Computations on historic masonry structures

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KEYWORDS: structural analysis, masonry, historical constructions, ... Telephone: +351 253 510209, fax: +351 253 510217, email: [email protected].
Computations of historical masonry constructions Paulo B. Lourenço1 University of Minho, Department of Civil Engineering, Guimarães, Portugal

ABSTRACT: Is modeling and analysis of historical masonry constructions needed? Is the experimental behavior of historical masonry constructions known? The answers seem to be yes and substantial developments occurred during the last decades in the challenging issues of conservation and restoration. A key issue is what type of analysis should be used. It seems to be possible to state that all methods are of interest, depending on the actual constraints of the engineering problem to solve. In this paper, the possibilities of analysis of historical constructions are addressed and a set of guidelines is proposed.

KEYWORDS: structural analysis, masonry, historical constructions, restoration

1

PhD, Associate Professor, Department of Civil Engineering, University of Minho, Azurém, 4800-058

Guimarães, Portugal. Telephone: +351 253 510209, fax: +351 253 510217, email: [email protected].

INTRODUCTION Time shows that many historical masonry constructions collapsed due to accidental actions, like earthquakes. Nevertheless, not only exceptional events affect historical constructions. Fatigue and strength degradation, accumulated damage due to traffic, wind and temperature loads, soil settlements and the lack of structural understanding of the original constructors are high risk factors for the architectural heritage. The analysis of historical masonry constructions is a complex task. Firstly, limited resources have been allocated to the study of the mechanical behavior of masonry, which includes non-destructive in situ testing, adequate laboratorial experimental testing and development of reliable numerical tools. Secondly, and most important, the difficulties in using the existing knowledge are inherent to the analysis of historical structures. Usually, salient aspects are: - Geometry data is missing; - Information about the inner core of the structural elements is also missing; - Characterization of the mechanical properties of the materials used is difficult and expensive; - Large variability of mechanical properties, due to workmanship and use of natural materials; - Significant changes in the core and constitution of structural elements, associated with long construction periods; - Construction sequence is unknown; - Existing damage in the structure is unknown; - Regulations and codes are non-applicable.

Conservation and restoration of historical constructions are disciplines that require specific training. The continuous changes in materials and construction techniques, that swiftly moved away from traditional practice, and the challenging technical and scientifical developments, which make new possibilities available for all the agents involved in the preservation of the architectural heritage, are key aspects in the division between the science of construction and the art of conservation and restoration. Modern principles of intervention seem to include aspects like: - Removability, as reversibility seems to be an outdated concept; - Unobtrusiveness, minimum repair and respect by the original conception; - Safety of the construction; - Durability and compatibility of the materials; - Balance between cost and available financial resources. The consideration of these aspects is complex and calls for qualified analysts that combine advanced knowledge in the area and engineering reasoning, as well as a careful, humble and, usually, time-consuming approach. Several methods and computational tools are available for the assessment of the mechanical behavior of historical constructions. The methods resort to different theories or approaches, resulting in: different levels of complexity (from simple graphical methods and hand calculations to complex mathematical formulations and large systems of non-linear equations), different availability for the practitioner (from readily available in any consulting engineer office to scarcely available in a few research oriented institutions and large consulting offices), different time requirements (from a few seconds of computer time to a few days of processing) and, of course, different costs. It should also be expected that results of different approaches might also be different, but this is not a sufficient reason to prefer one method from the other. In fact, a more complex analysis tool does not necessarily provide better results than a simplified tool. Key aspects to be considered include:

- Adequacy between the analysis tool and the sought information; - Analysis tools available to the practitioner involved in the project (it is of fundamental importance that the available engineering is compatible with the analysis tools); - Cost, available financial resources and time requirements. This paper addresses the possibilities of analysis of historical constructions, being advocated that most techniques of analysis are adequate, possibly for different applications, if combined with proper engineering reasoning. It is noted that only very recently the scientific community began to show interest in modern advanced testing (under displacement control) and advanced tools of analysis for historical constructions. The lack of experience in this field is notorious in comparison with more advanced research fields like concrete, soil, rock or composite mechanics. Therefore, it is also shown that a complete set of displacementcontrolled tests can be carried out, in order to obtain the properties necessary for advanced numerical models. With this information, the sophisticated and robust models for masonry structures that are currently available can be successfully used for the analysis of historical constructions.

BASIC CONCEPTS Architecture is a recent activity when compared to the history of mankind, approximately 10.000 years old [1]. The development of architecture is linked to the earliest civilization and simultaneously masonry arises as a building technique. The primitive savage endeavors of mankind to secure protection against the elements and from attack included seeking shelter in rock caves, learning how to build tents of bark, skins, turfs or brushwood and huts of wattleand-daub. Some of such types crystallized into houses of stone, clay or timber, materials that have been used as the two main materials available for building up to one hundred years ago. The evolution of mankind is thus linked to the history of architecture and the history of

building materials [2]. Due to the ephemeral nature of wood, masonry structures represent the vast majority of the architectural heritage of today. The mathematical development of structural analysis and the scientifical, technical and industrial instruction of architects and engineers occurred around from the middle eighteenth century (in France) to the second half of the nineteenth century (in other countries) [3,4]. Until this date the difference between what we call today “Architecture” and “Engineering” was blurred. Therefore, in the field of the architectural heritage, it seems debatable and inconsistent to dissociate architecture from engineering.

