Abstract
The formulation of an Euler–Bernoulli beam finite element with spatially varying uncertain properties is presented. Uncertainty is handled within a non-probabilistic framework resorting to a recently proposed interval field model able to quantify the dependency between adjacent values of an interval quantity that cannot differ as much as values that are further apart. Once the interval element stiffness matrix is defined, the set of linear interval equations governing the interval global displacements of the finite element model is derived by performing a standard assembly procedure. Then, the bounds of the interval displacements and bending moments are determined in approximate explicit form by applying a response surface approach in conjunction with the so-called improved interval analysis via extra unitary interval. For validation purposes, numerical results concerning both statically determinate and indeterminate beams with interval Young’s modulus are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Schmelzer, B., Oberguggenberger, M., Adam, C.: Efficiency of tuned mass dampers with uncertain parameters under stochastic excitation. Proc. Inst. Mech. Eng. O: J. Risk Reliab. 224(4), 297–308 (2010)
Adam, C., Heuer, R., Ziegler, F.: Reliable dynamic analysis of an uncertain compound bridge under traffic loads. Acta Mech. 223, 1567–1581 (2012)
Stefanou, G.: The stochastic finite element method: past, present and future. Comput. Methods Appl. Mech. Eng. 198, 1031–1051 (2009)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Ben-Haim, Y., Elishakoff, I.: Convex Models of Uncertainty in Applied Mechanics. Elsevier, Amsterdam (1990)
Qiu, Z., Elishakoff, I.: Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput. Methods Appl. Mech. Eng. 152, 361–372 (1998)
Köylüoğlu, H.U., Elishakoff, I.: A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties. Comput. Struct. 67, 91–98 (1998)
Muhanna, R.L., Mullen, R.L.: Uncertainty in mechanics problems-interval-based approach. J. Eng. Mech. ASCE 127, 557–566 (2001)
Degrauwe, D., Lombaert, G., De Roeck, G.: Improving interval analysis in finite element calculations by means of affine arithmetic. Comput. Struct. 88, 247–254 (2010)
Rama Rao, M.V., Mullen, R.L., Muhanna, R.L.: A new interval finite element formulation with the same accuracy in primary and derived variables. Int. J. Reliab. Saf. 5, 336–357 (2011)
Sofi, A., Romeo, E.: A novel interval finite element method based on the improved interval analysis. Comput. Methods Appl. Mech. Eng. 311, 671–697 (2016)
Moens, D., Vandepitte, D.: A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Eng. 194, 1527–1555 (2005)
Moens, D., Hanss, M.: Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: recent advances. Finite Elem. Anal. Des. 47, 4–16 (2011)
Muscolino, G., Sofi, A.: Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis. Probab. Eng. Mech. 28, 152–163 (2012)
Moens, D., De Munck, M., Desmet, W., Vandepitte, D.: Numerical dynamic analysis of uncertain mechanical structures based on interval fields. In: Belyaev, A.K., Langley, R.S. (eds.) IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, pp. 71–83. Springer, Dordrecht (2011)
Verhaeghe, W., Desmet, W., Vandepitte, D., Moens, D.: Interval fields to represent uncertainty on the output side of a static FE analysis. Comput. Methods Appl. Mech. Eng. 260, 50–62 (2013)
Vanmarcke, E.: Random Fields: Analysis and Synthesis. Revised and Expanded New Edition. World Scientific, Singapore (2010)
Imholz, M., Faes, M., Cerneels, J., Vandepitte, D., Moens, D.: On the comparison of two novel interval field formulations for the representation of spatial uncertainty. In: Freitag, S., Muhanna, R.L., Mullen, R.L. (eds.) Proceedings of the 7th International Workshop on Reliable Engineering Computing (REC2016), June 15–17, 2016. Ruhr University Bochum, Germany, pp. 367–378 (2016)
Faes, M., Cerneels, J., Vandepitte, D., Moens, D.: Identification and quantification of multivariate interval uncertainty in finite element models. Comput. Methods Appl. Mech. Eng. 315, 896–920 (2017)
Muscolino, G., Sofi, A., Zingales, M.: One-dimensional heterogeneous solids with uncertain elastic modulus in presence of long-range interactions: interval versus stochastic analysis. Comput. Struct. 122, 217–229 (2013)
Sofi, A., Muscolino, G.: Static analysis of Euler–Bernoulli beams with interval Young’s modulus. Comput. Struct. 156, 72–82 (2015)
Sofi, A., Muscolino, G., Elishakoff, I.: Static response bounds of Timoshenko beams with spatially varying interval uncertainties. Acta Mech. 226, 3737–3748 (2015)
Wu, D., Gao, W.: Hybrid uncertain static analysis with random and interval fields. Comput. Methods Appl. Mech. Eng. 315, 222–246 (2017)
Bucher, C.: Computational Analysis of Randomness in Structural Mechanics. Taylor & Francis, London (2009)
Dong, W., Shah, H.: Vertex method for computing functions of fuzzy variables. Fuzzy Sets Syst. 24, 65–78 (1987)
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Sofi, A.: Structural response variability under spatially dependent uncertainty: stochastic versus interval model. Probab. Eng. Mech. 42, 78–86 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sofi, A. Euler–Bernoulli interval finite element with spatially varying uncertain properties. Acta Mech 228, 3771–3787 (2017). https://doi.org/10.1007/s00707-017-1903-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-017-1903-7