Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26 357 I NTRODUCTION irst rational studies on the statics of masonry arches were blooming in 1700, with the development of so-called pre-elastic theories. In recent times, they have culminated, in the second half of 1900, in the modern reinterpretation and application of Limit Analysis to masonry constructions, according to the fundamental pioneering work by Jacques Heyman [1-4]. Specifically, Heyman stated three classical behavioural assumptions for masonry structures (1 - no tensile strength; 2 - infinite compressive strength; 3 - no sliding failure), and investigated the five-hinge purely-rotational collapse mode of continuous symmetric circular masonry arches under self-weight (taken as uniformly-distributed along the geometrical centreline of the arch, with self-weight per unit length w =  t d ), by providing analytical representations for the determination of the characteristic parameters of the masonry arch in the critical, least-thickness condition, such as (Figs. 1a,b): • angular position  of the rupture (radial) joint with inner intrados hinge at the haunches; • critical thickness to radius ratio  = t / r ; • non-dimensional horizontal thrust h = H /( w r ) acting in such a limit state of minimum thickness still available to sustain the arch ( Couplet-Heyman problem ). This classical least-thickness problem in the statics of masonry arches has been revisited by the present authors within a wide research project that has been considering different characteristic aspects, by employing both self-consistent analytical and numerical techniques [5-10]. Arches of a general half-angle of embrace 0 <  <  (including for under- complete and over-complete, horseshoe circular masonry arches) have been systematically analysed in analytical terms. Different solutions have been explicitly derived, and numerically explored, which appeared fully consistent with updated outcomes from a re-discussion by Heyman [4], and prior developments by Ochsendorf [11-12], as well as with classical earlier work by Milankovitch [13] (see Foce [14]), and several most recent attempts that meanwhile have appeared [15-24]. An earlier account on these developments was provided in SAHC10 conference paper [5]; later, a comprehensive analytical treatment with unprecedented closed-form explicit representations was provided in [6], while in [8], consistent comparisons were developed by a Discrete Element Method implementation, in the form of a Discontinuous Deformation Analysis (DDA) tool. Then, first new developments on the role of friction have been preliminarily investigated in such a research mainstream, as initially reported in SAHC12 conference work [7], prodromal to the present one, by releasing Heyman hypothesis 3 of no sliding failure, and accounting for both mixed sliding-rotational and purely-sliding collapse modes (Figs. 1c,d). Here, within that context, a new, complete, analytical treatment is developed, starting from the developed analytical solutions that had earlier been derived for purely-rotational collapse [6,8], with additional comprehensive numerical verification. Very recently, separate innovative numerical implementations as an optimisation problem to be solved by non-linear Mathematical Programming are being developed [9,10], with results that turn out truly consistent with the ones here derived and presented. Specifically, the limit values of Coulomb friction coefficient  = tan  (and friction angle  ), at the theoretical (radial) joints of a continuous arch , marking the transitions between the three collapse modes depicted in Figs. 1b-d are here explicitly derived, together with the determination of the analytical dependence of the mixed-mode collapse characteristics (Fig. 1c), as a function of friction coefficient  , namely  m (  ),  m (  ), h m (  ). It is newly shown that, at decreasing friction coefficient  at the (radial) joints of the arch, horizontal thrust h m (  ) under mixed mode becomes fixed (decreasing), by finite (reducing) friction. This induces, as a consequence, non-linear increasing dependencies of  (  ) and  (  ). In other words, at a decreasing friction coefficient at the joints of the arch, an increase in the critical value of least thickness is required to warrant the equilibrium of the self-standing masonry arch. Here, a main reference is made to complete semi-circular arches with a 2  =  opening (Fig. 1a), for a comprehensive illustration and discussion. For such a reference case, the friction range in which mixed-mode collapse is shown to appear turns out quite narrow, in practical terms, i.e. with friction angles  between around 22° and 17° (  = tan  between around 0.4 and 0.3). Despite, that the value of these findings shall appear to be more of a theoretical type (complete behaviour of a codified mechanical system), than of a practical nature (usually friction coefficients in masonry constructions shall safely be above that; arches are normally built with much higher margins in terms of thickness to radius ratios, with respect to the critical least-thickness condition; and so on), conditions that may be associated to a reducing friction, such as loosening joints, insertion of external materials among the blocks, non-firm or spreading supports, etc., may come up into the picture, leading to considerable interest in anyway reaching a full understanding on arch “stability” (say, equilibrium) at reducing F