Static instability analysis of an elastic band travelling in the gravitational field

Static instability analysis is performed for an axially moving elastic band, which is travelling at a constant velocity in a uniform gravitational field between two supports. The buckling of the band is investigated with the help of admitting small transverse deflections. The model of a thin elastic beam (panel) subjected to bending, centrifugal forces and nonhomogeneous tension (including a gravitational term) is used. Buckling analysis and estimation of the critical velocities of elastic instability are based on variational principles and variational inequalities. As a result, explicit formulas for upper and lower limits for critical velocities are found. It is shown analytically that a critical velocity always exists. The critical buckling modes are found, first, by solving the original differential equation directly, and, secondly, by energy minimization. The buckling modes and corresponding critical velocities are found and illustrated with some numerical examples. The gravitational force is shown to have a major effect on the buckled shape, but a minor effect on the critical velocity.