An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending

We present a numerical procedure, based upon a tangent approach, for evaluating the ultimate limit state (ULS) of reinforced concrete (RC) sections subject to axial force and biaxial bending. The RC sections are assumed to be of arbitrary polygonal shape and degree of connection; furthermore, it is possible to keep fixed a given amount of the total load and to find the ULS associated only with the remaining part which can be increased by means of a load multiplier. The solution procedure adopts two nested iterative schemes which, in turn, update the current value of the tentative ultimate load and the associated strain parameters. In this second scheme an effective integration procedure is used for evaluating in closed form, as explicit functions of the position vectors of the vertices of the section, the domain integrals appearing in the definition of the tangent matrix and of the stress resultants. Under mild hypotheses, which are practically satisfied for all cases of engineering interest, the existence and uniqueness of the ULS load multiplier is ensured and the global convergence of the proposed solution algorithm to such value is proved. An extensive set of numerical tests, carried out for rectangular, L-shaped and multicell sections shows the effectiveness of the proposed solution procedure.