Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus


  • Thuan Nguyen-Van Nha Trang University, Vietnam
  • Thanh Bui-Tien University of Transport and Communications, Vietnam
Volume: 13 | Issue: 4 | Pages: 11253-11257 | August 2023 |


This paper presents a stochastic finite element method to calculate the variation of eigenvalues and eigenvectors of functionally graded beams. The modulus of functionally graded material is assumed to have spatial uncertainty as a one-dimensional random field. The formulation of the stochastic finite element method for the functionally graded beam due to the randomness of the elastic modulus of the beam is given using the first-order perturbation approach. This approach was validated with Monte Carlo simulation in previous studies using spectral representation to generate the random field. The statistics of the beam responses were investigated using the first-order perturbation method for different fluctuations of the elastic modulus. A comparison of the results of the stochastic finite element method with the first-order perturbation approach and the Monte Carlo simulation showed a minimal difference.


perturbation method, FGM beam, FEM, eigenvector


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L. T. Ha and N. T. K. Kue, "Free vibration of functionally graded porous nano beams," Transport and Communications Science Journal, vol. 70, no. 2, pp. 95–103, 2019.

B. Uymaz, "Buckling Characteristics of FGM Plates Subjected to Linearly Varying In-Plane Loads," Mechanics of Composite Materials, vol. 57, no. 1, pp. 69–80, Mar. 2021.

J. K. Lee and B. K. Lee, "Coupled Flexural-Torsional Free Vibration of an Axially Functionally Graded Circular Curved Beam," Mechanics of Composite Materials, vol. 57, no. 6, pp. 833–846, Jan. 2022.

H. D. Ta and P.-C. Nguyen, "Perturbation based stochastic isogeometric analysis for bending of functionally graded plates with the randomness of elastic modulus," Latin American Journal of Solids and Structures, vol. 17, Sep. 2020, Art. no. e306.

S. Kumar, D. Prakash, M. Muthtamilselvan, B. Abdalla, and Q. M. Al-Mdallal, "Flexural Waves in an Electrically Short Hard Dielectric and Functionally Graded Piezoelectric Layer," Mechanics of Solids, vol. 57, no. 3, pp. 671–681, Jun. 2022.

D. T. Thuy, L. N. Ngoc, D. N. Tien, and H. V. Thanh, "An Analytical Solution for the Dynamics of a Functionally Graded Plate resting on Viscoelastic Foundation," Engineering, Technology & Applied Science Research, vol. 13, no. 1, pp. 9926–9931, Feb. 2023.

S. Sergey and S. Anastasia, "Structural Reliability Analysis Using Evidence Theory and Fuzzy Probability Distributions," Magazine of Civil Engineering, vol. 107, no. 7, 2021.

Yu. G. Matvienko, "A Simplified Probabilistic Approach to Estimating the Safety Factors of Crack Resistance," Journal of Machinery Manufacture and Reliability, vol. 50, no. 3, pp. 200–207, May 2021.

M. Koizumi, "FGM activities in Japan," Composites Part B: Engineering, vol. 28, no. 1, pp. 1–4, Jan. 1997.

R. M. Mahamood, E. T. Akinlabi, M. Shukla, and S. L. Pityana, "Functionally Graded Material: An overview," in Proceedings of the World Congress on Engineering 2012, London, UK, Jul. 2012.

V. Bhavar, P. Kattire, S. Thakare, S. Patil, and R. K. P. Singh, "A Review on Functionally Gradient Materials (FGMs) and Their Applications," IOP Conference Series: Materials Science and Engineering, vol. 229, no. 1, Jun. 2017, Art. no. 012021.

A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud, "Free vibration characteristics of a functionally graded beam by finite element method," Applied Mathematical Modelling, vol. 35, no. 1, pp. 412–425, Jan. 2011.

A. Chakraborty, S. Gopalakrishnan, and J. N. Reddy, "A new beam finite element for the analysis of functionally graded materials," International Journal of Mechanical Sciences, vol. 45, no. 3, pp. 519–539, Mar. 2003.

L.-L. Ke, J. Yang, and S. Kitipornchai, "An analytical study on the nonlinear vibration of functionally graded beams," Meccanica, vol. 45, no. 6, pp. 743–752, Dec. 2010.

