Mario Como

Equilibrium and Collapse Analysis of Masonry Bodies

This paper gives a general formulation of the statics of the masonry continuum within the conceptual framework set up by J. Heyman in his fundamental and pioneering studies of masonry arches and vaults. Here the masonry body will be represented by an assemblage of rigid particles of stones, held together only by compressive forces, and liable to crack as soon as tensile stresses begin to develop. The very small size of the stones, compared to the overall dimensions of the body, permits a treatment in terms of a continuum.

Minimum Thrust of Rounded Cross Vaults

This study analyzes the static behavior of the rounded cross vaults based on the limit analysis approach developed for masonry structures and adopts a rigid no-tension constitutive model with no sliding. The kinematic theorem of the limit analysis with a compatible tridimensional mechanism is applied on these structures with the aim of evaluating the minimum thrust. In this way it was possible to build some abaci in which the ratio between the minimum thrust and the weight is plotted versus the geometrical characteristics of the vaults. Finally, the proposed abaci are used to calculate the thrust of the main vault of the Diocletian Baths in Rome.

Statics of Bodies Made of a Compressionally Rigid No Tension Material

The paper gives the basics of Statics of masonry bodies and extends to the continuum the compressionally rigid no tension model, usually applied to systems of arches and piers. Admissible stress, strains and admissible crack openings are defined together with some appropriate stress-strain inequalities. In this framework the virtual work equation is properly formulated by using a new simple approach that considers the new boundary of the body including the cracks surfaces. Then a condition on the loads, necessary and sufficient to the existence of the admissible equilibrium, is proven; this condition is very useful to deal with the collapse analysis. The paper ends pointing out some specific features of the behaviour of masonry structures, as the no existence of self stresses or the no existence of failures due to the occurrence of costraints settlements.

Rocking in presence of cracking of masonry wall piers

The wall pier represents the vertical element of multi-storey walls with openings, the main resistant structural components of a masonry building. Structural systems of wall piers and spandrels are required to sustain the in-plane seismic actions acting on the wall, opposing with their weights to the action of horizontal forces. The behavior of masonry constructions results to be very far from the one characterizing ductile structures, because of the lack of energy dissipation during the deformation. A strength resource of masonry structures, properly reinforced in order to avoid early local failures, consists in exhibiting rocking behavior, until a failure condition is attained. An investigation on the dynamic behavior of masonry wall piers is carried out by following Housner’s studies and properly introducing the effect of diagonal cracks, shown by typical post-earthquake cracking patterns. As a consequence, the system is characterized by the detachment of a lower triangular region that becomes ineffective during the development of the mechanism and does not oppose with its weight to the overturning. Finally, it is shown that the occurrence of diagonal cracks can be prevented by the execution of suitable retrofit interventions.

Collapse state of multi-storey masonry walls reinforced by steel ties subjected to in-plane horizontal loads

The determination of the seismic strength of masonry building is strictly connected to the in-plane strength of masonry walls under the action of horizontal forces. Simplified criteria are currently available in literature, based on modelling of the structure as loaded by dead loads and by a gradually increasing distribution of horizontal forces, proportional to the mass of the building. According to this approach, called push-over method, the seismic strength of the building corresponds to the intensity of these gradually increasing horizontal loads, leading the building to the failure condition. This paper moves in the framework of the Limit Analysis, based on the Heyman’s masonry model (1966), rigid in compression with no tensile strength. The resistant model refers to a multi-storey wall with openings arranged in regular patterns, along both vertical and horizontal directions, reinforced at floor levels by steel ties. The in-plane failure of the regular multi-storey walls can occur with the development of various kinematically admissible mechanisms, characterized by the attainment of the yielding state in the steel ties. The proposed methodology consists in the definition of the mechanism along which the failure effectively occurs and in a subsequent check of the statical admissibility of the internal stress state at the limit load. Only in this case, the corresponding kinematical multiplier is the effective collapse multiplier. The presence of the panels situated above the openings strongly conditions the in-plane failure of the wall, acting as diagonal struts, causing different horizontal displacements between the piers at the floor levels and consequently engaging the horizontal ties in the mechanism. In order to ensure the development of the global failure, avoiding local brittle failures, steel strengths of the ties have thus to be suitably defined. Finally, a parametric investigation is carried out considering different geometries of masonry walls and varying the position of the piers self-weights and the horizontal forces distribution, constant or proportional to the height of the masses from the foundation level.

Seismic Strength of Reinforced Multi-Storey Masonry Walls

Analysis of force transmission through the various structural components is needed to the full understanding of the seismic behavior of masonry buildings. It is in fact necessary to identify the weak links of the chain and define the essential reinforcements to insert in the structure. In this context the Paper analyzes the strengths of masonry walls under the action of out of plane and in plane horizontal forces and compares the systems of reinforcement of the walls that can use steel or Fiber Reinforced Polymers (FRP).

