A peculiar set of problems in linear structural mechanics

Peculiar analytical results encountered m the hterature on continuously supported structures are pointed out and investigated. They are (1) for a fimte beam whxch rests on a Wlnkler foundatmn and is centrally loaded by a concentrated force P, the points of separatmn of beam and base are not affected by the magmtude of the load, and (2) according to the well known solution for an infinite (or a semi-infinite) beam attached to a Wmkler foundation and subjected to a concentrated load P, the location of the zero points for deflections and bending moments do not depend on P. Intumvely, it xs expected that these entrees should depend on the load P At first, it is shown that these peculiar results are a consequence of the llneanty of the respective formulations and that the same feature will also be exhxblted for other linear foundatxon models (for example, the Pasternak model or the elastic continuum). To clarify these analytxcal features, the above problems are re-analysed by including a non-lineanty m the Winkler foundatmn response To s~mplify the analyses, a bMinear response is used. It was found that (1) for the finite beam that rests on the base, the intensity of the load does affect the locatmn of the pomt of separation of beam and base; namely, that an increasing load and a "stiffening" base decrease the regmn of contact, (2) for the mfimte beam that is attached to ʹthe base the situation ~s similar, an increasing load and a "st~ffemng" of the base decrease the distances of the zero locations and reduce the maximum bending moment, whereas for a "softening" base these distances increase as compared to the linear case