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Abstract:

The methodology of Groebner bases in computational algebraic geometry was introduced in 1965 by Buchberger, an Austrian mathematician. With the increasing capability of symbolic computation in recent years, significant progress has been made in the area. Buchberger’s algorithm to determine Groebner bases has been implemented in many mathematical symbolic computational systems, such as Mathematica, Maple, etc. Because of its availability, the use of the Groebner basis methodology has become a unique option for many scientific and engineering applications.

 

The purpose of this study is to demonstrate the utility of the methodology of Groebner bases in one typical problem, namely, the nonlinear analysis of snap-through buckling of shallow arches (flat curved beams) with geometric imperfections. For simplicity, a sinusoidal shallow arch hinged at both ends under sinusoidal loading is employed in this analysis to obtain exact equilibrium equations with geometric imperfections and illustrate most of the features of snap-through buckling instability theory exactly. With the use of Groebner basis package in Maple, a set of uncoupled analytical expressions can be obtained from the original highly coupled nonlinear equilibrium equations, which allows the easy generation of load-deflection diagrams. Since the load-deflection functions are fully symbolic in all parameters (imperfection factors, geometric and material properties of the beam, etc.), it is convenient to observe the critical snapping point on the load-deflection equilibrium curve both with and without initial imperfections. In contrast, it is not a trivial task to observe the relationships between the parameters using finite element numerical methods. It is shown that with the fully symbolic forms, parametric studies can be performed with much less effort than is the case for numerical methods. Special emphasis is placed on not only how the initial imperfections, but also the geometric and material properties of the beam, affect the predictions of the critical snap-through buckling loads for engineering design. Furthermore, the numerical solutions calculated from quasi-statically loaded cases are discussed, which indicate that the critical load of snap buckling for an imperfect arch is less than the critical load of the corresponding perfect arch under practical circumstances. Moreover, this analysis also provides a general trend for other structural shapes that are more practical in engineering. Finally, it is expected that the Groebner basis methodology could potentially be just as effective for other nonlinear problems that occur across a variety of engineering disciplines.

Keywords: Snap-through buckling, Groebner basis, shallow arch, flat curved beam, nonlinear steady solution, quasi-static loading, computational algebraic geometry

Snap-through buckling, Groebner basis, shallow arch, flat curved beam, nonlinear steady solution, quasi-static loading, computational algebraic geometry