Stability Analysis of a Single-Degree-of Freedom Mechanical Model with Distinct Critical Points: I. Bifurcation Theory Approach

The buckling and post-buckling response of a single-degree-of-freedom mechanical model is re-examined in this work, within the context of nonlinear stability and bifurcation theory. This system has been reported in pioneer as well as in more recent literature to exhibit all kinds of distinct critical points. Its response is thoroughly discussed, the effect of all parameters involved is extensively examined, including imperfection sensitivity, and the results obtained lead to the important conclusion that the model is possibly associated with the butterfly singularity, a fact which will be validated by the contents of a companion paper, based on catastrophe theory.

Steel Beam-to-Column RBS Connections: FEM Analysis under Cyclic Loading

In a companion paper [1], an optimization scheme for extended-end-plate Reduced Beam Section (RBS) connections of steel-moment-frames was presented, based on the component method of Eurocode 3, on regression analysis and on principles of Mechanics under monotone loading. European beam and column profiles were utilized, in conjunction with geometric restrictions and constraints of North American and European Standards for prequalified radius-cut RBS. Τhe aforementioned method aimed for an excellent seismic performance, the verification and validation of which is the content of the present study. Using FEM modeling and accounting for the assumptions used in the optimum design, after calibration with existing experimental data, the optimum connections were numerically analyzed under cyclic loading, adopting a well-accepted displacement-based protocol. All optimum solutions exhibited an excellent cyclic response, and met very satisfactorily all the performance criteria for seismic design. Results in terms of hysteretic M-φcurves at three characteristic areas of the connections validate the whole analysis, a fact aiming to assist in incorporation the radius-cut RBS concept in European Steel Design Codes and engineering practice.

Stability Analysis of a SDOF Mechanical Model with Distinct Critical Points: II. Catastrophe Theory Approach

In a recent publication [1], the fully nonlinear stability analysis of a Single-Degree-of Freedom (SDOF) model with distinct critical points was dealt with on the basis of bifurcation theory, and it was demonstrated that this system is associated with the butterfly singularity. The present work is the companion one, tackling the problem via the Theory of Catastrophes. After Taylor expanding the original potential energy function and introducing Padè approximants of the trigonometric expression involved, the resulting truncated potential is a universal unfolding of the original one and an extended canonical form of the butterfly catastrophe potential energy function. Results in terms of equilibrium paths, bifurcation sets and manifold hyper-surface projections fully validate the whole analysis, being in excellent agreement with the findings obtained via bifurcation theory.