Large Structures Behaviour: the Past and the Future

The most relevant structures of the past centuries are the masonry structures, for instance large masonry arch bridges and monumental buildings. Today and in the future, cable supported bridges are the most challenging solutions to solve long span crossings

Cross and Cloister Vaults

The aim of this section is the study of Statics of cross and cloister vaults: they are related together in various respects, since from the point of view of their geometric generation. A brief introduction gives information about the historical development of these vaults: there are magnificent examples already in the Roman architecture. Membrane stresses in cross and cloister vaults are firstly thoroughly studied. The existence of tensile stresses is a necessary condition for the membrane equilibrium and cracking is thus nearly inevitable. A study of sliced models of both the vaults is thus pursued. In the sliced model of the cross vault, firstly proposed by Heyman, webs transmit vertical and horizontal loads to diagonal ribs or to groins that, in turn, convey these loads to piers and flying buttresses. A dual model for the cloister vault, that cracks along the diagonals and transmits loads on the side walls, is firstly proposed. A detailed analysis of the cracks patterns is conducted for both the vaults at their minimum thrust states. The chapter ends with the study of statics of some important examples of these vaults, as, in particular, the large cross vaults of the Diocletian Baths in Rome.

Fundamentals of Statics

Basics of Statics of masonry solids and structures are the subject of the chapter. Masonry behavior is strongly influenced by the dramatically lower strength in tension than in compression. Masonry structures can thus suffer cracks generating displacement fields, called mechanisms, which develop without any internal opposition of the material. Collapse can occur without any material failure. The Heyman masonry model, the idealized rigid in compression no tension material, is fruitfully assumed as basis of the approach followed in the chapter. The extension of this model to the masonry continuum is then developed. Strains and detachments occurring in a no tension masonry solid can thus obtain a suitable mathematical formulation together with the admissible equilibrium. A proper virtual work equation, that considers the boundary of the body including the crack surfaces, as a condition only on the loads, both necessary and sufficient to the existence of the masonry equilibrium, can be formulated. This last condition governs the collapse strength of masonry structures. The notion of the minimum thrust, from both static and kinematical approaches, is then introduced, widening the field of application of the Limit Analysis also to the study of the actual stress states. In this context, it follows that weight and geometry represent the essential elements in the strength of masonry structures. Further, it will be also proven that, if a structure under its own weight is stable, the k times magnified copy of the same structure will also be stable. This result, thoroughly discussed in the chapter, matches the so called theory of Proportions that has constantly ruled the Design in the history of Architecture. A critical analysis of the recent failure of the cathedral of Noto, in Sicily (Italy), useful to a better understanding of the above discussed mechanical concepts, ends the chapter.

Masonry Buildings Under Seismic Actions

This last chapter deals with the study of the seismic behaviour of historic masonry buildings. Starting point of the chapter is the remark that traditional masonry buildings have not been built to offer any resistance to horizontal actions. This is why most of the seismic damage occurs in old historic centres, as well as why there is currently such a great demand to determine the most suitable means to reinforce them. The first sections of the chapter are focused to point out that, contrariwise to steel or reinforced concrete structures, that can oppose the seismic action by using their ductility, masonry constructions don’t dissipate energy during their deformation, even if accompanied by cracks. If properly reinforced, to avoid early local failures, masonry constructions have the sole resource to escape the seismic action exhibiting rocking without failure, under alternate seismic action. A constant acceleration impulse, of a suitable duration, can represent the seismic action. A masonry pier wall, the basic resistant element of a masonry building, overturns under an acceleration impulse A o of suitable duration t o that turns out to be quite larger than the limit acceleration A L producing the statical collapse. The magnitude of the so-called reduced strength factor q = Ao/A L —the ratio between the above accelerations—can measure the actual capacity of the construction to follow the alternate seismic action exhibit rocking without overturning, over the whole duration of the quake. Due to the actual quite low values of this so defined reduced strength factor q, as shown in the chapter, the seismic protection of historic masonry constructions requires design criteria where strength has to be dominant. A first focal point is thus the analysis of the chain of transmission of the seismic forces along the resistant structure of the construction. The weak rings of this chain are thus pointed out: they are due to the natural lack of connection among the various components of the building structure. Suitable primary reinforcements, discussed in the chapter, have to be inserted in the structure to ensure that early local failures cannot occur. The out-plane and the in-plane strength of masonry walls is then evaluated, with new elaborations and inclusions. All the results presented have been obtained in the framework of the Limit Analysis of masonry structures, according to the approach followed in the book. Numerical examples and comparisons with Code prescriptions are given.