Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus
Received: 28 October 2022 | Revised: 3 December 2022 | Accepted: 4 December 2022 | Online: 5 February 2023
Corresponding author: Nguyen Van Thuan
Abstract
This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.
Keywords:
SFEM, free vibration, continuous beam, random fieldDownloads
References
P. C. Nguyen, "Nonlinear Inelastic Earthquake Analysis of 2D Steel Frames," Engineering, Technology & Applied Science Research, vol. 10, no. 6, pp. 6393–6398, Dec. 2020. DOI: https://doi.org/10.48084/etasr.3855
P. C. Nguyen, D. D. Pham, T. T. Tran, and T. Nghia-Nguyen, "Modified Numerical Modeling of Axially Loaded Concrete-Filled Steel Circular-Tube Columns," Engineering, Technology & Applied Science Research, vol. 11, no. 3, pp. 7094–7099, Jun. 2021. DOI: https://doi.org/10.48084/etasr.4157
P. C. Nguyen, B. Le-Van, and S. D. T. V. Thanh, "Nonlinear Inelastic Analysis of 2D Steel Frames : An Improvement of the Plastic Hinge Method," Engineering, Technology & Applied Science Research, vol. 10, no. 4, pp. 5974–5978, Aug. 2020. DOI: https://doi.org/10.48084/etasr.3600
D. X. Quy and N. V. Thi, "Static analysis of beam resting on elastic foundation by anisotropic beam-foundation element taking into account non-contact between beam and foundation," Transport and Communications Science Journal, vol. 72, no. 5, pp. 552–564, Jun. 2021.
J. G. R. Iniguez, M. L. Daza-Torres, A. P. Gonzalez, and A. Cros, "Natural frequency of a heavy flexible plate: power law evolution as a function of length," Latin American Journal of Solids and Structures, vol. 18, no. 5, Jun. 2021, Art. no. e377. DOI: https://doi.org/10.1590/1679-78256479
A. W. de Q. R. Reis, R. B. Burgos, and M. F. F. de Oliveira, "Nonlinear Dynamic Analysis of Plates Subjected to Explosive Loads," Latin American Journal of Solids and Structures, vol. 19, no. 1, Jan. 2022, Art. no. e422. DOI: https://doi.org/10.1590/1679-78256706
V. T. A. Ninh, "Fundamental frequencies of bidirectional functionally graded sandwich beams partially supported by foundation using different beam theories," Transport and Communications Science Journal, vol. 72, no. 4, pp. 452–467, 2021. DOI: https://doi.org/10.47869/tcsj.72.4.5
P. M. Phuc, "Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness," Transport and Communications Science Journal, vol. 71, no. 7, pp. 853–867, 2020. DOI: https://doi.org/10.25073/tcsj.71.7.10
Q.-H. Pham, V. K. Tran, T. T. Tran, P.-C. Nguyen, and P. Malekzadeh, "Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory," Alexandria Engineering Journal, vol. 61, no. 12, pp. 10025–10044, Dec. 2022. DOI: https://doi.org/10.1016/j.aej.2022.03.007
Q.-H. Pham, T. Thanh Tran, V. Ke Tran, P.-C. Nguyen, and T. Nguyen-Thoi, "Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element," Alexandria Engineering Journal, vol. 61, no. 3, pp. 1788–1802, Mar. 2022. DOI: https://doi.org/10.1016/j.aej.2021.06.082
Q.-H. Pham, V. K. Tran, T. T. Tran, T. Nguyen-Thoi, P.-C. Nguyen, and V. D. Pham, "A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation," Case Studies in Thermal Engineering, vol. 26, Aug. 2021, Art. no. 101170. DOI: https://doi.org/10.1016/j.csite.2021.101170
T. T. Tran, P.-C. Nguyen, and Q.-H. Pham, "Vibration analysis of FGM plates in thermal environment resting on elastic foundation using ES-MITC3 element and prediction of ANN," Case Studies in Thermal Engineering, vol. 24, Apr. 2021, Art. no. 100852. DOI: https://doi.org/10.1016/j.csite.2021.100852
Q.-H. Pham, P.-C. Nguyen, V.-K. Tran, and T. Nguyen-Thoi, "Finite element analysis for functionally graded porous nano-plates resting on elastic foundation," Steel and Composite Structures, vol. 41, no. 2, pp. 149–166, 2021.
