Some obstacle problems for partially hinged plates and related optimization issues
Elvise Berchio, Filomena Feo, Antonio Giuseppe Grimaldi
TL;DR
The paper investigates obstacle problems for partially hinged rectangular plates modeling bridge decks, distinguishing real obstacles to avoid from artificial obstacles intended to boost stability. It develops two parallel optimization frameworks: (i) a worst‑case density design that minimizes oscillations under potential loads by distributing material through a two‑phase reinforcement, and (ii) a worst‑case obstacle placement that minimizes the torsional gap via strategically placed artificial barriers. Existence results are established for both problems, and qualitative properties, including symmetry and Green function–driven heuristics, are derived to illuminate the location of optimal reinforcements and obstacles. The analysis highlights the role of higher‑order variational inequalities, positivity properties of the biharmonic Green function, and the gap function as a torsional stability measure. Overall, the work provides theoretical grounding for density‑based reinforcement and obstacle‑based stabilization strategies in plate‑governed bridge deck models.
Abstract
We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.