Structural analysis through time (“Ars sine scientia nihil est”) Structural design must balance the realities of construction practices and the discipline of structural engineering. The former is largely empirical, based on experience gained in building and the skills of the building crafts. The latter, usually expressed in mathematical terms, is founded on theoretical knowledge, experience and the profession’s responsibility for public safety. With this last concern, the wisdom of the engineer and decisions of governmental institutions define load values as well as partial safety factors for loads and materials. No such complexity is to be found in ancient times, when empirical knowledge of building crafts, taught by master to apprentices, provided the tradition and theory on which structural design was based. The Roman architect Vitruvius in his Ten Books of Architecture [5] compared the qualities of stone taken from different quarries and the wood of different trees. Although he wrote at length about the traditional proportions of columns and the spaces between them, little was said of structural considerations. Medieval masons during their apprenticeship were introduced to the geometrical techniques required to lay out plans and prepare the templates and models from which stonework would

be cut. The traditional methods and rules-of-thumb of their craft were imparted to them as part of the mysteries of the mason’s lodge and all guild members were sworn secrecy. The transformation of the massive stonework of Romanesque architecture into the delicate tracery of the Gothic presents clear evidence of the powerful logic of the trial and error methods employed by the medieval builders. To modern eyes, it is a breathtaking triumph of skill over probability. The documentary evidence from the cathedrals of Troyes and Milan disconcertingly shows that any coherent advanced plan was in practice ignored, and medieval buildings could apparently withstand substantial alterations to the fabric both during construction and after an interval of many years, see [6]. In Milan, Jean Mignot delivered the aphorism that made him immortal: Ars sine scientia nihil est. By this he meant that the practice (ars) of masonry is nothing without the theoretical knowledge (scientia), in this case the geometry. While craft traditions had sufficed for the remarkable traceries of the Gothic construction, theoretical explanations were sought in the Renaissance, see [7]. Leonardo da Vinci could have been the first to contend that the thrust followed a path that remained within the arch, but much of the lengthy study that commenced in the Renaissance focused on the construction of domes. In experiments, chains were draped to represent the curves that might be the inverted lines of thrusts and intricate graphic solutions attempted to follow forces from stone to stone. Long after Leonardo, in 1586, Simon Stevinus published a book on statics; its translation to Latin in 1608 as “Mathematicorum Hypomnemata de Statica” made his knowledge accessible to scientists and mathematicians throughout Europe, and provided the basis for nineteenth century in graphic statics, which enabled the solution of structural problems through drawings.

A BRIEF NOTE ON EXPERIMENTAL MODERN TESTING Masonry is a heterogeneous material that consists of units and joints. Units are such as bricks, blocks, ashlars, adobes, irregular stones and others. Mortar can be clay, bitumen, chalk, lime/cement based mortar, glue or other. The huge number of possible combinations generated by the geometry, nature and arrangement of units as well as the characteristics of mortars raises doubts about the accuracy of the term “masonry”. Still, the mechanical behavior of the different types of masonry has generally a common feature: a very low tensile strength. This property is so important that it has determined the shape of ancient constructions. The difficulties in performing advanced testing of ancient structures are quite large due to the innumerable variations of masonry, the variability of the masonry itself in a specific structure and the impossibility of reproducing it all in a specimen. Therefore, most of the advanced experimental research carried out in the last decades has concentrated in brick / block masonry and its relevance for design. Accurate modeling requires a thorough experimental description of the material. Next, special attention is given to displacement-controlled tests and the reader is referred to [8,9] for a more comprehensive discussion on these issues. A basic notion is softening, which is a gradual decrease of mechanical resistance under a continuous increase of deformation forced upon a material specimen or structure. It is a salient feature of soil, brick, mortar, ceramics, rock or concrete, which fail due to a process of progressive internal crack growth. For tensile failure this phenomenon has been well identified [10]. For shear failure, a softening process is also observed, associated with degradation of the cohesion in Coulomb friction models [11]. For compressive failure, softening behavior is highly dependent upon the boundary conditions in the experiments and the size of the specimen [12,13]. Experimental data seems to indicate that both local and continuum fracturing processes govern the behavior in uniaxial compression.

Properties of unit and mortar The properties of masonry are strongly dependent upon the properties of its constituents. Compressive strength tests are easy to perform and give a good indication of the general quality of the materials used. Experiments about the uniaxial post-peak behavior and about the biaxial behavior of bricks and blocks are lacking in the literature. It is also difficult to relate the tensile strength of the masonry unit to its compressive strength due to the different shapes and materials. Extensive information on the tensile strength and fracture energy of units can be found in [11,14]. For the mortar, standard test specimens are cast in steel moulds and the water absorption effect of the unit is ignored, being thus non-representative of the mortar inside the composite. Recently, investigations in mortar disks extracted from the masonry joints have been carried out in order to fully characterize the mortar behavior [15,16].

Properties of the unit-mortar interface Bond between unit and mortar is often the weakest link in masonry assemblages. The nonlinear response of the joints, which is then controlled by the unit-mortar interface, is one of the most relevant features of masonry behavior. Two different phenomena occur in the unitmortar interface, one associated with tensile failure (mode I) and the other associated with shear failure (mode II). Different test set-ups have been used for the characterization of the tensile behavior of the unit-mortar interface. These include (three-point, four-point, bond-wrench) flexural testing, [17], diametral compression (splitting test) [18] and direct tension testing [11]. For the purpose of numerical simulation, direct tension testing should be adopted because it allows for the full representation of the stress-displacement diagram and yield the correct strength value.

A discussion about the adequacy of different test configurations for shear testing will not be given here and the reader is referred to [19,20] for this purpose. Different test set-ups have been used, including direct shear or couplet testing [21] and triplet tests [22]. To obtain postpeak characteristics, the stress normal to the bed joint must be kept constant and the couplet test is more appropriate for displacement control. Adequate characterization of masonry shear behavior for numerical purposes is given in [11,19].

