Skip to main content
SpringerLink
Log in

Uncertainty quantification via λ-Neumann methodology of the stochastic bending problem of the Levinson–Bickford beam

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In stochastic structural mechanics, the treatment of uncertainty is often the main objective of numerical simulations to estimate the responses of a system or physical phenomenon. These predictions can form the basis for decision making, and therefore, a relevant issue to be studied is how reliable they are. Uncertainties, in general, are assessed under two aspects: from the available statistical information and considering the mathematical model that represents the problem numerically. The model identifies a set of relationships based on principles, conservation laws, and metrics of physical magnitude. For the stochastic bending problem of elastic and stationary beams, it is possible to associate randomness to the properties of the material, geometry, and loads on the structure. Thus, the estimates of the responses will be present in the field of stresses and deformations. In the present work, the variational formulation of the stochastic problem of the linear elliptical contour value with random coefficients is studied in light of the stochastic version of the Lax–Milgram lemma. The propagation and uncertainty quantification are investigated based on the recent numerical methodology of asymptotic complexity λ-Neumann–Monte Carlo. The results of the numerical simulation are obtained for the stochastic bending beam problem based on the Levinson–Bickford theory. The theory of high-order elastic bending has the advantage of meeting the condition of zero shears on the lateral surfaces of the beam. Numerical results are presented related to the accuracy, convergence, estimators of statistical moments, error estimate, and the processing time of the approximate solution for the high-order beam.

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
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    Book  Google Scholar 

  2. Babuska, I., Tempone, R., Zouraris, G.E.: Solving elliptic boundary value problems with uncertainty coefficients by the Finite Element method: the stochastic formulation. Comput. Methods Appl. Mech. Eng. 194(12–16), 1251–1294 (2005). https://doi.org/10.1016/j.cma.2004.02.026

    Article  MATH  Google Scholar 

  3. Sullivan, T.J.: Introduction to Uncertainty Quantification. Springer, Switzerland (2015)

    Book  Google Scholar 

  4. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley Interscience, New York (1973)

    Google Scholar 

  5. Rao, M.M., Swift, R.J.: Probability Theory with Applications, Mathematics and Its Applications, vol. 582. Springer, New York (2006)

    MATH  Google Scholar 

  6. Melchers, R.E., Beck, A.T.: Structural Reliability Analysis and Prediction, 3rd edn. Wiley, New York (2018)

    Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    Book  Google Scholar 

  8. Shinozuka, M., Astill, J.: Random eigenvalue problems in structural mechanics. Am. Inst. Aeronaut. Astronaut. J. 10(4), 456–462 (1972). https://doi.org/10.2514/3.50119

    Article  MATH  Google Scholar 

  9. Shinozuka, M., Leone, E.: A probabilistic model for spatial distribution of material properties. Eng. Fract. Mech. 8, 217–227 (1976). https://doi.org/10.1016/0013-7944(76)90087-4

    Article  Google Scholar 

  10. Adomian, G., Malakian, K.: Inversion of stochastic partial differential operators: the linear case. J. Math. Anal. Appl. 77, 309–327 (1980). https://doi.org/10.1016/0022-247X(80)90244-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Shinozuka, M.: Structural response variability. J. Eng. Mech., ASCE 113(6), 825–842 (1987). https://doi.org/10.1061/(ASCE)0733-9399(1987)113:6(825)

    Article  Google Scholar 

  12. Ávila, C.R.S.J., Beck, A.T., Rosa, E.D.: Solution of the stochastic beam bending problem by Galerkin method and the Askey-Wiener scheme. Latin Am. J. Solids Struct. 6(1), 51–72 (2009)

    Google Scholar 

  13. Yamazaki, F., Shinozuka, M., Dasgupta, G.: Neumann expansion for stochastic finite element analysis. J. Eng. Mech., ASCE 114(8), 1335–1354 (1988). https://doi.org/10.1061/(ASCE)0733-9399(1988)114:8(1335)

    Article  Google Scholar 

  14. Araújo, J.M., Awruch, A.M.: On stochastic finite elements for structural analysis. Comput. Struct. 52(3), 461–469 (1994). https://doi.org/10.1016/0045-7949(94)90231-3

    Article  MATH  Google Scholar 

  15. Chakraborty, S., Sarkar, S.K.: Analysis of a curved beam on uncertain elastic foundation. Finite Elem. Anal. Des. 36(1), 73–82 (2000). https://doi.org/10.1016/S0168-874X(00)00009-3

