Stability of Nonuniform Cracked Bars Under Arbitrarily Distributed Axial Loading

Abstract

This study is concerned with the stability analysis of nonuniform bars with arbitrary number of cracks and with arbitrary distribution of flexural stiffness or arbitrarily distributed axial loading. The homogeneous solutions of the governing differential equation for buckling of a nonuniform uncracked bar are derived for several important cases. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks. Then a new approach that combines the exact buckling solution of a nonuniform uncracked bar, the model of massless rotational spring, and the transfer matrix method is presented for the title problem. The main advantage of the proposed method is that the eigenvalue equation for buckling of a nonuniform bar with an arbitrary number of cracks, arbitrary distribution of flexural stiffness, or arbitrarily distributed axial loading can be conveniently determined from a second-order determinant. The decrease in the determinant order as compared with other methods leads to significant savings in the computational effort. A numerical example is given to illustrate the reliability of the proposed approach through comparisons with numerical solutions and to study the effect of cracks on the stability of a nonuniform bar. Nomenclature ai = depth of the ith crack Ci =fl exibility of the ith rotational spring Ci1, Ci2 = constants of integration 1F2(a1; a2; a3; a4) =h ypergeometric series f () =fl exibility function hi = height of cross section at xi Jν(x) = first Bessel function of the vth order Ki(x) =fl exural stiffness at x in the ith