Bending of Steel fibers on partly suppprted elastic foundation

Fiber reinforced cementitious composites are nowadays widely applied in civil engineering.The postcracking performance of this material depends on the interaction between a steel fiber, which isobliquely across a crack, and its surrounding matrix. While the partly debonded steel fiber is subjected topulling out from the matrix and simultaneously subjected to transverse force, it may be modelled as aBernoulli-Euler beam partly supported on an elastic foundation with non-linearly varying modulus. Thefiber bridging the crack may be cut into two parts to simplify the problem (Leung and Li 1992). Toobtain the transverse displacement at the cut end of the fiber (Fig. 1), it is convenient to directly solve thecorresponding differential equation. At the first glance, it is a classical beam on foundation problem.However, the differential equation is not analytically solvable due to the non-linear distribution of thefoundation stiffness. Moreover, since the second order deformation effect is included, the boundaryconditions become complex and hence conventional numerical tools such as the spline or differencemethods may not be sufficient. In this study, moment equilibrium is the basis for formulation of thefundamental differential equation for the beam (Timoshenko 1956). For the cantilever part of the beam,direct integration is performed. For the non-linearly supported part, a transformation is carried out toreduce the higher order differential equation into one order simultaneous equations. The Runge-Kuttatechnique is employed for the solution within the boundary domain. Finally, multi-dimensionaloptimization approaches are carefully tested and applied to find the boundary values that are of interest.The numerical solution procedure is demonstrated to be stable and convergent.