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This article describes a methodology for the non-linear analysis of existing masonry structures subjected to external yielding constraints, with particular attention to the historical and cultural heritage constructions. It is well known, indeed, that most of the arch and wall damages are often due to settlement of abutments, in the former case, and to settlement of foundations, in the latter one. The ability to observe and correctly analyze the cracking failure pattern, visible on such structures, is the main “diagnostic tool” for identifying its origin: the modification of load conditions over time, foundation settlements and earthquakes. The objective of this work is to identify a numeric modelling of masonry structures (such as walls, arches, vaults, ruins) under any load condition and subjected to inelastic settlements impressed to some external constraints. The purpose of the numerical procedure is to interpret the behaviour of such structures in order to assess both the peak settlement value and their specific failure mode in correspondence to a geometry which is very often compromised. Therefore, this procedure allows one to estimate the degree of the structures’ vulnerability, in order to prevent any future damage, both local and global. The iterative algorithm proposed in this article, developed in a calculation software, processes the structure considering, not only the properties of constitutive material, non-homogeneous and anisotropic, but also the change of the structure’s shape during the settlements increase. In this way a non-linear analysis is performed both materically and geometrically. Through a direct comparison between numerical and experimental results, obtained by testing some simple structural models in a laboratory, it was ascertained, both from a qualitative and quantitative point of view, the correctness and the efficacy of the proposed procedure, which will be explained below. Therefore, this numerical procedure demonstrates to be a useful “diagnostic tool” by which, starting from the input of the masonry structure to be studied and simulating a presumable event, one can trace the source of the causes that have generated a certain failure, comparing the cracking pattern of real structure with that plotted by the software.

A masonry structure is analyzed by means of a discrete model, previously proposed in other works, according to the early static theories of the mid-1700’s describing the behaviour of masonry arches. Such a model is still considered by J. Heyman [

Moreover, according to such a model, the shape of single blocks does not change when external loads are applied, so as their rigid displacements, due to detachments and/or sliding along the joints, which the contact devices allow, are prevailing in comparison with the deformability of the whole structural system. The main features of such a model consist of the inner coherence of the blocks and the presence of “discontinuity” surfaces along which such a coherence may also not be respected: the constitutive material is not able to support tensile stresses along the surfaces where its “inner structure” is detached, while it is coherent with and uncompressible in the other parts. Under these assumptions, the mechanical characterization of masonry, that is its generic constructive configuration, being both an arch and a wall, refers to a system of rigid blocks connected by unilateral contact and frictional [

As previously stated, the peculiar characteristic of the discrete model is the description of the masonry building by means of a rigid system in which the elasticity is totally thought to be concentrated in the mortar joints (such an approach has been proposed and developed in several works by the authors themselves [

The first model is designed to do a limit analysis of the structures; the second one, instead, for a kind of analysis which is able to furnish the actual stress state inside the structure, the position of the failure interfaces and also identify the shape and dimensions of the cracks.

Thus, the contact devices are described by a set of fictitious links, arranged orthogonal to the interface surfaces, capable of transmitting only compressive forces or, at most, weak tensile forces which do not exceed the assigned limit values, and, by an additional link, tangent to the interface surface, to transmit the shear force. In case of brittle-rigid joint only two normal links are strictly necessary. In the case of elastic-cracking joint it is better to consider at least four normal links in order to highlight the actual cracking pattern with the possibility of measuring the width and depth of the cracks inside the mortar joints.

Let us consider, therefore, the general problem of any masonry structure composed of a finite number of rigid blocks which are dry assembled or mortar layered. The system composed of n blocks and m interfaces, subjected to a load condition (represented by vector F) and inelastic displacements (represented by vector Δ) which are only located in the external joints, can be expressed through a system of equilibrium and elastic-kinematical equations, as follows:

sub, (1)

in the case of brittle-rigid contact joint, and:

sub, (2)

in the case of elastic-cracking contact joint.

In Equations (1) and (2) the coefficients of the subvectors and of vector represent the unknowns of the interface interactions related to the normal forces and to the tangential ones, to which the in equalities [

Referring to both the former and to the latter kind of joints, the solution () and the relevant distortions vectors [] or [] are computed by an iterative procedure whose starting point corresponds to the static solution obtained making use of the generalized inverse

of the equilibrium matrix [], whose uniqueness was demonstrated by Moore-Penrose, in the case of brittle-rigid model; while in the elastic-cracking model, the starting point corresponds to the standard solution obtained assuming the material is linear-elastic and bilateral. Thus, in the case of the brittle-rigid joint, it is possible to write:

; (3)

while in the case of elastic-cracking joint the equation is:

It is interesting to note that the initial solution [], reached under the hypothesis of brittle-rigid joint, corresponds exactly to the initial solution obtained under the assumption of strained joint in the case where the strain matrix [] is replaced by the identity matrix, that is assuming a constant und unity stiffness of all the links present in the model of the structure.

