# Safety Assessment of Masonry Arch Bridges Considering the Fracturing Benefit
## 3 Masonry Arch Bridge Assessment: Elastic-Fracture-Plastic Transitions  
Elastic and plastic analyses, currently exploited in order to evaluate the behaviour of arch or vault structures, often leave some doubts in the engineering practice. Elastic analysis, specifically [PERSON]'s method, is applicable to arch structures when the line of thrust lies within the inertial core of the arch cross-section, in order to prevent the onset of detrimental tensile stresses. [PERSON]'s studies, dating back to 1840, gained widespread recognition in the field of arch structures design: his method was based on the use of a graphic procedure in order to check the thrust line in agreement with the stress limitations identified by the theory of elasticity [29]. On the other hand, plastic theory, specifically the limit analysis, was firstly defined by [PERSON] [7] for masonry arch assessment. In his work, [PERSON] investigated arch structures using the principles of plastic design developed originally for steel frames. He applied in detail these principles to the analysis of the structural system of the gothic cathedral. The formation of a plastic hinge in masonry arches is acknowledged right where the thrust line is tangent to the arch edges. Three tangent points lead to the formation of three hinges: the limit to trigger a kinematic collapse mechanism generally lies in the formation of a fourth hinge [8]. Starting from these assumptions, none of the previous two analyses can capture the intermediate static regimes characterised by damage (see Figure 1): in other words, the elastic analysis describes the structure until the first non-linearity arises, whereas the plastic analysis defines the last condition of an arch structure before the final collapse occurs [10]. A more sophisticated analytical method is therefore needed, that allows the fracturing process between the elastic regime and the final collapse to be manifest [11]. In this context, a fracture mechanics-based analytical method with elastic-softening regime for masonry is suitable in order to study the global structural behaviour of masonry arches [19; 20; 21; 22]. An elastic-softening constitutive law for masonry is taken into account, exploiting an elastic constitutive law coupled with a fracturing process based on LEFM. As a matter of fact, crack initiation and growth in an arch structure with elastic-softening behaviour can influence the position of the line of thrust, redistributing the internal stresses similarly to an elastic-plastic structure when affected by the formation of plastic hinges [30]. In this way, the evolutionary analysis of fracturing processes assesses how the arch structural behaviour is affected by cracks formation, as well as by the internal stress redistribution, clarifying how the maximum admissible load evaluated by means of LEFM is larger than the load predicted by elasticity theory. Such an increase in terms of maximum admissible load can be defined \"fracturing benefit\", and it is analogous to the \"plastic benefit\" of the plastic limit analysis. On the other hand, the arch structure final condition determined by plastic analysis, i.e., the four hinges kinematic chain (Figure 1), can be foreseen by a fracture mechanics point of view: namely, while an uncracked cross-section of the arch structure carries out an internal action of perfectly fixed joint, a cracked cross section carries out an internal action of elastically fixed joint [19]. Such a stiffness loss can be represented by the so-called \"brittle hinge\" [11; 21; 30]. In the following, a comparison between elastic, plastic and fracture analyses of the structural behaviour of three monumental masonry arch bridges having different shallowness and slenderness ratios is performed. The purpose of the comparison is to highlight the different values of the maximum live load prior to failure that can be achieved using the three abovementioned analyses, with particular focus on the position of brittle hinges simulating the cracks, and the plastic ones triggering the kinematic collapse.