Numerical and experimental analysis of masonry arches strengthened ...
Università degli Studi di Cassino Facoltà di Ingegneria
Dottorato di Ricerca in Ingegneria Civile XX Ciclo
Numerical and experimental analysis of masonry arches strengthened with FRP materials
Maria Ricamato
Cassino, Novembre 2007
University of Cassino Department of Engineering Graduate School in Civil Engineering XX Cycle - November 2007
Numerical and experimental analysis of masonry arches strengthened with FRP materials
Maria Ricamato
Supervisor: Prof. Elio Sacco Coordinator: Prof.ssa Maura Imbimbo 2
To my mother and my father, with love
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Acknowledgements Three years ago this day seemed very distant, instead today I am working for concluding my PhD thesis. At the end of this experience, first the financial supports of the Italian National Research Council (CNR) and of The Laboratories University Network of Seismic Engineering (RELUIS) are gratefully acknowledged. My greatest thanks is for my “Scientific fathers”: Prof. Giovanni Romano for having directed me towards the research and Prof. Elio Sacco for giving me the opportunity to improve my scientific and technical knowledge, for the work done under his supervision and for his useful suggestions. I would like to thank Prof.ssa Sonia Marfia for the fruitful discussions and also Prof.ssa Maura Imbimbo and Prof. Raimondo Luciano for their disposal. I would like to express my gratitude to Prof. Olivier Allix of the ENS of Cachan (France), who gave me the opportunity to spend part of my PhD under his supervision. This period was very important for my experience, both from a professional and personal point of view. A special thanks to Ing. Pierre Gosselet, for his help to overcome many difficulties that I encountered with the multiscale methods and to the LMT who “hosted” me for 6 months, in particular thanks to Beatrice Faverjon who represented always a reference point also for human aspect. I would like to express my deep and sincere gratitude to my friends and colleagues Ernesto Grande and Veronica Evangelista, who spent several days with me working hardly. Many thanks to DIMSAT, LAPS and Geolab Sud staff. I would like to remember all my friends for their friendship whenever I needed it. Finally a great thank to my family: my mother Francesca and my father Lucio, for their love and support, my brother Nicandro for his humor that in times of distress has been able to give me the courage to continue. My gratitude to my Franco cannot be summarized in few rows: I will never forget his love, his patience and his continuous support in what I do... 4
INDEX Acknowledgements.......................................................................................................4 1. INTRODUCTION ....................................................................................................9 1.1. Early static theories of the arch.......................................................................12 1.2. Motivations of the research and outline of the thesis......................................15 2. MASONRY MATERIAL.......................................................................................18 2.1. Introduction.....................................................................................................18 2.2. Mechanical behavior .......................................................................................19 2.3. Masonry modelling .........................................................................................23 2.4. No-tension material model..............................................................................25 3. FRP COMPOSITE MATERIALS FOR STRENGTHENING MASONRY STRUCTURES......................................................................................28 3.1. Introduction.....................................................................................................28 3.2. Mechanical behavior .......................................................................................29 3.2.1. Alkaline ambient effects..........................................................................34 3.2.2. Humidity effects ......................................................................................35 3.2.3. Extreme temperature and thermal cycle effects ......................................35 3.2.4. Frost-thaw cycles effects .........................................................................35 3.2.5. Temperature effects.................................................................................35 3.2.6. Viscosity and relaxation effects ..............................................................36 3.2.7. Fatigue effects .........................................................................................36 3.3. Masonry structures reinforced with FRP materials.........................................36 3.4. Collapse mechanism for reinforced structures................................................38 4. EXPERIMENTAL PROGRAM .............................................................................40 4.1. Introduction.....................................................................................................40 4.2. Setup and instrumentations .............................................................................40 4.3. Preliminary experimental campaign ...............................................................42 4.4. Materials used in the experimental program...................................................44 4.5. Standard clay brick..........................................................................................44 5
4.5.1. Cubic compressive test............................................................................45 4.5.2. Indirect tensile test...................................................................................49 4.5.3. Elastic secant modulus ............................................................................53 4.6. Mortar..............................................................................................................59 4.6.1. Compressive tests ....................................................................................59 4.6.2. Elastic secant modulus ............................................................................62 4.7. Reinforcement material...................................................................................64 4.8. Experimental test on the arches ......................................................................69 4.9. Arch laying......................................................................................................69 4.10. Arch preparation ...........................................................................................72 4.11. Experimental campaign: Arch 1....................................................................74 4.11.1. Collapse mechanism description ...........................................................76 4.11.2. Load-displacements curves ...................................................................78 4.12. Experimental campaign: Arch 2....................................................................83 4.12.1. Collapse mechanism description ...........................................................84 4.12.2. Load-displacements curves ...................................................................85 4.13. Experimental campaign: Reinforced arch.....................................................88 4.13.1. Application of the FRP reinforcement ..................................................88 4.13.2. Test organization ...................................................................................91 4.13.3. Collapse mechanism description ...........................................................95 4.13.4. Load-displacement curves.....................................................................95 5. MODELING AND NUMERICAL PROCEDURES..............................................99 5.1. Introduction.....................................................................................................99 5.2. Masonry constitutive models ........................................................................100 5.2.1. Model 1..................................................................................................100 5.2.2. Model 2..................................................................................................103 5.3. FRP constitutive model.................................................................................105 5.4. Limit analysis................................................................................................107 5.5. Arch model....................................................................................................109 6
5.5.1. Governing equation of the arch .............................................................110 5.5.2. Kinematics of the arch...........................................................................111 5.5.3. Cross section..........................................................................................111 5.6. Stress formulation .........................................................................................114 5.6.1. Complementary energy .........................................................................117 5.6.2. Arc-length technique .............................................................................120 5.7. Displacement formulation.............................................................................125 5.7.1. Kinematics.............................................................................................127 5.7.2. Finite element implementation..............................................................128 5.8. Post-computation of the shear stresses..........................................................131 5.9. Numerical results ..........................................................................................135 5.9.1. Models and numerical procedures assessment......................................135 5.9.2. Experimental surveys numerical results................................................141 5.9.2.1. Comparison 1.................................................................................141 5.9.2.2. Comparison 2.................................................................................146 6. MULTISCALE APPROACHES ..........................................................................156 6.1. Introduction...................................................................................................156 6.2. Methods based on the homogenization.........................................................157 6.2.1. Theory of homogenization for periodic media......................................158 6.3. Methods based on the super-position............................................................159 6.3.1. Variational multiscale method...............................................................160 6.4. Methods based on the domain decomposition ..............................................160 6.4.1. Primal approach.....................................................................................161 6.4.2. FETI method..........................................................................................165 6.4.3. Mixed method: the micro-macro approach ...........................................166 6.5. Numerical results ..........................................................................................169 CONCLUSIONS ......................................................................................................175 APPENDIX: RELUIS SCHEDE ..............................................................................177 NOTATIONS............................................................................................................187 7
REFERENCES .........................................................................................................189
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1. INTRODUCTION Numerous ancient constructions are made of masonry material that is one of the oldest building material. Many ancient and historical masonry buildings are characterized by the presence of arches and vaults. In particular the arch is a fundamental constructive element having both load-bearing and ornamental function. The “false arch” was one of the first constructive elements. It was realized by flat stones placed on top of each other that created a stepwise arch. The constructive technique was refined during the centuries, also introducing the use of the mortar to joint the stones or the bricks. The Egyptian and the Babylonians introduced the use of arches in civil constructions, the Assyrians constructed the first vaults in masonry buildings, the Etruscans used arches in order to realize the first masonry bridges. The Romans made large use of masonry arches and vaults for the constructions, not only of buildings but also of roads, bridges, aqueducts and amphitheatres, as illustrated in Fig. 1.1.
Fig. 1.1: The Colosseum, Roma.
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On the contrary, cult buildings were made using columns and architraves, as for Greeks temples. One of the most representative cult building is the Pantheon of Roma, characterized by the presence of a very well-known vaulted structure (Fig. 1.2).
Fig. 1.2: The Pantheon, Roma.
Moreover, Romans constructed arches also as monuments, like the “triumphal arches”, e. g. the arch of Janus in Rome and the Triumphal arch in Paris (neoclassical version of the ancient triumphal arches of the Roman Empire).
Fig. 1.3: The arch of Janus in Rome and Triumphal arch in Paris.
During the Middle-Age both the Byzantine architecture in the East and the Romanic one in the West still adopted the Roman round arches. 10
The Goths, in the 13th century, substituted the semicircular arch with the pointed arch. A main characteristic of the Gothic structures is the lightness of the buildings, obtained by the introduction of flying buttresses and towers. The Cathedral of Milan is an example of Gothic structure, Fig. 1.4.
Fig. 1.4: Cathedral of Milan.
During the Renaissance, the churches assumed a great structural interest. In particular the Church of S. Maria del Fiore in Florence and the Basilica di S. Pietro, Fig. 1.5, represent great examples of regular shapes and geometrical symmetry due to the use of vaults.
Fig. 1.5: S. Maria del Fiore and Basilica di S. Pietro.
Then no relevant or innovative solution concerning the structural conception were developed, but still today the arch is a fundamental structural element and its use has 11
been extended to all types of construction by the use of “new” material, like reinforced concrete and steel.
1.1. Early static theories of the arch The arch is one of the most interesting structural elements in the construction history because of its intuitive static behavior. During the centuries, many studies were developed on the more appropriate shape of the arch, but only in the 17th century a static theory of the arch was proposed. The Romans used systematically the arch realizing structure both of great value and strong impact. The Roman scientist, Vitruvio, identified the main characteristics of the arch and wrote ten books De Architectura, in which both the theory and the practice concerning with the art of construction were presented. Vitruvio discussed about the presence of the thrust of the vault on the supporting columns and walls. He has also understood the functioning of the arched structures, suggesting to realize strong and massive supports in order to contrast the thrust of the arches and vaults. In the 13th century, Leon Battista Alberti wrote the De Re Aedificatoria, motivating the use of arched structures with the aim of increasing both of the spans and the bearing capability. A more refined theory was attributed to the constructors of the Middle-Age: its main characteristic is the approximation of the arch shape by the thrusts line. Also the geometrical rule to determine the thickness of the piers was attributed to them. This theory was the only respected during the Renaissance. Leonardo Da Vinci (1452 1519) developed some ideas and intuitions three centuries later. He asserted “…arco non è altro che una fortezza causata da due debolezze imporochè l’arco negli edifici è composto di due quarti di circolo, I quali quarti circoli ciascuno debolissimo per se desidera cadere e opponendosi alla ruina l’uno dell’altro, le due debolezze si 12
convertono in un’unica fortezza…l’arco non si romperà se la corda dell’archi di fori non toccherà l’arco di dentro…”.
Fig. 1.6: Leonardo’s intuitive static scheme for the arch.
This theory for which the arch is assimilate to two beams was reproposed by Caplet in the 18th century. The first significant theory of the arch was attributed to the mathematician astronomer Philippe de La Hire (1670 - 1718). In its treatise Traitè De Mécanique, posthumous published in 1730, he underlined the wedge mechanism of the arch. According to him, the arch results subdivided in blocks and each block can be considered like a piece of wedge incident on the mortar joints. Its model was the first approach in the static theory of the arch that considers the masonry structure like a rigid system of solids geometrically defined and with an own weight, neglecting the frictional phenomenon. Two problems were faced by de La Hire: the vaults equilibrium independent of its piers and the determination of the piers dimensions considering the vault thrusts. The first problem lead, in the years, to the method of polygon of the forces, while, concerning the second problem, he developed the basis of the limit analysis. In the 1785, Mascheroni, in the Nuove ricerche sull’equilibrio delle volte, proposed a collapse mechanism of the arch characterized by the formation of intrados cracks at key, of extrados cracks at springers and of intrados cracks at piers extremities, as schematically illustrated in Fig. 1.7. 13
Fig. 1.7: Mascheroni’s collapse mechanism.
In the 19th century the method of the successive resultants was diffused. It was adopted to study short span symmetric arches symmetrically loaded. It is based on the definition of the thrusts line contained inside the third medium. The thrusts line can be regarded as an indicator of stability: if it is not coincident with the center-line of the arch, there is eccentricity and the arch thickness must be such that the eccentricity remains inside the section. The early method characterized by a collapse analysis was the method of Mery. This method is based on the limit analysis and it is applicable only if the assumed collapse mechanism occurs. It can be used if the arch is semicircular and its thickness is constant, the maximum span of the arch is 8-10 metres, the arch is made of an homogeneous material in order to be schematized by a rigid body, the arch is symmetric and symmetrically loaded. The method of Mery can be applicable using the parallelogram rule. In order to verify the part of the arch included between the key and the springers, the arch must be subdivided in blocks of different dimensions. Established the loads agent on each block, the resultants of loads are determined and the thrusts line can be obtained applying the parallelogram rule again and again. In the 1833, Moseley in the On a new principle in static called the principle of least pressure enounced the least pressure principle for the determining the thrusts line of the arch. In 1867 Winkler wrote a treatise on the thrusts line of the arch based on the elasticity theory developed in those years. 14
Recently, in 1982, Heyman in The masonry arches enounced the safe theorem of the limit analysis particularized for the masonry arches. According to him, it is necessary to determine at least one line of thrusts contained inside the thickness of the arch to ensure that the structure is safe. On the other hand, it is sufficient a small variation in the position of the line of thrusts, e. g. caused by loading increase, to allow the formation of localized cracks. As consequence, the hinges formation can occur and a kinematical mechanism can be activate. Generally, the collapse mechanism occurs for formation of four hinges, two at extrados and two at intrados alternatively located.
1.2. Motivations of the research and outline of the thesis The preservation of historical and ancient buildings and monuments requires the definition of intervention methodologies for the maintenance and consolidation. The definition of these methods must reflect on one hand the structural safety, on the other hand the respect of the original structure. The masonry arch is essential and unique in the historic heritage. Some of the consolidation techniques of masonry arches, widely adopted in the recent years, can alter the nature and original structural working of the arch and they also introduce extraneous elements not compatible with the materials and traditional techniques. More recently, for the protection and maintenance of ancient and historical buildings, the use of innovative materials, such as composites, received great interest because of their possible advantages in terms of low weight, simplicity of application, high strength in the fiber directions, immunity to corrosion and reduced invasiveness. In particular, they appear particularly indicated for the maintenance and rehabilitation of ancient structures because they do not substantially violate the principles of the Carta di Venezia. After the earthquakes of 1997 (Umbria and Marche), an intensive research activity was developed for the definition of some rules for the design of the strengthening of 15
masonry arches by FRP. In 2003 the CNR, The National Council of Researches, established that it was necessary to elaborate a text containing the instructions for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, in order to give to engineers the guidelines for the use of fibercomposite materials for the reinforcement of concrete and masonry structures. In the year 2004 a full text “CNR DT/200” was published in Italian and then in English with the name. It could be emphasized that the DT200 is the first code in the world which contains a Chapter completely dedicated to the use of FRP for the strengthening of masonry structures. The present PhD thesis is aimed to derive and to develop some simple strategies to study the response of unreinforced and reinforced masonry arches. In particular, the aim is to formulate simple and effective procedures that the engineers can use for the design of the reinforcements of masonry arches, evaluating the safety of the structure. In order to validate the effectiveness of the developed models and procedures, an experimental campaign on un-reinforced and reinforced masonry arches is conduced. Moreover, a more sophisticate numerical procedure, based on the multiscale analysis, is developed. Finally, the dissertation concerns with three macro-arguments: the experimental program, the modeling and numerical procedures development and the multiscale analysis. In Chapter 2 a general overview on the modeling of masonry material is given. The different modeling strategies are also discussed, underlining the main advantages of each approach. Chapter 3 analyzes the FRP properties. In particular an excursus on FRP material “history” is made and its physical and mechanical characteristics are presented. Chapter 4 contains the description of the experimental program. In order to characterize the nonlinear behavior of the masonry material, the physical and mechanical properties of masonry material constituents are investigated through experimental test. Moreover an experimental campaigns is realized on unreinforced 16
and reinforced masonry arches. The adopted procedure for testing the arches is described and the experimental results are discussed. In Chapter 5 the modeling of masonry materials and FRP and the developed numerical procedures are illustrated. In particular the masonry material is assumed as a no-tensile material with a limited compressive strength, while the reinforcement is considered as an linear elastic material. Two different approaches are developed: a stress formulation, based on the complementary energy, and a displacement formulation, characterized by the implementation of a three node beam finite element based on the Timoshenko’s theory. Several numerical analysis are conduced in order to validate the models and the developed numerical procedure. The numerical results are also put in comparison with experimental results both available in literature and obtained during the experimental campaign. Chapter 6 illustrates a brief introduction to the multiscale methods; in particular the domain decomposition methods are analyzed. At the end, a summary and final conclusions, which can be deduced from this research, are given.
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2. MASONRY MATERIAL
2.1. Introduction The masonry material is one of the oldest building material, as confirmed by the historical heritage. The development of adequate stress analyses for masonry structures represents an important task not only to verify the stability of masonry constructions, as old buildings, historical town and monumental structures, but also to properly design effective strengthening and repairing works. In fact, many of masonry structures have been suffered from the accumulated effects of material degradation, aging, overloading and foundation settlements. For this reason, the rehabilitation and the maintenance of existing masonry structures represent an important topic. In the years several studies have been developed related to masonry structures, i.e. [1] - [38], mainly devoted to the development of new restoration technologies and, moreover, to the definition of effective computational procedure for reliable stress analyses. It could be emphasized that the analysis of masonry structures is not simple at least for several reasons: the masonry material can be considered as a composite material obtained by assembling bricks by means of mortar joints; the masonry material presents a strongly nonlinear behavior, so that linear elastic analyses generally cannot be considered as adequate; the structural schemes which can be adopted for the masonry structural analyses are more complex than that adopted for concrete or steel framed structures, as masonry elements are often modeled by two- or threedimensional elements. 18
As a consequence, the behavior and the analysis of masonry structures still represent one of the most important research field in civil engineering, receiving great attention from the scientific and professional community; for instance, in Reference [1] several specific problems related to the design and behavior of old and mainly new masonry constructions are discussed. In this chapter a brief discussion on the main aspects concerning the mechanical behavior of the masonry material, i.e. [2] - [5], is reported.
2.2. Mechanical behavior The mechanical behavior of the masonry material presents complex aspects due to the fact that it is a composite material made of units of natural or artificial origin (irregular stones, ashlars, adobes, bricks and blocks) jointed by dry or mortar joints (commonly clay, lime or cement based mortar). The units can be jointed together using mortar or just by simple superposition obtaining different combinations that can be classified [6] in stone masonry and bricks masonry, as illustrated in Fig. 2.1 and Fig. 2.2, respectively.
(a)
(b)
(c)
Fig. 2.1: Stone masonry (a) rubble masonry, (b) ashlar masonry, (c) coursed ashlar masonry.
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Fig. 2.2: Brick masonry, (a) common bond, (b) cross bond, (c) Flemish bond, (d) stack bond, (e) stretcher bond.
The heterogeneity of the masonry material, which depends on the assemblage of its constituents (brick and mortar, as previously seen), leads to a complex structural behavior. Generally, the behavior of the masonry is intermediate between the behavior of the brick and mortar, as shown in Fig. 2.3.
Mortar Masonry Brick
σ
ε
Fig. 2.3: Qualitative stress-strain diagram in uniaxial tension and compression.
In Tab. 2.1 the mechanical characteristics of the masonry constituents are reported. 20
Mortar 3.0 - 30.0
Brick 6.0 - 80.0
Compression Strength [MPa] 1.5 - 9.0 Tensile Strength 0.2 - 0.8 [MPa] Tensile Modulus (8.0 - 20.0) (15.0 - 25.0) 103.0 103.0 [MPa] Poisson’s coefficient 0.10 - 0.35 0.10 - 0.25 Tab. 2.1: Mortar and brick mechanical characteristics.
While the bricks properties are generally defined on the base of brick type, the mortar mechanical properties depend strongly as much on the natural materials of which it is constituted as on the procedures of manufacturing; indeed, the mortar strength is influenced a lot by the binder and the dosage. According to the Italian Code 20/11/1987 (Norme tecniche per la progettazione, esecuzione e collaudo degli edifici in muratura e loro consolidamento) and the previous and successive rules, four classes of mortar have been specified, as reported in Tab. 2.2. Class
Kind of Mortar
M4 M4
Grout Pozzolana mortar Cement lime Cement lime Mortar of cement Mortar of cement
M4 M3 M2 M1
Composition Cement Common Water Sand lime lime 1 3 1 -
Pozzolana 3
1 1 1
-
2 1 0.5
9 5 4
-
1
-
-
3
-
Tab. 2.2: Mortar classes.
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Subjected to a uniaxial load, the masonry material has a stress-strain curve that presents a brittle failure, characterized by a compression stress failure value greater than the tensile one, as illustrated in Fig. 2.4. In particular, it can be individuated the following characteristic features: compression OA that is essentially linear; AB characterized by a nonlinear behavior, increasing until the maximum value of the compression stress; BC, decreasing feature with nonlinear behavior and softening; tension OI very short feature that has a linear behavior and IL decreasing feature. Moreover, the point B represents the peak load and the point C represents the point in correspondence of which the masonry material collapses in compression.
σ
I O
L
ε
A C B
Fig. 2.4: Stress - strain masonry curve.
An important feature, common to all cohesive materials, is the occurrence of softening, which is defined as a progressive decrease of the mechanical strength under continuous imposed displacement, after the load peak. Softening behavior is experimentally observed in uniaxial compressive, tensile and shear failure.
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2.3. Masonry modelling The main problem in the development of accurate stress analysis for masonry structures is the definition and the use of suitable material constitutive laws. In the last twenty years several authors, (i.e. van Zijl [7], Berto et al. [8], Pietruszczak and Ushaksaraei [9]), have proposed different modelling strategies to predict the structural response of masonry structures and, consequently, to assess the safety level of existing buildings. Taking into account the heterogeneity of the masonry material, which results composed by blocks joined by mortar beds, the models proposed in literature can be framed in the three different classes briefly described below. Micro-models consider the units and the mortar joints separately, characterized by different constitutive laws; thus, the structural analysis is performed considering each constituent of the masonry material. The mechanical properties that characterize the models adopted for units and mortar joints, are obtained through experimental tests conducted on the single material components (compressive test, tension test, bending test, etc.). This approach leads to structural analyses characterized by great computational effort; in fact, in a finite element formulation framework, both the unit blocks and the mortar beds have to be discretized, obtaining a problem with a high number of nodal unknowns. Nevertheless, this approach can be successfully adopted for reproducing laboratory tests (i.e. Lofti and Shing [10], Giambanco and Di Gati [11], Alfano and Sacco [12]). Micro-macro models consider different constitutive laws for the units and the mortar joints; then, a homogenization procedure is performed obtaining a macro-model for masonry which is used to develop the structural analysis. Also in this case, the mechanical properties of units and mortar joints are obtained through experimental tests. The micro-macro models appear very appealing, as they allow to derive in a rational way the stress-strain 23
relationship of the masonry, accounting in a suitable manner for the mechanical properties of each material component. Moreover, this approach can lead to effective models, with reduced computational effort for a structural analysis (i.e. Luciano and Sacco [13], Milani et al. [14], [15]). On the other hand, the non-linear homogenization procedure required to recover a macro model could induce some theoretical or computational difficulties [16]. Macro-models are based on the use of phenomenological constitutive laws for the masonry material; i.e. the stress-strain relationships adopted for the structural analysis are derived performing tests on masonry, without distinguishing the blocks and the mortar behaviour. A phenomenological model could be unable to describe in a detailed manner some micromechanisms occurring in the damage evolution of masonry, but it is very effective from a computational point of view when structural analyses are performed [17], [18]. The linear elastic model is the simplest approach to the analysis of masonry structures. In the linear elastic model the material exhibits an infinite linear elastic behavior, both in compression and tension. The structural response obtained under the hypothesis of linear elastic behavior, although often not completely reliable for ancient constructions [19], can be of great help; in fact, the linear analysis requires few input data and it is less demanding, in terms of computer resources and engineering time used when compared with non-linear models. Moreover, for masonries characterized by significant tensile strength, linear analysis can provide a reasonable description of the process leading to the crack pattern.
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2.4. No-tension material model Because of the very low tensile strength of many masonries with respect to the compressive strength, a no-tension model is often adopted; it is based on the assumption of zero the tensile strength of the material, as illustrated in Fig. 2.5,. The no-tension material (NTM) model (i.e. [20] and [21]) leads to a realistic approximation for the evaluation of the mechanical response of the masonry material. In fact, the collapse mechanism of old masonry constructions is often due to the opening of cracks in tensile zones. The use of the no-tension model allows to compute the limit carrying load for masonry structures invoking the limit analysis theorems.
σ
ε O
Fig. 2.5: Linear elastic model with no tensile strength.
The no-tension material model is based on the fundamental hypothesis that the tensile strength is zero while it considers a linear elastic behavior in compression. The no-tension model presents the following very special properties: a convex strain energy function governing the stress-strain relationship exists, thus the constitutive law results to be reversible and there is no energy dissipation for the crack formation and evolution. 25
The question regarding the safety of the no-tension approach with respect the fracture mechanics solution was discussed by Bazant [22], who proved that the notension model is not always safe with respect to the fracture mechanics approach. The problems considered by Bazant concern the case of fractured rocks, characterized by the presence of a preexistent localized crack; for old masonry structures, which present sufficiently densely distributed microcracks, the no-tension model can be considered reliable. The no-tension material model received and still receives great attention by many researchers to study the behavior of old masonry structures. Indeed, the statement ”no tension material” was proposed by Zienkiewicz et al. [23] to study the behavior of fractured rocks. Then, several studies were developed regarding the NTM from a mechanical, i.e. [24] - [28], mathematical [29] and computational point of view, developing displacement, i.e. [30] - [32], as well as stress and mixed variational formulations, i.e. [33] - [35]. It could be emphasized that, although the NTM presents and apparent simplicity, its numerical treatment is not trivial. The assumption of a masonry linear elastic behavior in compression can be considered adequate only when the evaluation of the load carrying capacity of the structure occurs for a collapse mechanism accompanied by very low compressive stresses; on the contrary, when the compression strength plays a significant role in the evaluation of the structural collapse load, the no-tension model appears inadequate. This case may occur, for instance, for shear masonry panels, building walls and strengthened arches, where the presence of the reinforcement can prevent the formation of hinges. A first proposal of a no-tension model with limited compressive strength has been presented in Reference [36]. The model proposed by Lucchesi and coworkers is based on two fundamental assumptions: the stress-strain relation is again hyperelastic, so that the crushing of the material is considered to be reversible, and the inelastic strain in compression is always orthogonal to the fracture strain. Indeed, the crushing strain is quite irreversible in character and it could not also be 26
orthogonal to the fracture strain, during the whole loading history. As matter of fact, the compression failure is affected by progressive damage and inelastic irreversible strain. In order to derive a simple and effective model, Marfia and Sacco [18] developed a no-tension model which accounts for the inelastic behavior in compression, considering a plasticity model which neglect the damage and softening effects. The derived model appears appropriate for the description of the material crushing when limited values of the compressive strain arise. The elasto - plastic model is characterized by a first linear elastic feature OA and a plateau with a constant stress DE, as schematically illustrated in Fig. 2.6.
Fig. 2.6: Elasto - plastic model.
A delicate point is the determination of the point D: it can be fixed to avoid to underestimate the masonry stress and to ensure a safe state, far from point E. The possibility to determine the collapse load of masonry and the irreversible nature of strains in the plateau DE for cyclic load are the principal characteristics of this model [37].
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3. FRP COMPOSITE MATERIALS FOR STRENGTHENING MASONRY STRUCTURES
3.1. Introduction In the last decades the use of innovative materials, such as composites, received great interest because of their possible advantages in terms of low weight, simplicity of application, high strength in the fibers direction, immunity of corrosion and quite reduced invasiveness. The use of Fiber Reinforced Polymers (FRP), that are a class of composite materials characterized by the combination of high-strength fibers and a matrix, is growing in the different fields of the engineering. Initially adopted for applications in aircraft and space industries, FRP have been used in the medical, sporting goods, automotive and small ship industries (see Fig. 3.1) due to their high strength in the fibers direction.
Fig. 3.1: Ordinary FRP devices and appliances.
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The greater reduction of the fibers prices, due to their increasing use and to an optimization of the production processes, have recently concurred to their diffusion also in the field of the civil constructions. In particular, they appear particularly indicated for the maintenance and rehabilitation of ancient structures because they do not substantially violate the principles of the Carta di Venezia, as they can be considered (quite) reversible and distinguishable.
3.2. Mechanical behavior The FRP are composite materials constituted by two phases: polymeric matrix and high-strength fibers. The two phases are visible at microscope and they present mechanical and geometrical properties sufficiently different, as consequence the composite has mechanical properties different from those of the constituents ones. The nature of every phase influences significantly the final properties of the composite; however, in order to obtain a composite with a high mechanical resistance, it is not sufficient to use only resistant fibers: it is also necessary to guarantee a good adhesion between the matrix and the reinforcement. The adhesion is usually guaranteed by the employment of a third component, called interface or interphase, applied in a much thin layer on the surface of fibers, between fibers and matrix, as schematically illustrated in Fig. 3.2.
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INTERFACE
FIBER
MATRIX
Fig. 3.2: FRP phases.
The interphase, whose characteristics are fundamental for the good use of the material in structural applications, is usually a thin and monoatomic layer. In fact, the lack of adhesion between fiber and matrix is one of the causes of the structural yielding of the composite materials. The organic matrices guarantee the transfer of the stresses between the surrounding structure and the fibers embedded in it, protecting these last ones from the aggressions of the external agents and from mechanical hit. The matrices, more used for the fabrication of FRP, are the polymeric ones made up of thermosetting resins. These resins are available in shape partially polymerized and they are liquid or dense at ambient temperature. The resins, mixed with an opportune reagent, polymerize until becoming a vitreous solid material. The matrices have various advantages: they are characterized by capacity of impregnation of the fibers, by optimal adhesive properties, by good resistance to the chemical agents. Their main limitations are the temperatures of exercise, limited from the upper by the vitreous transition temperature, the brittle failure, the sensibility to the humidity in phase of application on the structure. The epoxy resins are the more utilized: they have a good resistance to the humidity and the to chemical agents and optimal adhesive property.
30
The fibers more used for composite materials employed in the applications of the civil engineering are: glass (Fig. 3.4), carbon (Fig. 3.3), and aramidic (Fig. 3.5) fibers.
Fig. 3.3: Carbon fibers at microscope.
Fig. 3.4: Glass fibers.
Fig. 3.5: Aramidic fiber.
The glass fibers have an elevated resistance to the corrosion, an elastic modulus lower than those of carbon and aramidic fibers, a quite reduced resistance to the abrasion, a discreet strength to plastic slip and to fatigue. In order to promote the adhesion between fibers and matrix and to protect fibers from the action of the alkaline agents and from the humidity, the fibers undergo special treatments. In the 31
operations of manipulation before the phase of impregnation great caution is necessary. For their easy damage during the treatments, they are covered from a protecting film that inhibit the installation of acid dioxides contained in the air, which, otherwise, would penetrate in the microscopic voids present on the surface. The aramidic fibers are of organic nature and they are characterized by an elevated resistance to the manipulation operations. The modulus and the tensile strength are intermediate between those of carbon and glass fibers, while the compressive resistance is approximately equal to 1/8 of the tensile one. They are characterized also by an elevated degree of anisotropy that favors the localized rupture with consequent instability. They can be degraded for extended exposure to the solar light and it is preferable not to use them at temperatures greater than 150°C for problems of material oxidation. Moreover, they are sensitive to the humidity. The carbon fibers are used for the fabrication of composite materials with elevated performances; they are distinguished for the high modulus and resistance. They exhibit a behavior with brittle failure. The crystalline structure of the graphite is hexagonal, with carbon atoms organized in structures essentially planar, tied from interaction transverse forces of van der Waals. The precursors of carbon fibers are the polyacrylonitrile (PAN) and the Rayon fibers. Starting from these, through a process of carbonization combined with thermal processes and the sizing process, two types of carbon fibers are produced: the High Strength (HS) and the High Modulus (HM). The carbon fibers are often dealt with epoxy materials that prevent the abrasion, increase the workability and realize a good compatibility with the matrices made up of epoxy resins. The principal properties, as tensile modulus and tensile strength, of some fibers used for composite materials are reported in Tab. 3.1.
32
Tensile Tensile strength Failure Coefficient of Density modulus [Gpa] [Mpa] strain [%] thermal expansion [g/cm^3] 72 - 80 3445 4.8 5 - 5.4 2.5 - 2.6 85 4585 5.4 1.6 - 2.9 2.46 - 2.49
Fiber E-glass Fiber S-glass Graphite fiber (high modulus)
390 - 760
2400 - 3400
0.5 - 0.8
-1.45
1.85 - 1.9
Graphite fiber (low modulus)
240 - 280
4100 - 5100
1.6 - 1.73
-0.6 - -0.9
1.75
Aramid fiber
62 - 180
3600 - 3800
1.9 - 5.5
-2
1.44 - 1.47
Polymeric matrix
2.7 - 3.6
40 - 82
1.4 - 5.2
30 - 54
1.10 - 1.25
206
250 - 400 (yield) 350 - 600 (failure)
20 - 30
10.4
7.8
Steel
Tab. 3.1: Properties of FRP constituents.
The most common shape for the composite materials is the laminate one. The laminates are constituted by two or more overlapped thin layers, called lamina, (see Fig. 3.6). z
x 2
x=1
1
1 2
X x
2
Y
Fig. 3.6: Laminate constituted by more laminas.
The main advantage of laminates is the maximum freedom in the disposition of fibers. In each plane, the direction of fibers can be chosen in order to obtain the desired physical and mechanical characteristics of the laminates. On the basis of the mechanical properties that have to be conferred to the laminate, different types of fibers can be adopted. For instance, hybrid laminates are obtained by assembling layers of epoxy resin reinforced by aramidic and carbon fibers, or by alternating layers of epoxy resin with aramidic or aluminum fibers. The orientation of fibers is one of the main aspects that determines the behavior of the composite material. A 33
disposition of unidirectional fibers, as schematically illustrated in Fig. 3.7, leads to an orthotropic response of the lamina.
Fig. 3.7: Laminate with unidirectional fibers.
With this type of disposition, the best mechanical properties is obtained in the direction of fibers. A bidirectional disposition confers to the composite mechanical characteristics which depends on the chosen fiber direction. Beyond to the orientation also the length, the shape, the composition and the percentage in volume of fibers, the mechanical properties of the resin and the interface influence the response of the composite. The mechanical properties (strength, strain, tension modulus) of some FRP systems degrade in presence of determined environmental conditions, i.e. alkaline ambient, extreme humidity, temperatures, thermal cycles.
3.2.1. Alkaline ambient effects The pores of the material that must be reinforced content water that can degrade the resin and the interphase. It is necessary that the resin complete the curing before the exposition to alkaline ambient.
34
3.2.2. Humidity effects The main effects connected to the absorption of humidity regard the resins; they are plasticization, reduction of vitreous transition temperature, strength and stiffness reduction. The absorption of humidity depends by the kind of resin, the composition and number of laminas, the curing conditions, the interphase and the processing.
3.2.3. Extreme temperature and thermal cycle effects The main effects of temperature are connected to the viscous answer of the composite. At the service temperature of most structures, the resins are stable, but when the temperature increases, the resin breaks down and evaporates; the composite performances strongly decrease when the temperature exceeds the vitreous transition one. The thermal cycle have not deleterious effects, even if they can favor the formation of micro-fractures.
3.2.4. Frost-thaw cycles effects The exposition to frost-thaw cycles do not influence the performance of the composites, but can reduce those of the resin and the interphase, because of the separation between fibers and matrix.
3.2.5. Temperature effects The increasing of the temperature involves a gradual degradation of the mechanical properties of composite in terms both of tensile strength and stiffness. 35
3.2.6. Viscosity and relaxation effects In a composite material, viscosity and relaxation depend from the properties of resin and fibers. The presence of fibers reduces the viscosity of the resin; the worse effect occurs when the load is applied in the direction orthogonal to the fibers or when the composite is characterized from one low percentage in volume of fibers. The viscosity can be reduced if it is assured a low stress in exercise.
3.2.7. Fatigue effects The performances of FRP under fatigue are very good and they are connected to the composition of matrix. In the unidirectional composites, fibers have got little defects and, consequently, they contrast the formation of fractures. Moreover the propagation of eventual fractures is limited from the action explicated from the fibers staying in the adjacent zones.
