From sequentially linear analysis to incremental sequentially linear analysis: Robust algorithms for solving the non-linear equations of structures of quasi-brittle materials

It is difficult to accurately predict the strength of masonry and concrete structures. The most widely used method for simulating their behaviour is finite element analysis with the Newton-Raphson method and arch length control. However, the Newton-Raphson method can diverge and not produce a result, for example in bifurcations or during snap-back. In order to enhance the robustness of solving non-linear problems, a new method – called incremental sequentially linear analysis (ISLA) – is proposed. The method is based on a combination of the Newton-Raphson method and a total approach called sequentially linear analysis.In ISLA, local damage is induced by reducing the material secant stiffness of the element that fails a unity check. The load is applied in force increments or displacement increments, which are adjusted to trace the complete structural response. It has been showed that ISLA can handle non-proportional loading, geometrically non-linear analysis and transient analysis. The robustness of ISLA has been demonstrated in four examples: a concrete beam with both prestress and vertical load; out-of-plane bending of a masonry wall with overburden; a differential settlement test on a pre-loaded masonry façade and a 3D pushover analysis of a masonry house.