Beam Embedded in a Winkler Soil with Clearances. Analytical Solution of the NonLineal Bending Problem

Abstract

The nonlinear bending problem of a beam embedded in a Winkler soil with clearances is addressed by means of an analytical approach. This problem is of interest, for instance, in the study of the drillstrings behavior under certain load conditions. Usually, a drillstring is modeled as a bar inside an outer cylinder (bore-hole wall) with clearances that add strong nonlinearities. This work is part of a wider studyon drillstringsand a paper on thenonlinearvibrationof thistypeof structurewaspresented in ENIEF 2006. Here the title problem is simplified to a plane bar making contact with a Winkler-type soil. The governing differential problem is derived using a minimal energy principle. The unknowns are the lateral displacement due to bending (as usual) and the length of contact. The consideration of the latter unknown leads to special restrictions among the admissible directions within the Calculus of Variation deriving in a particular statement of the problem. Once the differential problem is fully established, a solutionof pairs of load-lengthof contact is found. Two particular examples are worked out: a cantilever beam with a lateral tip load and a simply supported beam subjected to external end bending moments, in both cases with several soil stiffness values. The numerical results are compared with a finite element model. The availability of an analytical approach permits the calibration of other numerical solutions, e.g. the study of convergences issues. On the other hand other complexities, such as the consideration of axial loads including self-weight that leads to the inclusion of the second order effect, are under study at present.