Properties of the composite material The compressive strength of masonry in the direction normal to the bed joints has been traditionally regarded as the sole relevant structural material property. The RILEM test [23] seems to return the true uniaxial compressive strength of masonry. Since the pioneering work of Hilsdorf [24] it has been accepted by the masonry community that the difference in elastic properties of the unit and mortar is the precursor of failure. Uniaxial compression tests in the direction parallel to the bed joints have received substantially less attention from the masonry community. For tensile loading perpendicular to the bed joints, masonry strength can be generally equated to the tensile bond strength between the joint and the unit, or the tensile strength of the unit, whichever is the lowest. For tensile loading parallel to the bed joints, a sophisticated direct tension test program was set-up [25], where two different types of failure have been obtained: stepped cracks through head and bed joints or cracks running almost vertically through the units and head joints. The influence of the biaxial stress state has been investigated up to peak stress to provide a biaxial strength envelope, which cannot be described solely in terms of principal stresses because masonry is an anisotropic material. Basically, two different test set-ups have been

utilized, uniaxial compression oriented at a given angle with respect to the bed joints [26] and true biaxial loading at a given angle with respect to the bed joints [27,28].

STRATEGIES FOR THE NUMERICAL MODELING OF MASONRY STRUCTURES Masonry is a material exhibiting distinct directional properties due to the mortar joints, which act as planes of weakness. In general, the approach towards its numerical representation can focus on the micro modeling of the individual components, viz. unit (brick, block, etc.) and mortar, or the macro modeling of masonry as a composite [29]. Depending on the level of accuracy and the simplicity desired, it is possible to use the following modeling strategies, see Figure 1. •

Detailed micro-modeling - units and mortar in the joints are represented by continuum elements whereas the unit-mortar interface is represented by discontinuum elements;



Simplified micro-modeling - expanded units are represented by continuum elements whereas the behavior of the mortar joints and unit-mortar interface is lumped in discontinuum elements;



Macro-modeling - units, mortar and unit-mortar interface are smeared out in a homogeneous continuum.

In the first approach, Young's modulus, Poisson's ratio and, optionally, inelastic properties of both unit and mortar are taken into account. The interface represents a potential crack/slip plane with initial dummy stiffness to avoid interpenetration of the continuum. This enables the combined action of unit, mortar and interface to be studied under a magnifying glass. In the second approach, each joint, consisting of mortar and the two unit-mortar interfaces, is lumped into an average interface while the units are expanded in order to keep the geometry unchanged. Masonry is thus considered as a set of elastic blocks bonded by potential

fracture/slip lines at the joints. Accuracy is lost since Poisson's effect of the mortar is not included. The third approach does not make a distinction between individual units and joints but treats masonry as a homogeneous anisotropic continuum. One modeling strategy cannot be preferred over the other because different application fields exist for micro- and macromodels. Micro-modeling studies are necessary to give a better understanding about the local behavior of masonry structures. This type of modeling applies notably to structural details. Macro-models are applicable when the structure is composed of solid walls with sufficiently large dimensions so that the stresses across or along a macro-length will be essentially uniform. Clearly, macro modeling is more practice oriented due to the reduced time and memory requirements as well as a user-friendly mesh generation. This type of modeling is most valuable when a compromise between accuracy and efficiency is needed. It is noted that different levels of sophistication can also be adopted to create structural models. Next, a brief revision is made regarding analytical models using structural component models (a macro-modeling approach), finite element continua structural models (a macromodeling approach) and discontinuum structural models (a micro-modeling approach).

Structural component models The simplest approach to the modeling of complex historical buildings is given by the application of different structural elements resorting to truss, beam, panel, plate or shell elements to represent columns, piers, arches and vaults, with the assumption of homogeneous material behavior. Figure 2 illustrates various possibilities to model a wall with openings using structural component models, namely a lumped mass approach, a beam approach and a panel macromodel, see also [30]. The lumped approach or mass-spring-dashpot model of Figure 2b is at best a crude approximation of the actual geometry of the structure, using floor levels and

lumped parameters as structural components. The simplicity of the geometric model allows increased complexity on the loading side and in the non-linear dynamic response. It can be used to determine overall dynamic structural response to actual earthquake ground motion input but rely heavily on the correct definition of component hysteresis, which has to include material non-linearity and also effects resulting from the true geometry of the structure. Such a model cannot be used to predict local or global failure mechanisms or damage levels in individual structural components. The structural component model in Figure 2c approximates the actual structural geometry more accurately by using beams and joints as structural components. This approach allows to assess the system behavior with more detail. In particular, it is possible to determine the sequential formation of local, predefined failure mechanisms and overall collapse, both statically and dynamically. The increased geometric complexity (associated with a larger number of degrees-of-freedom) makes the use of non-linear dynamic time history analysis unwieldy. Many computer codes are readily available for the use of truss and beam elements, either in 2D or 3D. Simplified collapse load analysis can be carried out, such as in [31] for the analysis of infilled masonry frames. Rigorous non-linear behavior of the structural elements may be included [32], based on the use of curved 3D members of variable cross-section to analyze spatial skeletal structures, with elasto-plastic behavior in the compressive range and elastic-brittle tensile response, see also Figure 3 for an application [33]. Finally, the structural model in Figure 2d approximates the actual structural geometry even more accurately by panel macro-elements as structural components. Rigid or deformable macro-elements have been used extensively for modeling walls and wall panels, resulting in an overall 2D or 3D model with moderate number of degrees-of-freedom. Various formulations have been proposed, namely [34], in which square damageable rigid macroelements are adopted for incremental in plane loading, and [35], in which rigid blocks of

variable form are adopted for kinematic limit analysis. This last approach has a tremendous potential for simplified analysis and strengthening design of historical constructions in seismic areas, see Figure 4, and the reader is referred to [36] for a recent state of the art and further references. Even if important developments occurred regarding consistent formulation of compatibility and equilibrium along element boundaries, as well as, regarding validation of the proposed tools with respect to valuable experimental data, inadequate selection of structural component models might yield unacceptable large errors. Thus, it must be stressed that a lot of engineering intuition, structural understanding and experimental data must be employed to formulate and use reliable structural component models and associated analysis tools.

