This contribution presents a simple and effective numerical model for performing static analyses of beams in frictionless and bilateral contact with a three-dimensional (3D) elastic and isotropic half-space. Such a problem can suitably represent the behaviour of strip footings and shallow foundations in building structures.
A coupled finite element-boundary integral equation (FE-BIE) method, already introduced for beams on half-plane [1], is here extended to the case of beams on 3D half-space, by adopting a mixed variational formulation which assumes as independent fields both beam displacements, rotations and contact surface tractions. Mixed formulation includes the Green function of the 3D half-space [2]. The resulting numerical model makes use of locking-free \u201cmodified\u201d Hermitian shape functions [3] for the beam and piecewise constant function for the substrate tractions. For this purpose, only the contact surface underneath the foundation needs to be discretized with rectangular elements; furthermore, traction distribution in beam transverse direction is considered with an adequate mesh refinement of contact surface in both plane directions.
Numerical tests of both Euler-Bernoulli and Timoshenko beams subject to several load conditions are dealt with. Results in terms of displacements, contact tractions and bending moment are obtained, showing the effectiveness of the model and its convergence to existing analytical and numerical solutions [4].
References
[1] Tullini, N., Tralli, A., \u201cStatic analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral\u201d, Computational Mechanics, 45(2\u20133) pp. 211\u2013225 (2010).
[2] Johnson, K.L., Contact mechanics, Cambridge University Press, Cambridge (1985).
[3] Reddy, J.N., \u201cOn locking-free shear deformable beam finite elements\u201d, Computer Methods in Applied Mechanics and Engineering, 149(1-4), pp. 113\u2013132 (1997).
[4] Biot, M.A., \u201cBending of an infinite beam on an elastic foundation\u201d, Journal of Applied Mechanics, 4, pp. A1\u2013A7 (1937)