3.3. Masonry structures reinforced with FRP materials In the last years, a significant research activities has been performed to investigate on the possibility to adopt composite materials as reinforcement of the masonry buildings. Starting from the earliest works, Triantafillou and Fardis [38], several studies have been devoted to the evaluation of the advantages in terms of resistance and ductility, for the use of FRP for the strengthening of masonry constructions. Indeed, researches demonstrate that the use of FRP for the strengthening of masonry structures is very effective for different structural elements as masonry panels, but also arches and vaults. 36
Several researches have been oriented to the analysis of masonry walls reinforced by FRP sheets or laminates, subjected to in-plane and out-of-plane loads. The possibility of adopting FRP composites for strengthening of masonry was initially investigated by Croci et al. [39]. They presented the results of experimental tests performed on wall specimens reinforced by vertical FRP materials. Experimental investigations on the use of epoxy-bonded glass fabrics were developed by Saadatmanesh [40] and by Ehsani [41]. Luciano and Sacco [13], [42] and Marfia and Sacco [43] proposed micromechanical models to study the behavior of masonry elements reinforced with FRP sheets. Cecchi et al. [44] developed a homogenization technique to evaluate the overall behavior of reinforced masonry walls. Experimental tests, performed by Schwegler [45] and Laursen et al. [46], demonstrated the significant improvement of the in-plane shear capacity and the important increase of the ductility of masonry walls strengthened with FRP laminates. Triantafillou [47] and Velazquez et al. [48] developed experimental studies, showing that the flexural capacity of masonry walls can be drastically increased strengthening the panels with FRP laminates. Olivito and Zuccarello [49] presented the durability of masonry structures reinforced by FRP subjected to low cycle fatigue. In the last few years great interest was devoted to the reinforcement of masonry arches and vaults, probably as a result of the recent Umbria- Marche seismic events. In fact, aramidic fiber reinforced composites were adopted to restore the vaults of the Basilica di S. Francesco d’Assisi [50] and the Chiesa di San Filippo Neri, in Spoleto [51]. Como et al. [52] applied the limit analysis theorems in order to evaluate the collapse of reinforced arches. Olivito and Stumpo [53] proposed a numerical and experimental analysis of vaulted masonry structures subjected to moving load. Briccoli Bati and Rovero [54] and Aiello et al. [55] developed experimental investigations on reinforced masonry arches, emphasizing that the application of sheets or laminates of composite materials significantly increases the strength of the structure, modifying the collapse mechanism and the corresponding collapse load. 37
Chen [56] presented a method to calculate the limit load-bearing capacity of masonry arch bridges strengthened with FRP. Experimental tests and finite element analyses of masonry arches made of blocks in dry contact and reinforced by FRP materials have been developed by Luciano et al. [57], demonstrating the effectiveness of strengthening. Foraboschi [58] presented mathematical models for studying the possible failure modes of masonry arches and vaults with FRP reinforcement. Ianniruberto and Rinaldi [59] investigated on the influence of the presence of FRP to the collapse behavior of the structure when reinforcements are placed at the extrados or at the intrados of the arch. It can be emphasized that the collapse of masonry elements is generally induced by the opening of fractures due to the limited strength in tension. The presence of the FRP reinforcement, placed in the tensile zones of the masonry structure, inhibits the opening of the fractures; thus, a compression state can occur for bent elements, and the failure for crushing can be activated. As a consequence, a suitable masonry model for reinforced masonry should take into account the possibility of the collapse for compression, i.e. a limited compressive strength for the masonry material should be considered.
3.4. Collapse mechanism for reinforced structures When a masonry structure is reinforced, the collapse mechanism changes with respect to the unreinforced one. Indeed, the collapse of a reinforced masonry structure can occur for the activation of different failures: opening of cracks in the masonry for tensile stresses, crushing of masonry in compression, shear failure of the masonry, decohesion of the FRP from the masonry and failure of the reinforcement, i.e. [54], [60] and [61]. While the unreinforced masonry collapses generally for activation of mechanisms due to the very limited tensile strength of the masonry or for shear failure, for reinforced masonry the limited compressive strength of the 38
masonry and the delamination phenomenon can play fundamental roles in the overall collapse of the structure. Crushing of masonry in compression and reinforcement failure are strictly connected to the mechanical properties of masonry and reinforcement fibers respectively, while the decohesion phenomenon regards the interface masonry-FRP. The adhesion between masonry and composite is a very relevant factor in the masonry reinforcement by laminas or woven. The debonding can regard both laminas and woven applied on the extrados or intrados surface of the reinforced element. The understanding of the debonding mechanism is very important for the successful application of the external FRP reinforcement; it is necessary to know when debonding initiates and the parameters that influence it. The decohesion can be classified in Plate-end debonding (it initiates at a plate-end and propagates inwards) and Intermediate crack debonding (it initiates at a crack in the structure mid-span zone and then it propagates towards the nearest zones).
39
4. EXPERIMENTAL PROGRAM
4.1. Introduction The experimental program was realized at LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of the Geolab Sud of San Vittore del Lazio. The experimental tests were performed at the Geolab laboratory and part of the instrumentation was supplied by them. In order to determine the correct setup of the used instrumentation it was necessary to perform a preliminary experimental campaign on a steel beam.
4.2. Setup and instrumentations Several instrumentations were necessary to perform the experimental program; in particular, the devices to determine displacements, the strain gauges, the hydraulic jack to apply the external load, the load cells and the data acquisition system were used. Two instruments were adopted to determine the displacements: comparators and potentiometers. The comparator used in the experimental program is a dial gauge; the instrument bases its functionality on the displacement of a cylindrical rod that can be flow into a tubular guide for a maximum value of 100 mm. It is positioned on the interested surface, so the tracer point is in contact with the surface of the specimen subjected to the measurement. The potentiometer has the same performances of the dial gauge; it is composed by a cylindrical rod that can move into a tubular guide until 100 mm. On the extremity of 40
the rod there is a magnet that fixes the potentiometer to the interested surface on which a metallic element has been previously glued. The load was applied by an hydraulic jack and it was measured by two electric load cells. The load cells have a maximum value of 50 kN and 500 kN respectively; they are constituted by an inox steel body with an electronic device that allows to convert the mechanical tensile or compressive load into an electric signal. There is an optional plate that allows a more homogeneous load repartition on the body cell. The electronic device is constituted by resistive strain-gauges connected by an electric Wheatstone bridge. In the experimental program, two electric digital data acquisition systems, Leane and Wshay, were used. When it is subjected to load, each load cell emits an electric differential signal which is transmitted by a connector to the data acquisition system; the aim of data acquisition system is the data elaborations, i.e. the conversion of the electric signal into mechanical engineering quantities. So the data acquisition system allows the measurement by the manual or automatic data acquisition. The Washay is a model P3 Strain Indicator and Recorder; it is portable and alimented by battery; its data acquisition is manual. The measurements obtained by this data acquisition system were used to verify the correct working of the Leane data acquisition system. Leane is a portable data acquisition system characterized by electric and battery alimentation. The data acquisition system has seven modules and four channels for each one; in total it is possible to have 28 acquisitions at the same time. In the experimental program, Leane was used for the acquisition in continuous of the potentiometers and of the cell load of 5 t. The Leane acquisitions are transmitted to a PC by a cable and then, the results can be worked out by a software given by the Leane.
41
4.3. Preliminary experimental campaign This preliminary experimental campaign was necessary to validate the data acquisition system Leane, in particular to verify that the in continuous displacements acquisition did not depend on the potentiometers position on the data acquisition system channel and they were not different from the displacements measured using the comparators. It was necessary to calibrate a new load cell of 50 kN, called in the following as small load cell. The load values of the 50 kN load cell acquired with the Leane are in accordance with those measured by the 500 kN load cell, called in the following as great load cell, acquired with the Wshay. The specimen of preliminary tests campaign was a steel beam and the tests were organized in TEST A (potentiometer calibration and displacement acquisition crosscheck), TEST B (small load cell calibration) and TEST C (small load cell acquisition by Leane crosscheck).
TEST A The aim of the test A was the potentiometers calibration and the crosscheck of the correct displacements acquisition obtained by the potentiometer connection to the different channels of Leane. The potentiometers were connected to data acquisition system Leane to have the displacements in correspondence of each load variation, in continuous. As previously seen, with Leane it is possible to have 28 acquisitions; the steel beam was subjected to 6 load cycles, called Test 1, Test 2, Test 3, Test 4, Test 5 and Test 6, characterized by the same load steps. In every load cycle, the potentiometer position on the data acquisition system module was changed to validate the different acquisitions obtained for every module and to validate the acquisitions obtained for every different channel of each module. In the Test A it can be pointed out that the difference between the various displacement acquisitions is in all the cases lower than 0.1 mm. The channel 4 of the 42
module 1 does not work. The difference between the displacement values registered by potentiometers and comparators is satisfactory.
TEST B This campaign had the aim to calibrate the new small load cell. It was possible to put in comparison the acquisitions obtained from the small load cell and the acquisitions obtained by the great load cell, both connected with the data acquisition system Wshay. The maximum error of test resulted equal to 1%; thus, it can be pointed out that the new small cell works in good accordance with the normalized great one.
TEST C This campaign had the aim to verify the correct functionality of the small cell connected with the data acquisition system Leane. The Test C puts in evidence that the difference between the manual and automatic acquisition of the load is, on average, lower than 2%.
43
4.4. Materials used in the experimental program The determination of the physical and mechanical properties of the materials used in the experimental program is necessary to understand the behavior of reinforced masonry arches. In the following the properties of the masonry material constituents and of the reinforcement are presented. The masonry material is composed by standard clay bricks and mixed mortar. At LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of the Geolab Sud of San Vittore del Lazio, an experimental program both on standard clay brick and mortar was performed. For what concerns the reinforcement, it is composed by carbon fibers and epoxy matrix and their properties were given by the manufacturer.
4.5. Standard clay brick In the experimental program, standard clay bricks (Fig. 4.1) were used.
Fig. 4.1: Standard clay brick.
44
In order to determine its main mechanical properties, the standard clay brick was subjected to several experimental tests. In particular, a cubic compressive test, an indirect shear test and a test to individuate the elastic secant modulus were performed.
4.5.1. Cubic compressive test Standard clay brick cubic specimens were prepared in order to determinate the compressive strength, in accordance with UNI 8942/3. This code gives the guidelines for the determination of the unitary load of compressive failure strength, that has to be determined on a fixed number of specimens with prefixed geometric characteristics. According to the code, the tests have to be performed on cubic specimens with orthogonal faces and parallel plane of the bedding plane, as illustrated in Fig. 4.2.
Fig. 4.2: Cubic specimen extrapolated by standard clay brick.
The specimens were located on the Galdabini SUN 60 that is a universal testing machine with a 600 kN nominal capacity, used in displacement control. A series of pre-loading tests finalized to set the machine and to position the specimens into its slabs were realized before the compressive test. The failure load was obtained from 45
the yielding load of every specimen. Then the other parameters necessary to characterize the test results were determined: average compressive strength:
n
fb =
∑f i =1
bi
(4.1)
n
where fbi is the result of the single test and n is the number of test results; standard deviation:
∑( f n
s=
i =1
b
− fbi )
2
n
(4.2)
variation coefficient:
cv =
s fb
(4.3)
characteristic value: fbk = f b (1 − kcv )
(4.4)
where k is the fractile coefficient, fixed by normative in function of the number of tested specimens. The compressive test was realized on 6 cubic specimens extracted by one of the series of the standard clay brick; their dimensions are reported in Tab. 4.1. 46
Specimen [number] 1 2 3 4 5 6
Deep [mm] 55 55 56 55 55 55
Length [mm] 56 55 55 55 55 55
Heigth [mm] 55 54 55 55 54 55
Tab. 4.1: Specimens size.
A carton layer was interposed at the top of the specimen in order to distribute the compressive stress. The specimen was allocated into the press, Fig. 4.3.
Fig. 4.3: Specimen positioning.
The specimen was subjected to an axial load acting perpendicular to the bedding plane until its failure, Fig. 4.4.
47
Fig. 4.4: Typical failure of the specimen.
The failure load and the compressive strength were determined for each specimen, as reported in Tab. 4.2 Specimen [number] 1 2 3 4 5 6
Area [mm2] 3080 3025 3080 3025 3025 3025
Failure load [kN]
Compressive strength (fb)
127.883 107.918 125.144 104.944 110.894 126.239
[kN/mm2] 0.0415 0.0357 0.0406 0.0347 0.0366 0.0417
Tab. 4.2: Compressive test results.
The considered specimens exhibited a hourglass failure, Fig. 4.5, not perfectly symmetrical because of the heterogeneous nature of the bricks.
48
Fig. 4.5: Hourglass specimens failure.
The characteristic compressive strength was determined, using equations (4.1), (4.2), (4.3) and (4.4) and in accordance with the code for which k=2.33 if n=6; the results are reported in Tab. 4.3. Average compressive strength [N/mm2]
Standard deviation [N/mm2]
Variation coefficient
Characteristic compressive strength (fbk) [N/mmq]
38.5
7.47
0.23
14.9
Tab. 4.3: Characteristic compressive strength.
4.5.2. Indirect tensile test The indirect tensile test was realized in accordance with UNI 8942/3, which gives the guidelines for the determination of the yielding load of specimens subjected to a uniform load applied on the middle surface of the specimen, as schematically illustrated in Fig. 4.6.
49
F
Fig. 4.6: Indirect tensile test scheme.
The code prescribes that this test has to be performed on specimens with a low drilling percentage (the limit is fixed at 30%). The test was performed using the
Galdabini SUN 60 and it was executed with constant load increments until the failure. In order to diffuse the load two steel beam, whose dimensions were fixed by the code, were interposed between the specimen faces and the steel plates of the machine, as illustrated in Fig. 4.7.
Fig. 4.7: In direct tensile test particular.
The indirect tensile test was realized on 6 specimens whose dimensions are reported in Tab. 4.4.
50
Specimen [number]
Deep [mm]
Length [mm]
Heigth [mm]
1 2 3 4 5 6
117 117 117 117 117 117
255 255 255 255 255 255
55 55 55 55 55 55
Tab. 4.4: Specimen dimensions.
Initially a pre-loading was imposed to setup the Galdabini SUN 60 , then the constant load increments were applied. Each specimen was subjected to compression load up to failure. The failure occurred along the direction of load application considering the front view, as represented in Fig. 4.8.
Fig. 4.8: Specimen failure.
Analogously to the compressive test, the following quantities were determined: average tensile strength mean deviation variation coefficient characteristic value. The indirect tensile strength was determined for each specimen by formula:
51
fv =
2t π bh
(4.5)
where t is the external applied load in Newton; h and b are the specimen height and length respectively, expressed in mm. The failure load and the indirect tensile strength were determined for each specimen, as reported in Tab. 4.5. Specimen [number] 1 2 3 4 5 6
Area [mm2] 14025 14025 14025 14025 14025 14025
Failure load [kN]
Indirect tensile strength (fv)
38816 31800 45008 24739 37841 30543
[kN/mm2] 3.84 3.15 4.45 2.45 3.75 3.02
Tab. 4.5: Indirect tensile test results.
The characteristic indirect tensile strength was computed and the results are reported in Tab. 4.6. Average indirect tensile strength [N/mm2] 3.44
Mean deviation [N/mm2] 0.71
Variation coefficient
Characteristic indirect tensile strength (fvk)
0.21
[N/mm2] 1.79
Tab. 4.6: Characteristic indirect tensile strength.
52
4.5.3. Elastic secant modulus In order to evaluate the elastic secant modulus a test was realized in accordance with the prescription of UNI 6556 rule. The specimens extrapolated by standard clay bricks were prismatic; in fact, the rule prescribes that the tests has to be performed on cylindrical or prismatic with square base specimens. The test was realized using the universal testing machine Galdabini SUN 60. In order to evaluate the elastic secant modulus, the code prescribes the use of 3 + 3 specimens. In particular, 3 specimens were used for evaluating the compressive strength, and the others 3 to determine the elastic secant modulus. The test was organized in two phases. During the first phase, 3 specimens were obtained by standard clay brick and their size was 5x5x15 cm. Each specimen was allocated into the universal testing machine and it was loaded until its compressive load failure, as represented in Fig. 4.9.
Fig. 4.9: Positioning into the universal testing machine of the reference specimen.
The average failure load value of the i-th set of specimens was determined as:
53
3
N if =
∑N j =1
j f
(4.6)
3
After this test, the load values representing the extremes of the loading-unloading cycles were determined using the average failure load values, recovered by equation (4.6). In accordance with the UNI 6556 rule, the maximum load is N 3 = base load is N 0 = N 2 = ( 2 N1 − N 0 ) .
1 i N f , the 3
1 ⎛ N + 2 N0 ⎞ N 3 and the intermediate loads are N1 = ⎜ 3 ⎟ and 3 10 ⎝ ⎠ Consequently
the
load
cycles
are
defined
as
cycle 1: N 0 → N1 → N 0 , cycle 2: N 0 → N 2 → N 0 and cycle 3: N 0 → N 3 → N 0 . In the second phase, further 3 specimens were tested. The bricks for evaluating the elastic secant modulus were prepared. The brick surface was cleaned and the area, where the strain-gauge were applied, was dry sanded, removing all the eventually incrustations, as illustrated in Fig. 4.10.
Fig. 4.10: Brick surface preparation.
54
In order to simplify the strain-gauge application, guidelines were traced on the brick surface; then the resin was applied, as represented in Fig. 4.11, and the strain-gauge was positioned along the guidelines previously traced, as illustrated in Fig. 4.12.
Fig. 4.11: Resin application.
Fig. 4.12: Strain-gauge application.
Each specimen was allocated into the universal testing machine and all the straingauge was connected with the data acquisition system. The load cell also was connected to the data acquisition system to know the applied load at each loading step, as represented in Fig. 4.13.
55
Fig. 4.13: Specimen positionating.
For every specimen the elastic secant modulus was determined. The procedure can be schematically described as: 1. the base load N0 was fixed; 2. the base mean strain ε 0 was determined; 3. the maximum load of the cycle Ni was fixed; 4. loading phase was performed: N 0 → N i ; 5. the mean strain ε i in correspondence of the maximum load was determined; 6. unloading phase was performed: N i → N 0 ; 7. the elastic secant modulus was determined as Es =
σi =
σi −σ0 εi − ε0
where
Ni N , σ 0 = 0 and A is the specimen base area. A A
The results elaboration for all the specimens are reported in Tab. 4.7, Tab. 4.8 and Tab. 4.9.
56
Ni
σi
[N]
[N/mm2]
4150 16740 4230 4230 28350 4330 4330 41670 4230
1.6600 6.6960 1.6920 1.6920 11.3400 1.7320 1.7320 16.6680 1.6920
Δσ
ε
Δε
[N/mm2] 5.0360
9.6480
14.9360
0.000087 0.000351 0.000098 0.000098 0.000616 0.000112 0.000112 0.000947 0.000110
Es
Es
[N/mm2]
[N/mm2]
0.000264
19075.7576
0.000518
18625.4826
0.000835
17887.4251
18529.5551
Tab. 4.7: Results elaboration for specimen 1.
Ni [N] 4110 16510 4150 4150 28610 4070 4070 40630 4030
σi
Δσ 2
[N/mm ] 1.6440 6.6040 1.6600 1.6600 11.4440 1.6280 1.6280 16.2520 1.6120
ε
Δε
2
2
[N/mm ] 4.9600
9.7840
14.6240
Es
Es [N/mm ]
0.000163 0.000520 0.000182 0.000182 0.000912 0.000191 0.000191 0.001302 0.000192
0.000357
13893.5574
0.000730
13402.7397
0.001111
13162.9163
[N/mm2]
13486.40448
Tab. 4.8: Results elaboration for specimen 2.
Ni [N] 4090 16230 4090 4090 27420 4130 4130 40630 4070
σi
Δσ 2
[N/mm ] 1.6360 6.4920 1.6360 1.6360 10.9680 1.6520 1.6520 16.2520 1.6280
ε
Δε
2
2
[N/mm ] 4.8560
9.3320
14.6000
Es
Es [N/mm ]
0.000113 0.000407 0.000117 0.000117 0.000669 0.000115 0.000115 0.000977 0.000113
0.000294
16517.0068
0.000552
16905.7971
0.000862
16937.3550
Tab. 4.9: Results elaboration for specimen 3.
57
[N/mm2]
16786.7196
The elastic secant modulus of standard clay brick was obtained as the average value of the elastic secant modulus of every specimen and it is Es ≅ 16000 N / mm 2 .
58
4.6. Mortar The mortar used to realize the three arches belongs to the M3 class, in accordance with the Italian Code Ministerial Decree of 20/11/1987. The mortar is constituted by: 2800 g of pozzolana; 933 g of lime, Calcisernia, Contrada Tiegno, Isernia; 800 g of pozzolanico cement Duracem 32.5 R, Colleferro, Roma; 1.66 l of water.
In literature this mortar is classified as mixed because it is constituted by two binders: cement and lime.
4.6.1. Compressive tests The specimens were realized with the mortar described previously. Three specimens of 40x40x160 mm3 were prepared using a normalized sand, as represented in Fig. 4.14.
Fig. 4.14: Preparation of mortar.
59
The mortar was prepared by mechanical mixing and successively compacted using a normalized vibrating device as illustrated in Fig. 4.15.
Fig. 4.15: Device to mix the mortar and normalized vibrating device.
After 28 days of seasoning, the specimens were subjected to a bending test. The specimen was allocated into the universal testing machine with a lateral face on the support rollers and the longitudinal axis orthogonal to the supports. The vertical load was applied on the specimen lateral face and it was uniformly increased with a maximum ratio of 20 Kg / cm 2 s until the failure, as represented in Fig. 4.16.
60
Fig. 4.16: Bending failure of mortar specimen.
In this way two semi-prismatic specimens were obtained and they were successively subjected to the compressive test. In order to determine the compressive behavior of the mortar, the semi-prismatic specimen was tested, as shown in Fig. 4.17.
Fig. 4.17: Compressive test: test setup and typical compressive failure.
61
The tests were performed with the universal testing machine Galdabini SUN 60 and the results are reported in Tab. 4.10. 1 2 Specimen 4x4x16 4x4x16 Size [cm] 432.4 430.6 Weigth [g] Compressive strength 4.8094 4.7038 4.7438 4.7675 [N/mm2]
3 4x4x16 430.5 4.6219 4.7975
Tab. 4.10: Mortar compressive strength.
4.6.2. Elastic secant modulus The elastic secant modulus of the mortar was carried out. The procedure was exactly the same that was realized for the standard clay brick. Three reference specimens for each mixture were tested. Therefore, the final results are schematically reported in Tab. 4.11 and Tab. 4.12 for the mixtures one, two and three and for mixtures four, five and six, respectively.
62
ε0 [10^-6] -79.67 -88.00
N0 [N] 420.00 420.00
εi [10^-6] -258.67 -266.67
Ni [N] 1690.00 1690.00
ε0s [10^-6] -88.00 -103.00
N0s [N] 420.00 440.00
εp [10^-6] -8.33 -15.00
Em0 εe E E [10^-6] [N/mm^2] [N/mm^2] [N/mm^2] -250.33 3170.77 3137.54 -251.67 3104.30
-103.00 -137.33
440.00 420.00
-475.67 -502.33
3020.00 3000.00
-137.33 -158.33
420.00 460.00
34.33 -21.00
-510.00 -481.33
3186.27 3298.13
-150.67 -183.00 -58.67 -83.00
440.00 420.00 420.00 420.00
-699.00 -754.00 -247.67 -244.33
4230.00 4290.00 1690.00 1690.00
-183.00 -217.33 -83.00 -83.33
420.00 420.00 420.00 440.00
32.33 -34.33 -24.33 -0.33
-731.33 -719.67 -223.33 -244.00
3256.04 3360.93 3554.10 3201.84
-83.33 -139.33
420.00 400.00
-463.33 -500.33
3060.00 3060.00
-139.33 -157.67
400.00 420.00
-56.00 -18.33
-407.33 -482.00
4081.42 3423.24
-157.67 -223.00 -67.00 -74.50 -80.50
420.00 420.00 420.00 420.00 420.00
-713.67 -747.67 -283.50 -269.50 -283.00
4310.00 4310.00 1670.00 1670.00 1670.00
-223.00 -248.33 -74.50 -80.50 -90.50
420.00 420.00 420.00 420.00 420.00
-65.33 -25.33 -7.50 -6.00 -10.00
-648.33 -722.33 -276.00 -263.50 -273.00
3750.00 3365.83 2830.62 2964.90 2861.72
Cycle 2
-90.50 -101.50
420.00 420.00
-500.50 -514.00
2840.00 2840.00
-101.50 -142.75
420.00 420.00
-11.00 -41.25
-489.50 -472.75
3089.89 3199.37
3144.63
Cycle 3
-142.75 -212.75
420.00 420.00
-777.50 -847.00
4230.00 4230.00
-212.75 -250.00
420.00 420.00
-70.00 -37.25
-707.50 -809.75
3365.72 2940.72
3153.22
Mixture 1
Cycle Cycle 1
Cycle 2
Cycle 3
Mixture 2
Cycle 1
Cycle 2
Cycle 3
Mixture 3
Cycle 1
3242.20
3229
3308.49 3377.97
3752.33
3563
3557.91 2885.74
3061
Tab. 4.11: Results elaboration for mortar specimen 1.
ε0 [10^-6] -82.33 -95.00
N0 [N] 420.00 420.00
εi [10^-6] -330.67 -348.67
Ni [N] 1630.00 1630.00
ε0s [10^-6] -95.00 -108.33
N0s [N] 420.00 420.00
εp [10^-6] -12.67 -13.33
Em0 εe E E [10^-6] [N/mm^2] [N/mm^2] [N/mm^2] 318.00 2378.14 2316.68 335.33 2255.22
-108.33 -137.00
420.00 420.00
-570.00 -574.67
2820.00 2820.00
-137.00 -137.00
420.00 420.00
-28.67 0.00
541.33 574.67
2770.94 2610.21
-137.00 -175.67 -69.33 -98.33
420.00 420.00 420.00 420.00
-774.67 -806.33 -299.67 -296.00
4130.00 4130.00 1590.00 1590.00
-175.67 -199.00 -98.33 -114.83
420.00 420.00 420.00 420.00
-38.67 -23.33 -29.00 -16.50
736.00 783.00 270.67 279.50
3150.48 2961.37 2701.66 2616.28
-114.83 -135.67
420.00 420.00
-518.67 -533.00
2820.00 2820.00
-135.67 -154.00
420.00 420.00
-20.83 -18.33
497.83 514.67
3013.06 2914.51
-154.00 -184.00 -59.33 -67.83
420.00 420.00 420.00 420.00
-774.67 -784.67 -243.33 -256.00
4110.00 4110.00 1650.00 1650.00
-184.00 -184.00 -67.83 -77.00
420.00 420.00 420.00 420.00
-30.00 0.00 -8.50 -9.17
744.67 784.67 -234.83 -246.83
3097.02 2939.15 3273.60 3114.45
Cycle 2
-77.00 -108.00
420.00 420.00
-436.67 -473.33
2860.00 2880.00
-108.00 -139.00
420.00 460.00
-31.00 -31.00
-405.67 -442.33
3759.24 3475.89
3617.56
Cycle 3
-139.00 -232.67
460.00 440.00
-741.67 -781.00
4110.00 4110.00
-232.67 -263.33
440.00 420.00
-93.67 -30.67
-648.00 -750.33
3520.45 3056.97
3288.71
Mixture 4
Cycle Cycle 1
Cycle 2
Cycle 3
Mixture 5
Cycle 1
Cycle 2
Cycle 3
Mixture 6
Cycle 1
2690.57 3055.92 2658.97
2963.78
2880
3018.08 3194.02
Tab. 4.12: Results elaboration for mortar specimen 2.
The mortar elastic secant modulus results equal to Es ≅ 3100 N/mm 2 . 63
2688
3367
4.7. Reinforcement material In this experimental program the used reinforcement system is the woven SikaWrap300C NW. It is constituted by carbon fibers impregnated on-site with an epoxy resin
of type SikaDur 330. The woven was chosen because it can be easily adapted to the curvilinear surface of the arch. On the lateral surfaces the woven has a thin texture, that safeguards the fibers stability during the application process, made of thermoplastic material, as shown in Fig. 4.18.
Fig. 4.18: SikaWrap-300C NW
The fibers of the woven are unidirectional. In the following the properties of unidirectional carbon fiber provided by the manufacturer are reported.
64
65
66
The epoxy resin SikaDur 330 was used both as adhesive to the masonry arch and as matrix. The resin is constituted by two-component, it is 100% solid and grey color. The properties of the resin are reported below.
67
68
4.8. Experimental test on the arches The experimental campaign on masonry arches was conduced on a set of two arches having the same geometrical characteristics and realized with the same materials. The aims of this campaign is the evaluation of the mechanical response of unreinforced and reinforced arches. In particular, the main aim is the validation of the numerical model developed to individuate the behavior of the masonry arch reinforced by FRP.
4.9. Arch laying The arch was realized using the standard clay brick and mixed mortar previously seen. The laying of the arch was at LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of Geolab Sud of San Vittore del Lazio. The reference arch is an incomplete circular arch with the abutment angle Φ = 8o , the mean radius of 516 mm, a cross section of 120x250 mm2. It is loaded with a vertical increasing force applied in correspondence of the brick number 14, as schematically illustrated in Fig. 4.19.
69
F
5
6
7
8
11 12 13 14 9 10 15
16
17
4 3 2
r
1
O
18
19 20 21 22 23
X Y
Fig. 4.19: Reference arch.
The first step was the construction of a steel centering, Fig.4.20 and in Fig. 4.21.
Fig.4.20: Arch centering.
Fig. 4.21: Geometrical characteristics of centering.
70
The centering was positioned on a temporary support. The standard clay bricks were embedded in water and then they were put on the centering. Initially, the bricks at the extremities were positioned. Then, the springs were constrained, Fig. 4.22.
Fig. 4.22: Constraint of the springs.
The mortar was mixed and the others bricks were positioned on the centering, spaced out by mortar joints, Fig.4.23
Fig.4.23: Construction of the arch.
When the last brick was posed, the external surface of the arch was polished to eliminate the eventual excessive mortar. The realized arch was seasoned for 28 days.
71
4.10. Arch preparation The arch was positioned under the steel frame for the test. The springers were clamped. All the bricks were numbered from left to right. The instrumentation was positioned on the arch: the displacements acquisition was obtained by comparators and potentiometers, the applied load was read by load cells and all the data were registered by the data acquisition system Leane. In particular the potentiometer p2 (p_c) and p3 (p_c_o) were positioned at the arch key, in vertical and horizontal direction, respectively. The potentiometer p4 (p_f) was positioned in correspondence of the loaded section., with a vertical direction. Moreover two comparators were utilized, one in correspondence of the arch key (c_c), one in correspondence of the force application (c_f), as schematically illustrated in Fig. 4.24, in Fig. 4.25 and in Fig. 4.26.
Fig. 4.24: Arch front and back view.
72
Fig. 4.25: Arch extrados view.
Fig. 4.26: Displacements measurement system.
The load was applied by the hydraulic jack. On the surface of brick number 14 a plate was applied in order to make easier the positioning of the load cell and the application of the external load; for the unreinforced arch the small cell was used. The applied load and the displacements were acquired in continuous by their connection to the data acquisition system Leane. Moreover for every loading cycle the displacement values, at each fixed loading step, were acquired by both the potentiometers and comparators in order to validate the reliability of the data.
73
4.11. Experimental campaign: Arch 1 Three loading-unloading cycles, called Cycle I, II and III respectively, were performed applying the external load by the hydraulic jack in correspondence of the brick number 14. Two loading-unloading cycles, called Cycle IV and V respectively, were performed applying the external load by a normalized set of weights. The displacements were acquired by potentiometers and comparators and the applied load intensity was determined by the small cell. Summarizing, the test organization was been the following: cycle I: external load applied by the hydraulic jack; cycle II: external load applied by the hydraulic jack; cycle III: external load applied by the hydraulic jack; cycle IV: external load applied by normalized set of weights; cycle V: external load applied by normalized set of weights.
Cycle I.
During the first cycle the following hinges opening occurred: hinge at the extrados between bricks 13-14, interface 14, in correspondence
of a load value equal to F ≅ 350 N ; hinge at the intrados between bricks 8-9, interface 8, in correspondence of a
load value equal to F ≅ 400 N ; hinge at the extrados between bricks 1-2, interface 1, in correspondence of a
load value equal to F ≅ 500 N ; hinge at the intrados between bricks 18-19, interface 18, in correspondence
of a load value equal to F ≅ 550 N .
74
Fig. 4.27: Arch 1, cycle I, experimental test.
Fig. 4.28: Arch 1, cycle I, hinges formation.
Cycle II and Cycle III.
During the second and the third cycle the opening of hinges occurred in correspondence of an applied load lower than in the first cycle. The first cycle peak load equal to FPL ≅ 600 N decreased in the cycle II and cycle III to a value close to 400N . This reduction is due to a significant reduction of masonry tensile strength.
Cycle IV and cycle V.
Those cycles were done to verify the acquisition in continuous of the applied load intensity; therefore the load was applied by a normalized set of weights.
75
Fig. 4.29: Arch 1, cycle IV.
The limit load reached during the cycle IV was the same of the value obtained in the second and third cycle. In the cycle V the collapse mechanism occurred applying a load greater than 450N .
Fig. 4.30: Arch 1, cycle V, collapse mechanism.
4.11.1. Collapse mechanism description The collapse mechanism occurred consequently to the hinges formation. The hinges formation occurred in the points where the pressure curve overlaps the arch intrados or extrados, as schematically reported in Fig. 4.31.
76
Fig. 4.31: Arch 1, hinges formation scheme.
The hinges opening deternined the subdivision of the arch in blocks. The arch collapse mechanism was characterized by the displacements of the blocks, as schematically illustrated in Fig. 4.32 and in . Fig. 4.33.
Fig. 4.32: Arch 1, collapse mechanism scheme.
77
Fig. 4.33: Arch 1, cycle V, particular of the collapse mechanism.
4.11.2. Load-displacements curves The cycles I, II and III were acquired in continuous by the data acquisition system Leane and the acquired data are reported in Fig. 4.34.
-700.00
-600.00
Cycle I Cycle II Cycle III
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.00 0.00
240.00
480.00
720.00
960.00
1200.00
t [s]
Fig. 4.34: Arch 1, loading-unloading cycles.
78
1440.00
1680.00
The kinematical mechanism was confirmed by the data acquired by potentiometers. In Fig. 4.35 and in Fig. 4.36 the load-displacement curves relative to the arch key are reported.
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F[N]
-400.00
-300.00
-200.00
-100.00 2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0.00
w [mm]
Fig. 4.35: Arch 1, load-displacement curve for the horizontal key displacements.
79
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0.00
v [mm]
Fig. 4.36: Arch 1, load-displacement curve for the vertical key displacements.
A pushing action, characterized by a counter-clockwise spin and a vertical displacement towards the bottom, was exercised by the block 1 on the arch. This displacement was read by the potentiometer connected with the point in which the load was applied. In Fig. 4.37 the load-displacement curves relative to the vertical displacements of the point in which the load was applied are reported.
80
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.00 0.00
-0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
v [mm]
Fig. 4.37: Arch 1, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
The cycle I presents a peak load greater than the others cycles, as illustrated in Fig. 4.37. This reduction of the peak load could be a consequence of the decrease of the masonry tensile strength from the cycle I to the last one. In order to validate the potentiometers acquisitions, the measurements read by comparators were put in comparison with the measurements registered by the data acquisition system Leane, obviously in the same loading condition. The acquired data are reported in Tab. 4.13 and Tab. 4.14.
F [N] 0.00 196.12 294.18 392.24 490.30 588.36
p_c [mm] 74.73 74.70 74.72 74.77 74.81 74.89
c_c [mm] 25.53 25.52 25.54 25.62 25.67 25.82
v_p_c [mm] 0.00 -0.03 -0.01 0.04 0.08 0.16
v_c_c [mm] 0.00 -0.01 0.01 0.09 0.14 0.29
D_c [mm] 0.00 0.02 0.02 0.05 0.06 0.13
p_f [mm] 53.04 52.89 52.74 52.59 52.51 52.30
c_f [mm] 19.35 19.22 19.05 18.88 18.78 18.55
v_p_f [mm] 0.00 -0.15 -0.30 -0.45 -0.53 -0.74
v_c_f [mm] 0.00 -0.13 -0.30 -0.47 -0.57 -0.80
D_f [mm] 0.00 0.02 0.00 0.02 0.04 0.06
Error_p_c Error_p_f 0.00 0.67 2.00 1.25 0.75 0.81
Tab. 4.13: Arch 1, Cycle I, potentiometers and comparators data.
