Keywords: homogenization, composite beams, stochastic finite element method, finite element.

The main idea is to present the engineering aspects of application of the homogenization method in the Stochastic Finite Element analysis[1] of 1D periodic linear elastic beams with random parameters. As it is demonstrated below, the effective Young modulus of such a structure obtained in the closed form equation in deterministic approach can be applied in stochastic finite element analysis thanks to the second order perturbation second central probabilistic moment methodology. Since there is no need to solve any cell problems (does not matter deterministic nor stochastic), the described method can be relatively easily implemented in any FEM- based engineering CAD/CAM software and, on the other hand, may be linked with the Monte-Carlo simulation, for instance. Some other stochastic theoretical and computational approaches - weighted residuals method or stochastic spectral finite element techniques may be considered in further probabilistic implementations of the approach, too.

The entire procedure is illustrated with the computational experiments on the two-component beam and bar composite structures with Young moduli of the constituents defined deterministically (piecewise constant in structural components lengths in both cases) and probabilistically (as a random variable). Thanks to these experiments the computational algorithm for application of the homogenization method in a conjunction with the SFEM analysis is proposed by some necessary modifications with comparison to the theoretical approach. Considering practical engineering applications of the method, the proposed model can be used in the analysis of even traditional structures built up with different steel types or with various classes of timber structural elements as well as some periodic superconducting composite devices[2].

Finally, it is observed that having analytical expressions for effective Young modulus and their probabilistic moments, the presented model can be extended on the deterministic and stochastic structural sensitivity analysis for elastostatics or elastodynamics of the periodic composite bar structures. It can be done assuming perfect bonds between different homogeneous parts of the entire bar or even by introducing stochastic defects at the interface between them and the interphase model due to the considerations carried out in[3] or another related microstructural phenomena both in linear an nonlinear range. At the same time, starting from the deterministic description of the homogenized structure, one can obtain the effective behavior related to any external excitations described by the stochastic processes. The main advantage of the proposed approach is that any randomness in geometry or elasticity of the composite beams can be replaced by a single effective random variable of the Young modulus characterizing such structures. Hence, computational studies of engineering composites with different parameters being random variables by the use of equivalent homogeneous structure with deterministically defined geometry and effective probability density function of the Young modulus (elastic parameters) can be performed. It is observed that using analytical expression for the homogenized Young modulus, the randomness can be introduced in the periodicity cell geometry what results in random fluctuations of the effective parameter only. Furthermore, even if there is no periodicity in the composite structure, the results of homogenization application are quite satisfactory, i.e. probabilistic response of the structure homogenized approximate very well the real composite beam behavior. The main value of the proposed homogenization method is that the equations for expected values and covariances of effective characteristics do not depend on the PDF type of an input random fields, however, in case of greater values of higher order probabilistic moments related to the first two used above, the higher order version of the perturbation method is recommended. As it is demonstrated in the papers devoted to the wavelet analysis, this type of homogenization can be easily adopted for all these structures where the elastic or geometrical properties have the wavelet-based nature[4]. Starting from a finite set of the wavelets with probabilistically defined amplitude, the effective random Young modulus of the entire structural component can be determined what gives a result being different than that obtained in classical analysis[5].

1

M. Kleiber, T.D. Hien, "The Stochastic Finite Element Method". Wiley, 1992.

2

M. Lefik, B.A. Schrefler, "Homogenized material coefficients for 3D elastic analysis of superconducting coils". In: P. Ladeveze, O.C. Zienkiewicz, eds., "New Advances in Computational Structural Mechanics". 435-446, 1992.

3

M. Kaminski, M. Kleiber, "Numerical homogenization of n-phase composites including stochastic interface defects". International Journal of Numerical Methods in Engineering, 47, 1001-1025, 2000. doi:10.1002/(SICI)1097-0207(20000220)47:5<1001::AID-NME814>3.0.CO;2-V

4

M. Kaminski, Multiresolutional wavelet-based homogenization of random composites. Proc. 2nd European Conference on Computational Mechanics, Cracow 2001. doi:10.4203/ccp.75.138

5

B. Hassani, E. Hinton, "Homogenization and Structural Topology Optimization. Theory, Practice and Software". Springer-Verlag, 1999.

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