Static Upper/Lower Thrust and Kinematic Work Balance Stationarity for Least-Thickness Circular Masonry Arch Optimization
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Static Upper/Lower Thrust and Kinematic Work Balance Stationarity for Least-Thickness Circular Masonry Arch Optimization Giuseppe Cocchetti1,2
· Egidio Rizzi2
Received: 5 July 2020 / Accepted: 15 October 2020 / Published online: 28 November 2020 © The Author(s) 2020
Abstract This paper re-considers a recent analysis on the so-called Couplet–Heyman problem of least-thickness circular masonry arch structural form optimization and provides complementary and novel information and perspectives, specifically in terms of the optimization problem, and its implications in the general understanding of the Mechanics (statics) of masonry arches. First, typical underlying solutions are independently re-derived, by a static upper/lower horizontal thrust and a kinematic work balance, stationary approaches, based on a complete analytical treatment; then, illustrated and commented. Subsequently, a separate numerical validation treatment is developed, by the deployment of an original recursive solution strategy, the adoption of a discontinuous deformation analysis simulation tool and the operation of a new self-implemented Complementarity Problem/Mathematical Programming formulation, with a full matching of the achieved results, on all the arch characteristics in the critical condition of minimum thickness. Keywords Symmetric circular masonry arches · Couplet–Heyman problem · Minimum thickness · Structural form optimization · Purely rotational collapse mode
1 Introduction This work further investigates the issue of (symmetric) circular masonry arch form optimization (Couplet–Heyman problem), in the quest of a least-thickness evaluation under uniform self-weight (Figs. 1 and 2). The modern framing of such a problem
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Egidio Rizzi [email protected]
1
Politecnico di Milano, Dipartimento di Ingegneria Civile e Ambientale, piazza Leonardo da Vinci 32, 20133 Milan, Italy
2
Università degli studi di Bergamo, Dipartimento di Ingegneria e Scienze Applicate, viale Guglielmo Marconi 5, 24044 Dalmine, BG, Italy
123
708
Journal of Optimization Theory and Applications (2020) 187:707–757
Fig. 1 Sketch of a symmetric circular masonry arch subjected to self-weight (of specific weight per unit volume γ ) with all involved characteristic parameters (d: out of-plane depth of the arch)
Fig. 2 Five-hinge purely rotational collapse mode, with abutment-hinge reactions V and H (opposite of weight of half-arch and horizontal thrust)
relies in the contemporary contributions by Jacques Heyman [1–6] and the recent revisitation in earlier companion work [7], with therein extensive references, also to various historical and development perspectives on the subject. The present investigation belongs to a research project by the authors on the statics of masonry arches [7–14], where the following treatments have been attempted: analytical [7, 14]; analytical–numerical, accompanied by a Discrete Element Method (DEM) investigation, through an available Discontinuous Deformation Analysis (DDA) tool [8, 9, 15–17], and including reducing friction effec
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