Application of quantifier elimination to inverse buckling problems

作者：Ioakimidis Nikolaos I^*

来源：Acta Mechanica, 2017, 228(10): 3709-3724.

DOI：10.1007/s00707-017-1905-5

The inverse buckling problem for a column is the problem where both the loading and the buckling mode are defined in advance (the latter generally in a polynomial form), and the flexural rigidity of the column is sought in a similar form with the help of the related ordinary differential equation. This problem was proposed and studied in many buckling problems by Elishakoff and his collaborators. A serious difficulty in its solution is that the resulting flexural rigidity should be positive along the column. Here in order to check this positivity, the modern computational method of quantifier elimination is proposed and used inside the computational environment offered by the computer algebra system Mathematica and mainly based on the Collins cylindrical algebraic decomposition algorithm. At first, the simple inverse buckling problem of an inhomogeneous column under a concentrated load is studied with respect to the aforementioned positivity requirement. Next, the much more difficult problem concerning a variable distributed loading is also studied both in the case of one parameter and in the case of two parameters in this loading. Parametric rational and trigonometric forms of the flexural rigidity are also studied. Naturally, the resulting conditions for the positivity of the flexural rigidity are rather simple for one loading parameter, but they may become sufficiently complicated for two loading parameters. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions for the flexural rigidity of a column in a variety of inverse buckling problems.

• 出版日期2017-10