An asymptotic study of nonlinear coupled vibration of arch-foundation structural system

Nonlinear dynamic coupling between arch and elastic foundation is investigated by an asymptotic technique. Based upon a key weak foundation motion assumption, an arch-foundation coupled model is proposed via direct multiple scale method, with both quadratic and cubic nonlinearity being well characterized. The arch-foundation two-way coupling coefficients are analytically derived through construction of solvability condition of multi-scale expansions. It turns out that the foundation/arch mass ratio characterizes the coupling intensity and plays a key role in understanding arch-foundation coupled dynamics. Based upon the asymptotic model, arch-foundation's coupled responses with primarily resonant excitation on the arch are fully studied, with stability and bifurcation characteristics being determined, which are also compared with the ideal arch dynamics with fixed ends. All the approximately analytical results are verified by numerical ones produced by Galerkin truncated model.