Large displacements of FGSW beams in thermal environment using a finite element formulation


FGSW beam
total Lagrange formulation
reduced integration
thermal environment
large deflection analysis

How to Cite

Hoai, B. T. T., Kien, N. D., Huong, T. T. T., & Anh, L. T. N. (2020). Large displacements of FGSW beams in thermal environment using a finite element formulation. Vietnam Journal of Mechanics, 42(1), 43–61.


The large displacements of functionally graded sandwich (FGSW) beams in thermal environment  are studied using a finite element formulation. The beams are composed of three layers, a homogeneous core and two functionally graded face sheets with volume fraction of constituents following a power gradation law. The material properties of the beams are considered to be temperature-dependent.  Based on Antman beam model and the total Lagrange formulation, a two-node nonlinear beam element taking the effect of temperature rise into account  is formulated and employed in the study. The element with explicit expressions for the internal force vector and tangent stiffness matrix is derived using linear interpolations and reduced integration technique to avoid the shear locking. Newton-Raphson based iterative algorithm is employed in combination with the arc-length control method to compute the large displacement response of a cantilever FGSW beam subjected to end forces.  The accuracy of the formulated element is confirmed through a comparison study. The effects of the material inhomogeneity, temperature rise and layer thickness ratio on the large deflection response of the beam are examined and highlighted.


D. K. Nguyen. A non-linear element for analysing elastic frame structures at large deflections. Vietnam Journal of Mechanics, 22, (1), (2000), pp. 19–28.

D. K. Nguyen and Q. Q. Do. Large deflection analysis of frames by elements containing higher-order terms. Vietnam Journal of Mechanics, 25, (4), (2003), pp. 243–254.

R. D.Wood and O. C. Zienkiewicz. Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells. Computers & Structures, 7, (6), (1977), pp. 725–735.

D. K. Nguyen. Postbuckling behavior of beams on two-parameter elastic foundation. International Journal of Structural Stability and Dynamics, 4, (01), (2004), pp. 21–43.

M. Koizumi. FGM activities in Japan. Composites Part B: Engineering, 28, (1-2), (1997), pp. 1–4.

Y. A. Kang and X. F. Li. Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. International Journal of Non-Linear Mechanics, 44, (6), (2009), pp. 696–703.

Y. A. Kang and X. F. Li. Large deflections of a non-linear cantilever functionally graded beam. Journal of Reinforced Plastics and Composites, 29, (12), (2010), pp. 1761–1774.

T. Kocatürk, M. Şimşek, and Ş. D. Akbaş. Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material. Science and Engineering of Composite Materials, 18, (1-2), (2011), pp. 21–34.

C. A. Almeida, J. C. R. Albino, I. F. M. Menezes, and G. H. Paulino. Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation. Mechanics Research Communications, 38, (8), (2011), pp. 553–559.

S. V. Levyakov. Elastica solution for thermal bending of a functionally graded beam. Acta Mechanica, 224, (8), (2013), pp. 1731–1740.

S. V. Levyakov. Thermal elastica of shear-deformable beam fabricated of functionally graded material. Acta Mechanica, 226, (3), (2015), pp. 723–733.

D. G. Zhang. Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Composite Structures, 100, (2013), pp. 121–126.

D. K. Nguyen. Large displacement response of tapered cantilever beams made of axially functionally graded material. Composites Part B: Engineering, 55, (2013), pp. 298–305.

D. K. Nguyen. Large displacement behaviour of tapered cantilever Euler–Bernoulli beams made of functionally graded material. Applied Mathematics and Computation, 237, (2014), pp. 340–355.

D. K. Nguyen, T. H. Trinh, and T. H. Le. A co-rotational beam element for geometrically nonlinear analysis of plane frames. Vietnam Journal of Mechanics, 35, (1), (2013), pp. 51–65.

D. K. Nguyen, B. S. Gan, and T. H. Trinh. Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material. Structural Engineering and Mechanics, 49, (6), (2014), pp. 727–743.

T. H. Trinh, D. K. Nguyen, B. S. Gan, and S. Alexandrov. Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation. Structural Engineering and Mechanics, 58, (3), (2016), pp. 515–532.

D. K. Nguyen, K. V. Nguyen, B. S. Gan, S. Alexandrov, et al. Nonlinear bending of elastoplastic functionally graded ceramic-metal beams subjected to nonuniform distributed loads. Applied Mathematics and Computation, 333, (2018), pp. 443–459.

P. K. Masjedi, A. Maheri, and P. M. Weaver. Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation. Applied Mathematical Modelling, 76, (2019), pp. 938–957.

Y. Watanabe, Y. Inaguma, H. Sato, and E. Miura-Fujiwara. A novel fabrication method for functionally graded materials under centrifugal force: The centrifugal mixed-powder method. Materials, 2, (4), (2009), pp. 2510–2525.

T. P. Vo, H. T. Thai, T. K. Nguyen, A. Maheri, and J. Lee. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Engineering Structures, 64, (2014), pp. 12–22.

T. P. Vo, H. T. Thai, T. K. Nguyen, F. Inam, and J. Lee. A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Composite Structures, 119, (2015), pp. 1–12.

D. K. Nguyen and T. T. Tran. A corotational formulation for large displacement analysis of functionally graded sandwich beam and frame structures. Mathematical Problems in Engineering, 2016, (2016).

Y. S. Touloukian. Thermophysical properties of high temperature solid materials. Macmillan, New York, USA, (1967).

S. S. Antman. Nonlinear problems of elasticity. Springer-Verlag, New York, (1995).

C. Pacoste and A. Eriksson. Beam elements in instability problems. Computer Methods in Applied Mechanics and Engineering, 144, (1-2), (1997), pp. 163–197.

E. N. Lages, G. H. Paulino, I. F. M. Menezes, and R. R. Silva. Nonlinear finite element analysis using an object-oriented philosophy–application to beam elements and to the cosserat continuum. Engineering with Computers, 15, (1), (1999), pp. 73–89.

A. Mahi, E. A. A. Bedia, A. Tounsi, and I. Mechab. An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions. Composite Structures, 92, (8), (2010), pp. 1877–1887.

R. D. Cook, D. S. Malkus, and M. E. Plesha. Concepts and applications of finite element analysis. JohnWiley & Sons, New York, USA, 3rd edition, (1989).

M. A. Crisfield. Non-linear finite element analysis of solids and structures, Vol. 1: Essentials. John Wiley & Sons, Chichester, (1991).

H. S. Shen and Z. X. Wang. Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments. International Journal of Mechanical Sciences, 81, (2014), pp. 195–206.


Download data is not yet available.


Metrics Loading ...