Finite element models for continua (Macro-modeling) The difficulties related to a suitable representation of historical constructions in terms of structural component models, leads to the use of 2D and 3D finite element models. In fact, the complex geometry of historical structures involves usually massive structural parts, such as piers and buttresses, combined with arches and vaults. The higher computational effort required makes these models more appropriate for partial models, but rather large meshes are feasible with present computational tools, being the St. Marks’ Basilica in Venice a wellknown case-study [37]. Figure 5 illustrates a large 3D finite element model adopted for nonlinear static analysis of a building compound [38]. A large number of man-months is usually required for geometric discretization, analysis and processing of results. Several constitutive models have been developed for this purpose, mostly based on homogenization procedures or experimental data. Difficulties of conceiving and implementing macro-models for the analysis of masonry structures arise especially due to the fact that almost no comprehensive experimental results

are available (either for pre- and post-peak behavior), but also due to the intrinsic complexity of formulating anisotropic inelastic behavior. Only a reduced number of authors tried to develop specific models for the analysis of masonry structures [39-42]. Formulations of isotropic quasi-brittle materials behavior consider, generally, different inelastic criteria for tension and compression. The model introduced in [41], recently extended to accommodate shell masonry behavior [43], combines the advantages of modern plasticity concepts with a powerful representation of anisotropic material behavior, which includes different hardening/softening behavior along each material axis. Figure 6 shows the results of modeling a shear wall with an initial vertical pre-compression pressure. The horizontal force F drives the wall to failure and produces a horizontal displacement d at top. The wall is confined by two concrete slabs (top and bottom) and two masonry flanges (left and right). This confinement and the large size of the wall make it appropriate for continuum modeling. Initially, cracking occurs well distributed in the panel and finally concentrates in a single shear band from one corner of the panel to the other. The compressive stresses are well below the crushing strength of masonry, i.e. failure is dominated by tension. A complete discussion of the numerical results has been given in [41]. Figure 7 shows the results of modeling a panel with out-of-plane pressure. The panel is simply supported on two sides (left and right), fully clamped on one side (bottom) and free on the other (top). The central opening simulates a window and the panel was loaded with an air bag with a uniformly distributed load. The predicted form of collapse includes diagonal cracks from each lower corner of the panel up to the opening, which were also observed in the experiments. This form of yield line collapse does not mean that yield line design is safe due to the quasi-brittle behavior of the material. A complete discussion of the numerical results has been given in [43].

Modeling the behavior of regular masonry assemblages, the typical case being brick walls, may also be addressed by homogenization techniques. A first, powerful, approach is to handle the brickwork structure of masonry by considering the salient features of the discontinuum within the framework of a generalized / Cosserat continuum theory [44]. A second approach is to apply rigorously the homogenization theory for periodic media to the basic cell, i.e. to carry out a single step homogenization, with adequate boundary conditions and exact geometry [45]. The last, and most used, approach aims at substituting the complex geometry of the basic cell by a simplified geometry so that a close-form solution of the homogenization problem is possible [46,47]. The regularity of fabric, implied by homogenization techniques, is seldom found in historical constructions and the phenomenological approach based on experimental tests seems a more promising approach. A finite element representation with continuum finite elements was used e.g. in [48] to study to behavior of the façade of St. Peter’s Basilica in Rome, see Figure 8.

Discontinuum models (Micro-modeling) Masonry joints act as planes of weakness and the explicit representation of the joints and units in a numerical model seems a logical step towards a rigorous analysis tool. This kind of analysis is particularly adequate for small structures, subjected to states of stress and strain strongly heterogeneous, and demands the knowledge of each of the constituents of masonry (unit and mortar) as well as the interface. In terms of modeling, all the non-linear behavior can be concentrated in the joints and in straight potential vertical cracks in the centerline of all units. Applications can be carried out using finite elements [49,50], discrete elements [51,52] or limit analysis [53,54]. Naturally, a higher computational effort ensues, so this approach still has a wider application in research and in small models for localized analysis.

Page [55] performed the first attempt to use a micro-model for masonry structures, using interface elements. A complete micro-model must include all the failure mechanisms of masonry, namely, cracking of joints, sliding over one head or bed joint, cracking of the units and crushing of masonry, as in [50]. Figure 9 shows the results of modeling a shear wall with an initial vertical pre-compression pressure. The horizontal force F drives the wall to failure, keeping the top and bottom boundaries fully constrained, and produces a horizontal displacement d at top. Initially, two horizontal cracks develop at the top and bottom of the wall but at failure a diagonal stepped crack and crushing of the compressed toes are found. A complete discussion of the numerical results has been given in [50]. Figure 10 shows the results of modeling a pier-wall connection subjected to wind load. Initially, a uniformly distributed vertical load is applied, before a horizontal load at the top of the wall drives it to failure. A detailed comparison between experimental and numerical results can be found in [9]. In a first phase, the piers and wall are firmly glued, which yields a very stiff structure. At a certain stage, around peak load, separation between the two piers and the wall occurs. After this stage, a much lower load can be carried by the structure because piers and wall behave independently and slide over each other. The examples above demonstrate the power of modern numerical tools to represent the complex interaction between masonry components (units and joints). Both, the response of plane and three-dimensional structures controlled by the local behavior of masonry, and difficult phenomena observed in the experiments can be reproduced. A finite element representation with interface elements was used e.g. in [56] to study a pillar-arch stone structure, tested under pseudo dynamic loading, see Figure 11.

