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.
