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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 286
Viscoelastic Analysis of a Bernoulli-Navier Beam Resting on an Elastic Medium C. Floris and F.P. Lamacchia
Department of Structural Engineering, Politecnico di Milano, Milan, Italy Full Bibliographic Reference for this paper
C. Floris, F.P. Lamacchia, "Viscoelastic Analysis of a Bernoulli-Navier Beam Resting on an Elastic Medium", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 286, 2006. doi:10.4203/ccp.83.286
Keywords: viscoelasticity, Bernoulli Navier beam, elastic media, contact problem, numerical solution, Boussinesq model, Winkler model, analytical solution.
Summary
A Bernoulli Navier beam with viscoelastic behavior is considered. The beam is
subject to static loads and rests on an elastic medium. As regards the last, attention is
focused on Boussinesq's and Winkler's models. Many engineering materials exhibit a
constitutive law depending on time, that is they have a viscous behavior. This can
influence the static behaviour of a beam resting on an elastic medium. This is a problem that to the authors'
knowledge has had little attention.
As regards the beam, even if it is in a plane stress state and the shear
deformability is disregarded, from a theoretical point of view, if the beam is
homogeneous, the viscoelastic constitutive law requires the knowledge of two
functions, that is the volumetric By applying the usual assumptions of linear viscoelasticity and plane section law, the beam equilibrium is governed by the following integrodifferential equation: where ![]() ![]() ![]() where ![]() The compatibility between the displacements of the beam axis and the superior surface of the Boussinesq's elastic medium requires that the following integral equation is satisfied: where ![]() ![]() ![]() ![]() in which ![]() ![]() ![]() ![]() A Winkler's medium does not transmit shear, and the interface load is given by After substituting Equation (54) in (51), separating the variables, a solution is looked for in the form of an infinite series as
The functions
where ![]() ![]() ![]() ![]() ![]()
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