Analytical and numerical analysis on the collapse modes of least-thickness circular masonry arches at decreasing friction

Abstract

Departing from pioneering Heyman modern rational investigations on the purely-rotational collapse mode of least-thickness circular masonry arches, the hypothesis that joint friction shall be high enough to prevent inter-block sliding is here released. The influence of a reducing Coulomb friction coefficient on the collapse modes of the arch is explicitly inspected, both analytically and numerically, by tracing the appearance of purely-rotational, mixed sliding-rotational and purely-sliding modes. A classical doubly built-in, symmetric, complete semi-circular arch, with radial joints, under self-weight is specifically considered, for a main illustration. The characteristic values of the friction coefficient limiting the ranges associated to each collapse mode are first analytically derived and then numerically identified, by an independent self-implementation, with consistent outcomes. Explicit analytical representations are provided to estimate the geometric parameters defining the limit equilibrium states of the arch, specifically the minimum thickness to radius ratio, at reducing friction. These formulas, starting from the analysis of classical Heymanian instance of purely-rotational collapse, make new explicit reference to the mixed slidingrotational collapse mode, arising within a narrow range of limited friction coefficients (or friction angles). The obtained results are consistently compared to existing numerical ones from the competent literature. 1

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

“…The purely rotational collapse mode is anyway correctly reproduced (a high value of friction coefficient is set in the simulations, to avoid possible manifestations of any form of sliding [9], se also next subsection). This looks thus consistent with source Heyman hypothesis of no sliding failure, at the basis of the present theoretical treatment (finite friction effects are separately analysed in [10][11][12][13]). Figure 19 further gathers, on the top plot, the imposed values of angular inner-joint position β at variable half-opening values α of the arch, with fork values around CCR solution (Milankovitch solution is never much dissimilar, if not, a bit, for the last case of α 140°) and going through a maximum of β for an α at around 120°-130° [14].…”

“…142.504°, shall be prevented by μ ≥ μ lm 1.41527 (ϕ lm 54.7558°) [11], as also confirmed by the numerical analyses in [13].…”

“…Indeed, further subsequent investigations may consider different shapes of the masonry arch and possible true implications of the presence of finite friction, in terms of non-standard Limit Analysis, about the bearing capacity of the masonry arch and the resulting collapse mode, which may include sliding and other potential effects linked to the arising non-normality kinematics of the arch collapse, as initially investigated, analytically in [10,11], and numerically in [12,13].…”

“…The present investigation belongs to a research project by the authors on the statics of masonry arches [7][8][9][10][11][12][13][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][16][17], and including reducing friction effects and resulting mixed collapse modes [10,11]; analytical-numerical, by an innovative Complementarity Problem/Mathematical Programming formulation, truly accounting for finite friction implications [12,13]. This shall be framed within the relevant, updated literature, specifically considering the issue of minimum thickness evaluation , and dedicated attempts to reveal further and independent/complementary information on the analytical treatment, with additional separate validation in terms of numerical results.…”

Least-thickness symmetric circular masonry arch of maximum horizontal thrust

“…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).…”