Nonlinear analysis of shape memory alloy columns having variable cross-sections

Abstract

A new numerical scheme based on finite difference technique has been devised and used to trace the load-deflection curves (equilibrium configuration paths) of a column that has variable cross-sections along its span. Additionally, the column may have highly non-linear and prominent non-symmetric responses in tension and compression. As superelastic shape memory alloy (SMA) has typical non-linear and non-symmetric responses in tension and compression, the devised solution scheme had been applied for variable cross-section superelastic SMA columns. Theorems of Thompson and Hunt had been used as criteria to find buckling loads of the columns. Accordingly, column's load-deflection curves (mathematically, equilibrium configuration paths) had been traced continuously for increasing value of the loading parameter. When a distinct change in the mode of deformation is evident in those curves, the column is at a state of unstable equilibrium that leads to its buckling. Corresponding load (may it be a limit point, or a branching point, on the equilibrium configuration paths) is said to be the column's buckling load. Also the shape of column is largely deformed because of buckling. While solving the governing equation (that includes geometric nonlinearity) modifications are necessary to the stiffness modulus term at each load step since stresses may exceed proportional limit (since, material nonlinearity is also involved and Hooke's law can't be applied). Finally, a set of linear equations were solved to find response of the column. As mentioned, to make the study more comprehensive, a variable cross-section has been considered along the column's span. Column's crosssection is rectangular having a constant thickness and variable width along the span. Shape imperfections of the column have been also considered in addition to all of such points. The devised method can be used to calculate buckling loads of any column having variable cross-section together with material nonlinearity. Experimental buckling loads were obtained by rigorous tests for variable crosssection stainless steel (SS) columns. Later buckling loads were obtained for those using the devised numerical scheme. Comparison shows excellent agreement between the present results based on devised technique with those obtained from experiments and other studies.