Lateral-Torsional Stability Boundaries for Polygonally Depth-Tapered Strip Cantilevers Under Multi-Parameter Point Load Systems—An Analytical Approach

This paper reports an analytical study on the elastic lateral-torsional buckling behavior of strip cantilevers (i) whose depth is given by a monotonically decreasing polygonal function of the distance to the support and (ii) which are subjected to an arbitrary number of independent conservative point loads, all acting in the same “downward” direction. The study is conducted on the basis of a one-dimensional (beam) mathematical model. A specialized model problem, consisting of a two-segment cantilever acted by two loads, applied at the free end and at the junction between segments, is first considered in detail for it “contains all the germs of generality”. It is shown that the governing differential equations can be integrated in terms of confluent hypergeometric functions or Bessel functions (themselves special cases of confluent hypergeometric functions). This allows us to establish exactly the characteristic equation for this structural system, which implicitly defines its stability boundary. Moreover, it is shown that the methods used to solve the model problem also apply to the general problem. A couple of parametric illustrative examples are discussed. Some analytical solutions are compared with the results of shell finite element analyses—a good agreement is found.