Remembering that in the case of any compatible linear system, the whole system of solutions is expressed by, where is an arbitrary vector suitably chosen while is any inverse able to satisfy the condition [

in the case of brittle-rigid joint, and:

in the case of elastic-cracking joint.

The initial solution vector [] can result whenever it is coherent with the sign conditions expressed through the in Equalities (1) and (2): such a circumstance corresponds to the particular case of a masonry structure, subjected to any external action, whose joints are all compressed and satisfy the hypotheses about the friction. If any of the components of vector [] do not satisfy the imposed conditions, such a vector is then modified by an iterative procedure in which the choice of the suitable components of vector [] or [] follows the criterion, step by step, in search for the link with the highest tensile value; the vectors themselves preserve the values previously assumed. Then, the iterative procedure goes on searching for the tangential link whose force is greater than the friction force [

The choice of an iterative procedure represents, de facto, an effectiveness shortcut to the actual difficulty in finding the number and the index of the components of the distortions vector through the mathematically direct withdrawing of the full rank sub-matrix from the algebraic operator [], assuming it exists. To update the components of the distortions vectors, in both kinds of joint, the forms: or

is used at every step; in which indicates the vector whose components correspond to the internal forces in the joints which do not satisfy the conditions (1) and (2) above, while represents both the limit value for the tensile force in the normal links (L = 0) and the friction force for the tangential link.

The final static solution of the numerical procedure will provide a final vector solution in the form:

where, in correspondence of all the joints, it is:

and.

Such a solution, as computed above described, is the final one in the case of brittle-rigid joint and represents the limit equilibrium configuration of the structure. The nonzero components of vector [] correspond to the portions of the joints which still preserve the mutual contact and through which the interactions, which agree with the assumed hypotheses, are transferred. On the contrary, the zero components identify the portions in which the contact has been lost.

In the case of strained joint, since the final vector [] is generally different from the initial one [], it is necessary to restore the elastic-kinematical compatibility expressed through the second equation of system (2), keeping in mind the actual detachments in the interfaces. To restore the compatibility easily, it is better to consider a partition of all the matrices related to the system of elastickinematical equations; only the indices of the links whose normal and tangential component agree with the sign conditions at the end of the iterative procedure are considered. In this case, the vector containing the displacements of the centroids of the blocks, corresponding to the actual reacting structure, and the vector [] which is able to restore the compatibility of the system, are obtained through the following equations:

where the non-zero components of vector [] provide the position and the depth of the cracks.

The above described calculation procedure, while operating within the intrinsic nonlinearity characterizing the general problem of structures with unilateral constraints, refers to the initial geometric configuration of the masonry structure used to formulate both the equilibrium and compatibility equations. Only at the conclusion of the analysis, starting from the initial configuration, it will be meaningful to define a modified configuration of the structure which considers the joint strains and the possible dislocations between elements due to the presence of fractures.

On the contrary, in the case of behaviour analysis of masonry structures subjected to external inelastic displacement constraints, assuming they are applied by finite values, the geometrical nonlinearity, deriving from the modification of the initial geometry due to the subsequent increase of such displacements, should also be considered.

With the aim of defining the degree of vulnerability of a masonry structure [

The general procedure described in the previous paragraphs is still valid if considered inside a further process which depends on the subsequent increase of the nonzero components of vector [] of the inelastic settlements. At every step of such a procedure the general configuration matrix [] coefficients are updated as a function of the components of the displacements [] of the centroids of the blocks obtained in the previous step of the analysis. So it is possible to write:

Obviously such an equation can be used only in the case of structures whose joints are deformable [

It is not difficult to single out as belonging to this latter case the simple arch structure made of stone blocks and placed in mutual contact without interposition of mortar, which is in an initial configuration of stable equilibrium. As it is known, any settlement, even if small, allocated in correspondence with one of the abutments [

In the following, some numerical examples of masonry structures subjected to external settlements are described. The former series refers to the simple case of an arch, composed of rigid blocks placed in mutual contact without the presence of elastic joints, initially stable under an imposed load condition (

tlement, for both cases is perfectly verified, with a slight decrease (not a significant percentage) of the values in the case of the actual model due, of course, to the inevitable imperfections of the model itself.

The solution obtained for the same cases using the calculation model with elastic-cracking joint (assuming in this case a joint of minimum thickness and with very low deformation) provides the same results as the previous solution, approaching even more to the experimental result. This is probably due to the minimum deformation assumed in the joints which, from a physical point of view, can be compared with the influence of imperfecttions present in the actual model.

The latter group of examples deals with the analysis of the behaviour of a series of voussoir arches, subjected simply to self-weight, for which the spam and the number of blocks have been left unchanged, while the average radius of curvature and the angle of sets have been progressively modified, so as to obtain a decreasing rise.

Using the calculation model with elastic-cracking interfaces assuming self-weights, elastic moduli of the joint and geometry (whereand thick = 40 cm) compatible with a realistic condition, the peak horizontal settlements related to each configuration have been computed.