Keywords: exact solution, stochastic beam, statically indeterminate, bending flexibility, compound Poisson field, statistical moments.

The study of structures involving spatially random material and/or geometrical parameters, commonly referred to as "stochastic structures", has attracted the interest of many researchers in recent decades. The importance of this study is above all related to some structural problems, such as reliability assessment, for which neglecting the effective stochastic nature of structural parameters may not give accurate results. However, difficulties arise in obtaining exact solutions for stochastic structures since the governing equations are characterised by random coefficient functions and sometimes by random boundary conditions. As a consequence, several approximate methods have been developed to address the problem [1]. Most of them are based on perturbation techniques or series expansions and are applicable only to small variations of the random parameters.

It is worth noting that the search for closed-form solutions is desirable, since they may serve as benchmark solutions for comparison purposes with the aim of testing numerical methods, such as the stochastic finite element one.

For beam bending problems, both a spatially random material parameter (Young's modulus) and geometrical parameters (dimensions of the cross-section) can be combined into one parameter that is the bending stiffness, or its inverse, the bending flexibility. In 1994, Köylüoglu et al. [2] derived exact solutions for beams with stochastic stiffness under random loading by using the concept of Green's function and the spatial spectral densities. In 1995, Elishakoff et al. [3] gave an exact closed-form expressions for the mean and covariance functions for the displacement of statically determinate beams with spatially stochastic stiffness subjected to deterministic static loads. Later on, new exact solutions were formulated for stochastic shear beams under deterministic loads [4] and randomly loaded beams with stochastic flexibility [5]. In all the aforementioned papers, however, only statically determinate beams were considered. Analytical solutions for statically indeterminate beams comprising three elements with different random stiffness were derived in reference [6].

The aim of the present study is to establish exact solutions for both statically determinate and indeterminate stochastic beams under deterministic static loads. Bernoulli-Euler beams with spatially random bending flexibility represented by a compound Poisson field are considered. Starting from the equilibrium equation, the constitutive law and the kinematic relationship, exact closed-form expressions of the response random fields are derived. The key step consists in properly imposing the continuity conditions of slope and deflection at the random positions in which jumps of stochastic flexibility occur. Based on the knowledge of the exact response variability, closed-form expressions of response statistics are easily obtained for statically determinate beams, while a simulation technique is suggested in the case of statically indeterminate beams. Furthermore, it is shown that the formulation presented in the paper allows to solve problems involving stochastic beams subdivided into a prescribed number of elements, each characterised by a different probabilistic distribution of bending flexibility. In this case, response statistics can be derived analytically even for statically indeterminate beams.

To demonstrate the effectiveness of the present formulation, a cantilever beam and two statically indeterminate beams (clamped-clamped and clamped-simply supported) under a uniformly distributed load are studied.

1

M. Kleiber, T.D. Hien, "The stochastic finite element method", John Wiley & Sons, Chichester, 1992.

2

H.U. Köylüoglu, A.S. Camak, S.A.K. Nielsen, "Response of stochastically loaded Bernoulli-Euler beams with randomly varying bending stiffness", in: G.I. Schuëller, M. Shinozuka, J.T.P. Yao (Eds.), Structural Safety and Reliability, Balkema, Rotterdam, 1994.

3

I. Elishakoff, Y.J. Ren, M. Shinozuka, "Some exact solutions for the bending of beams with spatially stochastic stiffness", International Journal of Solids and Structures, 32(16), 2315-2327, 1995. doi:10.1016/0020-7683(94)00257-W

4

N. Impollonia, I. Elishakoff, "Exact and approximate solutions, and variational principles for stochastic shear beams under deterministic loading", International Journal of Solids and Structures, 35(24), 3151-3164, 1998. doi:10.1016/S0020-7683(98)00008-0

5

I. Elishakoff, N. Impollonia, Y.J. Ren, "New exact solutions for randomly loaded beams with stochastic flexibility", International Journal of Solids and Structures, 36, 2325-2340, 1999. doi:10.1016/S0020-7683(98)00113-9

6

O. Rollot, I. Elishakoff, "Large variation finite element method for beams with stochastic stiffness", Chaos Solitons & Fractals, 17, 749-779, 2003. doi:10.1016/S0960-0779(02)00470-8

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