MODELING ISSUES Independently of the approach used to simulate historical masonry structures, namely macromodeling (using continuum finite elements or structural component models) or micromodeling (using interface finite elements, discrete elements or limit analysis), two complementary aspects are of relevance for solving actual engineering problems: idealization of geometry and assumptions for structural behavior. These issues are briefly introduced next and the reader is referred to [57] for a comprehensive discussion.

Idealization of geometry Differently from modern structures, it is not straightforward to define the conditions under which a given idealization of the geometry is applicable. Usually, the geometry of historical masonry structural is rather complex as there is no distinction between decorative and structural elements. Therefore, as a first impression, it would seem reasonable to advise the use of three-dimensional elements. But this impression is erroneous! Given the difficulties addressed in the introduction, it is obvious that practical analysis of historical masonry structures entails severe simplifications. As a rule, the geometric idealization should be kept as simple as possible, as long as it can be considered adequate for the problem being analyzed. In particular, it is stressed that: •

Fully three-dimensional models are usually very time consuming with respect to preparation of the model, to perform the actual calculations and to analyze the results. In the case of the widely spread Finite Element Method, several authors are using eight-noded bricks, with one element over the thickness of the walls or vaults. The errors associated with such a discretization are very large already in the case of a linear elastic analysis, yielding meaningless results;



The results of models incorporating shell elements are reasonably difficult to analyze due to the variation of stresses along the thickness of the elements. In addition, the large thickness of the structural elements might yield a poor approximation of the actual state of stress;



Increasing the details and size of the model might result in a large amount of information that blurs the important aspects.

If possible, it seems preferable: (a) to use two-dimensional models than three-dimensional models, (b) to avoid using shell elements in areas important for the global behavior of the structure and (c) to model structural parts and details instead of modeling complete and large structures, see Figure 12 and Figure 13 for examples.

Idealization of structural behavior Common idealizations of the behavior used for analysis are elastic behavior (with or with-out redistribution of stresses), plastic behavior and non-linear behavior. Non-linear analysis is the most powerful method of analysis, able to trace the complete response of a structure from the elastic range, through cracking and crushing, up to complete failure. It is possible to include the construction sequence in the analysis. The effects of previous applications of loading and the way the intensity of loads are applied yield different results. Different types of non-linear behavior may be combined, namely, physical (related to the non-linearity of the material), geometrical (related to the fact that the point of application of loads changes with the increase of actions and structures buckle due to instability) and contact (related to the addition or removal of supports, or to the change of contact between bodies with the increase of actions). It may be used for both Ultimate Limit States (ULS) and Serviceability Limit States (SLS). Plastic analysis or limit analysis aims at evaluating the structural load at failure. It shall only be used for verification of ULS and, theoretically, the material must exhibit a ductile

response. This method can be assumed as adequate for the analysis of historical masonry structures if a zero tensile stress is assumed. The plastic analysis is either based on the lower bound (static) method or on the upper bound (kinematic) method. Thrust-line analysis is an example of the static method and yield hinges method for arches is an example of the kinematic method. The effects of previous applications of loading may generally be ignored and a monotonic increase of the intensity of actions may be assumed. Linear elastic analysis assumes that the material obeys Hooke’s law. This is hardly the case of masonry under tension, which cracks at very low stress levels. Linear elastic analysis, based on the theory of elasticity, has been widely used for the verification of both ULS and SLS of e.g. reinforced concrete structures. But, for serviceability limit states, a gradual evolution of cracking should be considered and, for ultimate limit states, careful detailing of the reinforcement to cover all zones where tensile stresses may appear is required. Linear elastic analysis with redistribution may be applied for the verification of ULS, assuming a reduced stiffness corresponding to fully cracked areas. Figure 14 illustrates the application of the different idealizations of the behavior. Nonlinear analysis is the reference analysis that should be primarily considered for understanding the behavior of historical masonry constructions. Several commercial finite element packages that consider non-linear behavior are available but one problem is that its use requires postgraduate structural analysts because the inherent complexity of the subject precludes mass teaching in civil engineering graduations. With respect to thrust-line analysis (plastic analysis), it must be stressed that its application to larger structures is rather cumbersome and the issue of structural safety is difficult to solve. This method seems mostly of interest as a pedagogical tool for engineering students. Quite on the contrary, collapse mechanism analysis (plastic analysis) is very useful for engineering purposes. The inherent difficulty in the use of this tool is the selection of the adequate

