Sharp bounds based on quantifier elimination in truss and other applied mechanics problems with uncertain, interval forces/loads and other parameters
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Ioakimidis, Nikolaos
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Abstract
The computational method of quantifier elimination in computer algebra for real numbers related to the elimination of the universal quantifier "for all" and/or the existential quantifier "exists" in quantified formulae has already been efficiently implemented in few computer algebra systems and also applied to several interesting problems of mathematics, physics and engineering. Here this method is applied to some simple problems of applied mechanics including truss problems in structural mechanics in the case of uncertain, interval parameters related to the applied loads and/or to the parameters of the structure. Therefore, the present results are related to classical interval analysis and they permit the determination of sharp bounds for the mechanical quantities of interest such as resultants of forces, reactions and displacements in truss problems. The implementation of quantifier elimination in the powerful and user-friendly computer algebra system Mathematica has been selected as the most efficient and appropriate tool for the present computational tasks and it offers one more computational possibility in simple applied mechanics problems under uncertainty that is described by interval parameters. The five applied mechanics problems studied in detail here concern (i) the resultant of three forces acting on a box, (ii) the resultant of four forces acting on a particle, (iii) a block resting and sliding on a horizontal plane, (iv) a three-member truss and, finally, (v) a six-member truss. All these problems were already proposed and solved with interval parameters by other researchers. The present results are in agreement with the already available related results and, moreover, they always provide sharp bounds for the interval quantities of interest.
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Keywords
Intervals, Interval analysis, Interval arithmetic, Interval parameters, Uncertainty, Uncertain variables, Uncertain parameters, Forces, Loads, Displacements, Structures, Trusses, Upper bounds, Lower bounds, Sharp bounds, Symbolic computations, Quantifiers, Universal quantifier, Existential quantifier, Quantified formulae, Quantified/free variables, Quantifier elimination, Quantifier-free formulae, Parametric interval linear algebraic systems, United solution set, Interval hull, Computer algebra, Computer algebra systems, Mathematica