81
0.00 0.13 0.00 0.04 0.08 0.08
F [N]
p_c [mm]
c_c [mm]
v_p_c [mm]
v_c_c [mm]
D_c [mm]
p_f [mm]
c_f [mm]
v_p_f [mm]
v_c_f [mm]
D_f [mm]
0.00 137.28 196.12 294.18 392.24 490.30
25.81 25.81 25.82 25.86 25.96 26.14
74.97 74.97 74.95 75.00 75.05 75.50
0.00 0.00 0.01 0.05 0.15 0.33
0.00 0.00 -0.02 0.03 0.08 0.53
0.00 0.00 0.03 0.02 0.07 0.20
18.86 18.76 18.60 18.44 18.07 17.13
52.64 52.57 52.41 52.28 51.95 51.13
0.00 -0.10 -0.26 -0.42 -0.79 -1.73
0.00 -0.07 -0.23 -0.36 -0.69 -1.51
0.00 0.03 0.03 0.06 0.10 0.22
Error_p_c Error_p_f
0.00 0.00 3.00 0.40 0.47 0.61
0.00 0.30 0.12 0.14 0.13 0.13
Tab. 4.14: Arch 1, cycle II, potentiometers and comparators data.
The errors in the displacements evaluation decreases with the increase of the displacement values. The potentiometer under the arch key p_c makes an error greater than the error made by potentiometer p_f because the achieved measurement and the error of the instrument have the same order of magnitude.
82
4.12. Experimental campaign: Arch 2 Three loading-unloading cycles, called cycle I, II and III respectively, were carried out applying the external load by the hydraulic jack connected with the brick number 14. The displacements were acquired by potentiometers and comparators; indeed the applied load intensity was acquired by the small cell. Summarizing, the test organization was the following: cycle I: external load applied by the hydraulic jack; cycle II: external load applied by the hydraulic jack; cycle III: external load applied by the hydraulic jack.
Cycle I.
During this cycle the following hinges opening occurred: hinge at the extrados between bricks 13-14, interface 14, in correspondence
of a load value equal to F ≅ 400 N ; hinge at the intrados between bricks 7-8, interface 7, in correspondence of a
load value equal to F ≅ 500 N
;
hinge at the extrados between bricks 1-2, interface 1, in correspondence of
a load value equal to F ≅ 500 N ; hinge at the intrados between bricks 19-20, interface 20, in correspondence
of a load value equal to F ≅ 550 N .
83
Fig. 4.38: Arch 2, cycle I, experimental test.
Fig. 4.39: Arch 2, cycle I, hinges formation.
Cycle II and cycle III.
During the second and the third cycle the opening of the hinges occurred in correspondence of an applied load lower than in the first cycle one. The peak load decreased significantly from the value equal to FPL ≅ 550 N obtained in the first cycle to a value equal to 450N obtained in the third cycle.
4.12.1. Collapse mechanism description The hinges formation determined the arch subdivision in blocks. The collapse mechanism occurred for relative displacements between the blocks. During the test the same collapse mechanism, characterizing the Arch 1, occurred: the raising of the second and third block, contrasting the pushing action towards the bottom.
84
4.12.2. Load-displacements curves The load related to cycles I, II and III were acquired in continuous by the data acquisition system Leane and the acquired data are reported in Fig. 4.40.
-700
Cycle I CycleII Cycle III
-600
-500
F [N]
-400
-300
-200
-100 0
240
480
720
960
1200
1440
1680
1920
2160
2400
2640
2880
3120
0
t [s]
Fig. 4.40: Arch 2, loading-unloading cycles.
In Fig. 4.41 and in Fig. 4.42 the load-displacement curves relative to the arch key are reported.
85
-600
Cycle I Cycle II Cycle III
-500
-400
F [N]
-300
-200
-100 2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0
100
w [mm]
Fig. 4.41: Arch 2, load-displacement curve for the horizontal key displacements. -600
Cycle I Cycle II Cycle III
-500
-400
F [N]
-300
-200
-100 1.00
0.80
0.60
0.40
0.20
0.00 0
100
v [mm]
Fig. 4.42: Arch 2, load-displacement curve for the vertical key displacements.
In Fig. 4.43 the load-displacement curves relative to the vertical displacements in the point in which the load was applied are reported.
86
-630
Cycle I CycleII Cycle III
-530
-430
F [N]
-330
-230
-130
0.00-30 -0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
-7.00
-7.50
70
v [mm]
Fig. 4.43: Arch 2, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
Because of the decrease of the masonry tensile strength, the peak load of the cycle I is greater than the one obtained in the others cycles, as illustrated in Fig. 4.43.
87
4.13. Experimental campaign: Reinforced arch The experimental campaign on reinforced arches was performed only on the arch 2 because of the ominous collapse of the arch 1. The instrumentations used for the experimental test on the reinforced arch was the same used for the unreinforced arches.
4.13.1. Application of the FRP reinforcement The FRP application was executed according to the codes (ACI 1999, fib TG 9.3 2001, JSCE 2001, etc.). The main phases of the reinforcement application were the following: Surface preparation: the surface where the reinforcement was applied was
suitable prepared. The arch surface was cleaned in order to remove every imperfection present on it and every contamination of chemical nature. Subsequently, it was necessary to fill the surface of the arch in order to render it flat; when the FRP was applied on the arch surface, it was completely clean, free from fats and oils and dry. At the end, the guides lines to install the FRP were traced. In order to avoid that the epoxy resin strewed over all the extrados surface, an adhesive tape was applied; it delimited the field of reinforcement application (Fig. 4.44).
88
Fig. 4.44: Surface preparation.
Epoxy resin preparation: the epoxy resin was constituted by two
component; the first one, called A, was white and the other, called B, was grey. Initially the two components were agitated separately, then they were mixed according with the technical card. The mixture was good, and therefore usable, when the colorful strips of the mixture were not more visible. In Fig. 4.45 this process is illustrated.
Fig. 4.45: Epoxy resin preparation.
Epoxy resin application: the epoxy resin was applied on the arch surface
using a roller, as illustrated in Fig. 4.46. 89
Fig. 4.46: Epoxy resin application.
FRP application: the woven was measured and pre-cut before its application
on the arch surface. It was placed on the surface and gently pressed into the epoxy resin, as illustrated in Fig. 4.47.
Fig. 4.47: FRP application.
Applying epoxy resin to FRP surface: a second coat of epoxy resin was
applied on the woven surface. Consolidation process control: after 48 hours the applied reinforcement was
examined to verify the presence of empties.
90
4.13.2. Test organization Three loading-unloading cycles, called cycle I, II and III respectively, were performed applying the external load by the hydraulic jack in correspondence with the brick number 14. The displacements were acquired by potentiometers and comparators; the applied load intensity was acquired by the small cell during the first cycle and by the great load cell in the other cases. Summarizing, the test organization was the following: cycle I: maximum external load applied F ≅ 5000 N ; cycle II: maximum external load applied F ≅ 25000 N ; cycle III: external load applied until the reinforced arch collapse.
Cycle I.
During this cycle the following phenomena occurred: FRP delamination in correspondence of the mortar joint between bricks 13-
14 for a load value equal to F ≅ 3000 N , as illustrated in Fig. 4.48; cracks under the brick 14, in correspondence of a load value equal to F ≅ 3500 N .
Fig. 4.48: Reinforced arch, cycle I, local delamination phenomenon at joint mortar.
91
Cycle II.
During this cycle the following aspects can be pointed out: increase of the crack opening in correspondence of the brick 14, as
illustrated in Fig. 4.49; formation of a mortar tooth between bricks 13-14 and bricks 14-15 in
correspondence of a load value equal to F ≅ 4000 N ; vertical sliding of brick 14 in correspondence of a load value equal to F ≅ 4500 N ;
breakaway of the bricks 14 for a load F ≅ 16000 N ; breakaway of the mortar tooth between bricks 14-15 in correspondence of a
load value equal to F ≅ 20000 N ; great increase of crack opening at the brick 14 in correspondence of a load
value equal to F ≅ 22000 N .
Fig. 4.49: Reinforced arch, cycle II, cracks on brick 14.
Cycle III.
During the third cycle the following aspects can be pointed out: great increase of crack opening at the brick 14 in correspondence of a load
value equal to F ≅ 39000 N ; cracks on the surface of the bricks 6 and 7 in correspondence of a load value
equal to F ≅ 42000 N , as represented in Fig. 4.50; 92
cracks on the surfaces of the bricks 8, 9, 10, 11 and 12 in correspondence of
a load equal to F ≅ 50000 N , as illustrated in Fig. 4.51; collapse of the reinforced arch in correspondence with a load greater than 50000N , as represented in Fig. 4.52 and Fig. 4.53.
Fig. 4.50: Reinforced arch, cycle III, cracks formation on bricks surface.
Fig. 4.51: Reinforced arch, cycle III, intrados and extrados arch surface view before the collapse.
93
Fig. 4.52: Reinforced arch, cycle III, arch collapse.
Fig. 4.53: Reinforced arch, after the collapse.
94
4.13.3. Collapse mechanism description The reinforcement prevents the classic masonry arch collapse mechanism; the FRP presence on the extrados surface, in fact, does not allow the hinges formation at the intrados because it prevents the cracks opening at the extrados. During the tests, the application of a concentrated load determined the presence of visible cracks on the surface of the brick 14 in correspondence of a load value little than the load for which the cracks on the other bricks occurred. The left part of the arch, from brick 1 to 14, was more significantly damaged than the right part of the arch, from brick 15 to 23. The collapse was preceded by the cracks opening on lateral and intrados surfaces of all the bricks from 2 to 14. When the collapse occurred, the FRP delamination under the bricks 13 and 14, the vertical sliding of the brick 14, the crush of bricks from 8 to 13 and the partial crush of bricks from 2 to 7 took place.
4.13.4. Load-displacement curves The data were acquired in continuous during the three loading-unloading cycles by the data acquisition system Leane. The acquired data are reported in Fig. 4.54.
95
-60000
Cycle I Cycle II Cycle III
-55000 -50000 -45000 -40000
F [N]
-35000 -30000 -25000 -20000 -15000 -10000 -5000
0
480
960
1440
1920
2400
2880
3360
3840
4320
4800
5280
5760
0
t [s]
Fig. 4.54: Reinforced arch, loading-unloading cycles.
The presence of the reinforcement induced an horizontal displacement in correspondence of the arch key, that became more significant in proximity of the collapse load. The load-displacement curves for the horizontal and the vertical key displacements are reported in Fig. 4.55 and in Fig. 4.56, respectively; while in Fig. 4.57 the load-displacement curve for the vertical displacement connected with the point of load application is reported.
96
-55000
Cycle I Cycle II Cycle III
-50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000 -15000 -10000
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-5000 -0.5 0
-1.0
-1.5
-2.0
5000
w [mm]
Fig. 4.55: Reinforced arch, load-displacement curve for the horizontal key displacement.
-55000 Cycle I Cycle II Cycle III
-50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000 -15000 -10000
2
1
0
-5000 -1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
5000
v [mm]
Fig. 4.56: Reinforced arch, load-displacement curve for the vertical key displacement.
97
-55000 -50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000
Cycle I Cycle II Cycle III
-15000 -10000
0
-5000 -1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
5000
v [mm]
Fig. 4.57:Reinforced arch, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
During the test, the first loading-unloading cycle was characterized by arrangement phase in which a push due to the bricks at extremities determined an horizontal sliding of the arch; unfortunately it was not possible to measure it because there was not the suitable instrumentation. The presence of reinforcement has increased a lot the load-bearing of the arch, in particular the arch strength is 100 times higher.
98
5. MODELING AND NUMERICAL PROCEDURES
5.1. Introduction In this chapter the modeling and the numerical procedures developed to study the behavior of unreinforced and reinforced masonry arches are presented. In particular, two approaches able to solve nonlinear problems are illustrated: the first one is based on the stress formulation and the second one is based on the displacement formulation. In the stress formulation, the structural problem is faced and solved developing a complementary energy approach. A numerical procedure, based on a new formulation of the arc-length method, proposed by Riks in the framework of the displacement approach [62], is developed. In the displacement approach, a three nodes finite element based on the Timoshenko’s theory is implemented into FEAP code [63]. The nonlinear problem is solved by the application of the finite element method. Moreover, a new post-computation technique of the stresses at the masonry-FRP interface is proposed, which takes into account the heterogeneity of the masonry material, responsible of local stress concentration. The proposed post-computation of the FRP-masonry interface stresses is based on a simplified approach of the multiscale method. In fact, once the stress analysis is performed on the homogenized model of the arch, a micromechanical study is developed, considering the different materials which constitute the masonry, i.e. the block, the mortar and the FRP reinforcement. Numerical applications are developed to assess the effectiveness of the proposed models. Numerical results are put in comparison with the experimental results available in the literature and with new experimental evidences obtained at the 99
LAPS. Moreover the unreinforced masonry arches numerical results are put in comparison with those obtained by the application of the limit analysis.
5.2. Masonry constitutive models The masonry material is modeled as a no-tension material with a limited compressive strength. In particular, a nonlinear elastic constitutive relationship is considered for the masonry material as the one proposed by Lucchesi et al. [36]. This approach is simple and it can be considered effective for monotonic loading condition, when local unloading does not occur in any point of the structure. When the loading cannot be considered monotonic, the use of an elasto-plastic no-tension model is required, as the one presented in Reference [18], where a model and a numerical procedure were proposed and several numerical applications for reinforced masonry panels and arches were developed. The structural behavior of the reinforced arch is determined developing a variational formulation based on the complementary energy, i.e. adopting a stress approach. In fact, for the analysis of elements made of no-tension material, the solution of the structural problem, if it exists, is unique in terms of stress, while it could be not unique in terms of displacements, as proved by Lucchesi et al. [36].
5.2.1. Model 1 The masonry material is modeled as a masonry like material, assuming a behavior characterized by no-tension response with limited strength in compression, Fig. 5.1. The lack of tensile response gives an admissibility condition for stresses:
σM ≤ 0 100
(5.1)
The condition σ M = 0 can be defined as a limit or collapse condition and in this case the strains have an indefinite non negative value. The compression strength is denoted as σy. The adopted model is simple, but very effective for a wide class of problems, as emphasized in literature.
σM ε
εy O
σy E
D
Fig. 5.1: Masonry constitutive model, model 1.
In Fig. 5.1 it can be individuated the following features: feature OD: linear elastic; feature DE: ideal plastic.
In terms of deformations, there are three distinctive features: feature OD: elastic reversible strain; feature DE: inelastic irreversible strain; point E: yield strain.
It is assumed the existence of the following form for the complementary energy density governing the stress-strain relationship:
ψ M (σ M , τ M ) =
1 1 2 σ M 2 + I Σ (σ M ) + τ (1 − h (σ M ) ) 2 EM 2GM 101
(5.2)
where EM and GM are the elastic and the shear modulus of masonry respectively, I Σ is the indicator function of the admissible set Σ = ⎡⎣σ y , 0 ⎤⎦ , assuming the following
values: I Σ = 0 if σ M ∈ Σ
(5.3)
I Σ = +∞ otherwise
and h (σ M ) is the Heaviside’s function, assuming the following values: h (σ M ) = 1 if σ M ≥ 0 h (σ M ) = 0 otherwise
ψ
σM
Fig. 5.2: Masonry complementary energy density.
102
(5.4)
As consequence of the existence of the potential (5.2), that is schematized in Fig. 5.2, the normal and tangential stresses, σ M and τ M , do not depend on the specific strain history and they can be determined by the formulas:
σ M = 0⎫ ⎬ τM = 0 ⎭
if ε > 0
σ M = EM ε ⎫ ⎬ τ M = GM γ ⎭
if ε y < ε ≤ 0
σ M = EM ε y ⎫⎪
(5.5)
if ε ≤ ε y
⎬
τ M = GM γ ⎪⎭
where ε y = σ y / EM is the limit strain in compression and σ y is the compressive strength. A simple shear stress-strain behavior is assumed; in fact, it is nonlinear elastic in the part of the cross-section in compression. In this way the possible collapse due to shear sliding cannot be reproduced.
5.2.2. Model 2 In order to improve the model previously presented, it is considered a constitutive law characterized, for small values of the strain, by a quadratic relationship between the stress and the strain, as the one proposed for the concrete by the Eurocode 2. Thus, for ε ≤ ε y , the stress-strain relationship is:
σ M = αε 2 + βε + μ The following conditions are imposed on the relationship (5.6): 103
(5.6)
ε ε ε ε
=0 =0 = εy = εy
, , , ,
σM σM σM σM
=0 '= E =σy '=0
(5.7)
Solving equations (5.7) with respect to α , β , μ and ε y , it results:
α =−
σy ε y2
β=
2σ y
εy
μ = 0 εy =
2σ y E
(5.8)
Substituting solution (5.8) into the equation (5.6):
σM = −
σy 2 ε + Eε ε y2
for ε y ≤ ε ≤ 0
(5.9)
Finally, the normal and tangential masonry stresses, σ M and τ M , can be determined by the formulas:
σ M = 0⎫ ⎬ τM = 0 ⎭
if ε > 0
σ M = αε 2 + βε ) ⎫⎪ ⎬ τ M = GM γ ⎪⎭
if ε y < ε ≤ 0
σ M = σ y ⎪⎫ ⎬ τ M = GM γ ⎪⎭
if ε ≤ ε y
(5.10)
in which α and β assume the values reported in expressions (5.8). Also in this case the possible collapse due to shear sliding cannot be captured. The constitutive relationship is schematically illustrated in Fig. 5.3.
104
σM εu
εy
ε O
σy
Fig. 5.3: Masonry constitutive model, Model 2.
5.3. FRP constitutive model Design of FRP reinforcement should ensure that the FRP system is always in tension. In fact, compression FRP is unable to increase the performance of the strengthened masonry member due to its small area compared to that of compressed masonry. Moreover, FRP in compression may be subjected to debonding due to local instability. A uniaxial linear elastic response is assumed for the FRP reinforcement, as illustrated in Fig. 5.4.
105
brittle failure
σ
ε
Fig. 5.4: FRP constitutive model.
The stress-strain relationship is:
σ R = ER ε
(5.11)
where ER is the elastic modulus of the FRP reinforcement. The corresponding complementary energy is:
ψ R (σ R ) =
1 σ R2 2 ER
106
(5.12)
5.4. Limit analysis In this section a brief discussion both on the plastic collapse theorems and limit analysis is presented. Numerous studies have been made on the theory of plasticity since the Hill’s [64], [65] and Prager’s and Hodge’s [66] works. The aim of the limit analysis is to evaluate the load capacity and the collapse mechanism of structures. Considering the limit behavior of the material, through a definition of a yield function ϕ in terms of stresses, it is assumed that if ϕ < 0 the material remains in the elastic phase, if ϕ = 0 the material becomes plastic and if
ϕ > 0 the stress state is inadmissible. The set ϕ = 0 is called the yield surface and the conditions ϕ ≤ 0 represent the admissible stresses. According to the definition of
ϕ , all points that are inside or on the yield surface are admissible, while all points located outside the yield surface are inadmissible. When the stress state belong to the yield surface and the plastic behavior is activated, it is necessary to define the flow direction. According to the classical limit analysis theory [67], the yield surface is convex and the flow direction is normal to the yield surface. The normality condition assures that the energy dissipated by the flow is the maximum possible. The normality condition is very important because it introduces great simplifications in the limit analysis theory and it is the base of the limit analysis theorems. For a structure, it is possible to define a statically admissible state (safe state) for which the internal stresses are in equilibrium with the external forces and the yield conditions are fulfilled in all the points. Making a proportional loading analysis, it is necessary to define q , that is the base variable load, λ , the definite positive load factor and λ q , the variable load applied on the structure. The applied load can be increased from zero until a limit value for the structure, through the use of the load factor. This limit value is called the safety factor. In the limit analysis theory, the
107
static and kinematical theorems are proved; moreover, the uniqueness theorem can be also proved. The static theorem, also called lower-bound theorem, affirms that the safety factor is the largest of all statically admissible load factors. In other words, if it is possible to find a statically admissible stress field for a given load factor. Consequently the structure is in a safe condition under that load level. The kinematical theorem, also called upper-bound theorem, ensures that the safety factor is the smallest of all the kinematically admissible load factors. Finally, for the uniqueness theorem the largest factor defined by the static theorem is equal to the smallest factor defined by the kinematical theorem. The use of those fundamental theorems requires the adoption of specific hypotheses, particularly for masonry structures. Among these, the non tensile strength, the infinite compressive strength, the absence of sliding failure and the small displacements. In the following the kinematical theorem is applied to the case of unreinforced masonry structures considering the no-tensile strength of the masonry.
108
5.5. Arch model The arch model is based on the theory of curvilinear beam. Several shear deformation beam theories are available in literature. In the following the Timoshenko’s beam theory, [68] and [69], is considered; it is widely used in structural analysis, as it accounts for the transverse shear deformation in a simple and effective manner. The compatibility, the equilibrium and the constitutive equations governing the problem of the arch are available in literature; herein, only the final results are reported. Two coordinate systems are introduced: a global system (O, x, y, z) and a local system (x*, y*, z*), with x* and y* rotated of an angle θ with respect to y, as schematically illustrated in Fig. 5.5.
Fig. 5.5: Arch global and local systems.
A typical infinitesimal part of the arch is reported in Fig. 5.6. The quantities in the local coordinate system are computed as function of the ones represented in the global system using a rotation matrix R. The local radius of the arch is indicated as R, while s denotes the curvilinear abscissa.
109
z
O
T M q*
N p* y*
y
s
z* T+dT M+dM N+dN
dθ
Fig. 5.6: Infinitesimal arch element.
5.5.1. Governing equation of the arch The arch is assumed subjected to a permanent and to a variable loading p and λ q , respectively, with λ the load multiplier. Thus, the equilibrium equations written in the local coordinate system take the form: Δc + p* + λq* = 0
where ⎡ d ⎢ ds ⎢ Δ=⎢ 0 ⎢ ⎢ ⎢− 1 ⎢⎣ R
0 d ds 0
1⎤ R⎥ ⎧ ps * ⎫ ⎧ qs * ⎫ ⎧N ⎫ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1⎥ c = ⎨ M ⎬ p* = ⎨0 ⎬ q* = ⎨0 ⎬ ⎥ ⎪ ⎪ ⎪p *⎪ ⎪q * ⎪ ⎥ ⎩T ⎭ ⎩ r ⎭ ⎩ r ⎭ d⎥ ds ⎥⎦
110
(5.13)
with N, M and T the stress resultants and ps*, qs* and pr*, qr* the tangential and radial distributed components of the loads. Note that p* and λ q* are the permanent and variable loads vectors represented in the local coordinate system.
5.5.2. Kinematics of the arch The kinematics of the typical cross section of the arch is defined by the transversal and the axial displacements, v* and w0* respectively, and by the rotation ϕ. The compatibility equations are: % * =d Δη
(5.14)
where ⎡ d ⎢ ds ⎢ %Δ = ⎢ 0 ⎢ ⎢ ⎢− 1 ⎣⎢ R
1⎤ R⎥ ⎧ w0* ⎫ ⎧ε 0 ⎫ ⎥ d ⎪ ⎪ ⎪ ⎪ * ⎥ 0 η = ⎨ϕ ⎬ d = ⎨ χ ⎬ ⎥ ds ⎪v* ⎪ ⎪γ ⎪ ⎥ ⎩ ⎭ ⎩ ⎭ d⎥ 1 ds ⎥⎦ 0
in which γ representing the shear deformation, ε0 the axial strain and χ the bending curvature.
5.5.3. Cross section The cross sections of the arches typically present very regular geometry; thus, without loosing in generality, a rectangular cross section is considered in the 111
following. In accordance with the local coordinate system introduced above, the section lies in the x*y* plane, with x* and y* principal inertial axes. In the determination of the overall behavior of the reinforced masonry arch section, a perfect adhesion between masonry and FRP is assumed. The strain at a point of the section is:
ε = ε 0 + y* χ
(5.15)
The section AM of the masonry is split in three parts: the no-reagent part for which ε > 0 , denoted as Ant; the compressed part for which ε y < ε ≤ 0 , denoted as Ae; the compressed part subjected to a constant stress for which ε ≤ ε y , denoted
as Ap. In order to determine the parts Ant, Ae and Ap, the neutral and the plasticity axes
y* = yn* and y* = ym* , for which the strain attains the values zero and ε y , respectively, are determined:
0 = ε 0 + yn* χ
ε y = ε0 + y χ * m
ε0 χ
⇒
yn* = −
⇒
ε −ε y = y 0 χ
(5.16)
* m
Two possible cases can occur, schematically illustrated in Fig. 5.7, depending on the sign of the curvature.
112
AFRP
AFRP
Fig. 5.7: Axes position for positive and negative curvature.
The axes defining the compressed parts of cross section are determined as:
χ ≥0
Ap : y1 ≤ y ≤ y3 Ae : y3 < y ≤ y2
⎧ h ⎪ y1 = − 2 ⎪ ⎪⎪ ⎧h ⎧ h * ⎫⎫ ⎨ y2 = min ⎨ , max ⎨− , yn ⎬⎬ ⎩ 2 ⎭⎭ ⎩2 ⎪ ⎪ ⎧h h ⎫⎫ ⎪ y3 = min ⎨ , max ⎧⎨− , ym* ⎬⎬ ⎪⎩ ⎩ 2 ⎭⎭ ⎩2
(5.17)
χ 0 and, as consequence of equations (5.5), σ M = 0 τ M = 0 ; Ae
where
εy < ε ≤ 0
and,
as
consequence
of
equations
(5.5),
σ M = EM ε τ M = GM γ ; in the compressed part of the cross-section, characterized by a linear stress-strain relation, the normal stress is represented as σ M = σ 0 + y *σ 1 ; Ap where ε ≤ ε y and σ M = EM ε y τ M = GM γ .
The resultants in the reinforced masonry are: ⎧ ⎫ ⎪ ∫ σ y dA + ∫ (σ 0 + y*σ 1 ) dA + N R ⎪ ⎪A p ⎪ Ae ⎪ ⎪ N ⎧ ⎫ ⎧ cˆ ⎫ ⎪ ⎪ ⎪ * ⎪ c = ⎨ ⎬ = ⎨ M ⎬ = ⎨ ∫ y σ y dA + ∫ y* (σ 0 + y *σ 1 ) dA + M R ⎬ ⎩T ⎭ ⎪ T ⎪ ⎪ Ap Ae ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ τ dA ⎪ A ∪∫ A M ⎪ ⎩e p ⎭
(5.19)
Developed, the formulas (5.19) become: ⎧σ y Ap + Aeσ 0 + Seσ 1 + N R ⎫ ⎪ ⎪ c = ⎨σ y S p + Seσ 0 + I eσ 1 + M R ⎬ ⎪ ⎪ ⎩τ M AS ⎭
(5.20)
where AS = χT ( Ae + Ap ) is the shear area, χT is the shear correction factor, Se and Sp are the static moments of elastic and plastic part, respectively, Ie is the inertial moment of elastic part, NR and MR are the stress resultants due to the reinforcement. 114
Because of the perfect adhesion between the masonry arch and the reinforcement, the resultants of the stresses at the top ( y * = − h / 2 ) and the bottom ( y * = h / 2 ) reinforcements are expressed as: h⎞ ⎛ N R + = n0 ⎜ σ 0 − σ 1 ⎟ AR + 2⎠ ⎝ h⎞ ⎛ N R − = n0 ⎜ σ 0 + σ 1 ⎟ AR − 2⎠ ⎝
(5.21)
with n0 = E R / EM the homogenization coefficient; so that the resultants of the reinforcements are:
N R = n0 (σ 0 AR + σ 1S R ) M R = n0 (σ 0 S R + σ 1 I R )
(5.22)
where
AR = AR+ + AR−
S R = h / 2 ( AR+ − AR− ) I R = ( h / 2)
2
(A
+ R
(5.23)
+ AR− )
Formulas (5.23) represent the area, the static and the moment of inertia, respectively, of the FRP. It can be emphasized that the parts Ant, Ae and Ap are not known a priori, but they have to be determined as functions of the kinematical quantities ε0 and χ. Setting:
115
) ⎧ Ap ⎫ ⎧σ ⎫ σ = ⎨ 0 ⎬ , Q = σ y ⎨ ⎬ , J = J% + J , ⎩σ 1 ⎭ ⎩S p ⎭
(5.24) ⎡A J% = ⎢ e ⎣ Se
Se ⎤ ) ⎡A , J = n0 ⎢ R ⎥ Ie ⎦ ⎣ SR
SR ⎤ I R ⎥⎦
from expression (5.20), taking into account equations (5.22), the values of σ 0 and
σ 1 are obtained as: σ = H ( cˆ − Q )
(5.25)
where H = J −1 . It can be emphasized that, because of expression (5.25), the stress in the masonry as well as the stress in the reinforcement can be expressed as function of the stress resultants N and M. In particular, the stress in the elastic part of the masonry section and on the reinforcements is determined as:
σ M = H ( cˆ − Q ) • Y N R + = n0 H ( cˆ − Q ) • Y + AR +
(5.26)
N R − = n0 H ( cˆ − Q ) • Y − AR − where Y = {1 y * } , Y + = {1 h / 2} and Y − = {1 − h / 2} , while the symbol • T
T
T
indicates the scalar product.
116
5.6.1. Complementary energy In this section the elastic problem is faced and solved developing an energy approach. The complementary energy of the structure is defined as: Ψ (σ , τ ) =
∫ ψ (σ ,τ ) dV + ∫ ψ (σ ) dV M
R
VM
VR
⎛ σ2 τ2 ⎞ σ2 = ∫⎜ + dV ⎟ dV + ∫ 2 EM 2GM ⎠ 2 ER VM ⎝ VR
(5.27)
where VM and VR are the masonry and the reinforcement volume, respectively. In particular, the complementary energy in the masonry and in the reinforcement domains can be written in the form: ⎡ 1 ⎛ ⎞⎤ 2 2 ⎢ ⎜ + σ dA σ dA M θf ∫A y ⎟⎟⎥⎥ ⎢ 2 EM ⎜⎝ A∫e p ⎠ ⎥ R dθ ∫V ψ M (σ M ,τ M ) dV = θ∫ ⎢ 1 ⎥ M i ⎢ τ M 2 dA ∫ ⎢ + 2G ⎥ M Ae ∪ Ap ⎣ ⎦ ⎡ 1 ⎤ H ( cˆ − Q ) ⊗ H ( cˆ − Q ) • J% ⎥ θf ⎢ 2 EM ⎥ R dθ =∫⎢ 2 2 ⎢ ⎥ A σ T y p θi S + ⎢+ ⎥ 2GM AS ⎣⎢ 2 EM ⎦⎥
(5.28)
θf
1 ∫ ψ (σ ) dV = θ∫ 2E ∫ σ R
R
VR
i
θf
=
∫
θi
2 R
dA ds
R AR
) n0 H ( cˆ − Q ) ⊗ H ( cˆ − Q ) • J R dθ 2 ER
117
(5.29)
with θi and θ f the angles defining the initial and the final section of the arch. Finally, the complementary energy is obtained as:
⎡ 1 σ y 2 Ap ⎤ ˆ ˆ H c Q H c Q J − ⊗ − • + ( ) ( ) ⎥ θf ⎢ 2 EM 2 EM ⎥ ⎢ R dθ Ψ ( cˆ , T ) = ∫ ⎢ ⎥ T2 θi ⎢+ ⎥ ⎢⎣ 2GM AS ⎥⎦
(5.30)
The solution of the problem is determined minimizing the complementary energy (5.30) under the equilibrium constraint. In fact, the equilibrated stress resultants admit the following representation form:
h
c = c p + λc q + ∑ xi ci i =1
h ⎧ ⎫ + + N λ N x N ∑ p q i i ⎪ ⎪ h ⎧ ⎫ ⎪ i =1 ⎪ ⎪cˆ p + λ cˆ q + ∑ xi cˆ i ⎪ ⎪ h ⎪ ⎪ ⎪ i =1 =⎨ ⎬ = ⎨ M p + λ M q + ∑ xi M i ⎬ h i =1 ⎪T + λT + x T ⎪ ⎪ ⎪ ∑ q i i h ⎪⎩ p ⎪ ⎪ ⎪ i =1 ⎭ ⎪ T p + λTq + ∑ xiTi ⎪ i =1 ⎩ ⎭
(5.31)
with c p a field of stress resultants equilibrated with permanent loads p, cq a field of stress resultants equilibrated with variable loads q, ci with i=1,..,h fields of selfequilibrated stress resultants and xi with i=1,..,h statically unknown parameters. The number h of self-equilibrated stresses depends on the structural system and on the constraint conditions. Substituting the representation form (5.31) into the complementary energy (5.30), the stationary condition of complementary energy with respect to the unknown minimum 118
parameters x1, x2,.., xh is enforced in order to solve the elastic problem. The typical stationary equation takes the form: h ⎡ 1 ⎤ ⎛⎛ ⎞ ⎞ H ⎜ ⎜ cˆ p + λcˆ q + ∑ xi cˆ i ⎟ − Q ⎟ • cˆ j ⎥ ⎢ i =1 θ f ⎢ EM ⎠ ⎝⎝ ⎠ ⎥ ∂Ψ ⎢ ⎥ R dθ h = 0= ∂x j θ∫i ⎢ T j ⎜⎛ Tp + λTq + ∑ xiTi ⎟⎞ ⎥ ⎢ ⎥ i =1 ⎝ ⎠ ⎢+ ⎥ GM AS ⎣ ⎦
(5.32)
which correspond to a kinematical compatibility equation. In fact, denoting as: θf
T jTp ⎞ ⎛ 1 sp = ∫ ⎜ H cˆ p • cˆ j + ⎟ R dθ GM AS ⎠ θ i ⎝ EM θf
T jTq ⎛ 1 sq = ∫ ⎜ Hcˆ q • cˆ j + GM AS θ i ⎝ EM
⎞ ⎟ R dθ ⎠
θf
T jTi ⎞ ⎛ 1 si = ∫ ⎜ Hcˆ i • cˆ j + ⎟ R dθ GM AS ⎠ θ i ⎝ EM θf
sQ =
1
∫ θ E i
(5.33)
Q • cˆ j R dθ
M
the compatibility equation (5.32) takes the form:
h
s p + λ s q + ∑ xi si − sQ = 0
(5.34)
i =1
where s p , s q , si and sQ are vectors of h components which assume the physical
meaning of the displacement associated to the permanent and variable loadings p and
119
q, to the self-equilibrated stress resultants ci and to the additive stresses due to the
inelastic behavior of the masonry material. It can be emphasized that, because of the considered nonlinear constitutive laws, the vectors s p , s q , si and sQ depend on the solution, in fact they depend on the partition of the section AM of the masonry into the no-reagent part Ant, the elastic part Ae and the plastic part Ap, i.e. s p = s p ( Ae , Ap ) ,
s q = s q ( Ae , Ap ) ,
si = si ( Ae , Ap ) and
sQ = sQ ( Ae , Ap ) .
The integration of equations (5.32) and (5.33) can be performed considering the arch composed of a number of nT parts in each of which the positions of the axes defined by y1, y2 and y3 are taken constant.
5.6.2. Arc-length technique The considered constitutive law for the masonry material, characterized by limited strength in compression with no-tensile response, leads to solve a nonlinear problem governed by equation (5.32). It could be remarked that, because of the elastic character of the constitutive equations for both the masonry and the reinforcement, the solution of the structural problem does not depend on the loading history, so that the stress state can be computed once the loading is assigned. Although the constitutive equations are not written in evolutive form, a numerical procedure able to solve the nonlinear problem (5.32) is developed considering the loading applied in several steps. In such a way, each loading step results easier to solve and, moreover, it is possible to define the behavior of the structure along the whole monotone loading path. In this context, the compatibility equation (5.34) can be written in the form:
120
s = s p ( Ae , Ap ) + ( λn + Δλ ) s q ( Ae , Ap ) + + ∑ ( xi ,n + Δxi ) si ( Ae , Ap ) − sQ ( Ae , Ap ) = 0 h
(5.35)
i =1
Furthermore, it is possible to define a limit load for the structure, i.e. a load multiplier λ which induces the collapse of the arch. In order to evaluate the whole nonlinear structural response of the arch and to compute the limit load, the arc-length technique is considered. The arc-length procedure is often developed in the framework of displacement or mixed formulation of the structural problem. In the following, a new version of the arc-length technique, based on the stress formulation, is proposed. In particular, the developed arc-length procedure is based on the kinematical compatibility equation (5.35) and on a constraint equation. The nonlinear equation (5.35) is solved developing an iterative procedure within each load step. Thus, denoting by the superscript k the solution at the k-th iteration, the new solution is determined developing equation (5.35) in Taylor series:
h
s = sk + ∑ i =1
∂s ∂xi
δ xi + s =s
k
∂s ∂λ
= s k + K k δ x + S k δλ = 0
where
121
δλ s =sk
(5.36)
(
) (
) (
s k = s p Ae k , Ap k + λn + Δλ k s q Ae k , Ap k h
(
) (
)
(
)
+ ∑ xi ,n + Δxi k si Ae k , Ap k − sQ Ae k , Ap k i =1
⎡ ∂s1 ⎢ ⎢ ∂x1 s =sk ⎢ . ⎢ Kk = ⎢ . ⎢ . ⎢ ⎢ ∂s ⎢ h ⎢⎣ ∂x1 s =sk
⎤ ⎥ s =s k ⎥ ⎥ . ⎥ .... . ⎥ ⎥ ⎥ ⎥ ∂sh .... ⎥ ∂xh s =sk ⎥⎦ ...