    Article  MATH  Google Scholar 

  16. Chakraborty, S., Bhattacharyya, B.: An efficient 3D stochastic finite element method. Int. J. Solids Struct. 39(9), 2465–2475 (2002). https://doi.org/10.1016/S0020-7683(02)00080-X

    Article  MATH  Google Scholar 

  17. Ávila, C.R.S.J., Beck, A.T.: A Fast convergence parameter for Monte Carlo–Neumann solution of linear stochastic systems. J. Risk Uncertain. Eng. Syst., Part B: Mech. Eng. 1, 1–9 (2015). https://doi.org/10.1115/1.4029741

    Article  Google Scholar 

  18. Ávila, C.R.S.J., Beck, A.T.: New method for efficient Monte Carlo–Neumann solution of linear stochastic systems. Probab. Eng. Mech. 40, 90–96 (2016). https://doi.org/10.1016/j.probengmech.2015.02.006

    Article  Google Scholar 

  19. Levinson, M., Stephens, N.G.: A Second order beam theory. J. Sound Vib. 67(3), 293–305 (1979). https://doi.org/10.1016/0022-460X(79)90537-6

    Article  MATH  Google Scholar 

  20. Bickford, W.B.: A consistent higher order beam theory. Dev Theoret. Appl. Mech. 11, 137–150 (1982)

    Google Scholar 

  21. Heyliger, P.R., Reddy, J.N.: A higher order beam finite element for bending and vibration problems. J. Sound Vib. 126(2), 309–326 (1988). https://doi.org/10.1016/0022-460X(88)90244-1

    Article  MATH  Google Scholar 

  22. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003). https://doi.org/10.1016/S0020-7683(02)00647-9

    Article  MATH  Google Scholar 

  23. Shi, G., Voyiadjis, G.: A sixth-order theory of shear deformable beams with variational consistent boundary conditions. J. Appl. Mech. 78, 1019 (2011). https://doi.org/10.1115/1.4002594

    Article  Google Scholar 

  24. Karttunen, A., von Hertzen, R.: Variational formulation of the static Levinson beam theory. Mech. Res. Commun. 66, 15–19 (2015). https://doi.org/10.1016/j.mechrescom.2015.03.006

    Article  Google Scholar 

  25. Reddy, J.N., Wang, C.M., Lim, G.T., Ng, K.H.: Bending solutions of Levinson beams and plates in terms of the classical theories. Int. J. Solids Struct. 328, 4701–4720 (2001). https://doi.org/10.1016/S0020-7683(00)00298-5

    Article  MATH  Google Scholar 

  26. Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D., Beskos, D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40(2), 385–400 (2003). https://doi.org/10.1016/S0020-7683(02)00522-X

    Article  MATH  Google Scholar 

  27. Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010). https://doi.org/10.1016/j.ijengsci.2010.09.020

    Article  MathSciNet  MATH  Google Scholar 

  28. Stefanou, G.: The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Eng. 198, 1031–1051 (2000). https://doi.org/10.1016/j.cma.2008.11.007

    Article  MATH  Google Scholar 

  29. Xiu, D., Shen, J.: Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys. 228, 266–281 (2009). https://doi.org/10.1016/j.jcp.2008.09.008

    Article  MathSciNet  MATH  Google Scholar 

  30. Grigoriu, M.: Stochastic Calculus Applications in Science and Engineering, p. 784. Springer, New York (2002)

    Book  Google Scholar 

  31. Gittelson, C.J.: Adaptive Galerkin methods for parametric and stochastic operator equations. Ph.D. Thesis, ETH Zurich, No. 19533 (2011)

Download references

Author information

Authors and Affiliations

  1. PPGEM, Federal University of Technology, Street Deputado Heitor Alencar Furtado, Curitiba, Paraná, 5000, Brazil

    Roberto M. F. Squarcio & Claudio R. Ávila da Silva Jr.

Authors
  1. Roberto M. F. Squarcio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudio R. Ávila da Silva Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto M. F. Squarcio.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Squarcio, R.M.F., da Silva, C.R.Á. Uncertainty quantification via λ-Neumann methodology of the stochastic bending problem of the Levinson–Bickford beam. Acta Mech 233, 3467–3480 (2022). https://doi.org/10.1007/s00707-022-03266-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03266-8

Search

Navigation