Skip to main content
SpringerLink
Log in

Spectral projection and linear regression approaches for stochastic flexural and vibration analysis of laminated composite beams

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

This paper presents a novel approach for assessing the uncertainty in vibration and static responses of laminated composite beams resulting from uncertainty in material properties and distributed loads. The proposed method utilizes surrogate models developed using polynomial chaos expansion (PCE) based on a relatively small sample size. These training samples are computed using a high-order shear beam model in which the governing equations are derived using Hamilton's principle, and solved by Ritz’s approach using a trigonometric series approximation. The proposed PCE model's coefficients are estimated using the spectral projection and linear regression techniques. The first four statistical moments and probability distributions of the mid-span displacement and the fundamental frequency of laminated composite beams are predicted. Global sensitivity analysis is also conducted to assess how material property variation and stochastic loads affect the beam's deflection and the fundamental frequency. The accuracy and efficiency of the proposed PCE models are compared with those from Monte Carlo simulation (MCS). A remarkable reduction in the computational cost of PCE models compared to MCS is observed without compromising the predictions' accuracy. As most real-world systems are subjected to multiple sources of uncertainty, this study provides a state-of-the-art method to quantify such uncertain parameters more efficiently and allow for a better reliability assessment in composite beam design.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Aydogdu, M.: Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Int. J. Mech. Sci. 47(11), 1740–1755 (2005)

    Google Scholar 

  2. Kant, T., Marur, S.R., Rao, G.S.: Analytical solution to the dynamic analysis of laminated beams using higher order refined theory. Compos. Struct. 40(1), 1–9 (1997)

    Google Scholar 

  3. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40(6), 1525–1546 (2003)

    Google Scholar 

  4. Khdeir, A.A., Reddy, J.N.: Free vibration of cross-ply laminated beams with arbitrary boundary conditions. Int. J. Eng. Sci. 32(12), 1971–1980 (1994)

    MathSciNet  Google Scholar 

  5. Nguyen, N.-D., Nguyen, T.-K., Thai, H.-T., Vo, T.P.: A Ritz type solution with exponential trial functions for laminated composite beams based on the modified couple stress theory. Compos. Struct. 191, 154–167 (2018)

    Google Scholar 

  6. Ebrahimi, F., Ahari, M.F.: Magnetostriction-assisted active control of the multi-layered nanoplates: effect of the porous functionally graded facesheets on the system’s behavior. Eng. Comput. 39(1), 269–283 (2023)

    Google Scholar 

  7. Dhore, N., Khalsa, L., Varghese, V.: Hygrothermoelastic analysis of non-simple nano-beam induced by ramp-type heating. Archive Appl. Mechan. 93, 3379–3393 (2023)

    Google Scholar 

  8. Vo, T.P., Lee, J.: Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory. Arch. Appl. Mech. 81(4), 419–435 (2011)

    Google Scholar 

  9. Zhang, Y., Zhang, L., Zhang, S.: Exact series solutions of composite beams with rotationally restrained boundary conditions: static analysis. Arch. Appl. Mech. 92(12), 3999–4015 (2022)

    Google Scholar 

  10. Ebrahimi, F., Hosseini, S.H.S.: Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study. Eng. Comput. 37(1), 395–408 (2021)

    Google Scholar 

  11. Ebrahimi, F., Karimiasl, M., Mahesh, V.: Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam. Eng. Comput. 37(2), 937–950 (2021)

    Google Scholar 

  12. Selvamani, R., Loganathan, R., Ebrahimi, F.: Nonlocal state-space strain gradient approach to the vibration of piezoelectric functionally graded nanobeam. Eng. Trans. 70(4), 319–338 (2022)

    Google Scholar 

  13. Selvamani, R., Rexy, J., Ebrahimi, F.: Vibration analysis of a magneto thermo electrical nano fiber reinforced with graphene oxide powder under refined beam model. J. Solid Mechan. 13(1), 80–94 (2021)

    Google Scholar 

  14. Ellali, M., Bouazza, M., Amara, K.: Thermal buckling of a sandwich beam attached with piezoelectric layers via the shear deformation theory. Arch. Appl. Mech. 92(3), 657–665 (2022)

