Stability to the finite disturbance of the critical equilibrium for three struts

Abstract

In this paper, sufficient conditions of stability for the critical equilibrium of Euler struts with three typical rigid boundaries, that is, one end fixed and the other clamped in rotation, fixed-free, pinned-pinned, are studied. A construction of a double orthonormal Fourier series for angle is presented for the three struts, respectively. By the double orthonormal series, the second variation of potential energy can be expressed in a diagonal quadratic form. The second variation of potential energy is proved to be semi-positive-definite. Sum of higher order variations than the second-order variation of potential energy is identically positive. The critical equilibrium is stable to disturbance with finite value, which is called “stable in the large” in the sense of Lagrange-Dirichlet stability criterion. Stability to disturbance with finite value includes the stability to disturbance with infinitesimal value in Koiter’s theory. In addition, there is a constraint on the angle for the pinned-pinned strut, which is not included in Koiter’s theory.