Mathematical Treatment of Bimodality for the Safety Factor in Riverbank Stability Analysis Using Cusp Catastrophe

Abstract

The safety factor (SF) in riverbank stability problems for noncohesive soils is treated mathematically in this article to unravel its inherent bimodality, as stipulated by the cusp catastrophe technique in a hyperspace. The developed methodology may be contrasted with traditional approaches delineating the SF space to failure and operational states. The emerging three states are a significant shift from traditional treatments overlooking bimodality. The mathematical treatment presented in the technical note incorporates classic soil equations for noncohesive soils into the cusp catastrophe technique, with clear formulations at multiple levels of a potential function, including the energy level. The integrated mathematical expressions use soil properties to express lower order properties such as catastrophe flags, for example, bifurcation sets, hysteresis, and bimodality delineating sudden and gradual change in the state of a system. These equations show that even the SF of operational states depends on soil properties and gravity, and therefore a safe use of the SF requires a deep knowledge of the SF hyperspace.Practical ApplicationsPractical applications of the new mathematical development presented in the technical note may be viewed in three steps. In Step 1, the tacit nature of the cusp catastrophe bimodality of safety factor in riverbank stability problems needs to be tested through experimental data. The laboratory tests and fieldwork can be designed to encompass the full range of cases comprising: (i) the failure region; (ii) the operational region; and (iii) their bimodal zone. In Step 2, the existence of the three cases is verified by various applications to underpin the dependency of the safety factor on soil parameters. In Step 3, this new knowledge is realized by wide applications to gain an insight into behaviors of safety factors in wide-ranging problems such as natural slopes, channels, embankments, riverbanks, and levees.