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 arc 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 sliding-rotational 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.
Highlights
The critical value of friction coefficient up to induce possible sliding may increase with the opening angle of the arch, possibly approaching the ranges of friction coefficients that may be encountered in practice and maybe leading to the potential appearance of sliding within the failure mode
There, the constitutive behaviour of the circular masonry arch is stated and equilibrium conditions are varied towards the determination of the critical least-thickness condition and of the attached collapse mode
S tarting from the classical analysis of purely-rotational collapse of circular masonry arches in the so-called Couplet-Heyman problem [1,2,3,4], the determination of collapse characteristics, h for a symmetric circular arch of general half-angle of embrace (Figs. 1a,b) may be stated by the solution of the following system of three characteristic equations, in terms of unknown non-dimensional horizontal thrust variable h [5,6,8]:
Summary
A n immediate, self-implemented numerical algorithm for the individuation of the collapse modes of symmetric circular masonry arches has been further created within commercial spreadsheet software Excel, to independently inspect and validate the previous analytical outcomes on the arch’s collapse characteristics, as revealed at reducing friction It makes use of an optimisation function named “Solver”, which allows for the selection of a GRG (Generalised Reduced Gradient) engine, towards the solution of smooth non-linear optimisation problems. Geometrical: arch width d, mean radius r (both nominally fixed to 1 m); material: limit tension stress t (set to zero, according to Heyman hypothesis 1), limit compression stress c (set to 1000 kN/m2, i.e. a high value apt to comply with Heyman hypothesis 2), variably-fixed friction coefficient , weight per unit volume (set to 25 kN/m3) Based on these input data, the trends of internal actions N( ), T( ), M( ) along the arch are recovered by equilibrium and confronted to the limit values that define section resistance, in terms of shear force T and moment M.
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