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

Abstract

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 flrst 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.

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

“…The hypergeometric function has earlier been adopted to solve the transverse vibration problem of non-uniform annular plates (Duan et al, 2005), the lateral-torsional buckling problem of linearly tapered cantilever (Challamel et al, 2007), the axisymmetric bending problem of micro/nanoscale circular plates , the buckling problem of columns with allowance for self-weight (Duan and Wang, 2008), and axisymmetric transverse vibration problem of circular cylindrical shells with variable thickness (Duan and Koh, 2008), the flexural-torsional buckling problem of cantilever (Challamel et al, 2010), and the lateral-torsional stability boundaries for polygonally depthtapered strip cantilevers (Andrade et al, 2012). The Frobenius companion matrix of a second kind Chebyshev polynomial U nÀ1 ðx=2Þ is given by (Horn and Johnson, 1990) …”