∂s1 ∂xh
)
⎧ ∂s1 ⎫ ⎪ ∂λ k ⎪ s =s ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ k S =⎨ . ⎬ ⎪ . ⎪ ⎪ ⎪ ⎪ ∂sh ⎪ ⎪ ⎪ ⎩ ∂λ s =sk ⎭
(5.37)
The new solution is determined solving equation (5.36):
δ x = − ( K k ) s k − ( K k ) S k δλ k = δ x k + δ x t k δλ −1
−1
(5.38)
so that the variation of statically unknown parameters results: Δx = Δx k + δ x = Δx k + δ x k + δ xt k δλ k
(5.39)
The updating of the geometrical quantities Ae and Ap is performed computing the neutral and the plasticity axes, solving equation (5.16), and then (5.17) or (5.18). Note that the kinematical parameters present in formulas (5.16) are evaluated solving the resultant constitutive equations in a typical section of the arch: ⎛ ⎡ Ae k ⎧Nk ⎫ = E ⎜⎢ k ⎨ k⎬ M ⎜ ⎩M ⎭ ⎝ ⎣ Se
Se k ⎤ ⎡A + n0 ⎢ R k ⎥ Ie ⎦ ⎣ SR 122
S R ⎤ ⎞ ⎧ε 0 k ⎫ ⎟⎨ ⎬ I R ⎥⎦ ⎟⎠ ⎩ χ k ⎭
(5.40)
The constraint equation required in the arc-length method is defined choosing a suitable control parameter. In particular, it is assumed as control parameter the maximum variation of bending curvature evaluated on all the cross sections of the arch. Thus, it is set:
χ% = max Δχ (θ )
(5.41)
θi ≤θ ≤θ f
and the constraint equation is assumed of the form:
χ% 2 − Δl 2 = 0
(5.42)
where Δl is a given length of the radius considered to follow the mechanical response path of the structure. For a fixed value of the angle θ , i.e. in the typical cross-section, solving equation (5.40) with definition (5.41), it is set: 2
( χ% )
2
h ⎡ k⎛ ⎞ ⎤ Δ Δxi N i ⎟ ⎥ H λ N + q ∑ ⎢ 21 ⎜ i =1 ⎝ ⎠ ⎥ =⎢ / EM2 h ⎢ ⎥ ⎛ ⎞ ⎢ + H 22 ⎜ Δλ M q +∑ Δxi M i ⎟ ⎥ i =1 ⎝ ⎠ ⎦⎥ ⎣⎢
(5.43)
On the other hand, as it is Δλ = Δλ k + δλ , equation (5.43) becomes:
( χ% )
2
(
= H 21k
) (δλ n 2
λ
(
+ n ) + H 22 k 2
) (δλ m 2
+2 H 21k H 22 k (δλ nλ + n )(δλ mλ + m ) where:
123
λ
+ m)
2
(5.44)
h
(
)
(
)
n = ∑ Δxi k + δ xi k N i + Δλ k N q i =1 h
m = ∑ Δxi k + δ xi k M i + Δλ k M q i =1
(5.45)
h
nλ = N q +∑ δ x N i k t ,i
i =1
h
mλ = M q +∑ δ xtk,i M i i =1
Equation (5.42), taking into account formula (5.44), can be written in the form:
δλ 2 a1 + δλ a2 + a3 = 0
(5.46)
where:
(
a1 = nλ 2 H 21k
)
(
2
a2 = 2nλ n H 21k +2n mλ H
(
a3 = H 21k
)
2
k 21
(
+ mλ 2 H 22 k
)
2
(
)
2
+ 2nλ mλ H 21k H 22 k
+ 2mλ m H 22 k
H 22
(
)
2
+2nλ m H 21k H 22 k
(5.47)
k
n 2 + H 22 k
)
2
m 2 + 2n m H 21k H 22 k − Δl 2
Equation (5.46) is solved with respect to δλ , leading to the determination of two roots, δλ1 and δλ2 . The solution is chosen according to the cylindrical method, such that δλ = δλ2 if c2 > c1 and it is δλ = δλ1 otherwise, with
124
h
(
a4 = ∑ Δxi k δ xi k + Δxi k i =1
)
h
a5 = ∑ Δxi k δ xi k
(5.48)
i =1
c1 = a4 + a5δλ1 c2 = a4 + a5δλ2 The load multiplier and the statically unknown parameters increments are updated setting: Δλ = Δλ k + δλ
(5.49)
Δx = Δx k + δ x = Δx k + δ x k + δλ δ xt k
The iteration process within each loading step is performed until the residual r = s is greater than a fixed tolerance.
5.7. Displacement formulation The displacement formulation approach has been developed considering both Model 1 and Model 2. In this section, only the Model 2 is described. The section AM of the masonry is specialized into: Ant where ε > 0 and, as consequence of equations (5.10), σ M = 0 τ M = 0 ; Ae
where
εy < ε ≤ 0
and,
as
consequence
σ M = αε 2 + βε τ M = GM γ ; Ap where ε ≤ ε y and σ M = σ y τ M = GM γ .
The resultants in the reinforced masonry are: 125
of
equations
(5.10),
⎧ ⎫ ⎪ ∫ σ y dA + ∫ (αε 2 + βε ) dA + N R ⎪ ⎪A p ⎪ Ae ⎪ ⎧N ⎫ ⎪ ⎧ cˆ ⎫ ⎪ ⎪ ⎪ * ⎪ * 2 c = ⎨ ⎬ = ⎨ M ⎬ = ⎨ ∫ y σ y dA + ∫ y (αε + βε ) dA + M R ⎬ ⎩T ⎭ ⎪ T ⎪ ⎪ Ap Ae ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ τ dA ⎪ A ∪∫ A M ⎪ ⎩e p ⎭
(5.50)
The axial resultant and bending moment are computed as:
N=
∫σ
dA +
y
Ap
∫ (αε
2
+ βε ) dA + N R
Ae
= σ y Ap +
∫ ⎡⎣⎢α (ε
+ y* χ ) + β (ε 0 + y* χ ) ⎤ dA + N R ⎦⎥ 2
0
Ae
(5.51)
= σ y Ap + A% ε 0 + S% χ + N R
M=
∫ yσ *
y
dA +
Ap
∫ y (αε *
2
+ βε ) dA + M R
Ae
= σ ySp +
∫y
Ae
*
⎡α ( ε + y * χ )2 + β ( ε + y* χ ) ⎤ dA + M 0 0 R ⎣⎢ ⎦⎥
(5.52)
= σ y S p + S%ε 0 + I% χ + M R
respectively, where A% =
∫ (αε
+α y* χ + β ) dA = (αε 0 + β ) A e +αχ Se
(5.53)
y* + 2α y*2 + β y* ) dA = (αε 0 + β ) Se + αχ I e
(5.54)
0
Ae
S% =
∫ (αε
0
Ae
126
I% =
∫ (α y
*3
)
χ + β y*2 + αε 0 y*2 ) dA = (αε 0 + β ) I e + αχ I
(5.55)
Ae
and AS = χT ( Ae + Ap ) is the shear area, χT is the shear correction factor, NR and MR are the stress resultants due to the reinforcement.
5.7.1. Kinematics The kinematics of the Timoshenko beam theory, schematically illustrated in Fig. 5.8, can be expressed as
u1 = 0 u2 = v
(5.56)
u3 = w + yϕ
Fig. 5.8: Timoshenko’s beam theory.
The strain field is given by
ε0 = w' χ =ϕ ' γ = v '+ ϕ 127
(5.57)
Note that the prime on a variable indicates its derivative with respect to z . The kinematics strain vector is introduced as ⎛d ⎜ ⎧ε 0 ⎫ ⎧ w ' ⎫ ⎜ dz ⎪ ⎪ ⎪ ⎪ d = ⎨χ ⎬ = ⎨ ϕ ' ⎬ = ⎜ 0 ⎜ ⎪ γ ⎪ ⎪v '+ ϕ ⎪ ⎜ ⎩ ⎭ ⎩ ⎭ ⎜ 0 ⎜ ⎝
0 0 d dz
⎞ 0 ⎟ ⎧ w⎫ ⎟ ⎧ w⎫ d ⎟⎪ ⎪ ⎪ ⎪ ⎨v ⎬ = L⎨v ⎬ ⎟ dz ⎪ ⎪ ⎪ϕ ⎪ ⎟ ⎩ϕ ⎭ ⎩ ⎭ 1 ⎟⎟ ⎠
(5.58)
5.7.2. Finite element implementation A discussion on displacement based or mixed formulation beam elements was presented by Crisfield [70], [71]. Reddy, [72]-[74], developed superconvergent (i.e. , yields exact nodal values) locking-free Timoshenko’s beam finite element based on the interdependent interpolation as well as assumed strain formulation. In the same paper, Reddy extended the procedure to develop a locking-free finite element for the third-order beam theory. A consistent beam finite element was proposed in reference [75]. The conventional finite element model of the Timoshenko’s beam is obtained by using Lagrange interpolation of v and ϕ . For example, linear interpolation of both v and ϕ is known to yield a finite element that exhibits locking. Reduced integration
of the shear stiffness alleviates this problem, but does not yield exact values of the displacements at the nodes without using a large number of elements. Here, we consider an alternative interpolation of the dependent variables that yields a lockingfree finite element. In particular, the transverse displacement v is approximated using the Hermite’s cubic interpolation, the rotation ϕ and the axial displacement w are approximated using Lagrange quadratic interpolation: 128
w = N1w w1 + N 2w w2 + N 3w w3 v = N1v v1 + N 2v v2 + N1θ θ1 + N 2θ θ 2
(5.59)
ϕ = N1ϕϕ1 + N 2ϕϕ2 + N 3ϕϕ3 where θ1 = −v '(0) and θ 2 = −v '( L) . Note that vi and θi are the transversal displacements and the slopes of the i − th node, respectively, with i = 1, 2 , and ϕi are the rotations of the cross-sections about the x axis, corresponding to the i − th node, with i = 1, 2, 3 , as schematically represented in Fig. 5.9.
ϕ3
x w1
v1
w2
z
v2
y
θ2 ϕ2
θ1 ϕ1 Fig. 5.9: Beam finite element.
In compact form, formulas (5.59) can be expressed as ⎧ w⎫ ⎪ ⎪ u = ⎨v ⎬ = NU ⎪ϕ ⎪ ⎩ ⎭
where
129
(5.60)
⎛ N1w ⎜ N=⎜ 0 ⎜ 0 ⎝
0 N1v 0
0 N1θ 0
0 0 N1ϕ
N 2w 0 0
0 N 2v 0
0 N 2θ 0
0 0 N 2ϕ
N 3w 0 0
0 0 0 ⎞ ⎟ 0 0 0 ⎟ 0 0 N 3ϕ ⎟⎠ (5.61)
U = {w1 v1 θ1 ϕ1
w2
v2 θ 2 ϕ 2
w3
0 0 ϕ3 }
T
with
v 1
N
N 2v θ
N1
N 2θ
1 N1w = ξ (ξ − 1) 2 1 N 2w = ξ (ξ + 1) 2 w N3 = 1 − ξ 2
(5.62)
(ξ + 2)(ξ − 1) 2 = 4 (ξ − 2)(ξ + 1) 2 =− 4 L(ξ + 1)(ξ − 1) 2 =− 8 L(ξ − 1)(ξ + 1) 2 =− 8
(5.63)
1 N1ϕ = ξ (ξ − 1) 2 1 N 2ϕ = ξ (ξ + 1) 2 ϕ N3 = 1 − ξ 2
(5.64)
L⎞ 2 ⎛ and ξ = ⎜ z − ⎟ , z ∈ [0, L] . 2⎠L ⎝ 130
The beam finite element was implemented in the code FEAP, developing an iterative numerical procedure able to solve the nonlinear arch problem. The proposed procedure is based on the secant stiffness method, and it allows to determine the solution of a nonlinear problem as solution of an opportune sequence of linear problems. A scheme of the numerical procedure is reported: 1) Initially the reagent section is all the geometrical section; 2) The nodal displacement are calculated; 3) At the generic cross section the deformations are noted; 4) Noted the deformations, the neutral and plasticity axes position can be valuated; 5) The new reagent section with its elastic and plastic part is defined; 6) Defined the reagent section the procedure comes back to step 2. The procedure iterates from step 2 to 6 until the residue, computed as the difference between the external forces and those determined by the state of deformation of the structure, is lower than a fixed tolerance.
5.8. Post-computation of the shear stresses The computation of the stresses at the FRP-masonry interface can be very important as they are responsible for the decohesion of the reinforcement from the masonry. It can be remarked that, because of the heterogeneity of the masonry material, the stresses at the interface can present local concentrations. Thus, the normal and shear stresses at the interface, σ d and τ d respectively, can be computed as the sum of two quantities: the first ones σ T and τ T are evaluated enforcing the equilibrium condition of the FRP for a typical infinitesimal element of the arch, the second ones σ h and τ h correspond to the local normal and shear stress concentration due to the masonry material heterogeneity. 131
With reference to Fig. 5.10, the normal and the shear stresses at the extrados and intrados masonry-FRP interfaces can be computed as:
σ T− =
2 N R− b ( 2R − h )
τ T− =
2 N R+ σ =− b ( 2R + h )
2 dN R− b ( 2 R − h ) dθ
dN R+ 2 τ =− b ( 2 R + h ) dθ
+ T
(5.65)
+ T
with evident meaning of the symbols.
N R+ bτ T+
R+
h 2
N R−
bτ T−
+ bσ T− bσ T
N R+ + dN R+
dθ
N R− + dN R−
O
R−
h 2
Fig. 5.10: Normal and shear stress at the masonry-FRP interface.
Indeed, because of the heterogeneity of the masonry, also when the stress resultants in a reinforcement is constant, stresses can occur at the interface. In order to evaluate the stresses profile due to the material heterogeneity, a micromechanical analysis of the reinforced masonry is developed. In particular, it is assumed that the masonry is a periodic heterogeneous material. Because of the symmetry of the repetitive cell with respect to the plane orthogonal to the local beam axes (see Fig. 5.11), the study can be limited to a half of the cell, 132
considering an elastic interface joining the FRP reinforcement to the masonry governed by the relationship: ⎧τ h ⎫ ⎡ Kτ ⎨ ⎬=⎢ ⎩σ h ⎭ ⎣ 0
0 ⎤ ⎧ sτ I ⎫ ⎨ ⎬ Kσ ⎦⎥ ⎩ sσ I ⎭
(5.66)
and Kσ are the tangential and normal stiffness respectively, while sτ I
where Kτ
and sσ I are the relative displacements in the tangential and normal direction.
Block
Mortar
Block
Mortar
ym
yn FRP
FRP Half cell
Half cell
Unit cell
Fig. 5.11: Stresses acting on the mortar joint of the masonry unit cell.
The unit cell is subjected to the normal stresses derived from the structural analysis; in particular, the stresses computed from the structural analysis are applied on the mortar joint, as illustrated in Fig. 5.11. A two-dimensional finite element analysis is performed of the unit cell in the framework of plane stress analysis, evaluating the normal and the shear stress profiles at the interface. As the post-computation of the stress profiles at the masonry FRP interface should be performed for several sections of the arch and for different values of the loading level, a simple numerical strategy is developed. Let nj be the number of nodes of the section where the stresses are applied, nj pre-analyses are developed determining the relative displacements occurring between the FRP and the masonry due to a unit 133
force acting on a single node. The results of the analyses are organized in an influence matrix G, whose i-th column represents the results of the micromechanical analysis due to a unit force applied at the i-th node. Once a loading step is selected and a section of the arch is considered, the structural analysis allows to compute the normal stress profile in the mortar joint, which is considered as an external distributed load for the unit cell. The distributed load is transformed in equivalent nj nodal forces following the classical finite element procedure, defining the vector F. The vector of the relative displacements s I due to the actual distribution of the normal stresses acting on the mortar joint is determined by the matrix product:
s I = GF
(5.67)
Thus, the normal and shear stresses can be computed substituting the values of the relative displacements obtained by expression (5.67) into equation (5.66). The obtained stresses must be added to the quantities determined by formulas (5.65).
134
5.9. Numerical results In this section, the obtained numerical results are presented. The numerical results deal with the stress formulation and the displacement formulation. The reliability of the nonlinear elastic constitutive law (Model 1), compared with Model 2 is tested. The effectiveness of the numerical procedure is also experienced. Moreover the results obtained by the application of the kinematical theorem of the limit analysis are illustrated. Finally a comparison between numerical and experimental results is proposed.
5.9.1. Models and numerical procedures assessment The aim of this analysis is the assessment of the stress and displacement formulations. A round clamped-clamped arch is considered, so that it results three times statically undetermined; the arch is subjected to a vertical downward distributed load of intensity p=10 N/mm and to a horizontal distributed load q=1 N/mm amplified by the multiplier λ. The geometry and the mechanical properties of the adopted materials are reported in the following: Geometry o masonry: round arch with radius R=5000 mm and rectangular crosssection with dimensions b=300 mm and h=1000 mm; o FRP: the thickness is assumed t=0.17 mm, which corresponds to one layer of composite, while the width is taken bFRP=200 mm. Materials o masonry: the Young’s modulus is set Em=15000 MPa, which corresponds to rock blocks, while the Poisson ration is n=0.2 and the shear modulus is Gm=6250 MPa; the limit strength in compression is set sy=7.5 MPa;
135
o FRP: the Young modulus of the carbon-fiber is EFRP=400000 MPa.
In order to assess the effectiveness of the proposed masonry model and to verify the robustness of the numerical procedure based on the complementary energy approach within the dual version of the arc-length technique, displacement finite element analyses are carried out considering the no-tension elasto-plastic masonry model implemented in the code FEAP. Thus, the numerical results obtained by the stress formulation of the nonlinear elastic model are put in comparison with the ones carried out by the displacement approach, based an elasto-plastic open-ended model. The computations are performed setting nT=300 for the stress approach, while a mesh of 60 elements is considered for the displacement finite element formulation. Initially, the response of the un-reinforced arch is studied. In Fig. 5.12, the value of the multiplier λ of the distributed horizontal load is plotted versus the horizontal displacement vk computed at the key of the arch. The results are reported adopting the following acronyms:
NT NTP EC FEM
no-tension material with unlimited compressive strength; no-tension material with limited compressive strength; complementary energy approach; elasto-plastic displacement finite element formulation.
136
3.50 3.00 2.50 2.00
λ 1.50 1.00 NT EC NT FEM NTP EC NTP FEM
0.50 0.00 0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
vk [mm]
Fig. 5.12: Un-reinforced arch modeled considering a no-tension material with unlimited and limited compressive strength.
It can be emphasized that all the computed solutions are in very good agreement. In particular, it can be remarked that there is not significant differences when unlimited or limited compressive strength is considered; in fact, because of the no-tensile capacity of the material, the collapse of the arch occurs for the formation and opening of hinges located at the extrados and at the intrados of the arch. As a consequence, the limited strength in compression does not play a significant role in the overall behavior of the arch. Moreover, the stress approach demonstrates to be effective and robust in the developed computations. Then, the case of arch reinforced at the extrados is studied. As in the previous analyses, several computations are performed and the obtained results are put in comparison. In Fig. 5.13, the plot of the multiplier λ of the horizontal load versus the horizontal displacement vk of the key section is reported.
137
18.00 16.00 14.00 12.00 10.00
λ 8.00 6.00
NT EC
4.00
NT FEM 2.00
NTP EC NTP FEM
0.00 0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
vk [mm]
Fig. 5.13: Reinforced arch modeled considering a no-tension material with unlimited and limited compressive strength.
Once again, it can be remarked the ability of the stress approach to reproduce the nonlinear response of the reinforced arch: the results obtained adopting the nonlinear elastic model does not significantly differs from the ones obtained considering the elasto-plastic model, so that it can be claimed that the loading history does not influence in this case the response of the arch. The differences of the results obtained considering unlimited or limited strength in compression can be remarked. In fact, when the arch is reinforced at the extrados, the hinges at the intrados cannot occur as the crack openings at the extrados are not allowed and the limited strength in compression of the masonry material plays a fundamental role in the response of the arch. Moreover, looking at Fig. 5.12 and Fig. 5.13, it can be noted the effectiveness of the reinforcement in the overall behavior of the arch. Then, computations are performed considering the arch reinforced by 1, 2, 5, 10 and 15 FRP layers, subjected to a vertical distributed load of intensity p = 100 N/mm and to the increasing distributed horizontal load, simulating the effect of an earthquake 138
on the structure. In Fig. 5.14 the curves concerning the horizontal load multiplier versus the horizontal displacement of the key section are reported for the unreinforced and for the reinforced arch.
80.00
15 FRP layers
70.00
10 FRP layers
60.00
5 FRP layers
50.00 2 FRP layers
λ 40.00
80
1 FRP layer 70
60
30.00 λ
50
20.00
30
Unreinforced
10.00
0.00 0.00
40
20
10
Compressive masonry failure Tensile FRP failure 50.00
100.00
150.00
0 0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
7000.00
8000.00
vk [m m ]
200.00
250.00
300.00
350.00
400.00
vk [mm]
Fig. 5.14: Collapse loading for the un-reinforced and reinforced arch.
In that figure, the diagram load multiplier vs horizontal displacement is reported considering two different scales for the displacements: in the main Figure the maximum displacement is set equal to 400 mm, while in the boxed image the maximum displacement is greater. Indeed, in both the considered scales, the reached displacements are much greater than the admissible ones obtained considering the limited compressive deformation of the masonry and the tensile strength of the FRP reinforcement. In fact, the boxed figure is reported only to evaluate the limit load multiplier in the case of un-reinforced and reinforced arch, λ≈13 and λ≈73 respectively, when no care is taken to the limited compressive deformation of the masonry and tensile strength of the FRP. In this case, the limit load for the un-reinforced and reinforced arch is due to the no-tensile response and to the plastic constitutive relationship of the 139
masonry, respectively. It can be remarked that the limit load for the reinforced arch does not depend on the number of FRP layers. Finally, it could be emphasized that the limit load is attained for very high values of the displacements, so that the classical assumption of small displacements and small deformations could be not valid anymore. As a consequence, from the developed numerical computations, it can be concluded that the classical limit analysis is not applicable to evaluate the maximum loading capability of reinforced arch. In the main scale of Fig. 5.14, the loading levels, with the corresponding values of the horizontal displacements, which induce the tensile failure of the FRP reinforcement and the reached limit compressive strain for the masonry, are reported with a square and a triangle, respectively. In particular, in the developed computations, the tensile failure of the FRP is set as σR,y=3500 MPa while the limit compressive strain for the masonry is taken εu=0.0035. When only 1 FRP layer is applied to reinforce the arch, the tensile failure of the reinforcement occurs before the compressive masonry collapse; increasing the number of FRP layers, a gradual inversion of the failure mechanism of the two materials can be noted. In correspondence of 15 FRP layers, the compressive collapse of masonry occurs before the FRP tensile failure. In particular, the following results are obtained: 1 FRP layer
λ≈23 masonry failure
λ≈15 FRP failure
2 FRP layer
λ≈29 masonry failure
λ≈21 FRP failure
5 FRP layer
λ≈42 masonry failure
λ≈35 FRP failure
10 FRP layer
λ≈49 masonry failure
λ≈48 FRP failure
15 FRP layer
λ≈62 masonry failure
λ≈71 FRP failure
It can be concluded that the reinforcement at the extrados of the arch is very effective from a structural viewpoint as its presence is able to enhance the performances of the arch with respect to horizontal loading.
140
5.9.2. Experimental surveys numerical results The numerical results are put in comparison with experimental results both available in literature and obtained by the experimental program realized at LAPS of Cassino.
5.9.2.1. Comparison 1 In this section the experimental evidences carried out by Briccoli Bati and Rovero [54] and numerical results obtained adopting the numerical procedures are illustrated. A reinforced masonry arch is studied and the geometry and mechanical properties of the adopted materials are: Geometry o masonry: arch with radius R=865 mm, rectangular cross-section b=100 mm and h=100 mm and clamped at 30o and 150o; the arch is composed assembling hollow clay masonry units of thickness 25 mm, joined by a mortar layer of 4 mm. o FRP: reinforcement applied at the whole intrados of the arch, with thickness t=0.17 mm, which corresponds to one layer of composite, and width bFRP=50 mm. Materials o masonry: the Young’s modulus of the hollow clay masonry units and of the mortar are 1785 MPa and 133 MPa, respectively, so that the overall modulus of the masonry is set Em=680 MPa, while the Poisson ration is n=0.2 and the shear modulus is Gm=283 MPa; o FRP: the Young modulus of the carbon-fiber is EFRP=230000 MPa.
The structure is subjected to an increasing concentrate force F applied at the key of the arch. The experimental test shows that the arch failure occurs for crushing of the masonry in compression. As consequence, the compressive strength of the masonry plays a fundamental role in the overall behavior of the arch.
141
In Reference [54], the compressive strength of masonry, strongly governed by the mortar strength, is evaluated testing some specimens and it results in the range between 7 and 8 MPa. It can be pointed out that this value is reasonably higher than the strength of the masonry constituting the arch, because of the possible imperfections occurring during the construction of the arch, mainly in the key. In particular, the thickness of the mortar bed between two blocks is not the same in whole experimental arch tested in Reference [54], and in correspondence of the key of the arch there is a great reduction of bed mortar. As a consequence, it is found that the arch tested in the laboratory presents some initial geometrical defects in the heterogeneities of the masonry, which induces localization phenomena that significantly reduced the compressive strength of the masonry material with respect to the value obtained on “perfect” specimens. Indeed, some difficulties arise in the evaluation of the limit compressive strength and the limit failure strain, which are necessary to study the arch and to understand its behavior. It can be reasonably assumed that the compressive strength in the masonry arch is reduced from 1/4 to 1/3 with respect to the strength deduced by the specimen tests. Thus, the masonry strength could be set in the range between 1.75 and 2.67 MPa In order to numerically reproduce the behavior of the arch, a parametric analysis is carried out considering different values of the limit compression strength σy in the range from 2.04 to 6.80 MPa. In Fig. 5.15, the comparison between the load-displacement curves obtained by experimental investigation and by numerical analyses is reported.
142
7.00 6.50 6.00 5.50 5.00 4.50 4.00
F [kN] 3.50 3.00 2.50
σy = 2.04 Mpa σy = 2.72 Mpa σy = 3.40 Mpa σy = 6.80 Mpa Experimental results
2.00 1.50 1.00 0.50 0.00 0.00
1.00
2.00
3.00
4.00
5.00
6.00
vk [mm]
Fig. 5.15: Comparison 1, reinforced arch subjected to a concentrate force.
For the different values of limit compression strength σy, it is computed the minimum strain επ/2 and the tensile stress in the reinforcement σR evaluated at the key section, i.e. for θ=π/2, when the experimental collapse load of the arch F=6.5 kN is reached. In particular, it results: επ/2=-0.02538
and σR=1387 MPa
when σy=2.04 MPa,
επ/2=-0.00784
and σR=1053 MPa
when σy=2.72 MPa,
επ/2=-0.00626
and σR=979 MPa
when σy=3.40 MPa,
επ/2=-0.00602
and σR=965 MPa
when σy=6.80 MPa.
It can be emphasized that, in any case, the FRP stress is lower than its failure strength assumed to be 3500 MPa. Looking at Fig. 5.15, the numerical results are in good agreement with the experimental ones when the strength of the masonry is set equal to 2.04 MPa, i.e. belonging to the range announced above taking into account the material imperfection.
143
As consequence of the above arguments, it can be concluded that the comparison between the numerical and experimental results can be considered satisfactory; in fact, it shows the effectiveness of the simple proposed model, which can be able to predict the collapse load. Of course, as the model does not consider any softening effect in compression, the post-critical behavior of the arch characterized by a quite brittle response once the maximum load is reached, cannot be numerically reproduced. As discussed in section 5.8, it can be very interesting to evaluate the normal and, mainly, the shear stresses at the interface between the FRP and the masonry, in order to predict the possibility of decohesion of the reinforcement. In Fig. 5.16, the shear stress profile evaluated close to the key section is reported when the external force is F=1 kN. In particular, as the key section is subjected to a concentrate force and, as consequence, to a very special stress state, the shear stresses are computed for a unit cell at a distance of about the section height from the key section.
0.60
τd τT τh
shear stress [MPa]
0.50
0.40
0.30
0.20
0.10
0.00 0.00
2.00
4.00
6.00
8.00
10.00
z* [mm]
Fig. 5.16: Shear stresses for F=1 kN.
144
12.00
14.00
In this Figure the decohesion shear stresses τ d and the single contributes τ T and τ h , due to the equilibrium condition of the FRP for a typical infinitesimal element of the arch and to the local shear stress concentration due to the masonry material heterogeneity, respectively, are reported. The curve of the decohesion shear stresses presents a maximum values in correspondence of the brick-mortar section because of the different consistency of the brick and the mortar and it shows that the contribute to the value of the shear stresses of the variation of the normal force into the reinforcement is less significant than the effect of the material heterogeneity. In Fig. 5.17, the comparison between the shear stresses obtained for different values of the applied force is illustrated. In particular, the considered force intensities are
F=1 kN, F=3 kN and F=5 kN; it is evident that increasing the value of the applied force, it corresponds a nonlinear increment of the shear stress.
2.00 1.80
F = 1 kN F = 3 kN F = 5 kN
1.60 1.40
shear stress [MPa]
1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
z* [mm]
Fig. 5.17: Shear stresses for different force intensities.
Moreover, for the considered case study, it can be noted that in correspondence of
F = 5 kN , the maximum shear stress reaches the value τ d ≈ 1.8 MPa in the brick. This value can be considered safe for the brick which, according to the experimental 145
investigations [54], is characterized by a limit strength in tension and in compression of 1.7 MPa and 17.39 MPa, respectively. On the other hand, experimental evidences demonstrated that the collapse of the arch occurred with no delamination effects. The decohesion of the FRP from the masonry support could be accounted for considering a suitable interface model a numerical procedure able to predict and reproduce the nonlinear phenomenon [12] and [16].
5.9.2.2. Comparison 2 In this section the numerical results are put in comparison with the experimental results obtained during the experimental campaign realized at LAPS of University of Cassino. The studied arch is schematically reported in Fig. 5.18.
146
5
6
7
8
11 12 13 14 9 10 15
16
17
18
4 3 2
19 20 21 22
1
23 O
Z
Y
Re Ri c f i
O
Z
Y
Fig. 5.18: Reinforced arch geometrical model.
In accordance with the introduced global system, the geometrical characteristics of the arch are:
147
R ext = 576.07 mm Rint = 456.07 mm R ext + Rint 2 ϑi = 8°, ϑ f = 172° RG =
(5.68)
ϑC = ϑ f − ϑi where R ext , Rint and RG are the external, internal and center-line radii respectively;
ϑi , ϑ f and ϑC are the initial, final and central angle, respectively. A very fine mesh is considered for the arch, which was subdivided in 100 linear finite elements. The mechanical properties of each masonry constituents have been reported in chapter 4. In the picture reported in Fig. 5.19 it can be noted the defects and the irregularities. In fact, a significant reduction of the size of the cross section of the arch is noted in the mortar.
Fig. 5.19: Mortar joints irregularities.
148
It was observed that the geometrical variation of mortar joints is comprised between 0.2 mm and 0.7 mm. Moreover, it was noted that the external part of the mortar was characterized by very low mechanical properties, as it was possible to damage the mortar by hands. Thus, the considered reagent masonry mortar section was reduced of 0.7 mm in height and width, with respect to the geometrical section. The size of homogenized reagent section is: bh = 70.5mm , hh = 106mm and lh = 241mm . The elastic modulus of homogenized masonry was determined assuming: 1) 2) 3) 4)
elastic-linear brick constitutive relationship: σ b = Ebε b ; elastic-linear mortar constitutive relationship: σ m = Emε m ; same stress for brick and mortar; elastic-linear masonry constitutive relationship: σ M = EM ε M .
Enforcing the equilibrium and congruence conditions:
σM = σb = σm
(5.69)
ε bbb + ε mbm = ε M (bb + bm )
(5.70)
Substituting the constitutive relationship into equation (5.69):
Ebε b = Emε m = EM ε M
(5.71)
which gives:
εb =
EM εM Eb
Substituting the expression (5.72) into equation (5.70) it results:
149
(5.72)
ε M ( bb + bm ) = ε bbb + ε mbm =
EM E ε M bb + M ε M bm Eb Em
(5.73)
Thus, the elastic modulus of masonry is:
EM =
( bb + bm )
(5.74)
⎛ bb bm ⎞ ⎜ + ⎟ ⎝ Eb Em ⎠
In this specific case, EM ≅ 8300 N mm 2 . This computed value for EM is greater than the true value of the elastic modulus of the masonry; in fact the mortar specimens realized and tested into the laboratories are characterized by mechanical properties significantly greater than those of the mortar joints. The shear modulus is calculated in classic way: GM =
EM ≅ 3400 N mm 2 . 2(1 + ν )
The arch was considered clamped at springs. It is subjected to the its weight and to an additional increasing force applied at section corresponding to the angle ϑ ≅ 70o . Initially the unreinforced arch is studied. The numerical results are obtained considering for masonry both Model 1 and Model 2. In Fig. 5.20 all the results are reported. It can be observed that there is not a substantial difference between the use of Model 1 and Model 2.
150
-800 -750 -700 -650 -600 -550 -500
F [N]
-450 -400 -350 -300
Experimental Arch 1 Experimental Arch 2 FEAP, model 1 FEAP, model 2 Safe theorem
-250 -200 -150 -100 -50 0.00 0
-0.20
-0.40
-0.60
-0.80
-1.00
-1.20
-1.40
-1.60
-1.80
vk [mm]
Fig. 5.20: Comparison 2, unreinforced arch load-displacement curve.
The numerical model approximates in a satisfactory manner the limit load of the masonry arch. The limit load of the arch was calculated also by applying of the kinematical theorem of the limit analysis, on the base of collapse mechanism characterized by the four hinges formation at extrados or at intrados. The hinges position was obtained by minimizing the lading factor. The position of the hinges is compared with the ones obtained from the results determined using the finite element approach. In Tab. 5.1 the hinges position is reported.
151
Element Node [number] [number] 39 20 40 41 87 44 88 89 143 72 144 145 199 100 200 201
Angle [°] [rad] 39.16 0.6831 40.80 78.52
Radius [mm]
39.9800
0.6974
51.6070
79.3400
1.3840
51.6070
125.2600
2.1851
51.6070
171.1800
2.9861
51.6070
0.7117 1.3697
80.16 124.44
1.3983 2.1708
126.08 170.36
2.1994 2.9718
172.00
Hinge angle [°] [rad]
3.0004
Tab. 5.1: Hinges position.
According to the hinges position, the arch was subdivided in three blocks, called block 1, block 2 and block 3. In Fig. 5.21 are schematically illustrated the position of hinges that coincides with the arch relative or absolute centers of rotation.
152
Fig. 5.21: Comparison 2, unreinforced arch kinematical mechanism.
Determined the position of arch relative and absolute centers, the vertical displacement components can be traced on the horizontal fundamental. The itself weight of each one block is: Block 1: P=187.82 N; Block 2: P=219.13 N; Block 3: P=219.13 N;
Applying the virtual work and considering that the external load must be equilibrated by the action of block 2 and 3, it has: Lve = − PIη I + PIIη II + PIIIη III = FLIMη LIM
(5.75)
Solving equation (5.75) the limit load is determined: Lve = 0 ⇒ FLIM = 591.87 N
153
(5.76)
In Fig. 5.20 it can observe that the limit load is not different from the limit load determined using the proposed finite element approach. Then, the reinforced arch was studied using the finite element formulation. The following further input data are considered: dimensions of FRP woven: height tFRP=0.17 mm and width LFRP=100 mm; mechanical properties of FRP: EFRP = 230000 N mm 2 ; homogenization coefficient (considering the perfect adhesion between
masonry and reinforcement): n0 =
EFRP ≅ 27 . EM
The kinematical mechanism of the unreinforced masonry arch is not possible for the reinforced arch. In fact, the presence of reinforcement does not allow the cracks opening on the reinforced side of the arch, consequently the hinges formation on the opposite side is prevented. In Fig. 5.22 the load-displacement curves for the reinforced arch are reported.
