Static Upper/Lower Thrust and Kinematic Work Balance Stationarity for Least-Thickness Circular Masonry Arch Optimization

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.

Least-thickness symmetric circular masonry arch of maximum horizontal thrust

“…Together with angular inner-hinge position β from the crown (0 ≤ β ≤ α, α half-opening angle of the symmetric circular arch), they form characteristic solution triplet (β, η, h) of least-thickness "Couplet-Heyman problem", and the attached determination of the associated purely rotational collapse mode (warranted by high friction, up to prevent any sliding), at variable α. The underlying governing equations linked to the analysis of the "Couplet-Heyman problem" of leastthickness optimization, in sought solution characteristic triplet (β, η, h), may be expressed, in terms of variable h, with appropriate on/off control δ flags, up to shift among possible different treatments, and attached solutions [7,9,11,15]:…”

“…Notice that g ≥ 0 for β ≥ 0; also, g − f = (1 − C)(β − S) ≥ 0 and f + g = (1 + C)(β + S) ≥ 0 for β ≥ 0. Functions f and g both vanish at β = 0 (see illustration in [15] and limit implications in [7,15]). The meaning of the stated governing equations goes as follows.…”

“…Within a previous mainstream of work by the present authors [7][8][9][10][11][12][13][14][15], such a classical optimization problem in the Mechanics (statics) of symmetric circular masonry arches has been revisited, and framed within the relevant, now updated, literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], by providing new, explicit analytical derivations and representations of the solution characteristics. Specifically, classical Heyman solution [3] has been shown to constitute a sort of approximation of the true solution (here labelled as "CCR" [7]), in Heyman assumption of uniform selfweight distribution (location of the centres of gravity along the geometrical centreline of the circular arch), while Milankovitch solution [35][36][37] may as well be consistently derived, in the consideration of the true self-weight distribution, though at the price of an increasing complexity in the explicit analytical handling of the governing equations (here newly resolved to a very end).…”

“…As per the treatment adopted in the paper, namely a full analytical one, the present study is framed between similar related investigations that have enquired the problem by analytical or semi-analytical approaches, between which specific important developments that shall be mentioned in sharing the vision and perspectives here pursued are those concisely outlined as follows: pioneering and fundamental Heyman contributions [1][2][3][4][5][6]; attempts by the present authors [7][8][9][10][11][12][13][14][15], specifically: analytical [7], analytical-numerical, accompanied by a Discrete Element Method (DEM) investigation (Discontinuous Deformation Analysis, (DDA) [8,9,15], and including reducing friction effects and resulting mixed collapse modes [10,11]; analytical-numerical, by a Complementarity Problem/Mathematical Programming formulation truly accounting for finite friction implications [12][13][14][15]; other studies on mainstream arch derivation and masonry arch mechanics after Heyman, specifically considering the issue of least thickness [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]…”

“…Thus, the here-mentioned literature setting does not seek an intent of completeness, just that of focusing on selfcoherent formulations that may consider further related, though different and complementary features, such as: different geometrical shapes of the masonry arch [22,25,[30][31][32]34,39,41], issues of "stereotomy" (i.e., the shape of the cut of the masonry blocks; here, a continuous circular arch with just potential rupture radial joints is considered) [6,28,33], dedicated use of "thrust-line analysis" or graphical-analytical methods, and so on, though all sharing the common characteristics of seeking and setting an analytical method of analysis, or being linked to the optimization target of acquiring the least-thickness condition. Numerical methods, which are massively employed in the analysis of masonry structures, including for masonry arches (vaults and domes), shall also be pursued and mentioned (see, for instance, about the adoption of the Discrete Element Method for the evaluation of the least-thickness condition [8,9,15,30,31], and therein quoted references), but these are here intentionally taken out from an explicit discussion. The study also focusses on the statics of masonry arches, not on the dynamics, which as well constitutes an important, though separate, topic, typically tied to numerical methods of analysis, and just for the self-supporting static condition under self-weight, though other external conditions or loadings may apply and be considered (e.g., lateral, as it may be linked to dynamical, seismic excitation conditions, for instance) [21,24,26,32].…”