Stability of a straight Kirchhoff elastic rod under the force screws

Abstract

Stability of a straight Kirchhoff elastic rod with circular cross section acted by a pair of force screws is studied. Cartesian coordinate and Cardan angle are used to express the position and attitude of a cross section of the rod. Special solution which is a straight equilibrium state of the rod is derived from Kirchhoff equation of the rod, and the linear perturbation equation on this special solution is further solved. The stability of the solution for the straight equilibrium state of the rod is discussed according to the existence of non-zero solution of integration constants at various kinds of boundary conditions of the rod, such as that with two joints ends, two fixed ends, a fixed end and a free end, or a joint end and a fixed end. The critical loads are deduced and the stable ranges are plotted. Greehill formula is extended to other cases, and Euler formula for compression rod becomes its special case.