An inequality constraint for the deflection of an elastic beam under a uniform distributed loading

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Ioakimidis, Nikolaos
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The problem of the deflection of a straight isotropic elastic beam under a uniform distributed loading during bending is reconsidered under the inequality constraint that this deflection should not exceed a critical value because of the existence of a rigid obstacle or because of strength or even aesthetic reasons. This problem reduces to the problem of positivity of an appropriate quartic polynomial along the beam, which is a computational quantifier elimination problem and can further be solved by using classical Sturm–Habicht sequences in the theory of polynomials. The final result is a logical combination of algebraic expressions including the parameters of the present beam problem, that is the deflections and the rotations at the beam ends, the constant distributed loading, the critical/maximum permissible deflection as well as the length and the flexural rigidity of the beam. More complicated loading conditions can also be considered by the same approach, which is also applicable to the classical finite element method in beam problems for each particular finite beam element.
Beams, Bending, Deflections, Elasticity, Finite elements, Inequality constraints, Obstacles, Quantifier elimination, Quantifier-free formulae, Sturm–Habicht sequences, Maple