    Google Scholar 

  15. Jun, L., Yuchen, B., Peng, H.: A dynamic stiffness method for analysis of thermal effect on vibration and buckling of a laminated composite beam. Arch. Appl. Mech. 87(8), 1295–1315 (2017)

    Google Scholar 

  16. Li, X., Yu, K., Zhao, R.: Thermal post-buckling and vibration analysis of a symmetric sandwich beam with clamped and simply supported boundary conditions. Arch. Appl. Mech. 88(4), 543–561 (2018)

    Google Scholar 

  17. Liu, L., Yang, W., Chai, Y., Zhai, G.: Vibration and thermal buckling analyses of multi-span composite lattice sandwich beams. Arch. Appl. Mech. 91(6), 2601–2616 (2021)

    Google Scholar 

  18. Nguyen, N.-D., Nguyen, T.-K., Nguyen, T.-N., Thai, H.-T.: New Ritz-solution shape functions for analysis of thermo-mechanical buckling and vibration of laminated composite beams. Compos. Struct. 184, 452–460 (2018)

    Google Scholar 

  19. Beheshti-Aval, S.B., Lezgy-Nazargah, M.: A coupled refined high-order global-local theory and finite element model for static electromechanical response of smart multilayered/sandwich beams. Arch. Appl. Mech. 82(12), 1709–1752 (2012)

    Google Scholar 

  20. Vo, T.P., Thai, H.-T., Inam, F.: Axial-flexural coupled vibration and buckling of composite beams using sinusoidal shear deformation theory. Arch. Appl. Mech. 83(4), 605–622 (2013)

    Google Scholar 

  21. Vo, T.P., Thai, H.-T.: Static behavior of composite beams using various refined shear deformation theories. Compos. Struct. 94(8), 2513–2522 (2012)

    Google Scholar 

  22. Nguyen, T.-K., Nguyen, N.-D., Vo, T., Thai, T.: Trigonometric-series solution for analysis of laminated composite beams. Compos. Struct. 160, 142–151 (2016)

    Google Scholar 

  23. Nguyen, N.-D., Nguyen, T.-K., Vo, T.P., Thai, H.-T.: Ritz-based analytical solutions for bending, buckling and vibration behavior of laminated composite beams. Int. J. Struct. Stab. Dyn. 18(11), 1850130 (2018)

    MathSciNet  Google Scholar 

  24. Khdeir, A.A., Reddy, J.N.: An exact solution for the bending of thin and thick cross-ply laminated beams. Compos. Struct. 37(2), 195–203 (1997)

    Google Scholar 

  25. Liu, X., Jiang, L., Xiang, P., Zhou, W., Lai, Z., Feng, Y.: Stochastic finite element method based on point estimate and Karhunen-Loéve expansion. Arch. Appl. Mech. 91(4), 1257–1271 (2021)

    Google Scholar 

  26. Stefanou, G.: The stochastic finite element method: past, present and future. Comput. Methods Appl. Mech. Eng. 198(9), 1031–1051 (2009)

    Google Scholar 

  27. Rahman, S., Rao, B.N.: A perturbation method for stochastic meshless analysis in elastostatics. Int. J. Numer. Meth. Eng. 50(8), 1969–1991 (2001)

    Google Scholar 

  28. Sahoo, R., Grover, N., Singh, B.N.: Random vibration response of composite–sandwich laminates. Arch. Appl. Mech. 91(9), 3755–3771 (2021)

    Google Scholar 

  29. Li, H.-S., Lü, Z.-Z., Yue, Z.-F.: Support vector machine for structural reliability analysis. Appl. Math. Mech. 27(10), 1295–1303 (2006)

    Google Scholar 

  30. Peng, X., Li, D., Wu, H., Liu, Z., Li, J., Jiang, S., Tan, J.: Uncertainty analysis of composite laminated plate with data-driven polynomial chaos expansion method under insufficient input data of uncertain parameters. Compos. Struct. 209, 625–633 (2019)

    Google Scholar 

  31. Nguyen, H.X., Duy Hien, T., Lee, J., Nguyen-Xuan, H.: Stochastic buckling behaviour of laminated composite structures with uncertain material properties. Aerosp. Sci. Technol. 66, 274–283 (2017)

