• Hoang Lan Ton-That Ho Chi Minh City University of Architecture, Department of Civil Engineering, 196 Pasteur Street, District 3, Ho Chi Minh City, Viet Nam; Ho Chi Minh City University of Technology and Education, Department of Civil Engineering, 01 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam




Combined strain, four-node quadrilateral element, first-order shear deformation theory, thermal environment


Functionally graded materials are commonly used in a thermal environment to change the properties of constituent materials. They inherently withstand high temperature gradients due to a low thermal conductivity, core ductility, low thermal expansion coefficient, and many others. It is essential to thoroughly study mechanical responses of them and to develop new effective approaches for an accurate prediction of solutions. In this paper, a new four-node quadrilateral element based on a combined strain strategy and first-order shear deformation theory is presented to achieve the behaviour of functionally graded plate/shell structures in a thermal environment. The main notion of the combined strain strategy is based on the combination of the membrane strain and the shear strain related to tying points as well as bending strain with respect to a cell-based smoothed finite element method. Due to the finite element analysis, the first-order shear deformation theory (FSDT) is simple to implement and apply for structures, but the shear correction factors are used to achieve the accuracy of solutions. The author assumes that the temperature distribution is uniform throughout the structure. The rule of mixtures is also considered to describe the variation of material compositions across the thickness. Many desirable characteristics and the enforcement of this element are verified and proved through various numerical examples. Numerical solutions and a comparison with other available solutions suggest that the procedure based on this new combined strain element is accurate and efficient.


N. Wattanasakulpong, G. B. Prusty, D. W. Kelly. Free and forced vibration analysis using improved third-order shear deformation theory for functionally graded plates under high temperature loading. Journal of Sandwich Structures & Materials 15(5):583 – 606, 2013. doi:10.1177/1099636213495751.

N. Wattanasakulpong, B. Gangadhara Prusty, D. W. Kelly. Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. International Journal of Mechanical Sciences 53(9):734 – 743, 2011. doi:10.1016/j.ijmecsci.2011.06.005.

X.-L. Huang, H.-S. Shen. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures 41(9):2403 – 2427, 2004. doi:10.1016/j.ijsolstr.2003.11.012.

J. Yang, H.-S. Shen. Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Composites Part B: Engineering 34(2):103 – 115, 2003. doi:10.1016/S1359-8368(02)00083-5.

T. Q. Bui, T. V. Do, L. H. T. Ton, et al. On the high temperature mechanical behaviors analysis of heated functionally graded plates using FEM and a new third-order shear deformation plate theory. Composites Part B: Engineering 92:218 – 241, 2016. doi:10.1016/j.compositesb.2016.02.048.

H. L. Ton-That, H. Nguyen-Van, T. Chau-Dinh. An improved four-node element for analysis of composite plate/shell structures based on twice interpolation strategy. International Journal of Computational Methods 17(06):1950020, 2020. doi:10.1142/S0219876219500208.

H. L. Ton That, H. Nguyen-Van, T. Chau-Dinh. Nonlinear bending analysis of functionally graded plates using sq4t elements based on twice interpolation strategy. Journal of Applied and Computational Mechanics 6(1):125 – 136, 2020. doi:10.22055/jacm.2019.29270.1577.

L. T. That-Hoang, H. Nguyen-Van, T. Chau-Dinh, C. Huynh-Van. Enhancement to four-node quadrilateral plate elements by using cell-based smoothed strains and higher-order shear deformation theory for nonlinear analysis of composite structures. Journal of Sandwich Structures & Materials 22(7):2302 – 2329, 2020. doi:10.1177/1099636218797982.

H. Nguyen-Van, H. L. Ton-That, T. Chau-Dinh, N. D. Dao. Nonlinear static bending analysis of functionally graded plates using misq24 elements with drilling rotations. In H. Nguyen-Xuan, P. Phung-Van, T. Rabczuk (eds.), Proceedings of the International Conference on Advances in Computational Mechanics 2017, pp. 461 – 475. Springer Singapore, Singapore, 2018. doi:10.1007/978-981-10-7149-2_31.

H. L. Ton-That. Finite element analysis of functionally graded skew plates in thermal environment based on the new third-order shear deformation theory. Journal of Applied and Computational Mechanics 6(4):1044 – 1057, 2020. doi:10.22055/jacm.2019.31508.1881.

M. Bayat, I. Alarifi, A. Khalili, et al. Thermo-mechanical contact problems and elastic behaviour of single and double sides functionally graded brake disks with temperature-dependent material properties. Scientific Reports 9:15317, 2019. doi:10.1038/s41598-019-51450-z.

S. Trabelsi, A. Frikha, S. Zghal, D. Fakhreddine. A modified FSDT-based four nodes finite shell element for thermal buckling analysis of functionally graded plates and cylindrical shells. Engineering Structures 178:444 – 459, 2018. doi:10.1016/j.engstruct.2018.10.047.

