Bending of Steel fibers on partly suppprted elastic foundation

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