P.-C. Nguyen, Q. H. Pham, T. T. Tran, and T. Nguyen-Thoi, "Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements," Ain Shams Engineering Journal, vol. 13, no. 3, May 2022, Art. no. 101615. DOI: https://doi.org/10.1016/j.asej.2021.10.010
Q.-H. Pham, P.-C. Nguyen, and T. Thanh Tran, "Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis," Composite Structures, vol. 290, p. 115565, Jun. 2022. DOI: https://doi.org/10.1016/j.compstruct.2022.115565
M. V. Shitikova and A. I. Krusser, "Force driven vibrations of nonlinear plates on a viscoelastic winkler foundation under the harmonic moving load," International Journal for Computational Civil and Structural Engineering, vol. 17, no. 4, pp. 161–180, Dec. 2021. DOI: https://doi.org/10.22337/2587-9618-2021-17-4-161-180
M. Barabash and P. Bogdan, "Modeling of the subway dynamic influance on the ground structure," International Journal for Computational Civil and Structural Engineering, vol. 17, no. 3, pp. 14–23, Sep. 2021. DOI: https://doi.org/10.22337/2587-9618-2021-17-3-14-23
Y. Karmi, Y. Khadri, S. Tekili, A. Daouadji, and E. M. Daya, "Dynamic Analysis of Composite Sandwich Beams with a Frequency-Dependent Viscoelastic Core under the Action of a Moving Load," Mechanics of Composite Materials, vol. 56, no. 6, pp. 755–768, Jan. 2021. DOI: https://doi.org/10.1007/s11029-021-09921-w
W.-R. Chen, "Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section," Latin American Journal of Solids and Structures, vol. 18, no. 7, Oct. 2021, Art. no. e397. DOI: https://doi.org/10.1590/1679-78256434
T. D. Hien and P.-C. Nguyen, "Evaluation of Response Variability of Euler–Bernoulli Beam Resting on Foundation Due to Randomness in Elastic Modulus," in International Conference on Sustainable Civil Engineering and Architecture, Ho Chi Minh, Vietnam, Oct. 2019, pp. 1087–1092. DOI: https://doi.org/10.1007/978-981-15-5144-4_105
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, no. 7, Sep. 2020, Art. no. e306. DOI: https://doi.org/10.1590/1679-78256066
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. DOI: https://doi.org/10.47869/tcsj.72.2.7
T.-T. Tran and D. Kim, "Uncertainty quantification for nonlinear seismic analysis of cabinet facility in nuclear power plants," Nuclear Engineering and Design, vol. 355, Dec. 2019, Art. no. 110309. DOI: https://doi.org/10.1016/j.nucengdes.2019.110309
T.-T. Tran, K. Salman, S.-R. Han, and D. Kim, "Probabilistic Models for Uncertainty Quantification of Soil Properties on Site Response Analysis," ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, vol. 6, no. 3, Sep. 2020, Art. no. 04020030. DOI: https://doi.org/10.1061/AJRUA6.0001079
W. K. Liu, T. Belytschko, and A. Mani, "Random field finite elements," International Journal for Numerical Methods in Engineering, vol. 23, no. 10, pp. 1831–1845, 1986. DOI: https://doi.org/10.1002/nme.1620231004
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. DOI: https://doi.org/10.1007/s11029-020-09897-z
R. G. Ghanem and P. D. Spanos, "Spectral Stochastic Finite‐Element Formulation for Reliability Analysis," Journal of Engineering Mechanics, vol. 117, no. 10, pp. 2351–2372, Oct. 1991. DOI: https://doi.org/10.1061/(ASCE)0733-9399(1991)117:10(2351)
O. H. Galal, W. El-Tahan, M. A. El-Tawil, and A. A. Mahmoud, "Spectral SFEM analysis of structures with stochastic parameters under stochastic excitation," Structural engineering and mechanics : An international journal, vol. 28, no. 3, pp. 281–294, 2008. DOI: https://doi.org/10.12989/sem.2008.28.3.281
T. D. Hien, B. T. Thanh, N. N. Long, N. Van Thuan, and D. T. Hang, "Investigation Into The Response Variability of A Higher-Order Beam Resting on A Foundation Using A Stochastic Finite Element Method," in 5th International Conference on Geotechnics, Civil Engineering Works and Structures, 2020, pp. 117–122. DOI: https://doi.org/10.1007/978-981-15-0802-8_15
C.-K. Choi and H.-C. Noh, "Weighted Integral SFEM Including Higher Order Terms," Journal of Engineering Mechanics, vol. 126, no. 8, pp. 859–866, Aug. 2000. DOI: https://doi.org/10.1061/(ASCE)0733-9399(2000)126:8(859)
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, vol. 9, no. 1, pp. 29–37, Apr. 2022.
S. Mohammadzadeh, M. Esmaeili, and M. Mehrali, "Dynamic response of double beam rested on stochastic foundation under harmonic moving load," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 38, no. 6, pp. 572–592, 2014. DOI: https://doi.org/10.1002/nag.2227
T.-P. Chang, "Dynamic finite element analysis of a beam on random foundation," Computers & Structures, vol. 48, no. 4, pp. 583–589, Aug. 1993. DOI: https://doi.org/10.1016/0045-7949(93)90251-8
D. T. Hang, X. T. Nguyen, and D. N. Tien, "Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements with Discretization Random Field by the Point Method," Engineering, Technology & Applied Science Research, vol. 12, no. 2, pp. 8458–8462, Apr. 2022. DOI: https://doi.org/10.48084/etasr.4819
J. Reddy, An Introduction to the Finite Element Method, 3rd ed. New York, NY, USA: McGraw-Hill Education, 2005.
M. Shinozuka and G. Deodatis, "Simulation of Stochastic Processes by Spectral Representation," Applied Mechanics Reviews, vol. 44, no. 4, pp. 191–204, Apr. 1991. DOI: https://doi.org/10.1115/1.3119501
P.-C. Nguyen, T. N. Van, and H. T. Duy, "Stochastic Free Vibration Analysis of Beam on Elastic Foundation with the Random Field of Young’s Modulus Using Finite Element Method and Monte Carlo Simulation," in 6th International Conference on Geotechnics, Civil Engineering and Structures, Ha Long, Vietnam, Oct. 2021, pp. 499–506. DOI: https://doi.org/10.1007/978-981-16-7160-9_50
Downloads
How to Cite
License
Copyright (c) 2022 Hien Ta Duy, Nguyen Dang Diem, Giap Van Tan, Vu Van Hiep, Nguyen Van Thuan
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