-55000 -50000 -45000 -40000
F [N]
-35000 -30000 -25000 -20000 Experimental reinforced arch
-15000
FEAP, model 1
Compressive masonry failure
-10000
FEAP, model 2 -5000 -1 0
-3
-5
-7
-9
-11
-13
-15
-17
-19
-21
-23
-25
-27
-29
-31
-33
vk [mm]
Fig. 5.22: Comparison 2, reinforced arch load-displacement curve.
154
-35
The numerical results agree very well with experimental curve, in particular when the Model 2 for the masonry material is adopted. The collapse of the reinforced arch determined during the laboratory tests occurred because of a shear mechanism; in fact, the sliding of the brick under the applied force occurred, leading to the crushing of masonry. This type of mechanisms is not accounted for in the proposed model, so that the limit collapse displacement cannot be numerically evaluated.
155
6. MULTISCALE APPROACHES
6.1. Introduction The aim of this chapter is to introduce multilevel strategies. Generally, we talk of multilevel strategies when we have a global “ macroscopic” problem, associated to a coarse solution, and a local “microscopic” problem, associated to a finer mesh in a limited zone of interest. In literature there are three families of multilevel approach: the methods based on the homogenization, these based on super-position and these based on decomposition of domain. The reference problem is a problem of classic mechanics: the quasi-static study of a little perturbation on the body Ω , subjected to an imposed displacement field u d on part ∂Ω1 of its surface, to a surface forces field Fd on the complementary part ∂Ω 2 = ∂Ω \ ∂Ω1 and to a volume force field called f d , as illustrated in Fig. 6.1.
Fd
∂Ω 2
fd
∂Ω1
Ω ud
Fig. 6.1: Multiscale approaches, reference problem.
156
We suppose the material is linear elastic, except explicit mention; K is the Hook’s tensor, u is the displacement field, σ is the Cauchy stress field and ε is the deformation field; thus the reference problem can be written as: Kinematics admissibility: u = u d on ∂1Ω ε=
1 ∇u + ∇uT ) in Ω ( 2
(6.1)
Static admissibility:
divσ + f d = 0 in Ω
σ n = Fd on ∂ 2 Ω
(6.2)
Constitutive relationship: σ = Kε in Ω
(6.3)
6.2. Methods based on the homogenization The most famous multiscale methods are based on the homogenization theory. The first works on this technique were analytic or semi-analytic studies on the macroscopic behavior structures from some “effective” medium quantities [77] [80]. These methods could not enable to analyze the local effects. Then the microscopic level was introduced by the “Unit cell methods” [81] - [83], in order to obtain a local solution into a Representative Volume Element. Finally, the periodic media theory [84] - [87], based on the asymptotic analysis has proposed a really 157
multiscale approach: it enables to obtain a local solution from one macroscale problem and one microscale problem.
6.2.1. Theory of homogenization for periodic media The homogenization technique is applied when a problem can be schematized by a repeated unit cell; indeed the fundamental hypothesis is the periodicity. This repeated unit cell is called RVE, Representative Volume Element and has shown in Fig. 6.2a and Fig. 6.2b; in particular Fig. 6.2a shows the generic body that can be schematized by its repeated part and Fig. 6.2b shows the RVE.
Fd
Y2
X2
Y1
ud
X1
RVE
Y3
X3 a)
b)
Fig. 6.2: Homogenization technique.
Two representative scales were introduced: a macroscale with the position vector Xi (i = 1, 2, 3) defined on the body Ω and a microscale with the position vector Yi (i = 1, 2, 3) defined on the RVE; the smallness parameter Z = X / Y put in comparison the two scales. The homogenization methods are based on the following assumptions: 158
Periodicity; Solution is periodic in statistic sense; The macroscopic fields are constant in the RVE.
Obviously these assumptions are not verified in the neighbourhood of boundaries and when the heterogeneities of the material are not small enough with respect to the dimensions of the macrostructure. The displacement solution is u = u(x, y ) . The idea is to develop the solution into asymptotic form as: uiε (x, y ) = uio (x, y ) + ε ui1 (x, y ) + ε 2 ui2 (x, y ) L
(6.4)
Analogously the asymptotic form of the stress field is: σ ijε (x, y ) = ε −1σ ij−1 (x, y ) + σ ijo (x, y ) + εσ ij1 (x, y ) + ε 2σ ij2 (x, y ) + L
(6.5)
Once injected inside the equilibrium and constitutive equations, these developments lead to a succession of problem at different orders. In this way the macroscopic field uio and the microscopic field ui1 can be determined.
6.3. Methods based on the super-position The homogenization approaches permit to pass from the macrolevel to the microlevel defining the macroscale problem and analyzing it at the microscale. The methods based on the superposition do a different thing: they superpose a microscopic enrichment into the interest zone for the solution of the macroscopic problem.
159
6.3.1. Variational multiscale method This method was initially proposed by Hughes [88]: all the elements problem are not soluble numerically, as Hughes said. The microscopic effects that are not “soluble” can not be represented by finite elements size superior to the microstructure size. Hughes has proposed a superposition principle that permit to consider the effects of the small cell at macroscopic level. Solving the local problem, the small elements effects are condensed to the macroscopic level, obtaining a quasi-exact solution for the macroscopic problem. The solution of the problem is decomposed into a macroscopic and microscopic part, called u M and u m respectively:
u = uM + um
(6.6)
The choice for the approximation of u m is very important; a good solution is to utilize the Green’s function.
6.4. Methods based on the domain decomposition When the microstructure is analyzed two cases can occur: the analysis of only an interest zone with a fine mesh, it is the case of a local-global analysis for which a natural separation between a coarse and fine zone occurs. The other case occurs when the fine mesh is for all the structure; it happens when the structure is strongly heterogeneous. Then the direct solution is very complex and it is necessary to apply domain decomposition by the subdivision in sub-structures of the initial structure. The presence of sub-structures permits the resolution of small interface problem. The decomposition domain methods are subdivided in three great families: the primal approaches (Balancing Domain Decomposition Method, BDDM, [89]), the dual approaches (Finite Element Tearing Interconnecting, FETI, [90]) and the mixed 160
theory (based on the Lagrange’s algorithm [91] or on the LATIN [92]. In every case the solution is based on the application of an iterative procedure. In order to obtain a quickly numerical solution the problem of the propagation of a global information must be solved. This implies the grew up of a coarse problem to verify the partial transmission condition into the sub-structures. In the first case (primal approaches) a force condition is imposed, for the dual approaches the condition is imposed on the partial verification of displacements, in the mixed approaches the conditions are imposed both forces and displacements.
6.4.1. Primal approach Considering the reference problem illustrated in section 6.1, let us consider a partition of the domain Ω in two substructures Ω(1) and Ω(2) , as illustrated in fig. 6.3.
Γ
Ω (2)
Ω(1)
Fig. 6.3: Two subdomain decomposition.
161
The interface between the two substructures is defined as Γ = ∂Ω(1) ∩ ∂Ω(2) and the equations governing the reference problem can be rewritten on the restrictions Ω(1) and Ω(2) of Ω : ⎧ divσ ( s ) + f ( s ) = 0 in Ω( s ) ⎪ (s) (s) (s) (s) ⎪ σ = a : ε(u ) in Ω ⎪ s = 1 o 2 ⎨ 2ε = ∇u ( s ) + ∇u ( s )T in Ω ( s ) ⎪σ ( s ) • n ( s ) = F ( s ) on ∂Ω ∩ ∂Ω( s ) d 1 ⎪ (s) (s) ⎪⎩ u = u d on ∂Ω 2 ∩ ∂Ω( s )
(6.7)
The interface connection conditions are the continuity of displacements:
u (1) = u (2)
on Γ
(6.8)
and the stresses equilibrium:
σ (1)n (1) + σ (2)n (2) = 0 on Γ
(6.9)
Of course the system constituted by equations (6.7), (6.8) and (6.9) is exactly the same as the system represented by equations (6.1), (6.2) and (6.3). In reality, a structure can be decomposed in N subdomains denoted Ω ( s ) . In this case three interfaces
are
defined:
the
interface
between
two
subdomains
Γ (i , j ) = Γ ( j ,i ) = ∂Ω( i ) ∩ ∂Ω( j ) , the complete interface of one subdomain (local
interface) Γ ( s ) = U j Γ ( s , j ) and the geometric interface at the complete structure scale (global interface) Γ = U s Γ ( s ) . Defined the interfaces, the reference problem can be discretized in order to obtain a classical finite element solution: 162
Ku = f
(6.10)
In order to rewrite equations (6.10) in a domain decomposed context, to introduce
λ ( s ) , the reaction imposed by neighbouring subdomains on subdomain (s). This reaction is defined in the whole subdomain, but it assumes non-zero value only on its interface. For every subdomain a local equilibrium condition is defined as: K ( s )u ( s ) = f ( s ) + λ ( s )
(6.11)
On the interface, a global equilibrium condition and a global condition of continuity of displacements are defined as:
∑A
t λ (s) = 0
(6.12)
t u(s) = 0
(6.13)
(s) (s)
s
∑A
(s) (s)
s
where t ( s ) and A( s ) are the local trace operator (restriction from Ω ( s ) to Γ ( s ) which permits to cast data from a complete subdomain to its interface) and the assembly operator (it is a strictly Boolean operator) respectively. The aim of this method is to write the interface problem in terms of one unique unknown: the interface displacement field ub. the problem can be solved introducing the primal Schur complement S p( s ) . The basic idea for every subdomain is to condense its behavior on its interface. Let consider the local equilibrium of a subdomain under interface loading:
K ( s )u ( s ) = λ ( s ) = t ( s ) λ b( s )
163
(6.14)
In order to separate the internal and boundary degree of freedom system (6.14) assumes the following form: ⎡ K ii( s ) ⎢ (s) ⎣ K bi
K ib( s ) ⎤ ⎧ui( s ) ⎫ ⎧ 0 ⎫ ⎨ (s) ⎬ = ⎨ (s) ⎬ (s) ⎥ K bb ⎦ ⎩ub ⎭ ⎩λb ⎭
(6.15)
from the first line it is obtained: ui( s ) = − K ii( s ) −1 K ib( s )ub( s )
(6.16)
Then the Gauss elimination of ui( s ) furnishes the primal Schur complement S p( s ) :
(K
(s) bb
− K bi( s ) K ii( s ) −1 K ib( s ) ) ub( s ) = S p( s )ub( s ) = λb( s )
(6.17)
Moreover if the subdomain is also loaded on internal degree of freedom the condensation of the equilibrium at the interfaces gives:
(s)
K u
(s)
= f
(s)
⇒
S p( s )ub( s ) = bp( s ) bp( s ) = f b( s ) − K bi( s ) K ii( s ) −1 f i ( s )
(6.18)
Using the primal Schur complement, the primal formulation of the interface problem assumes the form: S p ub = ( AS p◊ AT )ub = Ab ◊p = b p
Where the superscript
◊
(6.19)
denotes the row-block repetition of local vectors and the
diagonal-block repetition of matrices and S p = ∑ s A( s ) S p( s ) A
( s )T
Schur complement of the decomposed structure. 164
is the global primal
Primal approach comes with an efficient preconditioner, called the Neuman-Neuman preconditioner, enriched by a coarse problem associated to the well posedness of local problems with imposed forces on the interfaces. This preconditioner and its associated coarse problem is very similar to realizing a dual step as described in next section.
6.4.2. FETI method This method searches by an iterative procedure a field of forces that is continuous a priori at the interfaces to guarantee the continuity of displacements at convergence,
in other words the interface problem is formulated in terms of one unique unknown interface stresses field. The field of stresses must verify the equilibrium constrain of every sub-domain with the respect of external loads because of the local problem is well posed. Thus an iterative procedure is necessary; it is based on the conjugated gradient and its projection is associated to the equilibrium constraint of each subdomain. This projection can be considered as a macroscopic coarse problem that guarantees the rigid motions continuity at every iteration into the interface. For this reason the FETI method is considered as a multiscale strategy of calculus. But this macroscopic problem might be too poor because it is only associated to the rigid motions. In order to exceed this reef, the method was improved by the FETI 2 [93]: some additional constraints on the displacement field are introduced to guarantee the continuity at the interface middle. The projection becomes an enhanced macroscopic problem. A more recent version is called FETI-DP (Dual-Primal FETI Method [94]). The performances obtained with FETI method are very remarkable, but they are connected to the appropriate chosen of the pre-conditioner for the algorithm of resolution. This chosen depends essentially from the analyzed problem and it is possible because of the generality of the method. A classic chosen is that to consider as pre-conditioner a lumped or Dirichlet. For the heterogeneous structures the 165
Reference is [95]; for preconditioning and enhanced macroscopic problems based on the reuse of Krylov’s subspaces in a multiresolution context, References [96] and [97].
6.4.3. Mixed method: the micro-macro approach The principle of the mixed method is to rewrite the interface conditions in terms of a new local interface unknown, which is a linear combination of interfaces stresses and displacements. Mixed methods give a mechanical behavior to the interface and in the case of perfect interfaces it can be mechanically interpreted as the insertion of springs to connect the subdomains. The method is characterized by a micro-macro approach, i.e. [98] - [100]. The mixed formulation is indicated for the analysis of structures that are strongly heterogeneous, i.e. [101] and [102], and for the study of the nonlinearities, i.e. [103] and [104]. The characteristics of the method are the subdivision in sub-structures, the introduction of the multiscale effect only at interface level, where the stresses and the displacements are split into “macro” contributions and “micro” complements, and the satisfaction of the transmission conditions a priori by the interfaces macro stresses. So, the initial structure is subdivided in sub-structures subjected to the actions of the neighboring, as illustrated in Fig. 6.4.
166
FE F E’ ΩE
ΩE’
WE W E’ ΓEE’
Fig. 6.4: Sub-domains decomposition.
Consequently to the micro and macro separation, the force and displacement field are expressed in function of the micro and macro part: F = FEm + FEM
(6.20)
W = WEm + WEM
The micro and macro quantities must satisfy the splitting of the virtual work:
∫
Γ EE '
F • Wd Γ =
∫
FEm • WEm d Γ +
Γ EE '
∫
FEM • WEM d Γ
(6.21)
Γ EE '
At each interface the macro projector is defined as: F M = Π ΓMEE ' (F) F m = Π ΓmEE ' (F) W M = Π MΓ ( W)
(6.22)
EE '
W = Π Γ (W) m
m
EE '
The projector can be chosen in order to extract the linear part from the interface quantity. Finally the interface forces must satisfy the transmission conditions a priori
167
(in the case of perfect interfaces, macro-displacement can also be made continuous a
priori). To solve this problem the LATIN method [92] is applied. The LATIN method is an iterative resolution technique which takes into account the whole time interval studied. At each iteration, an homogenized macroproblem, defined over the whole time-space domain, is solved as well as a set of independent microproblems which are linear evolution problems defined within each substructure or at boundaries between the substructures. The LATIN method, illustrated in Fig. 6.5, is based on the idea of dealing the difficulties separately, splitting the equations in two subsets: ⎧Static and kinematic admissibility: ⎪ (σ , F ), (ε , W ) ∀M ∈ Ω ⎪ E E E E E Ad ⎨ Macro equilibrium: ⎪ ⎪⎩ FEM + FEM' = 0
(6.23)
⎧ Dissipation law of each substructure Γ⎨ ⎩The behavior and equilibrium at interfaces
(6.24)
Ad is the space of the global linear equations defined, while G is the space of the local nonlinear equations.
168
sn+1/2 E+ Esref.
sn+1
sn
Fig. 6.5: LATIN scheme for one iteration.
The solution of the problem is obtained by an iterative scheme. Each iteration consists of a local step and a linear step: the method permits to find a solution that verifies alternatively the equations of the first and second set of equations and at last it converges towards the solution sref.
6.5. Numerical results In this simple example a clamped beam, schematically illustrated in Fig. 6.6, subjected to an horizontal pressure was analyzed. Two tests were performed: the first one with the use of a single material and the second one with two different materials. The results obtained with the application of FETI method and of the mixed method were put in comparison. The beam properties are reported in the following: Geometrical characteristics: o Brick: rectangular cross section, b=100 mm and h=100 mm; o Mortar: rectangular cross section, b=4 mm and h=100 mm; o External load: horizontal pressure P=10 MPa.
169
Mechanical properties: o Brick: E=1785 MPa; n=0.2; o Mortar: E=1785 MPa; n=0.2.
P L Fig. 6.6: Beam scheme.
The mesh is reported in Fig. 6.7, while in Fig. 6.8 it is remarked the single sub-domain considered.
Fig. 6.7: Beam mesh.
Fig. 6.8: Beam sub-domain.
Two analysis are effected, with the application of the FETI method and the mixed method. In the following figures are reported the displacement and stress fields; the results are the same for the two methods.
170
Fig. 6.9: Case 1, displacement field for FETI.
Fig. 6.10: Case 1, displacement field for the mixed method.
Fig. 6.11: Case 1, stress field for FETI.
Fig. 6.12: Case 1, stress field for the mixed method.
171
The same clamped beam, schematically illustrated in Fig. 6.6, subjected to the same horizontal pressure was re-analyzed considering different mechanical properties for the two materials: Geometrical characteristics: o Brick: rectangular cross section with b=100 mm; h=100 mm; o Mortar: rectangular cross section with b=4 mm; h=100 mm; o External load: horizontal pressure P=10 MPa. Mechanical properties: o Brick: E=1785 MPa; n=0.2; o Mortar: E=113 MPa; n=0.2.
In this case it is evident that the second material is effectively more deformable and at interface some local effects are visible, as illustrated in the following.
Fig. 6.13: Case 2, displacement field for FETI.
Fig. 6.14: Case 2, stress field for FETI.
172
Also in this case the analysis was conduced by the mixed method and the results are perfectly the same, as illustrated in Fig. 6.15 and Fig. 6.16.
Fig. 6.15: Case 2, displacement field for the mixed method.
Fig. 6.16: Case 2, stress field for the mixed method.
Moreover these results were put in comparison with the classical theoretical results and with the results obtained by FEAP using the implemented three nodes finite element; in particular in Tab. 6.1 and in Tab. 6.2 are reported the axial displacement values for the first and second case, respectively. FETI w [mm]
≈5.58
MIXED FEAP THEORIC ≈5.88
≈5.83
≈5.83
Tab. 6.1: Case 1 results.
FETI w [mm]
≈11.16
MIXED FEAP THEORIC ≈9.07
≈12.53
Tab. 6.2: Case 2 results.
All the results are in good accordance. 173
≈12.53
The conclusion after the study of this simple example is the possibility of applying these methods to analyze reinforced masonry arches, reducing the solution of a great problem into the solution of a set of simplest problems. In fact, once defined the mesh, with a manual or automatic mesh generator, the subdivision in subdomains is immediate and the study of the single sub-structure could become simple enough, in terms both of numerical procedures and computational resource requirement. It must be emphasized that the domain decomposition methods are adapted to parallel processing, consisting of independent tasks having their own data that can be allocated to the various processors of the system. The DDM offer a framework where different design services can provide the models of their own parts of a structure, each assessed independently, and they can evaluate the behavior of the complete structure just setting specific behavior at interfaces. From an implementation point of view, often programming DDM can be added to existing solvers as an upper level of current code.
174
CONCLUSIONS The research activity presented in this thesis work has been focused on the experimental and numerical analysis of masonry arches strengthened with fiber reinforced plastic (FRP) materials. The experimental program regarded different tests performed on both masonry constituents (bricks and mortar) and structures (unstrengthened and FRPstrengthened arches). From the performed tests, important aspects concerning the effect of the FRP reinforcement on the structural response of masonry arches was observed. In fact, comparing the behavior of unstrengthened and FRP-strengthened arches, it was observed that the application of FRP at extrados surface of arches produces an increase both in terms of load-bearing capacity (strength) and in terms of ultimate displacement (ductility). These effects are related to the collapse mechanism. In fact, while in the case of unreinforced arches the collapse is due to the formation of the classic four hinges, the FRP-strengthening prevents the crack opening at the extrados, i.e. the presence of hinges at intrados, and leads to a collapse mechanism characterized by shear and crushing failure of masonry. On the basis of the experimental observations and in order to understand further aspects concerning the nonlinear response of FRP-strengthened masonry arches, in the second part of the thesis numerical analyses have been developed. In particular, two models have been considered for the masonry material; both the models assume the masonry material characterized by no-tension behavior and limited compressive strength. For reduced values of the compressive strain, the first one considers a linear stress strain relationship, while the second one considers a quadratic relationship. In order to solve the nonlinear unreinforced and reinforced masonry arch problem, a stress formulation, based on the complementary energy, and a displacement formulation, based on the implementation of a three nodes finite element into the FEAP code, have been developed. 175
Moreover, as the delamination phenomenon between the FRP and the masonry support can play an important role in terms of FRP-strengthening contribution, an effective procedure, based on a simplified approach of the multiscale method for the evaluation of the normal and tangential stresses at the interface has been developed. Moreover, in the context of the multiscale approaches, the domain decomposition methods are analyzed. Numerical applications based on the use of the proposed models have been developed with reference to the performed experimental tests. The comparison between the numerical and the experimental results demonstrated the ability of the proposed models to reproduce the global and local response of unstrengthened and FRP-strengthened arches. In particular, while in the case of the unstrengthened arches the two proposed models give the same results, in the case of the FRPstrengthened arch substantial differences occur between the two considered models. In fact, in this case the second model gives the best results both in terms of pre and post-peak behavior.
176
APPENDIX: RELUIS SCHEDE
MODELLO PER LA DESCRIZIONE SINTETICA DI PROVE SPERIMENTALI STRUTTURE/ELEMENTI STRUTTURALI IN MURATURA
SPERIMENTAZIONE DI ARCHI IN MURATURA CON E SENZA RINFORZO IN FRP
CANCELLIERE ILARIA, RICAMATO MARIA, SACCO ELIO
177
ISTRUZIONI Il modello è diviso in quattro sezioni: 1) Dati generali della prova sperimentale (G) 2) Descrizione elemento/ struttura testata (D) 3) Proprietà dei materiali (M) 4) Risultati della prova (R) La compilazione delle prime due sezioni richiede l’inserimento di figure con fotografie e/o disegni che si ritengono utili alla comprensione del setup di prova. La sezione materiali è basata su tabelle da compilare con i risultati disponibili per mattoni, malta e muratura. Cliccando sulle relative tabelle si apre una finestra di Excel in cui sono disponibili tutti i comandi dell’applicazione. In alternativa è possibile copiare nello stesso spazio una qualsiasi tabella dai contenuti analoghi a quella già predisposta. La sezione finale non ha un formato prestabilito e la sua compilazione è lasciata agli autori della prova. E’ possibile inserire in ciascuna sezione tutte le pagine necessarie alla descrizione della prova e dei risultati. La numerazione delle pagine ha il formato: codice sezione/N. totale di pagine della sezione – numero progressivo di pagina
178
DATI GENERALI DELLA PROVA SPERIMENTALE Arco • Oggetto della Prova • Autori
Portale
Volta
Cupola
Pannello Colonna
Altro
X Cancelliere Ilaria, Ricamato Maria, Sacco Elio
• Data di Esecuzione
Luglio - Settembre 2007
• Sede Laboratorio
Laboratorio di Progettazione Strutturale LaPS, Università di Cassino
• Riferimento Bibliografico
Cancelliere I. “Analisi numerica e sperimentale di archi in muratura rinforzati con FRP”, Tesi di Laurea specialistica in Ingegneria Civile, Università di Cassino (FR), Ottobre 2007 Ricamato M. “Numerical and experimental analysis of masonry arches strengthened with FRP materials”, Tesi di Dottorato, Università di Cassino, Novembre 2007
Messa in opera dell’arco
Esecuzione della prova sperimentale sull’arco non rinforzato
Esecuzione della prova sperimentale sull’arco rinforzato
179
DESCRIZIONE SETUP DI PROVA Geometria e Vincoli
Re Ri Rb Fi Ff Fc
8 172 164
576.07 [mm] 456.07 [mm] 516.07 [mm] [°] 0.1396 [°] 3.0004 [°] 2.8609
[rad] [rad] [rad]
Vincoli L’arco è stato fissato alla base tramite elementi di contrasto per la realizzazione di una condizione di incastro.
Note: Il carico è stato applicato mediante un martinetto idraulico disposto in posizione eccentrica. Tra l’estradosso dell’arco e il martinetto è stata posizionata la cella di carico. Sono stati utilizzati tre potenziometri e due comparatori: un potenziometro e un comparatore in direzione verticale in corrispondenza del martinetto; un potenziometro ed un comparatore in direzione verticale in chiave dell’arco; un potenziometro in direzione orizzontale posizionato in chiave.
180
PROPRIETA’ DEI MATERIALI MATTONI Descrizione (Tipo, Marca, Forno d'origine)
Dimensioni nell'elemento strutturale
Prova di Resistenza Resistenza misurata Norma di riferimento compressione D.M.20/11/1987
N. Prove
Dimensione campione Media 6 55x55x55 38,5 Mpa
Risultati sui singoli campioni Mattoni pieni in laterizio denominati "di Salerno". Cava e fornace ubicate nel comune di Salerno (Italia).
250x120x55 mm
3
Numero campione 1 2 3 4 5 6
181
Resistenza 41,5 35,7 40,06 34,7 36,6 41,7
NOTE rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra
Scarto 7,47
PROPRIETA’ DEI MATERIALI MALTA Descrizione
Spessore dei letti di malta nell'elemento strutturale
Prova di Resistenza Resistenza misurata flessione flessione compressione compressione
Norma di riferimento UNI-EN 196/1 UNI-EN 196/1 D.M.20/11/1987 D.M.20/11/1987
Numero campione 1 2 3
Resistenza 2,36 2,62 2,59
1 2 3
2,69 2,66 2,27
1 2 3 4 5 6
9,147 9,4475 8,1937 8,4806 9,008 8,22
1 2 3 4 5 6
9,83 9,85 8,355 8,87 8,53 8,75
N. Prove 3 3 6 6
Dimensione campione 40x40x160 40x40x160 40x40x80 40x40x80
Risultati sui singoli campioni
Malta bastarda: pozzolana, calce aerea, cemento pozzolanico (Duracem 32.5R) ed acqua
il valore medio misurato sulla struttura messa in opera è pari a 15 mm.
182
NOTE
Media 2,53 2,54 8,75 9,03
Scarto
PROPRIETA’ DEI MATERIALI MURATURA Prova di Resistenza Resistenza misurata (trazione, compressione, taglio)
Tipo di prova (compressione diagonale, compressione, etc..)
Norma di riferimento
N. Prove
Dimensione campione
Risultati sui singoli campioni Numero campione
NOTE (modalità di rottura, etc..)
Resistenza
Descrizione della prova (modalità di applicazione del carico, diagrammi carico spostamento, setup di prova, figure del campione, etc..)
183
Media
Scarto
PROPRIETA’ DEI MATERIALI FIBRE Descrizione: proprietà
unità di misura
Spessore (lamina)
0.17 mm
larghezza
100 mm
lunghezza
162 mm
Metodo di prova normativa di riferimento
Geometria della sezione (barre, cavi) mm2
Area nominale (barre, cavi) Perimetro nominale (barre, cavi)
mm
Colore
nero
densità
Contenuto in fibra
fibra
1.80 g/cm3
matrice
1.31 g/cm3
in peso
%
in volume
%
ISO 1183-1:2004 (E)
ISO 11667:1997 (E) ISO 11357-2:1999(E) (DSC) ISO11359-2:1999(E) (TMA)
Temperatura di transizione vetrosa della resina (Tg) Temperatura limite massima di utilizzo
°C
Conducibilità elettrica
S/m
Modulo di elasticità normale a trazione
230GPa
ISO 527-4,5:1997(E)
3900 MPa
ISO 527-4,5:1997(E)
Resistenza a trazione (valore caratteristico)
MPa
ISO 527-4,5:1997(E)
Deformazione a rottura a trazione
1.5 %
ISO 527-4,5:1997(E)
Modulo di elasticità normale a compressione (barre)
GPa
ISO 14126:1999(E)
Resistenza a compressione (barre) (valore medio)
MPa
ISO 14126:1999(E)
Resistenza a compressione (barre) (valore caratteristico)
MPa
ISO 14126:1999(E)
%
ISO 14126:1999(E)
Resistenza a trazione (valore medio)
Deformazione a rottura per compressione (barre) Resistenza a creep
ISO 899-1:2003(E)
Rilassamento (barre, cavi) Aderenza: tensione tangenziale (barre, cavi)
Prova di pull-out
184
Note
PROPRIETA’ DEI MATERIALI RESINA Descrizione resina: (nome commerciale, mono o bicomponente, pasta o liquida, tipologia di utilizzo ed ogni altra informazione generale ritenuta utile)
caratteristiche della resina non miscelata unità di misura
proprietà colore viscosità a 25°
comp. A
comp. B
miscela
bianco
grigio
grigio
note
ISO 2555:1989(E) ISO 3219:1993(E)
Pa s
indice di tissotropia densità rapporto di miscelazione condizioni di stoccaggio (contenitore siggillato)
metodo di prova
ASTM D2196-99
g/cm3 4:1 %
in volume in peso tempo
1.31
ISO 1675:1985(E)
mesi
temperatura
°C
caratteristiche della resina miscelata condizioni di miscelazione: condizioni di applicazione: proprietà tempo di lavorabilità (a 35°) a 5°C tempo di gelo a 20°C a 35°C temperatura minima di applicazione tempo picco esotermico temperatura tempo di completa a 5°C reticolazione (full a 20°C core) a 35°C
unità di misura 30 min min
metodo di prova normativa di riferimento
note
ISO 10364:1993(E) ISO 9396:1997 (E) ISO 2535:2001 (E) ISO 15040:1999 (E)
10°C min °C
ISO 12114:1997 (E)
min
ISO 12114:1997 (E)
proprietà della resina reticolata condizioni di stoccaggio: precauzioni d'uso e sicurezza: proprietà
unità di temperatura misura di prova
valore
metodo di prova normativa di riferimento
stagionato stagionato 5gg. a 22°C 1 ora a 70°C ritiro volumetrico coefficiente di dilatazioe termica
ISO 12114:1997 (E)
10-6 °C-1
ISO 11359-2:199 (E)
temperatura di transizione vetrosa, Tg
°C
ISO 11357-2:1999 (E) (DSC) ISO11359-2:1999(E) (TMA) ASTM E 1640 (DMA)
modulo di elasticità normale a trazione resistenza a trazione
Gpa Mpa
ISO 527:1993 (E) ISO 527:1993 (E)
185
RISULTATI DELLA PROVA F(v) in corrispondenza della forza -700.00 Ciclo I Ciclo II Ciclo III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.50
0.00 -0.50 0.00
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
-7.00
100.00
v [mm]
Cinematismo di collasso e curva forza - spostamento (relativa ai tre cicli di carico e scarico effettuati) per l’arco non rinforzato.
Immagini relative alle fasi di prova dell’arco rinforzato. F(v) in corrispondenza della forza -60000 Ciclo I Ciclo II Ciclo III
-50000
-40000
F [N]
-30000
-20000
-10000 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23
0
10000
v [mm]
Curva forza - spostamento (relativa ai tre cicli di carico e scarico effettuati) per l’arco rinforzato. Note
186
NOTATIONS
The following notation were used throughout the text:
Vectors and tensors Quantities
Scalar Vector Second order tensor Third order tensor
a a A && A
Inner product Vectorial product Dyadic product Gradient operator
• × ⊗ Δ
Operators
187
Matrices and columns Quantities Scalar
a
Column
⎧a1 ⎫ ⎪ ⎪ a=⎨M⎬ ⎪a ⎪ ⎩ i⎭
Line
a = {a1 L ai }
Matrix
A = ⎡⎣ Aij ⎤⎦
Operators
Matrix product
AB
Transposition
AT
Inversion
A −1
Any notation which has not been explicitly defined in this section will be explained at its first point of use.
188
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Dottorato di Ricerca in Ingegneria Civile XX Ciclo
Numerical and experimental analysis of masonry arches strengthened with FRP materials
Maria Ricamato
Cassino, Novembre 2007
University of Cassino Department of Engineering Graduate School in Civil Engineering XX Cycle - November 2007
Numerical and experimental analysis of masonry arches strengthened with FRP materials
Maria Ricamato
Supervisor: Prof. Elio Sacco Coordinator: Prof.ssa Maura Imbimbo 2
To my mother and my father, with love
3
Acknowledgements Three years ago this day seemed very distant, instead today I am working for concluding my PhD thesis. At the end of this experience, first the financial supports of the Italian National Research Council (CNR) and of The Laboratories University Network of Seismic Engineering (RELUIS) are gratefully acknowledged. My greatest thanks is for my “Scientific fathers”: Prof. Giovanni Romano for having directed me towards the research and Prof. Elio Sacco for giving me the opportunity to improve my scientific and technical knowledge, for the work done under his supervision and for his useful suggestions. I would like to thank Prof.ssa Sonia Marfia for the fruitful discussions and also Prof.ssa Maura Imbimbo and Prof. Raimondo Luciano for their disposal. I would like to express my gratitude to Prof. Olivier Allix of the ENS of Cachan (France), who gave me the opportunity to spend part of my PhD under his supervision. This period was very important for my experience, both from a professional and personal point of view. A special thanks to Ing. Pierre Gosselet, for his help to overcome many difficulties that I encountered with the multiscale methods and to the LMT who “hosted” me for 6 months, in particular thanks to Beatrice Faverjon who represented always a reference point also for human aspect. I would like to express my deep and sincere gratitude to my friends and colleagues Ernesto Grande and Veronica Evangelista, who spent several days with me working hardly. Many thanks to DIMSAT, LAPS and Geolab Sud staff. I would like to remember all my friends for their friendship whenever I needed it. Finally a great thank to my family: my mother Francesca and my father Lucio, for their love and support, my brother Nicandro for his humor that in times of distress has been able to give me the courage to continue. My gratitude to my Franco cannot be summarized in few rows: I will never forget his love, his patience and his continuous support in what I do... 4
INDEX Acknowledgements.......................................................................................................4 1. INTRODUCTION ....................................................................................................9 1.1. Early static theories of the arch.......................................................................12 1.2. Motivations of the research and outline of the thesis......................................15 2. MASONRY MATERIAL.......................................................................................18 2.1. Introduction.....................................................................................................18 2.2. Mechanical behavior .......................................................................................19 2.3. Masonry modelling .........................................................................................23 2.4. No-tension material model..............................................................................25 3. FRP COMPOSITE MATERIALS FOR STRENGTHENING MASONRY STRUCTURES......................................................................................28 3.1. Introduction.....................................................................................................28 3.2. Mechanical behavior .......................................................................................29 3.2.1. Alkaline ambient effects..........................................................................34 3.2.2. Humidity effects ......................................................................................35 3.2.3. Extreme temperature and thermal cycle effects ......................................35 3.2.4. Frost-thaw cycles effects .........................................................................35 3.2.5. Temperature effects.................................................................................35 3.2.6. Viscosity and relaxation effects ..............................................................36 3.2.7. Fatigue effects .........................................................................................36 3.3. Masonry structures reinforced with FRP materials.........................................36 3.4. Collapse mechanism for reinforced structures................................................38 4. EXPERIMENTAL PROGRAM .............................................................................40 4.1. Introduction.....................................................................................................40 4.2. Setup and instrumentations .............................................................................40 4.3. Preliminary experimental campaign ...............................................................42 4.4. Materials used in the experimental program...................................................44 4.5. Standard clay brick..........................................................................................44 5
4.5.1. Cubic compressive test............................................................................45 4.5.2. Indirect tensile test...................................................................................49 4.5.3. Elastic secant modulus ............................................................................53 4.6. Mortar..............................................................................................................59 4.6.1. Compressive tests ....................................................................................59 4.6.2. Elastic secant modulus ............................................................................62 4.7. Reinforcement material...................................................................................64 4.8. Experimental test on the arches ......................................................................69 4.9. Arch laying......................................................................................................69 4.10. Arch preparation ...........................................................................................72 4.11. Experimental campaign: Arch 1....................................................................74 4.11.1. Collapse mechanism description ...........................................................76 4.11.2. Load-displacements curves ...................................................................78 4.12. Experimental campaign: Arch 2....................................................................83 4.12.1. Collapse mechanism description ...........................................................84 4.12.2. Load-displacements curves ...................................................................85 4.13. Experimental campaign: Reinforced arch.....................................................88 4.13.1. Application of the FRP reinforcement ..................................................88 4.13.2. Test organization ...................................................................................91 4.13.3. Collapse mechanism description ...........................................................95 4.13.4. Load-displacement curves.....................................................................95 5. MODELING AND NUMERICAL PROCEDURES..............................................99 5.1. Introduction.....................................................................................................99 5.2. Masonry constitutive models ........................................................................100 5.2.1. Model 1..................................................................................................100 5.2.2. Model 2..................................................................................................103 5.3. FRP constitutive model.................................................................................105 5.4. Limit analysis................................................................................................107 5.5. Arch model....................................................................................................109 6
5.5.1. Governing equation of the arch .............................................................110 5.5.2. Kinematics of the arch...........................................................................111 5.5.3. Cross section..........................................................................................111 5.6. Stress formulation .........................................................................................114 5.6.1. Complementary energy .........................................................................117 5.6.2. Arc-length technique .............................................................................120 5.7. Displacement formulation.............................................................................125 5.7.1. Kinematics.............................................................................................127 5.7.2. Finite element implementation..............................................................128 5.8. Post-computation of the shear stresses..........................................................131 5.9. Numerical results ..........................................................................................135 5.9.1. Models and numerical procedures assessment......................................135 5.9.2. Experimental surveys numerical results................................................141 5.9.2.1. Comparison 1.................................................................................141 5.9.2.2. Comparison 2.................................................................................146 6. MULTISCALE APPROACHES ..........................................................................156 6.1. Introduction...................................................................................................156 6.2. Methods based on the homogenization.........................................................157 6.2.1. Theory of homogenization for periodic media......................................158 6.3. Methods based on the super-position............................................................159 6.3.1. Variational multiscale method...............................................................160 6.4. Methods based on the domain decomposition ..............................................160 6.4.1. Primal approach.....................................................................................161 6.4.2. FETI method..........................................................................................165 6.4.3. Mixed method: the micro-macro approach ...........................................166 6.5. Numerical results ..........................................................................................169 CONCLUSIONS ......................................................................................................175 APPENDIX: RELUIS SCHEDE ..............................................................................177 NOTATIONS............................................................................................................187 7
REFERENCES .........................................................................................................189
8
1. INTRODUCTION Numerous ancient constructions are made of masonry material that is one of the oldest building material. Many ancient and historical masonry buildings are characterized by the presence of arches and vaults. In particular the arch is a fundamental constructive element having both load-bearing and ornamental function. The “false arch” was one of the first constructive elements. It was realized by flat stones placed on top of each other that created a stepwise arch. The constructive technique was refined during the centuries, also introducing the use of the mortar to joint the stones or the bricks. The Egyptian and the Babylonians introduced the use of arches in civil constructions, the Assyrians constructed the first vaults in masonry buildings, the Etruscans used arches in order to realize the first masonry bridges. The Romans made large use of masonry arches and vaults for the constructions, not only of buildings but also of roads, bridges, aqueducts and amphitheatres, as illustrated in Fig. 1.1.