A. Frikha, S. Zghal, F. Dammak. Dynamic analysis of functionally graded carbon nanotubes-reinforced plate and shell structures using a double directors finite shell element. Aerospace Science and Technology 78:438 – 451, 2018. doi:10.1016/j.ast.2018.04.048.

S. Zghal, A. Frikha, D. Fakhreddine. Mechanical buckling analysis of functionally graded power-based and carbon nanotubes-reinforced composite plates and curved panels. Composites Part B: Engineering 150:165 – 183, 2018. doi:10.1016/j.compositesb.2018.05.037.

F. Tornabene, N. Fantuzzi, M. Bacciocchi, E. Viola. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Composites Part B: Engineering 89:187 – 218, 2016. doi:10.1016/j.compositesb.2015.11.016.

F. Tornabene, N. Fantuzzi, E. Viola. Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory. Composite Structures 119:67 – 89, 2015. doi:10.1016/j.compstruct.2014.08.005.

F. Tornabene, A. Liverani, G. Caligiana. FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations. International Journal of Mechanical Sciences 53:446 – 470, 2011. doi:10.1016/j.ijmecsci.2011.03.007.

F. Tornabene. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Computer Methods in Applied Mechanics and Engineering 198:2911 – 2935, 2009. doi:10.1016/j.cma.2009.04.011.

M. Avcar. Free vibration of imperfect sigmoid and power law functionally graded beams. Steel and Composite Structures 30:603 – 615, 2019. doi:10.12989/scs.2019.30.6.603.

M. Avcar, W. K. M. Mohammed. Free vibration of functionally graded beams resting on Winkler-Pasternak foundation. Arabian Journal of Geosciences 11:232, 2018. doi:10.1007/s12517-018-3579-2.

Ö. Civalek, M. Acar. Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations. International Journal of Pressure Vessels and Piping 84:527 – 535, 2007. doi:10.1016/j.ijpvp.2007.07.001.

Ö. Civalek. Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method. Composites Part B: Engineering 111:45 – 59, 2017. doi:10.1016/j.compositesb.2016.11.030.

Ö. Civalek, M. Avcar. Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Engineering with Computers pp. 1 – 33, 2020. doi:10.1007/s00366-020-01168-8.

A. Menasria, A. Kaci, A. A. Bousahla, et al. A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions. Steel and Composite Structures 36:355 – 367, 2020. doi:10.12989/scs.2020.36.3.355.

M. Rahmani, K. Abdelhakim, A. Bousahla, et al. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory. Computers and Concrete 25:225 – 244, 2020. doi:10.12989/cac.2020.25.3.225.

A. Zine, A. A. Bousahla, F. Bourada, et al. Bending analysis of functionally graded porous plates via a refined shear deformation theory. Computers and Concrete 26:63 – 74, 2020. doi:10.12989/cac.2020.26.1.063.

H. L. Ton-That. Improvement on eight-node quadrilateral element (IQ8) using twice-interpolation strategy for linear elastic fracture mechanics. Engineering Solid Mechanics 8:323 – 336, 2020. doi:10.5267/j.esm.2020.3.005.

D. K. Jha, T. Kant, R. K. Singh. A critical review of recent research on functionally graded plates. Composite Structures 96:833 – 849, 2013. doi:10.1016/j.compstruct.2012.09.001.

K. Yeongbin, P.-S. Lee, K.-J. Bathe. The MITC4+ shell element and its performance. Computers & Structures 169:57 – 68, 2016. doi:10.1016/j.compstruc.2016.03.002.

H. Nguyen-Van, N. Mai-Duy, T. Tran-Cong. A simple and accurate four-node quadrilateral element using stabilized nodal integration for laminated plates. Computers, Materials & Continua 6:159 – 176, 2007. doi:10.3970/cmc.2007.006.159.

K.-J. Bathe, E. N. Dvorkin. A formulation of general shell elements-the use of mixed interpolation of tensorial components. International journal for numerical methods in engineering 22:697 – 722, 1986. doi:10.1002/nme.1620220312.

D. D. Fox, J. C. Simo. A drill rotation formulation for geometrically exact shells. Computer Methods in Applied Mechanics and Engineering 98:329 – 343, 1992. doi:10.1016/0045-7825(92)90002-2.

J. N. Reddy. Analysis of functionally graded plates. International Journal for numerical methods in engineering 47:663 – 684, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8.

A. M. Zenkour. Generalized shear deformation theory for bending analysis of functionally graded plates. Applied Mathematical Modelling 30:67 – 84, 2006. doi:10.1016/j.apm.2005.03.009.

X. Zhao, Y. Lee, K. Liew. Thermoelastic and vibration analysis of functionally graded cylindrical shells. International Journal of Mechanical Sciences 51:694 – 707, 2009. doi:10.1016/j.ijmecsci.2009.08.001.