collapse mechanism for a given load combination. For traditional masonry constructions, such as the buildings in historical centers, the method is readily applicable to analysis and strengthening. For more complex and unique monumental structures, this method is still of interest to calculate strengthening, once the relevant collapse mechanisms are identified and the structural behavior is understood resorting to non-linear analysis. Finally, it is stressed that linear elastic finite element models have been widely used for analyzing historical constructions. Sometimes, cracks have been included in the model, by assuming a zero stiffness set of elements in areas where cracks already exist in the structure, or in areas where the linear elastic calculation predicts high tensile stresses. The use of linear elastic analysis seems debatable, taking into account the advanced tools today available to solve engineering problems. Essentially, three different aspects contribute for this statement: (a) Masonry possesses very limited tensile strength and it is usually not possible to make provisions to cover all zones where tensile stresses may appear; (b) The question “what is the maximum tensile stress admissible?” cannot be answered with a linear elastic analysis as the maximum admissible value depends on the distribution of stresses itself. If the area of the structure related to the crack propagation is rather large, the relevant stored energy might be too high and the structural behavior ends up quite brittle. If the relevant stored energy is low, the behavior can be rather ductile. Additionally, the linear elastic peak stresses in a structure are often meaningless quantities. Due to the application of concentrated loads and strong geometric discontinuities, e.g. openings in walls, the elastic peak stress is, theoretically, infinite and the value obtained in the finite element analysis depends only on the mesh discretization, namely element size; (c) Usually, the act of preparing a finite element model is very time consuming. Given the effort and costs involved in the preparation, the additional time requirements to carry out a non-linear static analysis are only marginal and the benefits for understanding the behavior of the structure are considerably high.

SAFETY ASSESSMENT OF A MASONRY ARCH Figure 15 illustrates a two-dimensional arch with a span of 5.0 m, a rise of 2.5 m, a thickness of 0.3 m, a width of 1.0 m, a radius of 2.5 m and a backfill up to 3.0 m height. The loads considered in the analyses include, as dead load, the weight of the arch (volumetric weight γ = 20 kN/m3) and fill (γ =15 kN/m3), and, as live load, a point load of 10 kN at quarter span. The dead load is applied first, followed by the monotonic application of the live load up to failure. The following analyses have been carried out for this structure: linear elastic finite element analysis, kinematic limit analysis, limit analysis for the calculation of the so-called geometric safety factor, non-linear physical finite element analysis and non-linear combined physical / geometrical finite element analysis.

Linear elastic finite element analysis Eight-noded plane stress elements, combined with six-noded line interface elements, were adopted for the analysis. The following properties have been assumed in the analyses: Unit – Young’s modulus of 10 × 103 N/mm2 and Poisson ratio of 0.2; Interface – Normal stiffness 2.4 × 103 N/mm3 and transverse stiffness of 1.0 × 103 N/mm3. The results of the analysis are shown in Figure 16. In order to establish the safety of the structures being considered, it is necessary to introduce the concept of “allowable maximum stress”. Here, a maximum allowable tensile stress of 0.2 N/mm2 and unlimited compressive stress were assumed. For the adopted mesh discretization, the obtained safety factor is 0.3. It is stressed that, as a general rule, the peak values of the stresses depend on the mesh discretization and the proposed procedure is debatable.

Limit blocky analysis The following properties have been assumed for the joints: zero tensile strength, unlimited compressive stress strength, friction angle of 37º and zero dilatancy. The results are given in Figure 17. Safety of the structures is automatically evaluated when using limit analysis, resulting in a kinematic safety factor of 1.8. Another popular concept is the so-called “geometric safety factor” [59], which represents the ratio between the actual thickness of the arch and the minimum thickness of an internal arch with the original span and able to resist the original applied load. This geometric safety factor is 1.2.

Non-linear finite element analysis Both physical and combined physical / geometrical non-linear behavior were assumed. The results of the physically non-linear analysis are shown in Figure 18. Due to cracking of the joints, compressive stresses are much higher than in Figure 16. For the sake of completeness, a second non-linear analysis has been carried out, adopting some limited, yet non-zero, strength. For this purpose, tensile strength ft of 0.2 N/mm2, cohesion of 0.3 N/mm2 and fracture energy of 0.1 N.mm/mm2 were assumed for the joints. Both physical and combined physical / geometrical non-linear behavior were again assumed. Safety of the structures is automatically evaluated when using non-linear analysis. For zero tensile strength and physically non-linear analysis, the ultimate load factor is the same as for limit analysis.

Comparison of results Figure 19 presents the load-displacement diagrams for the non-linear analyses calculations together with the ultimate load factors for kinematic limit analysis and Table 1 indicates the safety factors obtained in the different analyses. The results obtained allow to conclude that: - Linear elastic analysis requires information on the elastic properties of the materials and maximum allowable stresses, resulting in information on the deformational behavior and stress distribution of the structure. Limit analysis requires the strength of the materials, resulting in information on the failure mechanism of the structure. Non-linear analyses require the elastic properties, the strength of the materials and additional inelastic information (the stress-strain diagrams), resulting in information on the deformational behavior, stress distribution and on the failure mechanism of the structure. - The “safety factors” associated with a linear elastic analysis (and a maximum allowable stress) and with a static limit analysis (the so-called geometric safety factor) cannot be compared with the remaining safety factors. When such particular approaches are used, special care should be adopted if structural safety is a relevant issue; - For the simple structures presented, physically non-linear analysis and kinematic limit analysis yield the same failure mechanisms and safety factors, if a zero tensile strength is assumed. In complex structures, when using simple hand calculations, it might be difficult to find the correct failure mechanism by using limit analysis. Additionally, if geometrically nonlinear behavior is also included in the analysis, the safety factor is reduced by around 10%, for the arch studied; - The consideration of a non-zero, yet low and degrading, tensile strength increased the safety factors considerably. Therefore, when using non-zero tensile strength, special care might be necessary in real case applications, mainly because: (a) tensile strength is difficult to

assess and (b) tensile strength might be severely reduced at critical locations. It is noteworthy to stress that different failure mechanisms might be triggered in the analyses. In the particular case of the semi-circular arch, this is a minor difference and the non-zero tensile strength solution converges to the zero tensile strength solution, upon progressive tensile strength degradation. Finally, when using non-zero tensile strength, the consideration of geometrically non-linear behavior seems to affect only marginally the calculation of the safety factors; - The post-peak response obtained in a non-linear analysis is an important issue, when addressing safety factors. Indeed, brittle responses are dangerous and, from a reliability point of view, yield lower safety margins. In specific cases, it may be sensible to adopt as safety factor the residual plateau found in the physically non-linear analyses. Of course, in the case of combined physically and geometrically non-linear analysis, no plateau will be found; - The fact that different methods of analysis lead to different safety factors and different completeness of results is not a sufficient reason to select one method over the other.