Fig. 1.1: The Colosseum, Roma.
9
On the contrary, cult buildings were made using columns and architraves, as for Greeks temples. One of the most representative cult building is the Pantheon of Roma, characterized by the presence of a very well-known vaulted structure (Fig. 1.2).
Fig. 1.2: The Pantheon, Roma.
Moreover, Romans constructed arches also as monuments, like the “triumphal arches”, e. g. the arch of Janus in Rome and the Triumphal arch in Paris (neoclassical version of the ancient triumphal arches of the Roman Empire).
Fig. 1.3: The arch of Janus in Rome and Triumphal arch in Paris.
During the Middle-Age both the Byzantine architecture in the East and the Romanic one in the West still adopted the Roman round arches. 10
The Goths, in the 13th century, substituted the semicircular arch with the pointed arch. A main characteristic of the Gothic structures is the lightness of the buildings, obtained by the introduction of flying buttresses and towers. The Cathedral of Milan is an example of Gothic structure, Fig. 1.4.
Fig. 1.4: Cathedral of Milan.
During the Renaissance, the churches assumed a great structural interest. In particular the Church of S. Maria del Fiore in Florence and the Basilica di S. Pietro, Fig. 1.5, represent great examples of regular shapes and geometrical symmetry due to the use of vaults.
Fig. 1.5: S. Maria del Fiore and Basilica di S. Pietro.
Then no relevant or innovative solution concerning the structural conception were developed, but still today the arch is a fundamental structural element and its use has 11
been extended to all types of construction by the use of “new” material, like reinforced concrete and steel.
1.1. Early static theories of the arch The arch is one of the most interesting structural elements in the construction history because of its intuitive static behavior. During the centuries, many studies were developed on the more appropriate shape of the arch, but only in the 17th century a static theory of the arch was proposed. The Romans used systematically the arch realizing structure both of great value and strong impact. The Roman scientist, Vitruvio, identified the main characteristics of the arch and wrote ten books De Architectura, in which both the theory and the practice concerning with the art of construction were presented. Vitruvio discussed about the presence of the thrust of the vault on the supporting columns and walls. He has also understood the functioning of the arched structures, suggesting to realize strong and massive supports in order to contrast the thrust of the arches and vaults. In the 13th century, Leon Battista Alberti wrote the De Re Aedificatoria, motivating the use of arched structures with the aim of increasing both of the spans and the bearing capability. A more refined theory was attributed to the constructors of the Middle-Age: its main characteristic is the approximation of the arch shape by the thrusts line. Also the geometrical rule to determine the thickness of the piers was attributed to them. This theory was the only respected during the Renaissance. Leonardo Da Vinci (1452 1519) developed some ideas and intuitions three centuries later. He asserted “…arco non è altro che una fortezza causata da due debolezze imporochè l’arco negli edifici è composto di due quarti di circolo, I quali quarti circoli ciascuno debolissimo per se desidera cadere e opponendosi alla ruina l’uno dell’altro, le due debolezze si 12
convertono in un’unica fortezza…l’arco non si romperà se la corda dell’archi di fori non toccherà l’arco di dentro…”.
Fig. 1.6: Leonardo’s intuitive static scheme for the arch.
This theory for which the arch is assimilate to two beams was reproposed by Caplet in the 18th century. The first significant theory of the arch was attributed to the mathematician astronomer Philippe de La Hire (1670 - 1718). In its treatise Traitè De Mécanique, posthumous published in 1730, he underlined the wedge mechanism of the arch. According to him, the arch results subdivided in blocks and each block can be considered like a piece of wedge incident on the mortar joints. Its model was the first approach in the static theory of the arch that considers the masonry structure like a rigid system of solids geometrically defined and with an own weight, neglecting the frictional phenomenon. Two problems were faced by de La Hire: the vaults equilibrium independent of its piers and the determination of the piers dimensions considering the vault thrusts. The first problem lead, in the years, to the method of polygon of the forces, while, concerning the second problem, he developed the basis of the limit analysis. In the 1785, Mascheroni, in the Nuove ricerche sull’equilibrio delle volte, proposed a collapse mechanism of the arch characterized by the formation of intrados cracks at key, of extrados cracks at springers and of intrados cracks at piers extremities, as schematically illustrated in Fig. 1.7. 13
Fig. 1.7: Mascheroni’s collapse mechanism.
In the 19th century the method of the successive resultants was diffused. It was adopted to study short span symmetric arches symmetrically loaded. It is based on the definition of the thrusts line contained inside the third medium. The thrusts line can be regarded as an indicator of stability: if it is not coincident with the center-line of the arch, there is eccentricity and the arch thickness must be such that the eccentricity remains inside the section. The early method characterized by a collapse analysis was the method of Mery. This method is based on the limit analysis and it is applicable only if the assumed collapse mechanism occurs. It can be used if the arch is semicircular and its thickness is constant, the maximum span of the arch is 8-10 metres, the arch is made of an homogeneous material in order to be schematized by a rigid body, the arch is symmetric and symmetrically loaded. The method of Mery can be applicable using the parallelogram rule. In order to verify the part of the arch included between the key and the springers, the arch must be subdivided in blocks of different dimensions. Established the loads agent on each block, the resultants of loads are determined and the thrusts line can be obtained applying the parallelogram rule again and again. In the 1833, Moseley in the On a new principle in static called the principle of least pressure enounced the least pressure principle for the determining the thrusts line of the arch. In 1867 Winkler wrote a treatise on the thrusts line of the arch based on the elasticity theory developed in those years. 14
Recently, in 1982, Heyman in The masonry arches enounced the safe theorem of the limit analysis particularized for the masonry arches. According to him, it is necessary to determine at least one line of thrusts contained inside the thickness of the arch to ensure that the structure is safe. On the other hand, it is sufficient a small variation in the position of the line of thrusts, e. g. caused by loading increase, to allow the formation of localized cracks. As consequence, the hinges formation can occur and a kinematical mechanism can be activate. Generally, the collapse mechanism occurs for formation of four hinges, two at extrados and two at intrados alternatively located.
1.2. Motivations of the research and outline of the thesis The preservation of historical and ancient buildings and monuments requires the definition of intervention methodologies for the maintenance and consolidation. The definition of these methods must reflect on one hand the structural safety, on the other hand the respect of the original structure. The masonry arch is essential and unique in the historic heritage. Some of the consolidation techniques of masonry arches, widely adopted in the recent years, can alter the nature and original structural working of the arch and they also introduce extraneous elements not compatible with the materials and traditional techniques. More recently, for the protection and maintenance of ancient and historical buildings, the use of innovative materials, such as composites, received great interest because of their possible advantages in terms of low weight, simplicity of application, high strength in the fiber directions, immunity to corrosion and reduced invasiveness. In particular, they appear particularly indicated for the maintenance and rehabilitation of ancient structures because they do not substantially violate the principles of the Carta di Venezia. After the earthquakes of 1997 (Umbria and Marche), an intensive research activity was developed for the definition of some rules for the design of the strengthening of 15
masonry arches by FRP. In 2003 the CNR, The National Council of Researches, established that it was necessary to elaborate a text containing the instructions for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, in order to give to engineers the guidelines for the use of fibercomposite materials for the reinforcement of concrete and masonry structures. In the year 2004 a full text “CNR DT/200” was published in Italian and then in English with the name. It could be emphasized that the DT200 is the first code in the world which contains a Chapter completely dedicated to the use of FRP for the strengthening of masonry structures. The present PhD thesis is aimed to derive and to develop some simple strategies to study the response of unreinforced and reinforced masonry arches. In particular, the aim is to formulate simple and effective procedures that the engineers can use for the design of the reinforcements of masonry arches, evaluating the safety of the structure. In order to validate the effectiveness of the developed models and procedures, an experimental campaign on un-reinforced and reinforced masonry arches is conduced. Moreover, a more sophisticate numerical procedure, based on the multiscale analysis, is developed. Finally, the dissertation concerns with three macro-arguments: the experimental program, the modeling and numerical procedures development and the multiscale analysis. In Chapter 2 a general overview on the modeling of masonry material is given. The different modeling strategies are also discussed, underlining the main advantages of each approach. Chapter 3 analyzes the FRP properties. In particular an excursus on FRP material “history” is made and its physical and mechanical characteristics are presented. Chapter 4 contains the description of the experimental program. In order to characterize the nonlinear behavior of the masonry material, the physical and mechanical properties of masonry material constituents are investigated through experimental test. Moreover an experimental campaigns is realized on unreinforced 16
and reinforced masonry arches. The adopted procedure for testing the arches is described and the experimental results are discussed. In Chapter 5 the modeling of masonry materials and FRP and the developed numerical procedures are illustrated. In particular the masonry material is assumed as a no-tensile material with a limited compressive strength, while the reinforcement is considered as an linear elastic material. Two different approaches are developed: a stress formulation, based on the complementary energy, and a displacement formulation, characterized by the implementation of a three node beam finite element based on the Timoshenko’s theory. Several numerical analysis are conduced in order to validate the models and the developed numerical procedure. The numerical results are also put in comparison with experimental results both available in literature and obtained during the experimental campaign. Chapter 6 illustrates a brief introduction to the multiscale methods; in particular the domain decomposition methods are analyzed. At the end, a summary and final conclusions, which can be deduced from this research, are given.
17
2. MASONRY MATERIAL
2.1. Introduction The masonry material is one of the oldest building material, as confirmed by the historical heritage. The development of adequate stress analyses for masonry structures represents an important task not only to verify the stability of masonry constructions, as old buildings, historical town and monumental structures, but also to properly design effective strengthening and repairing works. In fact, many of masonry structures have been suffered from the accumulated effects of material degradation, aging, overloading and foundation settlements. For this reason, the rehabilitation and the maintenance of existing masonry structures represent an important topic. In the years several studies have been developed related to masonry structures, i.e. [1] - [38], mainly devoted to the development of new restoration technologies and, moreover, to the definition of effective computational procedure for reliable stress analyses. It could be emphasized that the analysis of masonry structures is not simple at least for several reasons: the masonry material can be considered as a composite material obtained by assembling bricks by means of mortar joints; the masonry material presents a strongly nonlinear behavior, so that linear elastic analyses generally cannot be considered as adequate; the structural schemes which can be adopted for the masonry structural analyses are more complex than that adopted for concrete or steel framed structures, as masonry elements are often modeled by two- or threedimensional elements. 18
As a consequence, the behavior and the analysis of masonry structures still represent one of the most important research field in civil engineering, receiving great attention from the scientific and professional community; for instance, in Reference [1] several specific problems related to the design and behavior of old and mainly new masonry constructions are discussed. In this chapter a brief discussion on the main aspects concerning the mechanical behavior of the masonry material, i.e. [2] - [5], is reported.
2.2. Mechanical behavior The mechanical behavior of the masonry material presents complex aspects due to the fact that it is a composite material made of units of natural or artificial origin (irregular stones, ashlars, adobes, bricks and blocks) jointed by dry or mortar joints (commonly clay, lime or cement based mortar). The units can be jointed together using mortar or just by simple superposition obtaining different combinations that can be classified [6] in stone masonry and bricks masonry, as illustrated in Fig. 2.1 and Fig. 2.2, respectively.
(a)
(b)
(c)
Fig. 2.1: Stone masonry (a) rubble masonry, (b) ashlar masonry, (c) coursed ashlar masonry.
19
Fig. 2.2: Brick masonry, (a) common bond, (b) cross bond, (c) Flemish bond, (d) stack bond, (e) stretcher bond.
The heterogeneity of the masonry material, which depends on the assemblage of its constituents (brick and mortar, as previously seen), leads to a complex structural behavior. Generally, the behavior of the masonry is intermediate between the behavior of the brick and mortar, as shown in Fig. 2.3.
Mortar Masonry Brick
σ
ε
Fig. 2.3: Qualitative stress-strain diagram in uniaxial tension and compression.
In Tab. 2.1 the mechanical characteristics of the masonry constituents are reported. 20
Mortar 3.0 - 30.0
Brick 6.0 - 80.0
Compression Strength [MPa] 1.5 - 9.0 Tensile Strength 0.2 - 0.8 [MPa] Tensile Modulus (8.0 - 20.0) (15.0 - 25.0) 103.0 103.0 [MPa] Poisson’s coefficient 0.10 - 0.35 0.10 - 0.25 Tab. 2.1: Mortar and brick mechanical characteristics.
While the bricks properties are generally defined on the base of brick type, the mortar mechanical properties depend strongly as much on the natural materials of which it is constituted as on the procedures of manufacturing; indeed, the mortar strength is influenced a lot by the binder and the dosage. According to the Italian Code 20/11/1987 (Norme tecniche per la progettazione, esecuzione e collaudo degli edifici in muratura e loro consolidamento) and the previous and successive rules, four classes of mortar have been specified, as reported in Tab. 2.2. Class
Kind of Mortar
M4 M4
Grout Pozzolana mortar Cement lime Cement lime Mortar of cement Mortar of cement
M4 M3 M2 M1
Composition Cement Common Water Sand lime lime 1 3 1 -
Pozzolana 3
1 1 1
-
2 1 0.5
9 5 4
-
1
-
-
3
-
Tab. 2.2: Mortar classes.
21
Subjected to a uniaxial load, the masonry material has a stress-strain curve that presents a brittle failure, characterized by a compression stress failure value greater than the tensile one, as illustrated in Fig. 2.4. In particular, it can be individuated the following characteristic features: compression OA that is essentially linear; AB characterized by a nonlinear behavior, increasing until the maximum value of the compression stress; BC, decreasing feature with nonlinear behavior and softening; tension OI very short feature that has a linear behavior and IL decreasing feature. Moreover, the point B represents the peak load and the point C represents the point in correspondence of which the masonry material collapses in compression.
σ
I O
L
ε
A C B
Fig. 2.4: Stress - strain masonry curve.
An important feature, common to all cohesive materials, is the occurrence of softening, which is defined as a progressive decrease of the mechanical strength under continuous imposed displacement, after the load peak. Softening behavior is experimentally observed in uniaxial compressive, tensile and shear failure.
22
2.3. Masonry modelling The main problem in the development of accurate stress analysis for masonry structures is the definition and the use of suitable material constitutive laws. In the last twenty years several authors, (i.e. van Zijl [7], Berto et al. [8], Pietruszczak and Ushaksaraei [9]), have proposed different modelling strategies to predict the structural response of masonry structures and, consequently, to assess the safety level of existing buildings. Taking into account the heterogeneity of the masonry material, which results composed by blocks joined by mortar beds, the models proposed in literature can be framed in the three different classes briefly described below. Micro-models consider the units and the mortar joints separately, characterized by different constitutive laws; thus, the structural analysis is performed considering each constituent of the masonry material. The mechanical properties that characterize the models adopted for units and mortar joints, are obtained through experimental tests conducted on the single material components (compressive test, tension test, bending test, etc.). This approach leads to structural analyses characterized by great computational effort; in fact, in a finite element formulation framework, both the unit blocks and the mortar beds have to be discretized, obtaining a problem with a high number of nodal unknowns. Nevertheless, this approach can be successfully adopted for reproducing laboratory tests (i.e. Lofti and Shing [10], Giambanco and Di Gati [11], Alfano and Sacco [12]). Micro-macro models consider different constitutive laws for the units and the mortar joints; then, a homogenization procedure is performed obtaining a macro-model for masonry which is used to develop the structural analysis. Also in this case, the mechanical properties of units and mortar joints are obtained through experimental tests. The micro-macro models appear very appealing, as they allow to derive in a rational way the stress-strain 23
relationship of the masonry, accounting in a suitable manner for the mechanical properties of each material component. Moreover, this approach can lead to effective models, with reduced computational effort for a structural analysis (i.e. Luciano and Sacco [13], Milani et al. [14], [15]). On the other hand, the non-linear homogenization procedure required to recover a macro model could induce some theoretical or computational difficulties [16]. Macro-models are based on the use of phenomenological constitutive laws for the masonry material; i.e. the stress-strain relationships adopted for the structural analysis are derived performing tests on masonry, without distinguishing the blocks and the mortar behaviour. A phenomenological model could be unable to describe in a detailed manner some micromechanisms occurring in the damage evolution of masonry, but it is very effective from a computational point of view when structural analyses are performed [17], [18]. The linear elastic model is the simplest approach to the analysis of masonry structures. In the linear elastic model the material exhibits an infinite linear elastic behavior, both in compression and tension. The structural response obtained under the hypothesis of linear elastic behavior, although often not completely reliable for ancient constructions [19], can be of great help; in fact, the linear analysis requires few input data and it is less demanding, in terms of computer resources and engineering time used when compared with non-linear models. Moreover, for masonries characterized by significant tensile strength, linear analysis can provide a reasonable description of the process leading to the crack pattern.
24
2.4. No-tension material model Because of the very low tensile strength of many masonries with respect to the compressive strength, a no-tension model is often adopted; it is based on the assumption of zero the tensile strength of the material, as illustrated in Fig. 2.5,. The no-tension material (NTM) model (i.e. [20] and [21]) leads to a realistic approximation for the evaluation of the mechanical response of the masonry material. In fact, the collapse mechanism of old masonry constructions is often due to the opening of cracks in tensile zones. The use of the no-tension model allows to compute the limit carrying load for masonry structures invoking the limit analysis theorems.
σ
ε O
Fig. 2.5: Linear elastic model with no tensile strength.
The no-tension material model is based on the fundamental hypothesis that the tensile strength is zero while it considers a linear elastic behavior in compression. The no-tension model presents the following very special properties: a convex strain energy function governing the stress-strain relationship exists, thus the constitutive law results to be reversible and there is no energy dissipation for the crack formation and evolution. 25
The question regarding the safety of the no-tension approach with respect the fracture mechanics solution was discussed by Bazant [22], who proved that the notension model is not always safe with respect to the fracture mechanics approach. The problems considered by Bazant concern the case of fractured rocks, characterized by the presence of a preexistent localized crack; for old masonry structures, which present sufficiently densely distributed microcracks, the no-tension model can be considered reliable. The no-tension material model received and still receives great attention by many researchers to study the behavior of old masonry structures. Indeed, the statement ”no tension material” was proposed by Zienkiewicz et al. [23] to study the behavior of fractured rocks. Then, several studies were developed regarding the NTM from a mechanical, i.e. [24] - [28], mathematical [29] and computational point of view, developing displacement, i.e. [30] - [32], as well as stress and mixed variational formulations, i.e. [33] - [35]. It could be emphasized that, although the NTM presents and apparent simplicity, its numerical treatment is not trivial. The assumption of a masonry linear elastic behavior in compression can be considered adequate only when the evaluation of the load carrying capacity of the structure occurs for a collapse mechanism accompanied by very low compressive stresses; on the contrary, when the compression strength plays a significant role in the evaluation of the structural collapse load, the no-tension model appears inadequate. This case may occur, for instance, for shear masonry panels, building walls and strengthened arches, where the presence of the reinforcement can prevent the formation of hinges. A first proposal of a no-tension model with limited compressive strength has been presented in Reference [36]. The model proposed by Lucchesi and coworkers is based on two fundamental assumptions: the stress-strain relation is again hyperelastic, so that the crushing of the material is considered to be reversible, and the inelastic strain in compression is always orthogonal to the fracture strain. Indeed, the crushing strain is quite irreversible in character and it could not also be 26
orthogonal to the fracture strain, during the whole loading history. As matter of fact, the compression failure is affected by progressive damage and inelastic irreversible strain. In order to derive a simple and effective model, Marfia and Sacco [18] developed a no-tension model which accounts for the inelastic behavior in compression, considering a plasticity model which neglect the damage and softening effects. The derived model appears appropriate for the description of the material crushing when limited values of the compressive strain arise. The elasto - plastic model is characterized by a first linear elastic feature OA and a plateau with a constant stress DE, as schematically illustrated in Fig. 2.6.
Fig. 2.6: Elasto - plastic model.
A delicate point is the determination of the point D: it can be fixed to avoid to underestimate the masonry stress and to ensure a safe state, far from point E. The possibility to determine the collapse load of masonry and the irreversible nature of strains in the plateau DE for cyclic load are the principal characteristics of this model [37].
27
3. FRP COMPOSITE MATERIALS FOR STRENGTHENING MASONRY STRUCTURES
3.1. Introduction In the last decades the use of innovative materials, such as composites, received great interest because of their possible advantages in terms of low weight, simplicity of application, high strength in the fibers direction, immunity of corrosion and quite reduced invasiveness. The use of Fiber Reinforced Polymers (FRP), that are a class of composite materials characterized by the combination of high-strength fibers and a matrix, is growing in the different fields of the engineering. Initially adopted for applications in aircraft and space industries, FRP have been used in the medical, sporting goods, automotive and small ship industries (see Fig. 3.1) due to their high strength in the fibers direction.
Fig. 3.1: Ordinary FRP devices and appliances.
28
The greater reduction of the fibers prices, due to their increasing use and to an optimization of the production processes, have recently concurred to their diffusion also in the field of the civil constructions. In particular, they appear particularly indicated for the maintenance and rehabilitation of ancient structures because they do not substantially violate the principles of the Carta di Venezia, as they can be considered (quite) reversible and distinguishable.
3.2. Mechanical behavior The FRP are composite materials constituted by two phases: polymeric matrix and high-strength fibers. The two phases are visible at microscope and they present mechanical and geometrical properties sufficiently different, as consequence the composite has mechanical properties different from those of the constituents ones. The nature of every phase influences significantly the final properties of the composite; however, in order to obtain a composite with a high mechanical resistance, it is not sufficient to use only resistant fibers: it is also necessary to guarantee a good adhesion between the matrix and the reinforcement. The adhesion is usually guaranteed by the employment of a third component, called interface or interphase, applied in a much thin layer on the surface of fibers, between fibers and matrix, as schematically illustrated in Fig. 3.2.
29
INTERFACE
FIBER
MATRIX
Fig. 3.2: FRP phases.
The interphase, whose characteristics are fundamental for the good use of the material in structural applications, is usually a thin and monoatomic layer. In fact, the lack of adhesion between fiber and matrix is one of the causes of the structural yielding of the composite materials. The organic matrices guarantee the transfer of the stresses between the surrounding structure and the fibers embedded in it, protecting these last ones from the aggressions of the external agents and from mechanical hit. The matrices, more used for the fabrication of FRP, are the polymeric ones made up of thermosetting resins. These resins are available in shape partially polymerized and they are liquid or dense at ambient temperature. The resins, mixed with an opportune reagent, polymerize until becoming a vitreous solid material. The matrices have various advantages: they are characterized by capacity of impregnation of the fibers, by optimal adhesive properties, by good resistance to the chemical agents. Their main limitations are the temperatures of exercise, limited from the upper by the vitreous transition temperature, the brittle failure, the sensibility to the humidity in phase of application on the structure. The epoxy resins are the more utilized: they have a good resistance to the humidity and the to chemical agents and optimal adhesive property.
30
The fibers more used for composite materials employed in the applications of the civil engineering are: glass (Fig. 3.4), carbon (Fig. 3.3), and aramidic (Fig. 3.5) fibers.
Fig. 3.3: Carbon fibers at microscope.
Fig. 3.4: Glass fibers.
Fig. 3.5: Aramidic fiber.
The glass fibers have an elevated resistance to the corrosion, an elastic modulus lower than those of carbon and aramidic fibers, a quite reduced resistance to the abrasion, a discreet strength to plastic slip and to fatigue. In order to promote the adhesion between fibers and matrix and to protect fibers from the action of the alkaline agents and from the humidity, the fibers undergo special treatments. In the 31
operations of manipulation before the phase of impregnation great caution is necessary. For their easy damage during the treatments, they are covered from a protecting film that inhibit the installation of acid dioxides contained in the air, which, otherwise, would penetrate in the microscopic voids present on the surface. The aramidic fibers are of organic nature and they are characterized by an elevated resistance to the manipulation operations. The modulus and the tensile strength are intermediate between those of carbon and glass fibers, while the compressive resistance is approximately equal to 1/8 of the tensile one. They are characterized also by an elevated degree of anisotropy that favors the localized rupture with consequent instability. They can be degraded for extended exposure to the solar light and it is preferable not to use them at temperatures greater than 150°C for problems of material oxidation. Moreover, they are sensitive to the humidity. The carbon fibers are used for the fabrication of composite materials with elevated performances; they are distinguished for the high modulus and resistance. They exhibit a behavior with brittle failure. The crystalline structure of the graphite is hexagonal, with carbon atoms organized in structures essentially planar, tied from interaction transverse forces of van der Waals. The precursors of carbon fibers are the polyacrylonitrile (PAN) and the Rayon fibers. Starting from these, through a process of carbonization combined with thermal processes and the sizing process, two types of carbon fibers are produced: the High Strength (HS) and the High Modulus (HM). The carbon fibers are often dealt with epoxy materials that prevent the abrasion, increase the workability and realize a good compatibility with the matrices made up of epoxy resins. The principal properties, as tensile modulus and tensile strength, of some fibers used for composite materials are reported in Tab. 3.1.
32
Tensile Tensile strength Failure Coefficient of Density modulus [Gpa] [Mpa] strain [%] thermal expansion [g/cm^3] 72 - 80 3445 4.8 5 - 5.4 2.5 - 2.6 85 4585 5.4 1.6 - 2.9 2.46 - 2.49
Fiber E-glass Fiber S-glass Graphite fiber (high modulus)
390 - 760
2400 - 3400
0.5 - 0.8
-1.45
1.85 - 1.9
Graphite fiber (low modulus)
240 - 280
4100 - 5100
1.6 - 1.73
-0.6 - -0.9
1.75
Aramid fiber
62 - 180
3600 - 3800
1.9 - 5.5
-2
1.44 - 1.47
Polymeric matrix
2.7 - 3.6
40 - 82
1.4 - 5.2
30 - 54
1.10 - 1.25
206
250 - 400 (yield) 350 - 600 (failure)
20 - 30
10.4
7.8
Steel
Tab. 3.1: Properties of FRP constituents.
The most common shape for the composite materials is the laminate one. The laminates are constituted by two or more overlapped thin layers, called lamina, (see Fig. 3.6). z
x 2
x=1
1
1 2
X x
2
Y
Fig. 3.6: Laminate constituted by more laminas.
The main advantage of laminates is the maximum freedom in the disposition of fibers. In each plane, the direction of fibers can be chosen in order to obtain the desired physical and mechanical characteristics of the laminates. On the basis of the mechanical properties that have to be conferred to the laminate, different types of fibers can be adopted. For instance, hybrid laminates are obtained by assembling layers of epoxy resin reinforced by aramidic and carbon fibers, or by alternating layers of epoxy resin with aramidic or aluminum fibers. The orientation of fibers is one of the main aspects that determines the behavior of the composite material. A 33
disposition of unidirectional fibers, as schematically illustrated in Fig. 3.7, leads to an orthotropic response of the lamina.
Fig. 3.7: Laminate with unidirectional fibers.
With this type of disposition, the best mechanical properties is obtained in the direction of fibers. A bidirectional disposition confers to the composite mechanical characteristics which depends on the chosen fiber direction. Beyond to the orientation also the length, the shape, the composition and the percentage in volume of fibers, the mechanical properties of the resin and the interface influence the response of the composite. The mechanical properties (strength, strain, tension modulus) of some FRP systems degrade in presence of determined environmental conditions, i.e. alkaline ambient, extreme humidity, temperatures, thermal cycles.
3.2.1. Alkaline ambient effects The pores of the material that must be reinforced content water that can degrade the resin and the interphase. It is necessary that the resin complete the curing before the exposition to alkaline ambient.
34
3.2.2. Humidity effects The main effects connected to the absorption of humidity regard the resins; they are plasticization, reduction of vitreous transition temperature, strength and stiffness reduction. The absorption of humidity depends by the kind of resin, the composition and number of laminas, the curing conditions, the interphase and the processing.
3.2.3. Extreme temperature and thermal cycle effects The main effects of temperature are connected to the viscous answer of the composite. At the service temperature of most structures, the resins are stable, but when the temperature increases, the resin breaks down and evaporates; the composite performances strongly decrease when the temperature exceeds the vitreous transition one. The thermal cycle have not deleterious effects, even if they can favor the formation of micro-fractures.
3.2.4. Frost-thaw cycles effects The exposition to frost-thaw cycles do not influence the performance of the composites, but can reduce those of the resin and the interphase, because of the separation between fibers and matrix.
3.2.5. Temperature effects The increasing of the temperature involves a gradual degradation of the mechanical properties of composite in terms both of tensile strength and stiffness. 35
3.2.6. Viscosity and relaxation effects In a composite material, viscosity and relaxation depend from the properties of resin and fibers. The presence of fibers reduces the viscosity of the resin; the worse effect occurs when the load is applied in the direction orthogonal to the fibers or when the composite is characterized from one low percentage in volume of fibers. The viscosity can be reduced if it is assured a low stress in exercise.
3.2.7. Fatigue effects The performances of FRP under fatigue are very good and they are connected to the composition of matrix. In the unidirectional composites, fibers have got little defects and, consequently, they contrast the formation of fractures. Moreover the propagation of eventual fractures is limited from the action explicated from the fibers staying in the adjacent zones.
3.3. Masonry structures reinforced with FRP materials In the last years, a significant research activities has been performed to investigate on the possibility to adopt composite materials as reinforcement of the masonry buildings. Starting from the earliest works, Triantafillou and Fardis [38], several studies have been devoted to the evaluation of the advantages in terms of resistance and ductility, for the use of FRP for the strengthening of masonry constructions. Indeed, researches demonstrate that the use of FRP for the strengthening of masonry structures is very effective for different structural elements as masonry panels, but also arches and vaults. 36
Several researches have been oriented to the analysis of masonry walls reinforced by FRP sheets or laminates, subjected to in-plane and out-of-plane loads. The possibility of adopting FRP composites for strengthening of masonry was initially investigated by Croci et al. [39]. They presented the results of experimental tests performed on wall specimens reinforced by vertical FRP materials. Experimental investigations on the use of epoxy-bonded glass fabrics were developed by Saadatmanesh [40] and by Ehsani [41]. Luciano and Sacco [13], [42] and Marfia and Sacco [43] proposed micromechanical models to study the behavior of masonry elements reinforced with FRP sheets. Cecchi et al. [44] developed a homogenization technique to evaluate the overall behavior of reinforced masonry walls. Experimental tests, performed by Schwegler [45] and Laursen et al. [46], demonstrated the significant improvement of the in-plane shear capacity and the important increase of the ductility of masonry walls strengthened with FRP laminates. Triantafillou [47] and Velazquez et al. [48] developed experimental studies, showing that the flexural capacity of masonry walls can be drastically increased strengthening the panels with FRP laminates. Olivito and Zuccarello [49] presented the durability of masonry structures reinforced by FRP subjected to low cycle fatigue. In the last few years great interest was devoted to the reinforcement of masonry arches and vaults, probably as a result of the recent Umbria- Marche seismic events. In fact, aramidic fiber reinforced composites were adopted to restore the vaults of the Basilica di S. Francesco d’Assisi [50] and the Chiesa di San Filippo Neri, in Spoleto [51]. Como et al. [52] applied the limit analysis theorems in order to evaluate the collapse of reinforced arches. Olivito and Stumpo [53] proposed a numerical and experimental analysis of vaulted masonry structures subjected to moving load. Briccoli Bati and Rovero [54] and Aiello et al. [55] developed experimental investigations on reinforced masonry arches, emphasizing that the application of sheets or laminates of composite materials significantly increases the strength of the structure, modifying the collapse mechanism and the corresponding collapse load. 37
Chen [56] presented a method to calculate the limit load-bearing capacity of masonry arch bridges strengthened with FRP. Experimental tests and finite element analyses of masonry arches made of blocks in dry contact and reinforced by FRP materials have been developed by Luciano et al. [57], demonstrating the effectiveness of strengthening. Foraboschi [58] presented mathematical models for studying the possible failure modes of masonry arches and vaults with FRP reinforcement. Ianniruberto and Rinaldi [59] investigated on the influence of the presence of FRP to the collapse behavior of the structure when reinforcements are placed at the extrados or at the intrados of the arch. It can be emphasized that the collapse of masonry elements is generally induced by the opening of fractures due to the limited strength in tension. The presence of the FRP reinforcement, placed in the tensile zones of the masonry structure, inhibits the opening of the fractures; thus, a compression state can occur for bent elements, and the failure for crushing can be activated. As a consequence, a suitable masonry model for reinforced masonry should take into account the possibility of the collapse for compression, i.e. a limited compressive strength for the masonry material should be considered.
3.4. Collapse mechanism for reinforced structures When a masonry structure is reinforced, the collapse mechanism changes with respect to the unreinforced one. Indeed, the collapse of a reinforced masonry structure can occur for the activation of different failures: opening of cracks in the masonry for tensile stresses, crushing of masonry in compression, shear failure of the masonry, decohesion of the FRP from the masonry and failure of the reinforcement, i.e. [54], [60] and [61]. While the unreinforced masonry collapses generally for activation of mechanisms due to the very limited tensile strength of the masonry or for shear failure, for reinforced masonry the limited compressive strength of the 38
masonry and the delamination phenomenon can play fundamental roles in the overall collapse of the structure. Crushing of masonry in compression and reinforcement failure are strictly connected to the mechanical properties of masonry and reinforcement fibers respectively, while the decohesion phenomenon regards the interface masonry-FRP. The adhesion between masonry and composite is a very relevant factor in the masonry reinforcement by laminas or woven. The debonding can regard both laminas and woven applied on the extrados or intrados surface of the reinforced element. The understanding of the debonding mechanism is very important for the successful application of the external FRP reinforcement; it is necessary to know when debonding initiates and the parameters that influence it. The decohesion can be classified in Plate-end debonding (it initiates at a plate-end and propagates inwards) and Intermediate crack debonding (it initiates at a crack in the structure mid-span zone and then it propagates towards the nearest zones).
39
4. EXPERIMENTAL PROGRAM
4.1. Introduction The experimental program was realized at LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of the Geolab Sud of San Vittore del Lazio. The experimental tests were performed at the Geolab laboratory and part of the instrumentation was supplied by them. In order to determine the correct setup of the used instrumentation it was necessary to perform a preliminary experimental campaign on a steel beam.
