Stability of a horizontal elastic bar

Stability of a horizontal bar lying on an absolutely rigid base in the gravity force is investigated. The base prevents the bar from deflection by the force of gravity and, in particular, excludes the initial deflection. The bar can be bent only upward, against the gravity force. In the absence of the supporting plane, the gravity force increases the bend of the bar, which makes the initial rectilinear form of balance impossible; in the presence of the supporting plane, the gravity force is directed against the deflection and promotes the stability of the rectilinear form of balance.
The possibility of the curved bar balance forms adjacent to a rectilinear form is considered.
It is shown that Euler's force is not the lower bound value of the compressing force, sufficient for transformation of the bar from a rectilinear form of balance to a curved form of balance. The value of a critical force which makes such transition possible is obtained. The critical force significantly exceeds the corresponding value of the force calculated by Euler's formula and depends not only on the stress-related properties of bar material but also on its density. The critical force is determined from the condition of the equality of the compressing force work and total potential energy of the curved bar, including the elastic energy and gravitational energy.
The bend of the bar in process of stability loss in some part of the bar length, while the other part remains rectilinear and horizontal, is investigated. Conditions under which such bend becomes possible are found. It is shown that the length of the curved part of the bar increases with an increase in the compressing force.