Session: 07-23-01: 100th Anniversary of the Timoshenko-Ehrenfest Beam Model

Paper Number: 96729

96729 - Response and Constitutive Identification of Random Axial Functionally Gradient Timoshenko-Ehrenfest Beams 

This work aims to propose a simplified method for the random analysis of axial functionally graded (AFG) Timoshenko-Ehrenfest beams. The direct and inverse problem are addressed. The first consists in finding the stochastic properties of the kinematic mechanical response of the beam once known the stochastic properties of the constitutive parameters. The second means to perform a stochastic identification. Semi-closed form solutions are derived in terms of probability density functions (PDFs) via the probability transformation method (PTM) and Monte Carlo (MC) simulations.

The choice of studying random axial functionally gradient beams lies into many reasons: the growing importance of functionally graded (FG) materials within the scientific panorama; the generic nature of the mathematical model, which proves useful to study beams where the assumption of constant and deterministic stiffnesses along the axis does not hold anymore. As an example, we can refer to the bleeding phenomenon closely related to the unpredictable movement of water in concrete structures.

Several publications in the literature present numerical approaches where just the Young modulus, E, is supposed to be random and the stochastic properties of the shear deformable modulus, G, are derived by means of the classical relationship between G and E. The Poisson coefficient, υ, and the shear correction factor, χ, are usually supposed to be deterministic physical quantities. The listed choices are subjected to criticism since the lack of knowledge about the Poisson coefficient and the shear correction factor themselves. Moreover, this kind of models, developed in the field of classical elasticity, are hardly applicable in the field of Cosserat (or micropolar) continua.

During the 20^th century, the Cosserat theory was rediscovered and its capability of predicting the mechanical response of granular and composite materials is still being investigated. Timoshenko-Ehrenfest beam models based on Cosserat continua were applied in the analyses of nano- and micro-electromechanical systems (NEMS and MEMS). The latter involve new constitutive parameters, inside the bending stiffness, able to consider small scale effects. Clearly, to assume the new constitutive parameters, related to the meso-structure of the beams, as deterministic physical quantities is a strong approximation.

In light of all the above remarks, the main objective of the present paper is to develop a semi-closed procedure for the direct and inverse stochastic analysis of a Timoshenko-Ehrenfest beam model. Three different cases are analyzed. Firstly, the entire shear and bending stiffnesses are assumed to be random along the axis and stochastically identified. Secondly, the constitutive parameters are assumed to be random along the axis, except for υ, and the stochastic properties of χ(z) are derived. Then, by assuming χ as a deterministic quantity, the random field υ(z) is identified. All random fields are approximated up by the authors as stepwise stochastic fields by means of the Heaviside function. Hence, the generalized functions properties are largely applied to derive approximated relationships between the transversal displacements and the random stiffnesses under different conditions of constraints. The derived input-output transformations allow to solve both direct and inverse stochastic problems. In the case of simply supported beams, it is shown how the inverse stochastic problem can be easily reduced to an equivalent linear one. Finally, the application of the PTM leads to the PDFs of all the random variables linked to the displacements and axially variable stiffnesses. The obtained results are compared to the ones derived via MC simulations.

Presenting Author: Gabriele La Valle Università degli Studi di Messina

Presenting Author Biography: Gabriele La Valle is a Phd student at University of Messina in Engineering and Chemistry of Materials and Buildings. His research interests are primarily in micropolar theories and stochastic analyses. His last works deal with a new relative rotation tensor for describing granular materials and a new version of the probability transformation method for nonlinear input-output transformations between random variables. Recently, he addressed the mechanical response of random metamaterials. He wrote 5 papers in Acta Mechanica, Communications in Nonlinear Science and Numerical Simulation, Continuum Mechanics and Termodynamics, Mechanics Research Communications. Others two are about to be published in Zeitschrift für angewandte Mathematik und Physik and in Mathematics and Mechanics of Solids.

Authors:

Gabriele La Valle Università degli Studi di Messina
Giovanni Falsone Università degli Studi di Messina