## 1 Introduction  
Following the well-known studies carried out by [PERSON] [1, 2], the mechanism model was widely used for the limit analysis of stone and masonry arches. [PERSON] particularized the limit design, developed for steel structures, to masonry arches made of rigid voussoirs that are laid dry without any mortar, with the assumption that the material has no tensile strength, but infinite strength in compression, and that sliding failure cannot occur. The model was developed based on the previous work performed by [PERSON] [3] that pointed out how stone arches fail by forming pin joints, as experimental studies also demonstrated [4]. The collapse of a stone arch is a geometrical issue rather than a problem of the strength of the material, as is already known from the studies of the eighteenth century performed by eminent scientists, including [PERSON], [PERSON], [PERSON] and [PERSON]. The limit analysis of voussoir arches was widely applied for the evaluation of stone arch bridges [5] and often preferred to a more sophisticated analysis [6, 7, 8, 9, 10] due to its simplicity and speed. The plasticity theorems allowed for the investigation of the collapse mechanisms and the influence of the geometrical and loading parameters on the limit behaviour [11]. Experimental tests confirmed the expected behaviour [12, 13]. The mechanism model was also used to analyse the dynamic behaviour of stone arches under seismic actions. [PERSON] [14] discussed the failure conditions of a circular arch under a horizontal longitudinal acceleration. [PERSON] [15] analysed the dynamic behaviour of an arch without backfill under sinusoidal base acceleration and focused on the influence of the frequency content and the amplitude of the input and the initial conditions. Experimental studies conducted on shaking tables confirmed the hypothesis that masonry arches fail by forming pin joints and allowed for the establishment of a correlation between the input acceleration and the damage in the structure [16]. Successive developments referred to the impact condition during oscillations for a circular arch, in the hypotheses that the hinge locations in the four-link mechanism do not vary, but they reflect when the motion is inverted [17]. This hypothesis was sometimes considered only for the internal hinges, while two hinges at the springing sections were fixed [18]. Recently, the dynamic identification of masonry arch bridges was proposed as a support technique to finite element modelling [19]. Finally, the behaviour of single and multi-span masonry arch bridges under seismic actions was also analysed [20; 21; 22; 23], and retrofit techniques were proposed [24; 25; 26]. The effects of an earthquake on an elastic structure are related to the correspondence between the resonance frequencies of the structure and the frequency content of the seismic acceleration at its base. Things are quite different for the model of an arch made of rigid voussoirs. Because of its infinite rigidity, a stone arch does not show any relative motion with respect to its springing points until the amplitude of the external load is sufficient to turn it into a mechanism. When the mechanism is put in action, the arch could return to its natural configuration after some oscillations. From a technical point of view, the ratio between the minimum acceleration necessary to turn the structure into a mechanism and the design peak ground acceleration at the site can be assumed as a safety index. In this way, the safety check of a masonry arch under seismic actions becomes a static issue. On the basis of these considerations and by using [PERSON]'s hypotheses, [PERSON] and [PERSON] [27] carried out a detailed analysis of stone voussoir arches under horizontal longitudinal loadings, determining the collapse mechanism and the related load factor. Both the parabolic and circular shapes were considered, subject to their self-weight and backfill.