4.2. Setup and instrumentations Several instrumentations were necessary to perform the experimental program; in particular, the devices to determine displacements, the strain gauges, the hydraulic jack to apply the external load, the load cells and the data acquisition system were used. Two instruments were adopted to determine the displacements: comparators and potentiometers. The comparator used in the experimental program is a dial gauge; the instrument bases its functionality on the displacement of a cylindrical rod that can be flow into a tubular guide for a maximum value of 100 mm. It is positioned on the interested surface, so the tracer point is in contact with the surface of the specimen subjected to the measurement. The potentiometer has the same performances of the dial gauge; it is composed by a cylindrical rod that can move into a tubular guide until 100 mm. On the extremity of 40
the rod there is a magnet that fixes the potentiometer to the interested surface on which a metallic element has been previously glued. The load was applied by an hydraulic jack and it was measured by two electric load cells. The load cells have a maximum value of 50 kN and 500 kN respectively; they are constituted by an inox steel body with an electronic device that allows to convert the mechanical tensile or compressive load into an electric signal. There is an optional plate that allows a more homogeneous load repartition on the body cell. The electronic device is constituted by resistive strain-gauges connected by an electric Wheatstone bridge. In the experimental program, two electric digital data acquisition systems, Leane and Wshay, were used. When it is subjected to load, each load cell emits an electric differential signal which is transmitted by a connector to the data acquisition system; the aim of data acquisition system is the data elaborations, i.e. the conversion of the electric signal into mechanical engineering quantities. So the data acquisition system allows the measurement by the manual or automatic data acquisition. The Washay is a model P3 Strain Indicator and Recorder; it is portable and alimented by battery; its data acquisition is manual. The measurements obtained by this data acquisition system were used to verify the correct working of the Leane data acquisition system. Leane is a portable data acquisition system characterized by electric and battery alimentation. The data acquisition system has seven modules and four channels for each one; in total it is possible to have 28 acquisitions at the same time. In the experimental program, Leane was used for the acquisition in continuous of the potentiometers and of the cell load of 5 t. The Leane acquisitions are transmitted to a PC by a cable and then, the results can be worked out by a software given by the Leane.
41
4.3. Preliminary experimental campaign This preliminary experimental campaign was necessary to validate the data acquisition system Leane, in particular to verify that the in continuous displacements acquisition did not depend on the potentiometers position on the data acquisition system channel and they were not different from the displacements measured using the comparators. It was necessary to calibrate a new load cell of 50 kN, called in the following as small load cell. The load values of the 50 kN load cell acquired with the Leane are in accordance with those measured by the 500 kN load cell, called in the following as great load cell, acquired with the Wshay. The specimen of preliminary tests campaign was a steel beam and the tests were organized in TEST A (potentiometer calibration and displacement acquisition crosscheck), TEST B (small load cell calibration) and TEST C (small load cell acquisition by Leane crosscheck).
TEST A The aim of the test A was the potentiometers calibration and the crosscheck of the correct displacements acquisition obtained by the potentiometer connection to the different channels of Leane. The potentiometers were connected to data acquisition system Leane to have the displacements in correspondence of each load variation, in continuous. As previously seen, with Leane it is possible to have 28 acquisitions; the steel beam was subjected to 6 load cycles, called Test 1, Test 2, Test 3, Test 4, Test 5 and Test 6, characterized by the same load steps. In every load cycle, the potentiometer position on the data acquisition system module was changed to validate the different acquisitions obtained for every module and to validate the acquisitions obtained for every different channel of each module. In the Test A it can be pointed out that the difference between the various displacement acquisitions is in all the cases lower than 0.1 mm. The channel 4 of the 42
module 1 does not work. The difference between the displacement values registered by potentiometers and comparators is satisfactory.
TEST B This campaign had the aim to calibrate the new small load cell. It was possible to put in comparison the acquisitions obtained from the small load cell and the acquisitions obtained by the great load cell, both connected with the data acquisition system Wshay. The maximum error of test resulted equal to 1%; thus, it can be pointed out that the new small cell works in good accordance with the normalized great one.
TEST C This campaign had the aim to verify the correct functionality of the small cell connected with the data acquisition system Leane. The Test C puts in evidence that the difference between the manual and automatic acquisition of the load is, on average, lower than 2%.
43
4.4. Materials used in the experimental program The determination of the physical and mechanical properties of the materials used in the experimental program is necessary to understand the behavior of reinforced masonry arches. In the following the properties of the masonry material constituents and of the reinforcement are presented. The masonry material is composed by standard clay bricks and mixed mortar. At LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of the Geolab Sud of San Vittore del Lazio, an experimental program both on standard clay brick and mortar was performed. For what concerns the reinforcement, it is composed by carbon fibers and epoxy matrix and their properties were given by the manufacturer.
4.5. Standard clay brick In the experimental program, standard clay bricks (Fig. 4.1) were used.
Fig. 4.1: Standard clay brick.
44
In order to determine its main mechanical properties, the standard clay brick was subjected to several experimental tests. In particular, a cubic compressive test, an indirect shear test and a test to individuate the elastic secant modulus were performed.
4.5.1. Cubic compressive test Standard clay brick cubic specimens were prepared in order to determinate the compressive strength, in accordance with UNI 8942/3. This code gives the guidelines for the determination of the unitary load of compressive failure strength, that has to be determined on a fixed number of specimens with prefixed geometric characteristics. According to the code, the tests have to be performed on cubic specimens with orthogonal faces and parallel plane of the bedding plane, as illustrated in Fig. 4.2.
Fig. 4.2: Cubic specimen extrapolated by standard clay brick.
The specimens were located on the Galdabini SUN 60 that is a universal testing machine with a 600 kN nominal capacity, used in displacement control. A series of pre-loading tests finalized to set the machine and to position the specimens into its slabs were realized before the compressive test. The failure load was obtained from 45
the yielding load of every specimen. Then the other parameters necessary to characterize the test results were determined: average compressive strength:
n
fb =
∑f i =1
bi
(4.1)
n
where fbi is the result of the single test and n is the number of test results; standard deviation:
∑( f n
s=
i =1
b
− fbi )
2
n
(4.2)
variation coefficient:
cv =
s fb
(4.3)
characteristic value: fbk = f b (1 − kcv )
(4.4)
where k is the fractile coefficient, fixed by normative in function of the number of tested specimens. The compressive test was realized on 6 cubic specimens extracted by one of the series of the standard clay brick; their dimensions are reported in Tab. 4.1. 46
Specimen [number] 1 2 3 4 5 6
Deep [mm] 55 55 56 55 55 55
Length [mm] 56 55 55 55 55 55
Heigth [mm] 55 54 55 55 54 55
Tab. 4.1: Specimens size.
A carton layer was interposed at the top of the specimen in order to distribute the compressive stress. The specimen was allocated into the press, Fig. 4.3.
Fig. 4.3: Specimen positioning.
The specimen was subjected to an axial load acting perpendicular to the bedding plane until its failure, Fig. 4.4.
47
Fig. 4.4: Typical failure of the specimen.
The failure load and the compressive strength were determined for each specimen, as reported in Tab. 4.2 Specimen [number] 1 2 3 4 5 6
Area [mm2] 3080 3025 3080 3025 3025 3025
Failure load [kN]
Compressive strength (fb)
127.883 107.918 125.144 104.944 110.894 126.239
[kN/mm2] 0.0415 0.0357 0.0406 0.0347 0.0366 0.0417
Tab. 4.2: Compressive test results.
The considered specimens exhibited a hourglass failure, Fig. 4.5, not perfectly symmetrical because of the heterogeneous nature of the bricks.
48
Fig. 4.5: Hourglass specimens failure.
The characteristic compressive strength was determined, using equations (4.1), (4.2), (4.3) and (4.4) and in accordance with the code for which k=2.33 if n=6; the results are reported in Tab. 4.3. Average compressive strength [N/mm2]
Standard deviation [N/mm2]
Variation coefficient
Characteristic compressive strength (fbk) [N/mmq]
38.5
7.47
0.23
14.9
Tab. 4.3: Characteristic compressive strength.
4.5.2. Indirect tensile test The indirect tensile test was realized in accordance with UNI 8942/3, which gives the guidelines for the determination of the yielding load of specimens subjected to a uniform load applied on the middle surface of the specimen, as schematically illustrated in Fig. 4.6.
49
F
Fig. 4.6: Indirect tensile test scheme.
The code prescribes that this test has to be performed on specimens with a low drilling percentage (the limit is fixed at 30%). The test was performed using the
Galdabini SUN 60 and it was executed with constant load increments until the failure. In order to diffuse the load two steel beam, whose dimensions were fixed by the code, were interposed between the specimen faces and the steel plates of the machine, as illustrated in Fig. 4.7.
Fig. 4.7: In direct tensile test particular.
The indirect tensile test was realized on 6 specimens whose dimensions are reported in Tab. 4.4.
50
Specimen [number]
Deep [mm]
Length [mm]
Heigth [mm]
1 2 3 4 5 6
117 117 117 117 117 117
255 255 255 255 255 255
55 55 55 55 55 55
Tab. 4.4: Specimen dimensions.
Initially a pre-loading was imposed to setup the Galdabini SUN 60 , then the constant load increments were applied. Each specimen was subjected to compression load up to failure. The failure occurred along the direction of load application considering the front view, as represented in Fig. 4.8.
Fig. 4.8: Specimen failure.
Analogously to the compressive test, the following quantities were determined: average tensile strength mean deviation variation coefficient characteristic value. The indirect tensile strength was determined for each specimen by formula:
51
fv =
2t π bh
(4.5)
where t is the external applied load in Newton; h and b are the specimen height and length respectively, expressed in mm. The failure load and the indirect tensile strength were determined for each specimen, as reported in Tab. 4.5. Specimen [number] 1 2 3 4 5 6
Area [mm2] 14025 14025 14025 14025 14025 14025
Failure load [kN]
Indirect tensile strength (fv)
38816 31800 45008 24739 37841 30543
[kN/mm2] 3.84 3.15 4.45 2.45 3.75 3.02
Tab. 4.5: Indirect tensile test results.
The characteristic indirect tensile strength was computed and the results are reported in Tab. 4.6. Average indirect tensile strength [N/mm2] 3.44
Mean deviation [N/mm2] 0.71
Variation coefficient
Characteristic indirect tensile strength (fvk)
0.21
[N/mm2] 1.79
Tab. 4.6: Characteristic indirect tensile strength.
52
4.5.3. Elastic secant modulus In order to evaluate the elastic secant modulus a test was realized in accordance with the prescription of UNI 6556 rule. The specimens extrapolated by standard clay bricks were prismatic; in fact, the rule prescribes that the tests has to be performed on cylindrical or prismatic with square base specimens. The test was realized using the universal testing machine Galdabini SUN 60. In order to evaluate the elastic secant modulus, the code prescribes the use of 3 + 3 specimens. In particular, 3 specimens were used for evaluating the compressive strength, and the others 3 to determine the elastic secant modulus. The test was organized in two phases. During the first phase, 3 specimens were obtained by standard clay brick and their size was 5x5x15 cm. Each specimen was allocated into the universal testing machine and it was loaded until its compressive load failure, as represented in Fig. 4.9.
Fig. 4.9: Positioning into the universal testing machine of the reference specimen.
The average failure load value of the i-th set of specimens was determined as:
53
3
N if =
∑N j =1
j f
(4.6)
3
After this test, the load values representing the extremes of the loading-unloading cycles were determined using the average failure load values, recovered by equation (4.6). In accordance with the UNI 6556 rule, the maximum load is N 3 = base load is N 0 = N 2 = ( 2 N1 − N 0 ) .
1 i N f , the 3
1 ⎛ N + 2 N0 ⎞ N 3 and the intermediate loads are N1 = ⎜ 3 ⎟ and 3 10 ⎝ ⎠ Consequently
the
load
cycles
are
defined
as
cycle 1: N 0 → N1 → N 0 , cycle 2: N 0 → N 2 → N 0 and cycle 3: N 0 → N 3 → N 0 . In the second phase, further 3 specimens were tested. The bricks for evaluating the elastic secant modulus were prepared. The brick surface was cleaned and the area, where the strain-gauge were applied, was dry sanded, removing all the eventually incrustations, as illustrated in Fig. 4.10.
Fig. 4.10: Brick surface preparation.
54
In order to simplify the strain-gauge application, guidelines were traced on the brick surface; then the resin was applied, as represented in Fig. 4.11, and the strain-gauge was positioned along the guidelines previously traced, as illustrated in Fig. 4.12.
Fig. 4.11: Resin application.
Fig. 4.12: Strain-gauge application.
Each specimen was allocated into the universal testing machine and all the straingauge was connected with the data acquisition system. The load cell also was connected to the data acquisition system to know the applied load at each loading step, as represented in Fig. 4.13.
55
Fig. 4.13: Specimen positionating.
For every specimen the elastic secant modulus was determined. The procedure can be schematically described as: 1. the base load N0 was fixed; 2. the base mean strain ε 0 was determined; 3. the maximum load of the cycle Ni was fixed; 4. loading phase was performed: N 0 → N i ; 5. the mean strain ε i in correspondence of the maximum load was determined; 6. unloading phase was performed: N i → N 0 ; 7. the elastic secant modulus was determined as Es =
σi =
σi −σ0 εi − ε0
where
Ni N , σ 0 = 0 and A is the specimen base area. A A
The results elaboration for all the specimens are reported in Tab. 4.7, Tab. 4.8 and Tab. 4.9.
56
Ni
σi
[N]
[N/mm2]
4150 16740 4230 4230 28350 4330 4330 41670 4230
1.6600 6.6960 1.6920 1.6920 11.3400 1.7320 1.7320 16.6680 1.6920
Δσ
ε
Δε
[N/mm2] 5.0360
9.6480
14.9360
0.000087 0.000351 0.000098 0.000098 0.000616 0.000112 0.000112 0.000947 0.000110
Es
Es
[N/mm2]
[N/mm2]
0.000264
19075.7576
0.000518
18625.4826
0.000835
17887.4251
18529.5551
Tab. 4.7: Results elaboration for specimen 1.
Ni [N] 4110 16510 4150 4150 28610 4070 4070 40630 4030
σi
Δσ 2
[N/mm ] 1.6440 6.6040 1.6600 1.6600 11.4440 1.6280 1.6280 16.2520 1.6120
ε
Δε
2
2
[N/mm ] 4.9600
9.7840
14.6240
Es
Es [N/mm ]
0.000163 0.000520 0.000182 0.000182 0.000912 0.000191 0.000191 0.001302 0.000192
0.000357
13893.5574
0.000730
13402.7397
0.001111
13162.9163
[N/mm2]
13486.40448
Tab. 4.8: Results elaboration for specimen 2.
Ni [N] 4090 16230 4090 4090 27420 4130 4130 40630 4070
σi
Δσ 2
[N/mm ] 1.6360 6.4920 1.6360 1.6360 10.9680 1.6520 1.6520 16.2520 1.6280
ε
Δε
2
2
[N/mm ] 4.8560
9.3320
14.6000
Es
Es [N/mm ]
0.000113 0.000407 0.000117 0.000117 0.000669 0.000115 0.000115 0.000977 0.000113
0.000294
16517.0068
0.000552
16905.7971
0.000862
16937.3550
Tab. 4.9: Results elaboration for specimen 3.
57
[N/mm2]
16786.7196
The elastic secant modulus of standard clay brick was obtained as the average value of the elastic secant modulus of every specimen and it is Es ≅ 16000 N / mm 2 .
58
4.6. Mortar The mortar used to realize the three arches belongs to the M3 class, in accordance with the Italian Code Ministerial Decree of 20/11/1987. The mortar is constituted by: 2800 g of pozzolana; 933 g of lime, Calcisernia, Contrada Tiegno, Isernia; 800 g of pozzolanico cement Duracem 32.5 R, Colleferro, Roma; 1.66 l of water.
In literature this mortar is classified as mixed because it is constituted by two binders: cement and lime.
4.6.1. Compressive tests The specimens were realized with the mortar described previously. Three specimens of 40x40x160 mm3 were prepared using a normalized sand, as represented in Fig. 4.14.
Fig. 4.14: Preparation of mortar.
59
The mortar was prepared by mechanical mixing and successively compacted using a normalized vibrating device as illustrated in Fig. 4.15.
Fig. 4.15: Device to mix the mortar and normalized vibrating device.
After 28 days of seasoning, the specimens were subjected to a bending test. The specimen was allocated into the universal testing machine with a lateral face on the support rollers and the longitudinal axis orthogonal to the supports. The vertical load was applied on the specimen lateral face and it was uniformly increased with a maximum ratio of 20 Kg / cm 2 s until the failure, as represented in Fig. 4.16.
60
Fig. 4.16: Bending failure of mortar specimen.
In this way two semi-prismatic specimens were obtained and they were successively subjected to the compressive test. In order to determine the compressive behavior of the mortar, the semi-prismatic specimen was tested, as shown in Fig. 4.17.
Fig. 4.17: Compressive test: test setup and typical compressive failure.
61
The tests were performed with the universal testing machine Galdabini SUN 60 and the results are reported in Tab. 4.10. 1 2 Specimen 4x4x16 4x4x16 Size [cm] 432.4 430.6 Weigth [g] Compressive strength 4.8094 4.7038 4.7438 4.7675 [N/mm2]
3 4x4x16 430.5 4.6219 4.7975
Tab. 4.10: Mortar compressive strength.
4.6.2. Elastic secant modulus The elastic secant modulus of the mortar was carried out. The procedure was exactly the same that was realized for the standard clay brick. Three reference specimens for each mixture were tested. Therefore, the final results are schematically reported in Tab. 4.11 and Tab. 4.12 for the mixtures one, two and three and for mixtures four, five and six, respectively.
62
ε0 [10^-6] -79.67 -88.00
N0 [N] 420.00 420.00
εi [10^-6] -258.67 -266.67
Ni [N] 1690.00 1690.00
ε0s [10^-6] -88.00 -103.00
N0s [N] 420.00 440.00
εp [10^-6] -8.33 -15.00
Em0 εe E E [10^-6] [N/mm^2] [N/mm^2] [N/mm^2] -250.33 3170.77 3137.54 -251.67 3104.30
-103.00 -137.33
440.00 420.00
-475.67 -502.33
3020.00 3000.00
-137.33 -158.33
420.00 460.00
34.33 -21.00
-510.00 -481.33
3186.27 3298.13
-150.67 -183.00 -58.67 -83.00
440.00 420.00 420.00 420.00
-699.00 -754.00 -247.67 -244.33
4230.00 4290.00 1690.00 1690.00
-183.00 -217.33 -83.00 -83.33
420.00 420.00 420.00 440.00
32.33 -34.33 -24.33 -0.33
-731.33 -719.67 -223.33 -244.00
3256.04 3360.93 3554.10 3201.84
-83.33 -139.33
420.00 400.00
-463.33 -500.33
3060.00 3060.00
-139.33 -157.67
400.00 420.00
-56.00 -18.33
-407.33 -482.00
4081.42 3423.24
-157.67 -223.00 -67.00 -74.50 -80.50
420.00 420.00 420.00 420.00 420.00
-713.67 -747.67 -283.50 -269.50 -283.00
4310.00 4310.00 1670.00 1670.00 1670.00
-223.00 -248.33 -74.50 -80.50 -90.50
420.00 420.00 420.00 420.00 420.00
-65.33 -25.33 -7.50 -6.00 -10.00
-648.33 -722.33 -276.00 -263.50 -273.00
3750.00 3365.83 2830.62 2964.90 2861.72
Cycle 2
-90.50 -101.50
420.00 420.00
-500.50 -514.00
2840.00 2840.00
-101.50 -142.75
420.00 420.00
-11.00 -41.25
-489.50 -472.75
3089.89 3199.37
3144.63
Cycle 3
-142.75 -212.75
420.00 420.00
-777.50 -847.00
4230.00 4230.00
-212.75 -250.00
420.00 420.00
-70.00 -37.25
-707.50 -809.75
3365.72 2940.72
3153.22
Mixture 1
Cycle Cycle 1
Cycle 2
Cycle 3
Mixture 2
Cycle 1
Cycle 2
Cycle 3
Mixture 3
Cycle 1
3242.20
3229
3308.49 3377.97
3752.33
3563
3557.91 2885.74
3061
Tab. 4.11: Results elaboration for mortar specimen 1.
ε0 [10^-6] -82.33 -95.00
N0 [N] 420.00 420.00
εi [10^-6] -330.67 -348.67
Ni [N] 1630.00 1630.00
ε0s [10^-6] -95.00 -108.33
N0s [N] 420.00 420.00
εp [10^-6] -12.67 -13.33
Em0 εe E E [10^-6] [N/mm^2] [N/mm^2] [N/mm^2] 318.00 2378.14 2316.68 335.33 2255.22
-108.33 -137.00
420.00 420.00
-570.00 -574.67
2820.00 2820.00
-137.00 -137.00
420.00 420.00
-28.67 0.00
541.33 574.67
2770.94 2610.21
-137.00 -175.67 -69.33 -98.33
420.00 420.00 420.00 420.00
-774.67 -806.33 -299.67 -296.00
4130.00 4130.00 1590.00 1590.00
-175.67 -199.00 -98.33 -114.83
420.00 420.00 420.00 420.00
-38.67 -23.33 -29.00 -16.50
736.00 783.00 270.67 279.50
3150.48 2961.37 2701.66 2616.28
-114.83 -135.67
420.00 420.00
-518.67 -533.00
2820.00 2820.00
-135.67 -154.00
420.00 420.00
-20.83 -18.33
497.83 514.67
3013.06 2914.51
-154.00 -184.00 -59.33 -67.83
420.00 420.00 420.00 420.00
-774.67 -784.67 -243.33 -256.00
4110.00 4110.00 1650.00 1650.00
-184.00 -184.00 -67.83 -77.00
420.00 420.00 420.00 420.00
-30.00 0.00 -8.50 -9.17
744.67 784.67 -234.83 -246.83
3097.02 2939.15 3273.60 3114.45
Cycle 2
-77.00 -108.00
420.00 420.00
-436.67 -473.33
2860.00 2880.00
-108.00 -139.00
420.00 460.00
-31.00 -31.00
-405.67 -442.33
3759.24 3475.89
3617.56
Cycle 3
-139.00 -232.67
460.00 440.00
-741.67 -781.00
4110.00 4110.00
-232.67 -263.33
440.00 420.00
-93.67 -30.67
-648.00 -750.33
3520.45 3056.97
3288.71
Mixture 4
Cycle Cycle 1
Cycle 2
Cycle 3
Mixture 5
Cycle 1
Cycle 2
Cycle 3
Mixture 6
Cycle 1
2690.57 3055.92 2658.97
2963.78
2880
3018.08 3194.02
Tab. 4.12: Results elaboration for mortar specimen 2.
The mortar elastic secant modulus results equal to Es ≅ 3100 N/mm 2 . 63
2688
3367
4.7. Reinforcement material In this experimental program the used reinforcement system is the woven SikaWrap300C NW. It is constituted by carbon fibers impregnated on-site with an epoxy resin
of type SikaDur 330. The woven was chosen because it can be easily adapted to the curvilinear surface of the arch. On the lateral surfaces the woven has a thin texture, that safeguards the fibers stability during the application process, made of thermoplastic material, as shown in Fig. 4.18.
Fig. 4.18: SikaWrap-300C NW
The fibers of the woven are unidirectional. In the following the properties of unidirectional carbon fiber provided by the manufacturer are reported.
64
65
66
The epoxy resin SikaDur 330 was used both as adhesive to the masonry arch and as matrix. The resin is constituted by two-component, it is 100% solid and grey color. The properties of the resin are reported below.
67
68
4.8. Experimental test on the arches The experimental campaign on masonry arches was conduced on a set of two arches having the same geometrical characteristics and realized with the same materials. The aims of this campaign is the evaluation of the mechanical response of unreinforced and reinforced arches. In particular, the main aim is the validation of the numerical model developed to individuate the behavior of the masonry arch reinforced by FRP.
4.9. Arch laying The arch was realized using the standard clay brick and mixed mortar previously seen. The laying of the arch was at LAPS, Laboratories of Structural Analysis and Design of University of Cassino, with the collaboration of Geolab Sud of San Vittore del Lazio. The reference arch is an incomplete circular arch with the abutment angle Φ = 8o , the mean radius of 516 mm, a cross section of 120x250 mm2. It is loaded with a vertical increasing force applied in correspondence of the brick number 14, as schematically illustrated in Fig. 4.19.
69
F
5
6
7
8
11 12 13 14 9 10 15
16
17
4 3 2
r
1
O
18
19 20 21 22 23
X Y
Fig. 4.19: Reference arch.
The first step was the construction of a steel centering, Fig.4.20 and in Fig. 4.21.
Fig.4.20: Arch centering.
Fig. 4.21: Geometrical characteristics of centering.
70
The centering was positioned on a temporary support. The standard clay bricks were embedded in water and then they were put on the centering. Initially, the bricks at the extremities were positioned. Then, the springs were constrained, Fig. 4.22.
Fig. 4.22: Constraint of the springs.
The mortar was mixed and the others bricks were positioned on the centering, spaced out by mortar joints, Fig.4.23
Fig.4.23: Construction of the arch.
When the last brick was posed, the external surface of the arch was polished to eliminate the eventual excessive mortar. The realized arch was seasoned for 28 days.
71
4.10. Arch preparation The arch was positioned under the steel frame for the test. The springers were clamped. All the bricks were numbered from left to right. The instrumentation was positioned on the arch: the displacements acquisition was obtained by comparators and potentiometers, the applied load was read by load cells and all the data were registered by the data acquisition system Leane. In particular the potentiometer p2 (p_c) and p3 (p_c_o) were positioned at the arch key, in vertical and horizontal direction, respectively. The potentiometer p4 (p_f) was positioned in correspondence of the loaded section., with a vertical direction. Moreover two comparators were utilized, one in correspondence of the arch key (c_c), one in correspondence of the force application (c_f), as schematically illustrated in Fig. 4.24, in Fig. 4.25 and in Fig. 4.26.
Fig. 4.24: Arch front and back view.
72
Fig. 4.25: Arch extrados view.
Fig. 4.26: Displacements measurement system.
The load was applied by the hydraulic jack. On the surface of brick number 14 a plate was applied in order to make easier the positioning of the load cell and the application of the external load; for the unreinforced arch the small cell was used. The applied load and the displacements were acquired in continuous by their connection to the data acquisition system Leane. Moreover for every loading cycle the displacement values, at each fixed loading step, were acquired by both the potentiometers and comparators in order to validate the reliability of the data.
73
4.11. Experimental campaign: Arch 1 Three loading-unloading cycles, called Cycle I, II and III respectively, were performed applying the external load by the hydraulic jack in correspondence of the brick number 14. Two loading-unloading cycles, called Cycle IV and V respectively, were performed applying the external load by a normalized set of weights. The displacements were acquired by potentiometers and comparators and the applied load intensity was determined by the small cell. Summarizing, the test organization was been the following: cycle I: external load applied by the hydraulic jack; cycle II: external load applied by the hydraulic jack; cycle III: external load applied by the hydraulic jack; cycle IV: external load applied by normalized set of weights; cycle V: external load applied by normalized set of weights.
Cycle I.
During the first cycle the following hinges opening occurred: hinge at the extrados between bricks 13-14, interface 14, in correspondence
of a load value equal to F ≅ 350 N ; hinge at the intrados between bricks 8-9, interface 8, in correspondence of a
load value equal to F ≅ 400 N ; hinge at the extrados between bricks 1-2, interface 1, in correspondence of a
load value equal to F ≅ 500 N ; hinge at the intrados between bricks 18-19, interface 18, in correspondence
of a load value equal to F ≅ 550 N .
74
Fig. 4.27: Arch 1, cycle I, experimental test.
Fig. 4.28: Arch 1, cycle I, hinges formation.
Cycle II and Cycle III.
During the second and the third cycle the opening of hinges occurred in correspondence of an applied load lower than in the first cycle. The first cycle peak load equal to FPL ≅ 600 N decreased in the cycle II and cycle III to a value close to 400N . This reduction is due to a significant reduction of masonry tensile strength.
Cycle IV and cycle V.
Those cycles were done to verify the acquisition in continuous of the applied load intensity; therefore the load was applied by a normalized set of weights.
75
Fig. 4.29: Arch 1, cycle IV.
The limit load reached during the cycle IV was the same of the value obtained in the second and third cycle. In the cycle V the collapse mechanism occurred applying a load greater than 450N .
Fig. 4.30: Arch 1, cycle V, collapse mechanism.
4.11.1. Collapse mechanism description The collapse mechanism occurred consequently to the hinges formation. The hinges formation occurred in the points where the pressure curve overlaps the arch intrados or extrados, as schematically reported in Fig. 4.31.
76
Fig. 4.31: Arch 1, hinges formation scheme.
The hinges opening deternined the subdivision of the arch in blocks. The arch collapse mechanism was characterized by the displacements of the blocks, as schematically illustrated in Fig. 4.32 and in . Fig. 4.33.
Fig. 4.32: Arch 1, collapse mechanism scheme.
77
Fig. 4.33: Arch 1, cycle V, particular of the collapse mechanism.
4.11.2. Load-displacements curves The cycles I, II and III were acquired in continuous by the data acquisition system Leane and the acquired data are reported in Fig. 4.34.
-700.00
-600.00
Cycle I Cycle II Cycle III
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.00 0.00
240.00
480.00
720.00
960.00
1200.00
t [s]
Fig. 4.34: Arch 1, loading-unloading cycles.
78
1440.00
1680.00
The kinematical mechanism was confirmed by the data acquired by potentiometers. In Fig. 4.35 and in Fig. 4.36 the load-displacement curves relative to the arch key are reported.
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F[N]
-400.00
-300.00
-200.00
-100.00 2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0.00
w [mm]
Fig. 4.35: Arch 1, load-displacement curve for the horizontal key displacements.
79
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0.00
v [mm]
Fig. 4.36: Arch 1, load-displacement curve for the vertical key displacements.
A pushing action, characterized by a counter-clockwise spin and a vertical displacement towards the bottom, was exercised by the block 1 on the arch. This displacement was read by the potentiometer connected with the point in which the load was applied. In Fig. 4.37 the load-displacement curves relative to the vertical displacements of the point in which the load was applied are reported.
80
-700.00
Cycle I Cycle II Cycle III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.00 0.00
-0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
v [mm]
Fig. 4.37: Arch 1, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
The cycle I presents a peak load greater than the others cycles, as illustrated in Fig. 4.37. This reduction of the peak load could be a consequence of the decrease of the masonry tensile strength from the cycle I to the last one. In order to validate the potentiometers acquisitions, the measurements read by comparators were put in comparison with the measurements registered by the data acquisition system Leane, obviously in the same loading condition. The acquired data are reported in Tab. 4.13 and Tab. 4.14.
F [N] 0.00 196.12 294.18 392.24 490.30 588.36
p_c [mm] 74.73 74.70 74.72 74.77 74.81 74.89
c_c [mm] 25.53 25.52 25.54 25.62 25.67 25.82
v_p_c [mm] 0.00 -0.03 -0.01 0.04 0.08 0.16
v_c_c [mm] 0.00 -0.01 0.01 0.09 0.14 0.29
D_c [mm] 0.00 0.02 0.02 0.05 0.06 0.13
p_f [mm] 53.04 52.89 52.74 52.59 52.51 52.30
c_f [mm] 19.35 19.22 19.05 18.88 18.78 18.55
v_p_f [mm] 0.00 -0.15 -0.30 -0.45 -0.53 -0.74
v_c_f [mm] 0.00 -0.13 -0.30 -0.47 -0.57 -0.80
D_f [mm] 0.00 0.02 0.00 0.02 0.04 0.06
Error_p_c Error_p_f 0.00 0.67 2.00 1.25 0.75 0.81
Tab. 4.13: Arch 1, Cycle I, potentiometers and comparators data.
81
0.00 0.13 0.00 0.04 0.08 0.08
F [N]
p_c [mm]
c_c [mm]
v_p_c [mm]
v_c_c [mm]
D_c [mm]
p_f [mm]
c_f [mm]
v_p_f [mm]
v_c_f [mm]
D_f [mm]
0.00 137.28 196.12 294.18 392.24 490.30
25.81 25.81 25.82 25.86 25.96 26.14
74.97 74.97 74.95 75.00 75.05 75.50
0.00 0.00 0.01 0.05 0.15 0.33
0.00 0.00 -0.02 0.03 0.08 0.53
0.00 0.00 0.03 0.02 0.07 0.20
18.86 18.76 18.60 18.44 18.07 17.13
52.64 52.57 52.41 52.28 51.95 51.13
0.00 -0.10 -0.26 -0.42 -0.79 -1.73
0.00 -0.07 -0.23 -0.36 -0.69 -1.51
0.00 0.03 0.03 0.06 0.10 0.22
Error_p_c Error_p_f
0.00 0.00 3.00 0.40 0.47 0.61
0.00 0.30 0.12 0.14 0.13 0.13
Tab. 4.14: Arch 1, cycle II, potentiometers and comparators data.
The errors in the displacements evaluation decreases with the increase of the displacement values. The potentiometer under the arch key p_c makes an error greater than the error made by potentiometer p_f because the achieved measurement and the error of the instrument have the same order of magnitude.
82
4.12. Experimental campaign: Arch 2 Three loading-unloading cycles, called cycle I, II and III respectively, were carried out applying the external load by the hydraulic jack connected with the brick number 14. The displacements were acquired by potentiometers and comparators; indeed the applied load intensity was acquired by the small cell. Summarizing, the test organization was the following: cycle I: external load applied by the hydraulic jack; cycle II: external load applied by the hydraulic jack; cycle III: external load applied by the hydraulic jack.
Cycle I.
During this cycle the following hinges opening occurred: hinge at the extrados between bricks 13-14, interface 14, in correspondence
of a load value equal to F ≅ 400 N ; hinge at the intrados between bricks 7-8, interface 7, in correspondence of a
load value equal to F ≅ 500 N
;
hinge at the extrados between bricks 1-2, interface 1, in correspondence of
a load value equal to F ≅ 500 N ; hinge at the intrados between bricks 19-20, interface 20, in correspondence
of a load value equal to F ≅ 550 N .
83
Fig. 4.38: Arch 2, cycle I, experimental test.
Fig. 4.39: Arch 2, cycle I, hinges formation.
Cycle II and cycle III.
During the second and the third cycle the opening of the hinges occurred in correspondence of an applied load lower than in the first cycle one. The peak load decreased significantly from the value equal to FPL ≅ 550 N obtained in the first cycle to a value equal to 450N obtained in the third cycle.
4.12.1. Collapse mechanism description The hinges formation determined the arch subdivision in blocks. The collapse mechanism occurred for relative displacements between the blocks. During the test the same collapse mechanism, characterizing the Arch 1, occurred: the raising of the second and third block, contrasting the pushing action towards the bottom.
84
4.12.2. Load-displacements curves The load related to cycles I, II and III were acquired in continuous by the data acquisition system Leane and the acquired data are reported in Fig. 4.40.
-700
Cycle I CycleII Cycle III
-600
-500
F [N]
-400
-300
-200
-100 0
240
480
720
960
1200
1440
1680
1920
2160
2400
2640
2880
3120
0
t [s]
Fig. 4.40: Arch 2, loading-unloading cycles.
In Fig. 4.41 and in Fig. 4.42 the load-displacement curves relative to the arch key are reported.
85
-600
Cycle I Cycle II Cycle III
-500
-400
F [N]
-300
-200
-100 2.00
1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00 0
100
w [mm]
Fig. 4.41: Arch 2, load-displacement curve for the horizontal key displacements. -600
Cycle I Cycle II Cycle III
-500
-400
F [N]
-300
-200
-100 1.00
0.80
0.60
0.40
0.20
0.00 0
100
v [mm]
Fig. 4.42: Arch 2, load-displacement curve for the vertical key displacements.
In Fig. 4.43 the load-displacement curves relative to the vertical displacements in the point in which the load was applied are reported.
86
-630
Cycle I CycleII Cycle III
-530
-430
F [N]
-330
-230
-130
0.00-30 -0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
-7.00
-7.50
70
v [mm]
Fig. 4.43: Arch 2, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
Because of the decrease of the masonry tensile strength, the peak load of the cycle I is greater than the one obtained in the others cycles, as illustrated in Fig. 4.43.
87
4.13. Experimental campaign: Reinforced arch The experimental campaign on reinforced arches was performed only on the arch 2 because of the ominous collapse of the arch 1. The instrumentations used for the experimental test on the reinforced arch was the same used for the unreinforced arches.