CONCLUSIONS AND FUTURE ISSUES All the numerical techniques reviewed in this paper have advantages and shortcomings. Constraints to be considered in the use of advanced modeling are the cost, the need of an experienced user / engineer, the level of accuracy required, the availability of input data, the need for validation and the use of the results. Cost and the need of an experienced used seem straightforward arguments. The level of accuracy and the availability of input data are more difficult to quantify, whereas the need of validation is a key issue, meaning that the results of a complex analysis, using a complex model, might be useless if the results are not validated against in situ observations, such as cracking, crushing, displacements, flat-jack tests, etc. For this purpose recommendations regarding adequate strategies to tackle historical structures are available [60-62].

Obtained results are usually important for understanding the structural behavior of the constructions. But it should not be concluded that non-linear analysis is the sole numerical tool to be used for all constructions, by all engineers. As a rule, advanced modeling is a necessary means for understanding the behavior and damage of (complex) historical constructions but this requires specialized consulting engineers and it is less effective for designing strengthening. On the other hand, simplified modeling, such as limit analysis (kinematic method), is a great tool for everyday constructions, such as the buildings in historical centers. The key message for analysts might be “prefer simplicity over complexity and adopt an analysis tool, that can be validated and assessed”. General recommendations are: - It is better to model structural parts than complete structures; - Do not use full structure three-dimensional models unless it is necessary; - Avoid making linear elastic calculations for historical constructions.

Future Issues Even tough large advances in the field of structural analysis of historical constructions have been observed in the last decades, a successful and cost-effective intervention remains a true challenge. Numerical models can be used as a numerical laboratory, where the sensitivity of the results to input material parameters, boundary conditions and actions is studied, and may be invaluable in the conception and understanding of in situ testing and monitoring. Once the fundamental mechanics are fully understood, design of the intervention can be carried out with simplified analysis tools. From a fundamental point of view, developments in the numerical tools are still required. For this purpose, carefully controlled experiments are needed and even more detailed

analysis, at a level of refinement lower than traditional unit-joint micro-models, should be addressed [13,63]. Possible future issues of research are: •

Creep and fatigue, which seem to be related to recent collapses of historical constructions under sustained heavy load. Also the distribution of stresses in multileaf walls are severely affected by long term effects;



Cyclic static loading and dynamic loading, which have received far less attention from the research community. Due to the mathematical difficulties, seismic analysis of historical constructions is still a true challenge;



Adequate representation of irregular masonry, so that the behavior of historical multileaf walls can be predicted both in plane and out of plane. In the author’s opinion, the assessment and strengthening of multi-leaf walls under compression is one of the most complex issues for modern practitioners.

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TABLES AND FIGURES CAPTIONS Table 1: Safety factors for the different analyses considered. Figure 1: Modeling strategies for masonry structures: (a) detailed micro-modeling; (b) simplified micro-modeling; (c) macro-modeling. Figure 2: Examples of structural component models for (a) wall with openings: (b) lumped parameters; (c) beam elements; (d) macro-elements (rigid or deformable). Figure 3: Tapered beam models elaborated for Barcelona Cathedral: (a) cimborio and (b) typical bay of the nave [33]. Figure 4: Typical seismic failure mechanisms adopted for limit analysis, depending both on the position of the house in the urban texture and the location of the openings in the external wall [36]. Figure 5: Finite element model for a block compound in Lisbon: (a) finite element mesh with 200.000 degrees-of-freedom. Six man-months were required for mesh discretization and five non-linear analyses; (b) results for seismic analysis using equivalent static loading [38] (shading indicates damage levels). Figure 6: Results for an analysis of a masonry shear wall (macro-modeling): (a) loaddisplacement diagram; (b,c) predicted cracking pattern at peak and ultimate load. Figure 7: Results for an analysis of a panel subjected to uniform out-of-plane loading (macromodeling): predicted cracking pattern at (a) bottom and (b) top face of the panel. Figure 8: Model of the entire façade of S. Peter’s Basilica in Rome [48]. Figure 9: Results for an analysis of a shear wall (micro-modeling): (a) force-displacement diagram; (b,c) deformed meshes at peak and ultimate load. Figure 10: Results for an analysis of a pier-wall connection (micro-modeling): (a) geometry and load; (b,c,d) incremental deformed meshes (only ½ of the wall is shown) before, during and after separation. Figure 11: Monastery S. Vicente de For a, in Lisbon: (a) full-scale model of the façade and (b) deformation pattern using interface elements [51] Figure 12: Cloister of a Cistercian monastery in Salzedas, Portugal (eighteenth century). The barrel vaults of the second level exhibit severe longitudinal cracking. The two-dimensional model (a) was initially adopted for performing a non-linear analysis aiming at assessing the safety of the structure. The three-dimensional model (b) was later used for academic purposes. The difference in displacements and stresses at specific control points was less than 10% [58].

