Analytical Model for Fictitious Crack Propagation in Concrete Beams

An analytical model for load-displacement curves of unreinforced notched and un-notched concrete beams is presented. The load displacement-curve is obtained by combining two simple models. The fracture is modelled by a fictitious crack in an elastic layer around the mid-section of the beam. Outside the elastic layer the deformations are modelled by the Timoshenko beam theory. The state of stress in the elastic layer is assumed to depend bi-lineary on local elongation corresponding to a linear softening relation for the fictitious crack. For different beam size results from the analytical model is compared with results from a more accurate model based on numerical methods. The analytical model is shown to be in good agreement with the numerical results if the thickness of the elastic layer is taken as half the beam depth. Several general results are obtained. It is shown that the point on the load-displacement curve where the fictitious crack starts to develope, and the point where the real crack starts to grow will always correspond to the same bending moment. Closed from solutions for the maximum size of the fracture zone and the minimum slope on the load-displacement curve is given. The latter result is used for derivation of a general snap-back criterion depending only on beam geometry.