    Google Scholar 

  32. Elishakoff, I., Archaud, E.: Modified Monte Carlo method for buckling analysis of nonlinear imperfect structures. Arch. Appl. Mech. 83(9), 1327–1339 (2013)

    Google Scholar 

  33. da Claudio, S.R.Á., Squarcio, R.M.F.: The Neumann-Monte Carlo methodology applied to the quantification of uncertainty in the problem stochastic bending of the Levinson-Bickford beam. Archive Appl. Mechan. 93(5), 2009–2024 (2023)

    Google Scholar 

  34. Naskar, S., Mukhopadhyay, T., Sriramula, S., Adhikari, S.: Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties. Compos. Struct. 160, 312–334 (2017)

    Google Scholar 

  35. Li, J., Tian, X., Han, Z., Narita, Y.: Stochastic thermal buckling analysis of laminated plates using perturbation technique. Compos. Struct. 139, 1–12 (2016)

    Google Scholar 

  36. Onkar, A.K., Upadhyay, C.S., Yadav, D.: Stochastic finite element buckling analysis of laminated plates with circular cutout under uniaxial compression. J. Appl. Mech. 74(4), 798–809 (2006)

    Google Scholar 

  37. Verma, V.K., Singh, B.N.: Thermal buckling of laminated composite plates with random geometric and material properties. Int. J. Struct. Stabil. Dyn. 09(02), 187–211 (2009)

    Google Scholar 

  38. Chandra, S., Sepahvand, K., Matsagar, V.A., Marburg, S.: Stochastic dynamic analysis of composite plate with random temperature increment. Compos. Struct. 226, 111159 (2019)

    Google Scholar 

  39. Chakraborty, S., Mandal, B., Chowdhury, R., Chakrabarti, A.: Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion. Compos. Struct. 135, 236–249 (2016)

    Google Scholar 

  40. Bhattacharyya, B.: On the use of sparse Bayesian learning-based polynomial chaos expansion for global reliability sensitivity analysis. J. Comput. Appl. Math. 420, 114819 (2023)

    MathSciNet  Google Scholar 

  41. Reddy, J.N.: Mechanics of laminated composite plates and shells: theory and analysis, 2nd edn. CRC Press, Boca Raton (2003)

    Google Scholar 

  42. Xiu, D., Karniadakis, G.E.: The wiener-askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    MathSciNet  Google Scholar 

  43. Dalbey, K., Eldred, M.S., Geraci, G., Jakeman, J.D., Maupin, K.A., Monschke, J.A., Seidl, D.T., Swiler, L.P., Tran, A., Menhorn, F., Zeng, X.: Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 Theory Manual. 2020, ; Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). p. Medium: ED; Size, p 128

  44. Sobol, I.M.: Sensitivity estimates for nonlinear mathematical models. Mathe. Modell. Comput. Exp. 1, 407 (1993)

    MathSciNet  Google Scholar 

  45. Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., Tarantola, S.: Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput. Phys. Commun. 181(2), 259–270 (2010)

    MathSciNet  Google Scholar 

  46. Saltelli, A., Ratto, M., Tarantola, S., Campolongo, F.: Update 1 of: sensitivity analysis for chemical models. Chem. Rev. 112(5), PR1–PR21 (2012)

    Google Scholar 

  47. Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc City, Ho Chi Minh City, Viet Nam

    Xuan-Bach Bui

  2. Faculty of International Education, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc City, Ho Chi Minh City, Viet Nam

    Phong T. T. Nguyen

  3. CIRTech Institute, HUTECH University, 475A Dien Bien Phu Street, Binh Thanh District, Ho Chi Minh City, Viet Nam

    Trung-Kien Nguyen

Authors
  1. Xuan-Bach Bui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Phong T. T. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Trung-Kien Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Trung-Kien Nguyen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bui, XB., Nguyen, P.T.T. & Nguyen, TK. Spectral projection and linear regression approaches for stochastic flexural and vibration analysis of laminated composite beams. Arch Appl Mech 94, 1021–1039 (2024). https://doi.org/10.1007/s00419-024-02565-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-024-02565-x

Keywords

Search

Navigation