Elastic instability of beams subjected to a partially tangential force

The influence of the tangency coefficient and the elastically restrained boundary conditions on the elastic instability of an uniform Bernoulli-Euler beam is investigated. It is shown that at both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient has no influence on the critical load of the beam, and the coefficient may either increase or reduce the stability of a clamped-translational and rotational elastic spring supported beam. The boundary curves for the flutter and divergence instability of the beam in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant, and in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant are shown. It is found that, in general, the boundary curves can be divided into four sections by three critical points. When the tangency coefficient, the translational elastic spring constant and the rotational elastic spring constant are increased to cross over the boundary curves, except the critical points, the instability mechanism changes and the critical load makes a jump. The jump phenomenon of critical load owing to the change of instability mechanism is explored.