Formulas for an Infinitely Long Bernoulli-Euler Beam on the Pasternak Model

Abstract

The Winkler model is the simplest mechanical model of a continuum and the easiest for mathematical treatment. However, it has some shortcomings involving discontinuity of adjacent spring displacements. Many researchers have proposed improved soil models to overcome these problems. These models are often called "two-parameter models", because they have the second parameter which presents the continuity of adjacent springs in addition to the first parameter. Pasternak's model, the most reasonable and generalized two-parameter model, can account for the actual shearing effect of soils in the vertical direction. Pasternak used his model to show the analytical solutions of an infinitely long beam on the model and compared the displacements and stresses of the beam with those on the Winkler model. However, his work did not involve whole cases and its actual usefulness has not been made clear yet. This paper, therefore, presents a complete set of formulas to calculate the displacements and stresses on an infinitely long Bernoulli-Euler beam on the Pasternak model. We then carried out numerical case studies on mechanical quantities of the beam and the shear layer. The resulting effects of shear stiffness of the Pasternak model on these quantities are discussed in comparison with those of the Winkler model. Then, the applicability and usefulness of the Pasternak model are shown.