H. T. Duy, N. D. Diem, G. V. Tan, V. V. Hiep, and N. V. Thuan, "Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus," Engineering, Technology & Applied Science Research, vol. 13, no. 1, pp. 9985–9990, Feb. 2023.

M. S. M. Noori and R. M. Abbas, "Reliability Analysis of an Uncertain Single Degree of Freedom System Under Random Excitation," Engineering, Technology & Applied Science Research, vol. 12, no. 5, pp. 9252–9257, Oct. 2022.

H. Naderpour, M. Maddahi, and M. Gerami, "Effect of uncertainties of shear wall on reliability of rehabilitated structure," Magazine of Civil Engineering, vol. 117, no. 1, pp. 11701–11701, 2023.

N. T. Nguyen, H. D. Ta, T. N. Van, and T. N. Dao, "Stochastic finite element analysis of the free vibration of non-uniform beams with uncertain material," Journal of Materials and Engineering Structures «JMES», vol. 9, no. 1, pp. 29–37, 2022.

T. D. Hien, N. D. Hung, N. T. Kien, and H. C. Noh, "The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters," Applied Mathematical Modelling, vol. 67, pp. 676–687, Mar. 2019.

H. C. Noh, "Effect of multiple uncertain material properties on the response variability of in-plane and plate structures," Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 19, pp. 2697–2718, Apr. 2006.

L. L. Graham and G. Deodatis, "Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties," Probabilistic Engineering Mechanics, vol. 16, no. 1, pp. 11–29, Jan. 2001.

T. Takada, "Weighted integral method in stochastic finite element analysis," Probabilistic Engineering Mechanics, vol. 5, no. 3, pp. 146–156, Sep. 1990.

C. Eckert, M. Beer, and P. D. Spanos, "A polynomial chaos method for arbitrary random inputs using B-splines," Probabilistic Engineering Mechanics, vol. 60, Apr. 2020, Art. no. 103051.

N. V. Thuan and T. D. Hien, "Variablitity in frequencies of vehicle vibration anlysis with muiltiple random variables," Transport and Communications Science Journal, vol. 72, no. 2, pp. 215–226, 2021.

M. Mohammadi, M. Eghtesad, and H. Mohammadi, "Stochastic analysis of dynamic characteristics and pull-in instability of FGM micro-switches with uncertain parameters in thermal environments," International Journal of Mechanics and Materials in Design, vol. 14, no. 3, pp. 417–442, Sep. 2018.

H.-C. Noh, "A formulation for stochastic finite element analysis of plate structures with uncertain Poisson’s ratio," Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 45, pp. 4857–4873, Nov. 2004.

T. D. Hien, "A static analysis of nonuniform column by stochastic finite element method using weighted integration approach," Transport and Communications Science Journal, vol. 71, no. 4, pp. 359–367, 2020.

K. Li, W. Gao, D. Wu, C. Song, and T. Chen, "Spectral stochastic isogeometric analysis of linear elasticity," Computer Methods in Applied Mechanics and Engineering, vol. 332, pp. 157–190, Apr. 2018.

S. N. Thakur, S. Chakraborty, and C. Ray, "Reliability analysis of laminated composite shells by response surface method based on HSDT," Structural Engineering and Mechanics, An Int'l Journal, vol. 72, no. 2, pp. 203–216, 2019.

N. V. Thuan and H. C. Noh, "Variability of Mid-plane Symmetric Functionally Graded Material Beams in Free Vibration," Journal of the Computational Structural Engineering Institute of Korea, vol. 31, no. 3, pp. 127–132, 2018.

N. V. Thuan and T. D. Hien, "Stochastic Perturbation-Based Finite Element for Free Vibration of Functionally Graded Beams with an Uncertain Elastic Modulus," Mechanics of Composite Materials, vol. 56, no. 4, pp. 485–496, Sep. 2020.

M. Shinozuka and G. Deodatis, "Simulation of Stochastic Processes by Spectral Representation," Applied Mechanics Reviews, vol. 44, no. 4, pp. 191–204, Apr. 1991.


How to Cite

T. Nguyen-Van and T. Bui-Tien, “Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus”, Eng. Technol. Appl. Sci. Res., vol. 13, no. 4, pp. 11253–11257, Aug. 2023.


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