Figure 13: Refectory of Monastery of Jerónimos, Lisbon. In a modal analysis, the fully threedimensional model (a,b) yields similar results to the simplified shell model (c,d) for the first two vibration modes. Nevertheless, the need for correcting the thickness of walls, to take into account the additional stiffness provided by the intersection of the thick walls, precludes any non-linear analysis using the shell model. Figure 14: General load-displacement diagram for a structural analysis. Figure 15: Geometry and loading in masonry arch. Figure 16: Results of linear elastic finite element analysis: (a) maximum and (b) minimum principal stresses. Figure 17: Results of limit analysis: (a) failure mechanism and (b) thrust line for “geometric load factor (internal dotted arch). Figure 18: Results of non-linear elastic finite element analysis: minimum principal stresses and collapse mechanism. Figure 19: Load-displacements diagrams for the different non-linear analyses and limit analysis safety factor.

TABLES

Table 1: Safety factors for the different analyses considered.

Approach/Analysis type Allowable stresses (fta=0.2 N/mm2) Kinematic limit analysis Geometric safety factor ft = 0, Physically non-linear ft = 0, Physically and geometrically nonlinear ft = 0.2 N/mm2, Physically non-linear ft = 0.2 N/mm2, Phys. and geomet. non-linear

Safety Factor 0.31 1.8 1.2 1.8 1.7 2.5 2.5

FIGURES

Mortar

Unit

Interface Unit/Mortar

“Unit” “Joint”

Composite

Figure 1: Modeling strategies for masonry structures: (a) detailed micro-modeling; (b) simplified micro-modeling; (c) macro-modeling.

(a)

Mass M2

Mass M1

(b)

Rigid elements

(c)

(d) Figure 2: Examples of structural component models for (a) wall with openings: (b) lumped parameters; (c) beam elements; (d) macro-elements (rigid or deformable).

(a)

(b) Figure 3: Tapered beam models elaborated for Barcelona Cathedral: (a) cimborio and (b) typical bay of the nave [33].

Figure 4: Typical seismic failure mechanisms adopted for limit analysis, depending both on the position of the house in the urban texture and the location of the openings in the external wall [36].

(a)

(b) Figure 5: Finite element model for a block compound in Lisbon: (a) finite element mesh with 200.000 degrees-of-freedom. Six man-months were required for mesh discretization and five non-linear analyses; (b) results for seismic analysis using equivalent static loading [38] (shading indicates damage levels).

F

Experimental Numerical d (a)

(b)

(c) Figure 6: Results for an analysis of a masonry shear wall (macro-modeling): (a) loaddisplacement diagram; (b,c) predicted cracking pattern at peak and ultimate load.

(a)

(b) Figure 7: Results for an analysis of a panel subjected to uniform out-of-plane loading (macromodeling): predicted cracking pattern at (a) bottom and (b) top face of the panel.

Figure 8: Model of the entire façade of S. Peter’s Basilica in Rome [48].

F

Experimental Numerical (a)

(b)

d

(c) Figure 9: Results for an analysis of a shear wall (micro-modeling): (a) force-displacement diagram; (b,c) deformed meshes at peak and ultimate load.

(a)

(b)

(c)

(d) Figure 10: Results for an analysis of a pier-wall connection (micro-modeling): (a) geometry and load; (b,c,d) incremental deformed meshes (only ½ of the wall is shown) before, during and after separation.

(a)

(b) Figure 11: Monastery S. Vicente de For a, in Lisbon: (a) full-scale model of the façade and (b) deformation pattern using interface elements [51]

(a)

(b) Figure 12: Cloister of a Cistercian monastery in Salzedas, Portugal (eighteenth century). The barrel vaults of the second level exhibit severe longitudinal cracking. The two-dimensional model (a) was initially adopted for performing a non-linear analysis aiming at assessing the safety of the structure. The three-dimensional model (b) was later used for academic

purposes. The difference in displacements and stresses at specific control points was less than 10% [58].

(a)

(b)

(c)

(d) Figure 13: Refectory of Monastery of Jerónimos, Lisbon. In a modal analysis, the fully threedimensional model (a,b) yields similar results to the simplified shell model (c,d) for the first two vibration modes. Nevertheless, the need for correcting the thickness of walls, to take into account the additional stiffness provided by the intersection of the thick walls, precludes any non-linear analysis using the shell model.

Load

Non-linear analysis Linear elastic analysis Plastic analysis

Control displacement at a selected point Figure 14: General load-displacement diagram for a structural analysis.

2.5

3

1.25

10 kN

5

Figure 15: Geometry and loading in masonry arch.

Peak value : 0.64 N/mm2 (a)

Peak value : -1.0 N/mm2 (b) Figure 16: Results of linear elastic finite element analysis: (a) maximum and (b) minimum principal stresses.

Kinematic load factor : 1.8 (a)

Geometric load factor : 1.2 (b) Figure 17: Results of limit analysis: (a) failure mechanism and (b) thrust line for “geometric load factor (internal dotted arch).

Peak value : -5.4 N/mm2

Figure 18: Results of non-linear elastic finite element analysis: minimum principal stresses and collapse mechanism.

3.0

Load factor

2.5 2.0 1.5 Limit analysis ft = 0, Physically non-linear ft = 0, Physically / Geometrically non-linear ft = 0.2 N/mm2, Physically non-linear ft = 0.2 N/mm2, Physically / Geometrically non-linear

1.0 0.5 0.0 0

2

4 6 8 10 12 Vertical displacement at quarter span (mm)

14

Figure 19: Load-displacements diagrams for the different non-linear analyses and limit analysis safety factor.