalized differential quadrature method. Arch. Proc. Inst. Mech. Eng.
C J. Mech. Eng. Sci. 203–210, 1989–1996 (2016)
7 Mathematical Models of Functionally Graded Beams in Temperature Field
55. Janghorban, M., Zare, A.: Free vibration analysis of functionally graded carbon nanotubes
with variable thickness by differential quadrature method. Phys. E 43, 1602–1604 (2011)
56. Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A., Ahmadian, M.T.: A strain gradient functionally graded Euler-Bernoulli beam formulation. Int. J. Eng. Sci. 52, 65–76 (2012)
57. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent
functionally graded microbeams. Int. J. Eng. Sci. 50(1), 256–267 (2012)
58. Kiani, K.: Longitudinal and transverse instabilities of moving nanoscale beam-like structures
made of functionally graded materials. Compos. Struct. 107, 610–619 (2014)
59. Nateghi, A., Salamat-Talab, M., Rezapour, J., Daneshian, B.: Size dependent buckling analysis
of functionally graded micro beams based on modified couple stress theory. Appl. Math.
Model. 36, 4971–4987 (2012)
60. Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams.
Mech. Phys. Solids 59, 2382–2399 (2011)
61. Simsek, M., Yurtcu, H.H.: Analytical solutions for bending and buckling of functionally
graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97,
378–386 (2013)
62. Simsek, M.: Nonlocal effects in the free longitudinal vibration of axially functionally graded
tapered nanorods. Comput. Mater. Sci. 61, 257–265 (2012)
63. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T.: The modified couple
stress functionally graded Timoshenko beam formulation. Mater. Des. 32, 1435–1443 (2011)
64. Arbind, A., Reddy, J.N., Srinivasa, A.R.: Modified couple stress-based third-order theory
for nonlinear analysis of functionally graded beams. Lat. Am. J. Solids Struct. 11, 459–487
(2014)
65. Ansari, R., Sahmani, S., Rouhi, H.: Rayleigh-Ritz axial buckling analysis of single-walled
carbon nanotubes with different boundary conditions. Phys. Lett. A 375, 1255–1263 (2011)
66. Anjomshoa, A.: Application of Ritz functions in buckling analysis of embedded orthotropic
circular and elliptical micro/nano-plates based on nonlocal elasticity theory. Meccanica 48,
1337–1353 (2013)
67. Rahmani, O., Jandaghian, A.A.: Buckling analysis of functionally graded nanobeams based
on a nonlocal third-order shear deformation theory. Appl. Phys. A 119, 1019–1032 (2015)
68. Ansari, R., Gholami, R., Sahmani, S.: Size-dependent vibration of functionally graded curved
microbeams based on the modified strain gradient elasticity theory. Arch. Appl. Mech. 83,
1439–1449 (2013)
69. Liu, Y.P., Reddy, J.N.: A nonlocal curved beam model based on a modified coupled stress
theory. Int. J. Struct. Stab. Dyn. 11, 495–512 (2011)
70. Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L.: A novel size-dependent functionally graded
curved microbeam model based on the strain gradient elasticity theory. Compos. Struct. 106,
374–392 (2013)
71. Xiang, H.J., Yang, J.: Free and forced vibration of a laminated FGM Timoshenko beam of
variable thickness under heat conduction. Compos. Part B: Eng. 39(2), 292–303 (2008)
72. Dehrouyeh-Semnani, A.M., Dehrouyeh, M., Torabi-Kafshgari, M., Nikkhah-Bahrami, M.: A
damped sandwich beam model based on symmetric-deviatoric couple stress theory. Int. J.
Eng. Sci. 92, 83–94 (2015)
73. Chen, W.J., Li, X.P.: Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Arch. Appl. Mech. 83(3), 431–444
(2013)
74. Chen, W.J., Li, X.P., Ma, X.: A modified couple stress model for bending analysis of composite
laminated beams with first order shear deformation. Compos. Struct. 93(11), 2723–2737
(2011)
75. Mohammadimehr, M., Shahedi, S., Navi, B.R.: Nonlinear vibration analysis of FG-CNTRC
sandwich Timoshenko beam based on modified couple stress theory subjected to longitudinal
magnetic field using generalized differential quadrature method. Arch. Proc. Inst. Mech. Eng.
C J. Mech. Eng. Sci. 203–210, 1989–1996 (2016)