Buckling load optimization for heavy elastic columns: a perturbation approach

Buckling lengths of heavy column with various end conditions

“…Since this paper, very little but considerable works have been done on the buckling analysis of heavy columns: Grishcoff (1930) studied buckling loads for the combined effect of both the self-weight and an axial load for the cantilever column by using an infinite series; Wang and Drachman (1981) investigated the buckling of the cantilever heavy column with an end load, based on the second order differential equation in terms of the arc length of buckled column. Interestingly, this paper included the column hanging from its fixed end, i.e., inverted cantilever column, subjected to an upward end load; Wang and Ang (1988) derived buckling formulas for the heavy column which is subjected to an axial load and is restrained by several elastic braces or internal supports; Chai and Wang (2006) determined the critical buckling load of axially compressed heavy columns with various end conditions using the differential transformation technique; Duan and Wang (2008) derived analytical exact solutions for the elastic buckling of heavy column with various of end conditions, in terms of generalized hypergeometric functions; and concerning shape optimizations of the heavy column, tallest columns with the variable cross-section and constant volume have been investigated by Keller and Niordson (1966), Atanackovic and Glavardanov (2004) and Sadiku (2008).…”

Buckling of Tapered Heavy Columns with Constant Volume

“…Lee and Lee [8] studied the buckling of a prismatic heavy column under various end conditions, where the buckling length of the column was calculated by considering only its self-weight (without any axial compressive load). Regarding the optimization of heavy columns, tall columns with variable cross-sections and constant volumes were investigated by Keller and Niordson [9], Atanackovic and Glavardanov [10], and Sadiku [11].…”

Free vibration and buckling of heavy column with regular polygon cross-section

“…This extends to the case where the column is suspended from the foundation, completing the stability boundary of the primary mode and examining the higher mode buckling; Wang 9 developed some useful approximate formulas for the stability criterion of the first mode, which is the most important mode in structural mechanics; Wang and Ang 10 studied the buckling loads of heavy columns which are restrained by several internal elastic braces. Here the buckling loads was determined by minimizing the generalized Rayleigh quotient subject to lateral restraints; Bjorhovde 11 reported the effect of self-weight on the maximum strength of wide-flanged steel columns with large cross-sectional dimensions; Atanackovic and Glavardanov 12 used Pontryagin's maximum principle to determine the optimal shape of a heavy compression rod that is stable against buckling; Chai and Wang 13 studied the buckling loads of axially compressed heavy columns using a differential transformation technique that converts the governing differential equations into algebraic recursive equations; Duan and Wang 14 investigated the analytical solutions for the elastic buckling of heavy columns with various combinations of end conditions derived in terms of generalized hypergeometric functions; Sadiku 15 discussed the issue of achieving a weight distribution along the height of the column that optimizes the buckling load of the column, taking into account the column's own weight; Wang 16 determined the optimum location of the intermediate support for the maximum load-bearing capacity of a standing column with clamped or pinned base and free or sliding tip; Okay et al 7 applied the variational iteration method for finding buckling loads and mode shapes of a heavy column; Wei et al 17 studied the stability of tapered heavy columns with concentrated end loads. Here the governing equation is transformed into an integral equation according to the relevant boundary conditions of the Euler-Bernoulli column; and Lee and Lee 18 studied the buckling length of heavy columns under various end conditions, where the buckling length and its buckling mode shape according to their own weight were reported.…”