Nonlinear forced dynamics of planar arches

"The nonlinear dynamics of planar elastic arches, under resonant vertical harmonic tip force and different sag-to-span ratios, are considered. An analytical model based on the polar continuum, curved rod theory, is formulated. Different values of initial curvature are considered, ranging from non-shallow to shallow conditions. The nonlinear change in curvature is expressed in terms of displacement components. The hypothesis of vanishing axial strain is assumed when dealing with non-shallow cases, while the Mettler theory, based on a constant strain, is usually used for shallow arches. The PDEs of the motion obtained through the extended Hamilton principle, are projected on the reduced basis constituted by the two first linear modes or analogous meaningful functions. Regions of instability of the one-mode solution are numerically detected, and coupled regular and non-regular motions are described using standard complexity indicators. Experimental tests are realized on two companion laboratory steel prototypes of arches, in order to compare and validate the results of the prevision of the analytical model. The behavior charts of the analytical and experimental problems and the nonlinear frequency-response curves show good agreement in a wide range of amplitude and frequency of the external excitation. The 2D model recently proposed by the authors seems to be the only adequate when dealing with real arches."