Regularization of inverse problems in reinforced concrete fracture

Reinforced concrete beams with flexural cracks are simulated by the bridged crack model. The weight function method of determining stress intensity factors has been followed to derive a transformation between the crack bridging force (the rebar force) and the crack opening displacements (CODs). The matrix of the transformation is then approximated by its finite difference equivalent within finite dimensional vector spaces. Direct problem of the transformation solves for CODs, which require a known rebar force. Alternatively, the inverse problem works out the rebar force from known CODs. However, the inverse transformations of such convolution type integral equations become ill-posed if input CODs are perturbed. The Tikhonov regularization method is followed in its numerical form to regularize the linear ill-posed inverse problem. Restoration of mathematical stability and consistency are demonstrated by specific examples, where the results of the direct and the corresponding inverse problem are cross checked. Results of the direct problem (i.e., the analytical CODs) are deliberately perturbed by adding machine generated random numbers of a certain width. The inverse problems are solved with these CODs to simulate practical situations, where measured CODs data will inevitably be noisy. Computations reveal that the inverse analysis of CODs satisfactorily determines the rebar force without cross-section information.