A PECULIAR SET OF PROBLEMS IN LINEAR STRUCTURAL MECHANICS

Peculiar analytical results encountered in the literature on continuou sly supported structures are pointed out and investigated. They are: ( 1) for a finite beam which rests on a Winkler foundation and is centra lly loaded by a concentrated force P, the points of separation of beam and base are not affected by the magnitude of the load, and (2) accor ding to the well known solution for an infinite (or a semi-infinite) b eam attached to a Winkler foundation and subjected to a concentrated l oad P, the location of the zero points for deflections and bending mom ents do not depend on P. Intuitively, it is expected that these entiti es should depend on the load P. At first, it is shown that these pecul iar results are a consequence of the linearity of the respective formu lations and that the same feature will also be exhibited for other lin ear foundation models (for example, the Pasternak model or the elastic continuum). To clarify these analytical features, the above problems are re-analysed by including a non-linearity in the Winkler foundation response. To simplify the analyses, a bi-linear response is used. It was found that: (1) for the finite beam that rests on the base, the in tensity of the load does affect the location of the point of separatio n of beam and base; namely, that an increasing load and a ''stiffening '' base decrease the region of contact; (2) for the infinite beam that is attached to the base the situation is similar; an increasing load and a ''stiffening'' of the base decrease the distances of the zero lo cations and reduce the maximum bending moment, whereas for a ''softeni ng'' base these distances increase as compared to the linear case.