4.13.1. Application of the FRP reinforcement The FRP application was executed according to the codes (ACI 1999, fib TG 9.3 2001, JSCE 2001, etc.). The main phases of the reinforcement application were the following: Surface preparation: the surface where the reinforcement was applied was
suitable prepared. The arch surface was cleaned in order to remove every imperfection present on it and every contamination of chemical nature. Subsequently, it was necessary to fill the surface of the arch in order to render it flat; when the FRP was applied on the arch surface, it was completely clean, free from fats and oils and dry. At the end, the guides lines to install the FRP were traced. In order to avoid that the epoxy resin strewed over all the extrados surface, an adhesive tape was applied; it delimited the field of reinforcement application (Fig. 4.44).
88
Fig. 4.44: Surface preparation.
Epoxy resin preparation: the epoxy resin was constituted by two
component; the first one, called A, was white and the other, called B, was grey. Initially the two components were agitated separately, then they were mixed according with the technical card. The mixture was good, and therefore usable, when the colorful strips of the mixture were not more visible. In Fig. 4.45 this process is illustrated.
Fig. 4.45: Epoxy resin preparation.
Epoxy resin application: the epoxy resin was applied on the arch surface
using a roller, as illustrated in Fig. 4.46. 89
Fig. 4.46: Epoxy resin application.
FRP application: the woven was measured and pre-cut before its application
on the arch surface. It was placed on the surface and gently pressed into the epoxy resin, as illustrated in Fig. 4.47.
Fig. 4.47: FRP application.
Applying epoxy resin to FRP surface: a second coat of epoxy resin was
applied on the woven surface. Consolidation process control: after 48 hours the applied reinforcement was
examined to verify the presence of empties.
90
4.13.2. Test organization Three loading-unloading cycles, called cycle I, II and III respectively, were performed applying the external load by the hydraulic jack in correspondence with the brick number 14. The displacements were acquired by potentiometers and comparators; the applied load intensity was acquired by the small cell during the first cycle and by the great load cell in the other cases. Summarizing, the test organization was the following: cycle I: maximum external load applied F ≅ 5000 N ; cycle II: maximum external load applied F ≅ 25000 N ; cycle III: external load applied until the reinforced arch collapse.
Cycle I.
During this cycle the following phenomena occurred: FRP delamination in correspondence of the mortar joint between bricks 13-
14 for a load value equal to F ≅ 3000 N , as illustrated in Fig. 4.48; cracks under the brick 14, in correspondence of a load value equal to F ≅ 3500 N .
Fig. 4.48: Reinforced arch, cycle I, local delamination phenomenon at joint mortar.
91
Cycle II.
During this cycle the following aspects can be pointed out: increase of the crack opening in correspondence of the brick 14, as
illustrated in Fig. 4.49; formation of a mortar tooth between bricks 13-14 and bricks 14-15 in
correspondence of a load value equal to F ≅ 4000 N ; vertical sliding of brick 14 in correspondence of a load value equal to F ≅ 4500 N ;
breakaway of the bricks 14 for a load F ≅ 16000 N ; breakaway of the mortar tooth between bricks 14-15 in correspondence of a
load value equal to F ≅ 20000 N ; great increase of crack opening at the brick 14 in correspondence of a load
value equal to F ≅ 22000 N .
Fig. 4.49: Reinforced arch, cycle II, cracks on brick 14.
Cycle III.
During the third cycle the following aspects can be pointed out: great increase of crack opening at the brick 14 in correspondence of a load
value equal to F ≅ 39000 N ; cracks on the surface of the bricks 6 and 7 in correspondence of a load value
equal to F ≅ 42000 N , as represented in Fig. 4.50; 92
cracks on the surfaces of the bricks 8, 9, 10, 11 and 12 in correspondence of
a load equal to F ≅ 50000 N , as illustrated in Fig. 4.51; collapse of the reinforced arch in correspondence with a load greater than 50000N , as represented in Fig. 4.52 and Fig. 4.53.
Fig. 4.50: Reinforced arch, cycle III, cracks formation on bricks surface.
Fig. 4.51: Reinforced arch, cycle III, intrados and extrados arch surface view before the collapse.
93
Fig. 4.52: Reinforced arch, cycle III, arch collapse.
Fig. 4.53: Reinforced arch, after the collapse.
94
4.13.3. Collapse mechanism description The reinforcement prevents the classic masonry arch collapse mechanism; the FRP presence on the extrados surface, in fact, does not allow the hinges formation at the intrados because it prevents the cracks opening at the extrados. During the tests, the application of a concentrated load determined the presence of visible cracks on the surface of the brick 14 in correspondence of a load value little than the load for which the cracks on the other bricks occurred. The left part of the arch, from brick 1 to 14, was more significantly damaged than the right part of the arch, from brick 15 to 23. The collapse was preceded by the cracks opening on lateral and intrados surfaces of all the bricks from 2 to 14. When the collapse occurred, the FRP delamination under the bricks 13 and 14, the vertical sliding of the brick 14, the crush of bricks from 8 to 13 and the partial crush of bricks from 2 to 7 took place.
4.13.4. Load-displacement curves The data were acquired in continuous during the three loading-unloading cycles by the data acquisition system Leane. The acquired data are reported in Fig. 4.54.
95
-60000
Cycle I Cycle II Cycle III
-55000 -50000 -45000 -40000
F [N]
-35000 -30000 -25000 -20000 -15000 -10000 -5000
0
480
960
1440
1920
2400
2880
3360
3840
4320
4800
5280
5760
0
t [s]
Fig. 4.54: Reinforced arch, loading-unloading cycles.
The presence of the reinforcement induced an horizontal displacement in correspondence of the arch key, that became more significant in proximity of the collapse load. The load-displacement curves for the horizontal and the vertical key displacements are reported in Fig. 4.55 and in Fig. 4.56, respectively; while in Fig. 4.57 the load-displacement curve for the vertical displacement connected with the point of load application is reported.
96
-55000
Cycle I Cycle II Cycle III
-50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000 -15000 -10000
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-5000 -0.5 0
-1.0
-1.5
-2.0
5000
w [mm]
Fig. 4.55: Reinforced arch, load-displacement curve for the horizontal key displacement.
-55000 Cycle I Cycle II Cycle III
-50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000 -15000 -10000
2
1
0
-5000 -1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
5000
v [mm]
Fig. 4.56: Reinforced arch, load-displacement curve for the vertical key displacement.
97
-55000 -50000 -45000 -40000 -35000
F [N]
-30000 -25000 -20000
Cycle I Cycle II Cycle III
-15000 -10000
0
-5000 -1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
5000
v [mm]
Fig. 4.57:Reinforced arch, load-displacement curve for the vertical displacements in correspondence with the point in which the load has been applied.
During the test, the first loading-unloading cycle was characterized by arrangement phase in which a push due to the bricks at extremities determined an horizontal sliding of the arch; unfortunately it was not possible to measure it because there was not the suitable instrumentation. The presence of reinforcement has increased a lot the load-bearing of the arch, in particular the arch strength is 100 times higher.
98
5. MODELING AND NUMERICAL PROCEDURES
5.1. Introduction In this chapter the modeling and the numerical procedures developed to study the behavior of unreinforced and reinforced masonry arches are presented. In particular, two approaches able to solve nonlinear problems are illustrated: the first one is based on the stress formulation and the second one is based on the displacement formulation. In the stress formulation, the structural problem is faced and solved developing a complementary energy approach. A numerical procedure, based on a new formulation of the arc-length method, proposed by Riks in the framework of the displacement approach [62], is developed. In the displacement approach, a three nodes finite element based on the Timoshenko’s theory is implemented into FEAP code [63]. The nonlinear problem is solved by the application of the finite element method. Moreover, a new post-computation technique of the stresses at the masonry-FRP interface is proposed, which takes into account the heterogeneity of the masonry material, responsible of local stress concentration. The proposed post-computation of the FRP-masonry interface stresses is based on a simplified approach of the multiscale method. In fact, once the stress analysis is performed on the homogenized model of the arch, a micromechanical study is developed, considering the different materials which constitute the masonry, i.e. the block, the mortar and the FRP reinforcement. Numerical applications are developed to assess the effectiveness of the proposed models. Numerical results are put in comparison with the experimental results available in the literature and with new experimental evidences obtained at the 99
LAPS. Moreover the unreinforced masonry arches numerical results are put in comparison with those obtained by the application of the limit analysis.
5.2. Masonry constitutive models The masonry material is modeled as a no-tension material with a limited compressive strength. In particular, a nonlinear elastic constitutive relationship is considered for the masonry material as the one proposed by Lucchesi et al. [36]. This approach is simple and it can be considered effective for monotonic loading condition, when local unloading does not occur in any point of the structure. When the loading cannot be considered monotonic, the use of an elasto-plastic no-tension model is required, as the one presented in Reference [18], where a model and a numerical procedure were proposed and several numerical applications for reinforced masonry panels and arches were developed. The structural behavior of the reinforced arch is determined developing a variational formulation based on the complementary energy, i.e. adopting a stress approach. In fact, for the analysis of elements made of no-tension material, the solution of the structural problem, if it exists, is unique in terms of stress, while it could be not unique in terms of displacements, as proved by Lucchesi et al. [36].
5.2.1. Model 1 The masonry material is modeled as a masonry like material, assuming a behavior characterized by no-tension response with limited strength in compression, Fig. 5.1. The lack of tensile response gives an admissibility condition for stresses:
σM ≤ 0 100
(5.1)
The condition σ M = 0 can be defined as a limit or collapse condition and in this case the strains have an indefinite non negative value. The compression strength is denoted as σy. The adopted model is simple, but very effective for a wide class of problems, as emphasized in literature.
σM ε
εy O
σy E
D
Fig. 5.1: Masonry constitutive model, model 1.
In Fig. 5.1 it can be individuated the following features: feature OD: linear elastic; feature DE: ideal plastic.
In terms of deformations, there are three distinctive features: feature OD: elastic reversible strain; feature DE: inelastic irreversible strain; point E: yield strain.
It is assumed the existence of the following form for the complementary energy density governing the stress-strain relationship:
ψ M (σ M , τ M ) =
1 1 2 σ M 2 + I Σ (σ M ) + τ (1 − h (σ M ) ) 2 EM 2GM 101
(5.2)
where EM and GM are the elastic and the shear modulus of masonry respectively, I Σ is the indicator function of the admissible set Σ = ⎡⎣σ y , 0 ⎤⎦ , assuming the following
values: I Σ = 0 if σ M ∈ Σ
(5.3)
I Σ = +∞ otherwise
and h (σ M ) is the Heaviside’s function, assuming the following values: h (σ M ) = 1 if σ M ≥ 0 h (σ M ) = 0 otherwise
ψ
σM
Fig. 5.2: Masonry complementary energy density.
102
(5.4)
As consequence of the existence of the potential (5.2), that is schematized in Fig. 5.2, the normal and tangential stresses, σ M and τ M , do not depend on the specific strain history and they can be determined by the formulas:
σ M = 0⎫ ⎬ τM = 0 ⎭
if ε > 0
σ M = EM ε ⎫ ⎬ τ M = GM γ ⎭
if ε y < ε ≤ 0
σ M = EM ε y ⎫⎪
(5.5)
if ε ≤ ε y
⎬
τ M = GM γ ⎪⎭
where ε y = σ y / EM is the limit strain in compression and σ y is the compressive strength. A simple shear stress-strain behavior is assumed; in fact, it is nonlinear elastic in the part of the cross-section in compression. In this way the possible collapse due to shear sliding cannot be reproduced.
5.2.2. Model 2 In order to improve the model previously presented, it is considered a constitutive law characterized, for small values of the strain, by a quadratic relationship between the stress and the strain, as the one proposed for the concrete by the Eurocode 2. Thus, for ε ≤ ε y , the stress-strain relationship is:
σ M = αε 2 + βε + μ The following conditions are imposed on the relationship (5.6): 103
(5.6)
ε ε ε ε
=0 =0 = εy = εy
, , , ,
σM σM σM σM
=0 '= E =σy '=0
(5.7)
Solving equations (5.7) with respect to α , β , μ and ε y , it results:
α =−
σy ε y2
β=
2σ y
εy
μ = 0 εy =
2σ y E
(5.8)
Substituting solution (5.8) into the equation (5.6):
σM = −
σy 2 ε + Eε ε y2
for ε y ≤ ε ≤ 0
(5.9)
Finally, the normal and tangential masonry stresses, σ M and τ M , can be determined by the formulas:
σ M = 0⎫ ⎬ τM = 0 ⎭
if ε > 0
σ M = αε 2 + βε ) ⎫⎪ ⎬ τ M = GM γ ⎪⎭
if ε y < ε ≤ 0
σ M = σ y ⎪⎫ ⎬ τ M = GM γ ⎪⎭
if ε ≤ ε y
(5.10)
in which α and β assume the values reported in expressions (5.8). Also in this case the possible collapse due to shear sliding cannot be captured. The constitutive relationship is schematically illustrated in Fig. 5.3.
104
σM εu
εy
ε O
σy
Fig. 5.3: Masonry constitutive model, Model 2.
5.3. FRP constitutive model Design of FRP reinforcement should ensure that the FRP system is always in tension. In fact, compression FRP is unable to increase the performance of the strengthened masonry member due to its small area compared to that of compressed masonry. Moreover, FRP in compression may be subjected to debonding due to local instability. A uniaxial linear elastic response is assumed for the FRP reinforcement, as illustrated in Fig. 5.4.
105
brittle failure
σ
ε
Fig. 5.4: FRP constitutive model.
The stress-strain relationship is:
σ R = ER ε
(5.11)
where ER is the elastic modulus of the FRP reinforcement. The corresponding complementary energy is:
ψ R (σ R ) =
1 σ R2 2 ER
106
(5.12)
5.4. Limit analysis In this section a brief discussion both on the plastic collapse theorems and limit analysis is presented. Numerous studies have been made on the theory of plasticity since the Hill’s [64], [65] and Prager’s and Hodge’s [66] works. The aim of the limit analysis is to evaluate the load capacity and the collapse mechanism of structures. Considering the limit behavior of the material, through a definition of a yield function ϕ in terms of stresses, it is assumed that if ϕ < 0 the material remains in the elastic phase, if ϕ = 0 the material becomes plastic and if
ϕ > 0 the stress state is inadmissible. The set ϕ = 0 is called the yield surface and the conditions ϕ ≤ 0 represent the admissible stresses. According to the definition of
ϕ , all points that are inside or on the yield surface are admissible, while all points located outside the yield surface are inadmissible. When the stress state belong to the yield surface and the plastic behavior is activated, it is necessary to define the flow direction. According to the classical limit analysis theory [67], the yield surface is convex and the flow direction is normal to the yield surface. The normality condition assures that the energy dissipated by the flow is the maximum possible. The normality condition is very important because it introduces great simplifications in the limit analysis theory and it is the base of the limit analysis theorems. For a structure, it is possible to define a statically admissible state (safe state) for which the internal stresses are in equilibrium with the external forces and the yield conditions are fulfilled in all the points. Making a proportional loading analysis, it is necessary to define q , that is the base variable load, λ , the definite positive load factor and λ q , the variable load applied on the structure. The applied load can be increased from zero until a limit value for the structure, through the use of the load factor. This limit value is called the safety factor. In the limit analysis theory, the
107
static and kinematical theorems are proved; moreover, the uniqueness theorem can be also proved. The static theorem, also called lower-bound theorem, affirms that the safety factor is the largest of all statically admissible load factors. In other words, if it is possible to find a statically admissible stress field for a given load factor. Consequently the structure is in a safe condition under that load level. The kinematical theorem, also called upper-bound theorem, ensures that the safety factor is the smallest of all the kinematically admissible load factors. Finally, for the uniqueness theorem the largest factor defined by the static theorem is equal to the smallest factor defined by the kinematical theorem. The use of those fundamental theorems requires the adoption of specific hypotheses, particularly for masonry structures. Among these, the non tensile strength, the infinite compressive strength, the absence of sliding failure and the small displacements. In the following the kinematical theorem is applied to the case of unreinforced masonry structures considering the no-tensile strength of the masonry.
108
5.5. Arch model The arch model is based on the theory of curvilinear beam. Several shear deformation beam theories are available in literature. In the following the Timoshenko’s beam theory, [68] and [69], is considered; it is widely used in structural analysis, as it accounts for the transverse shear deformation in a simple and effective manner. The compatibility, the equilibrium and the constitutive equations governing the problem of the arch are available in literature; herein, only the final results are reported. Two coordinate systems are introduced: a global system (O, x, y, z) and a local system (x*, y*, z*), with x* and y* rotated of an angle θ with respect to y, as schematically illustrated in Fig. 5.5.
Fig. 5.5: Arch global and local systems.
A typical infinitesimal part of the arch is reported in Fig. 5.6. The quantities in the local coordinate system are computed as function of the ones represented in the global system using a rotation matrix R. The local radius of the arch is indicated as R, while s denotes the curvilinear abscissa.
109
z
O
T M q*
N p* y*
y
s
z* T+dT M+dM N+dN
dθ
Fig. 5.6: Infinitesimal arch element.
5.5.1. Governing equation of the arch The arch is assumed subjected to a permanent and to a variable loading p and λ q , respectively, with λ the load multiplier. Thus, the equilibrium equations written in the local coordinate system take the form: Δc + p* + λq* = 0
where ⎡ d ⎢ ds ⎢ Δ=⎢ 0 ⎢ ⎢ ⎢− 1 ⎢⎣ R
0 d ds 0
1⎤ R⎥ ⎧ ps * ⎫ ⎧ qs * ⎫ ⎧N ⎫ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1⎥ c = ⎨ M ⎬ p* = ⎨0 ⎬ q* = ⎨0 ⎬ ⎥ ⎪ ⎪ ⎪p *⎪ ⎪q * ⎪ ⎥ ⎩T ⎭ ⎩ r ⎭ ⎩ r ⎭ d⎥ ds ⎥⎦
110
(5.13)
with N, M and T the stress resultants and ps*, qs* and pr*, qr* the tangential and radial distributed components of the loads. Note that p* and λ q* are the permanent and variable loads vectors represented in the local coordinate system.
5.5.2. Kinematics of the arch The kinematics of the typical cross section of the arch is defined by the transversal and the axial displacements, v* and w0* respectively, and by the rotation ϕ. The compatibility equations are: % * =d Δη
(5.14)
where ⎡ d ⎢ ds ⎢ %Δ = ⎢ 0 ⎢ ⎢ ⎢− 1 ⎣⎢ R
1⎤ R⎥ ⎧ w0* ⎫ ⎧ε 0 ⎫ ⎥ d ⎪ ⎪ ⎪ ⎪ * ⎥ 0 η = ⎨ϕ ⎬ d = ⎨ χ ⎬ ⎥ ds ⎪v* ⎪ ⎪γ ⎪ ⎥ ⎩ ⎭ ⎩ ⎭ d⎥ 1 ds ⎥⎦ 0
in which γ representing the shear deformation, ε0 the axial strain and χ the bending curvature.
5.5.3. Cross section The cross sections of the arches typically present very regular geometry; thus, without loosing in generality, a rectangular cross section is considered in the 111
following. In accordance with the local coordinate system introduced above, the section lies in the x*y* plane, with x* and y* principal inertial axes. In the determination of the overall behavior of the reinforced masonry arch section, a perfect adhesion between masonry and FRP is assumed. The strain at a point of the section is:
ε = ε 0 + y* χ
(5.15)
The section AM of the masonry is split in three parts: the no-reagent part for which ε > 0 , denoted as Ant; the compressed part for which ε y < ε ≤ 0 , denoted as Ae; the compressed part subjected to a constant stress for which ε ≤ ε y , denoted
as Ap. In order to determine the parts Ant, Ae and Ap, the neutral and the plasticity axes
y* = yn* and y* = ym* , for which the strain attains the values zero and ε y , respectively, are determined:
0 = ε 0 + yn* χ
ε y = ε0 + y χ * m
ε0 χ
⇒
yn* = −
⇒
ε −ε y = y 0 χ
(5.16)
* m
Two possible cases can occur, schematically illustrated in Fig. 5.7, depending on the sign of the curvature.
112
AFRP
AFRP
Fig. 5.7: Axes position for positive and negative curvature.
The axes defining the compressed parts of cross section are determined as:
χ ≥0
Ap : y1 ≤ y ≤ y3 Ae : y3 < y ≤ y2
⎧ h ⎪ y1 = − 2 ⎪ ⎪⎪ ⎧h ⎧ h * ⎫⎫ ⎨ y2 = min ⎨ , max ⎨− , yn ⎬⎬ ⎩ 2 ⎭⎭ ⎩2 ⎪ ⎪ ⎧h h ⎫⎫ ⎪ y3 = min ⎨ , max ⎧⎨− , ym* ⎬⎬ ⎪⎩ ⎩ 2 ⎭⎭ ⎩2
(5.17)
χ 0 and, as consequence of equations (5.5), σ M = 0 τ M = 0 ; Ae
where
εy < ε ≤ 0
and,
as
consequence
of
equations
(5.5),
σ M = EM ε τ M = GM γ ; in the compressed part of the cross-section, characterized by a linear stress-strain relation, the normal stress is represented as σ M = σ 0 + y *σ 1 ; Ap where ε ≤ ε y and σ M = EM ε y τ M = GM γ .
The resultants in the reinforced masonry are: ⎧ ⎫ ⎪ ∫ σ y dA + ∫ (σ 0 + y*σ 1 ) dA + N R ⎪ ⎪A p ⎪ Ae ⎪ ⎪ N ⎧ ⎫ ⎧ cˆ ⎫ ⎪ ⎪ ⎪ * ⎪ c = ⎨ ⎬ = ⎨ M ⎬ = ⎨ ∫ y σ y dA + ∫ y* (σ 0 + y *σ 1 ) dA + M R ⎬ ⎩T ⎭ ⎪ T ⎪ ⎪ Ap Ae ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ τ dA ⎪ A ∪∫ A M ⎪ ⎩e p ⎭
(5.19)
Developed, the formulas (5.19) become: ⎧σ y Ap + Aeσ 0 + Seσ 1 + N R ⎫ ⎪ ⎪ c = ⎨σ y S p + Seσ 0 + I eσ 1 + M R ⎬ ⎪ ⎪ ⎩τ M AS ⎭
(5.20)
where AS = χT ( Ae + Ap ) is the shear area, χT is the shear correction factor, Se and Sp are the static moments of elastic and plastic part, respectively, Ie is the inertial moment of elastic part, NR and MR are the stress resultants due to the reinforcement. 114
Because of the perfect adhesion between the masonry arch and the reinforcement, the resultants of the stresses at the top ( y * = − h / 2 ) and the bottom ( y * = h / 2 ) reinforcements are expressed as: h⎞ ⎛ N R + = n0 ⎜ σ 0 − σ 1 ⎟ AR + 2⎠ ⎝ h⎞ ⎛ N R − = n0 ⎜ σ 0 + σ 1 ⎟ AR − 2⎠ ⎝
(5.21)
with n0 = E R / EM the homogenization coefficient; so that the resultants of the reinforcements are:
N R = n0 (σ 0 AR + σ 1S R ) M R = n0 (σ 0 S R + σ 1 I R )
(5.22)
where
AR = AR+ + AR−
S R = h / 2 ( AR+ − AR− ) I R = ( h / 2)
2
(A
+ R
(5.23)
+ AR− )
Formulas (5.23) represent the area, the static and the moment of inertia, respectively, of the FRP. It can be emphasized that the parts Ant, Ae and Ap are not known a priori, but they have to be determined as functions of the kinematical quantities ε0 and χ. Setting:
115
) ⎧ Ap ⎫ ⎧σ ⎫ σ = ⎨ 0 ⎬ , Q = σ y ⎨ ⎬ , J = J% + J , ⎩σ 1 ⎭ ⎩S p ⎭
(5.24) ⎡A J% = ⎢ e ⎣ Se
Se ⎤ ) ⎡A , J = n0 ⎢ R ⎥ Ie ⎦ ⎣ SR
SR ⎤ I R ⎥⎦
from expression (5.20), taking into account equations (5.22), the values of σ 0 and
σ 1 are obtained as: σ = H ( cˆ − Q )
(5.25)
where H = J −1 . It can be emphasized that, because of expression (5.25), the stress in the masonry as well as the stress in the reinforcement can be expressed as function of the stress resultants N and M. In particular, the stress in the elastic part of the masonry section and on the reinforcements is determined as:
σ M = H ( cˆ − Q ) • Y N R + = n0 H ( cˆ − Q ) • Y + AR +
(5.26)
N R − = n0 H ( cˆ − Q ) • Y − AR − where Y = {1 y * } , Y + = {1 h / 2} and Y − = {1 − h / 2} , while the symbol • T
T
T
indicates the scalar product.
116
5.6.1. Complementary energy In this section the elastic problem is faced and solved developing an energy approach. The complementary energy of the structure is defined as: Ψ (σ , τ ) =
∫ ψ (σ ,τ ) dV + ∫ ψ (σ ) dV M
R
VM
VR
⎛ σ2 τ2 ⎞ σ2 = ∫⎜ + dV ⎟ dV + ∫ 2 EM 2GM ⎠ 2 ER VM ⎝ VR
(5.27)
where VM and VR are the masonry and the reinforcement volume, respectively. In particular, the complementary energy in the masonry and in the reinforcement domains can be written in the form: ⎡ 1 ⎛ ⎞⎤ 2 2 ⎢ ⎜ + σ dA σ dA M θf ∫A y ⎟⎟⎥⎥ ⎢ 2 EM ⎜⎝ A∫e p ⎠ ⎥ R dθ ∫V ψ M (σ M ,τ M ) dV = θ∫ ⎢ 1 ⎥ M i ⎢ τ M 2 dA ∫ ⎢ + 2G ⎥ M Ae ∪ Ap ⎣ ⎦ ⎡ 1 ⎤ H ( cˆ − Q ) ⊗ H ( cˆ − Q ) • J% ⎥ θf ⎢ 2 EM ⎥ R dθ =∫⎢ 2 2 ⎢ ⎥ A σ T y p θi S + ⎢+ ⎥ 2GM AS ⎣⎢ 2 EM ⎦⎥
(5.28)
θf
1 ∫ ψ (σ ) dV = θ∫ 2E ∫ σ R
R
VR
i
θf
=
∫
θi
2 R
dA ds
R AR
) n0 H ( cˆ − Q ) ⊗ H ( cˆ − Q ) • J R dθ 2 ER
117
(5.29)
with θi and θ f the angles defining the initial and the final section of the arch. Finally, the complementary energy is obtained as:
⎡ 1 σ y 2 Ap ⎤ ˆ ˆ H c Q H c Q J − ⊗ − • + ( ) ( ) ⎥ θf ⎢ 2 EM 2 EM ⎥ ⎢ R dθ Ψ ( cˆ , T ) = ∫ ⎢ ⎥ T2 θi ⎢+ ⎥ ⎢⎣ 2GM AS ⎥⎦
(5.30)
The solution of the problem is determined minimizing the complementary energy (5.30) under the equilibrium constraint. In fact, the equilibrated stress resultants admit the following representation form:
h
c = c p + λc q + ∑ xi ci i =1
h ⎧ ⎫ + + N λ N x N ∑ p q i i ⎪ ⎪ h ⎧ ⎫ ⎪ i =1 ⎪ ⎪cˆ p + λ cˆ q + ∑ xi cˆ i ⎪ ⎪ h ⎪ ⎪ ⎪ i =1 =⎨ ⎬ = ⎨ M p + λ M q + ∑ xi M i ⎬ h i =1 ⎪T + λT + x T ⎪ ⎪ ⎪ ∑ q i i h ⎪⎩ p ⎪ ⎪ ⎪ i =1 ⎭ ⎪ T p + λTq + ∑ xiTi ⎪ i =1 ⎩ ⎭
(5.31)
with c p a field of stress resultants equilibrated with permanent loads p, cq a field of stress resultants equilibrated with variable loads q, ci with i=1,..,h fields of selfequilibrated stress resultants and xi with i=1,..,h statically unknown parameters. The number h of self-equilibrated stresses depends on the structural system and on the constraint conditions. Substituting the representation form (5.31) into the complementary energy (5.30), the stationary condition of complementary energy with respect to the unknown minimum 118
parameters x1, x2,.., xh is enforced in order to solve the elastic problem. The typical stationary equation takes the form: h ⎡ 1 ⎤ ⎛⎛ ⎞ ⎞ H ⎜ ⎜ cˆ p + λcˆ q + ∑ xi cˆ i ⎟ − Q ⎟ • cˆ j ⎥ ⎢ i =1 θ f ⎢ EM ⎠ ⎝⎝ ⎠ ⎥ ∂Ψ ⎢ ⎥ R dθ h = 0= ∂x j θ∫i ⎢ T j ⎜⎛ Tp + λTq + ∑ xiTi ⎟⎞ ⎥ ⎢ ⎥ i =1 ⎝ ⎠ ⎢+ ⎥ GM AS ⎣ ⎦
(5.32)
which correspond to a kinematical compatibility equation. In fact, denoting as: θf
T jTp ⎞ ⎛ 1 sp = ∫ ⎜ H cˆ p • cˆ j + ⎟ R dθ GM AS ⎠ θ i ⎝ EM θf
T jTq ⎛ 1 sq = ∫ ⎜ Hcˆ q • cˆ j + GM AS θ i ⎝ EM
⎞ ⎟ R dθ ⎠
θf
T jTi ⎞ ⎛ 1 si = ∫ ⎜ Hcˆ i • cˆ j + ⎟ R dθ GM AS ⎠ θ i ⎝ EM θf
sQ =
1
∫ θ E i
(5.33)
Q • cˆ j R dθ
M
the compatibility equation (5.32) takes the form:
h
s p + λ s q + ∑ xi si − sQ = 0
(5.34)
i =1
where s p , s q , si and sQ are vectors of h components which assume the physical
meaning of the displacement associated to the permanent and variable loadings p and
119
q, to the self-equilibrated stress resultants ci and to the additive stresses due to the
inelastic behavior of the masonry material. It can be emphasized that, because of the considered nonlinear constitutive laws, the vectors s p , s q , si and sQ depend on the solution, in fact they depend on the partition of the section AM of the masonry into the no-reagent part Ant, the elastic part Ae and the plastic part Ap, i.e. s p = s p ( Ae , Ap ) ,
s q = s q ( Ae , Ap ) ,
si = si ( Ae , Ap ) and
sQ = sQ ( Ae , Ap ) .
The integration of equations (5.32) and (5.33) can be performed considering the arch composed of a number of nT parts in each of which the positions of the axes defined by y1, y2 and y3 are taken constant.
5.6.2. Arc-length technique The considered constitutive law for the masonry material, characterized by limited strength in compression with no-tensile response, leads to solve a nonlinear problem governed by equation (5.32). It could be remarked that, because of the elastic character of the constitutive equations for both the masonry and the reinforcement, the solution of the structural problem does not depend on the loading history, so that the stress state can be computed once the loading is assigned. Although the constitutive equations are not written in evolutive form, a numerical procedure able to solve the nonlinear problem (5.32) is developed considering the loading applied in several steps. In such a way, each loading step results easier to solve and, moreover, it is possible to define the behavior of the structure along the whole monotone loading path. In this context, the compatibility equation (5.34) can be written in the form:
120
s = s p ( Ae , Ap ) + ( λn + Δλ ) s q ( Ae , Ap ) + + ∑ ( xi ,n + Δxi ) si ( Ae , Ap ) − sQ ( Ae , Ap ) = 0 h
(5.35)
i =1
Furthermore, it is possible to define a limit load for the structure, i.e. a load multiplier λ which induces the collapse of the arch. In order to evaluate the whole nonlinear structural response of the arch and to compute the limit load, the arc-length technique is considered. The arc-length procedure is often developed in the framework of displacement or mixed formulation of the structural problem. In the following, a new version of the arc-length technique, based on the stress formulation, is proposed. In particular, the developed arc-length procedure is based on the kinematical compatibility equation (5.35) and on a constraint equation. The nonlinear equation (5.35) is solved developing an iterative procedure within each load step. Thus, denoting by the superscript k the solution at the k-th iteration, the new solution is determined developing equation (5.35) in Taylor series:
h
s = sk + ∑ i =1
∂s ∂xi
δ xi + s =s
k
∂s ∂λ
= s k + K k δ x + S k δλ = 0
where
121
δλ s =sk
(5.36)
(
) (
) (
s k = s p Ae k , Ap k + λn + Δλ k s q Ae k , Ap k h
(
) (
)
(
)
+ ∑ xi ,n + Δxi k si Ae k , Ap k − sQ Ae k , Ap k i =1
⎡ ∂s1 ⎢ ⎢ ∂x1 s =sk ⎢ . ⎢ Kk = ⎢ . ⎢ . ⎢ ⎢ ∂s ⎢ h ⎢⎣ ∂x1 s =sk
⎤ ⎥ s =s k ⎥ ⎥ . ⎥ .... . ⎥ ⎥ ⎥ ⎥ ∂sh .... ⎥ ∂xh s =sk ⎥⎦ ...
∂s1 ∂xh
)
⎧ ∂s1 ⎫ ⎪ ∂λ k ⎪ s =s ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ k S =⎨ . ⎬ ⎪ . ⎪ ⎪ ⎪ ⎪ ∂sh ⎪ ⎪ ⎪ ⎩ ∂λ s =sk ⎭
(5.37)
The new solution is determined solving equation (5.36):
δ x = − ( K k ) s k − ( K k ) S k δλ k = δ x k + δ x t k δλ −1
−1
(5.38)
so that the variation of statically unknown parameters results: Δx = Δx k + δ x = Δx k + δ x k + δ xt k δλ k
(5.39)
The updating of the geometrical quantities Ae and Ap is performed computing the neutral and the plasticity axes, solving equation (5.16), and then (5.17) or (5.18). Note that the kinematical parameters present in formulas (5.16) are evaluated solving the resultant constitutive equations in a typical section of the arch: ⎛ ⎡ Ae k ⎧Nk ⎫ = E ⎜⎢ k ⎨ k⎬ M ⎜ ⎩M ⎭ ⎝ ⎣ Se
Se k ⎤ ⎡A + n0 ⎢ R k ⎥ Ie ⎦ ⎣ SR 122
S R ⎤ ⎞ ⎧ε 0 k ⎫ ⎟⎨ ⎬ I R ⎥⎦ ⎟⎠ ⎩ χ k ⎭
(5.40)
The constraint equation required in the arc-length method is defined choosing a suitable control parameter. In particular, it is assumed as control parameter the maximum variation of bending curvature evaluated on all the cross sections of the arch. Thus, it is set:
χ% = max Δχ (θ )
(5.41)
θi ≤θ ≤θ f
and the constraint equation is assumed of the form:
χ% 2 − Δl 2 = 0
(5.42)
where Δl is a given length of the radius considered to follow the mechanical response path of the structure. For a fixed value of the angle θ , i.e. in the typical cross-section, solving equation (5.40) with definition (5.41), it is set: 2
( χ% )
2
h ⎡ k⎛ ⎞ ⎤ Δ Δxi N i ⎟ ⎥ H λ N + q ∑ ⎢ 21 ⎜ i =1 ⎝ ⎠ ⎥ =⎢ / EM2 h ⎢ ⎥ ⎛ ⎞ ⎢ + H 22 ⎜ Δλ M q +∑ Δxi M i ⎟ ⎥ i =1 ⎝ ⎠ ⎦⎥ ⎣⎢
(5.43)
On the other hand, as it is Δλ = Δλ k + δλ , equation (5.43) becomes:
( χ% )
2
(
= H 21k
) (δλ n 2
λ
(
+ n ) + H 22 k 2
) (δλ m 2
+2 H 21k H 22 k (δλ nλ + n )(δλ mλ + m ) where:
123
λ
+ m)
2
(5.44)
h
(
)
(
)
n = ∑ Δxi k + δ xi k N i + Δλ k N q i =1 h
m = ∑ Δxi k + δ xi k M i + Δλ k M q i =1
(5.45)
h
nλ = N q +∑ δ x N i k t ,i
i =1
h
mλ = M q +∑ δ xtk,i M i i =1
Equation (5.42), taking into account formula (5.44), can be written in the form:
δλ 2 a1 + δλ a2 + a3 = 0
(5.46)
where:
(
a1 = nλ 2 H 21k
)
(
2
a2 = 2nλ n H 21k +2n mλ H
(
a3 = H 21k
)
2
k 21
(
+ mλ 2 H 22 k
)
2
(
)
2
+ 2nλ mλ H 21k H 22 k
+ 2mλ m H 22 k
H 22
(
)
2
+2nλ m H 21k H 22 k
(5.47)
k
n 2 + H 22 k
)
2
m 2 + 2n m H 21k H 22 k − Δl 2
Equation (5.46) is solved with respect to δλ , leading to the determination of two roots, δλ1 and δλ2 . The solution is chosen according to the cylindrical method, such that δλ = δλ2 if c2 > c1 and it is δλ = δλ1 otherwise, with
124
h
(
a4 = ∑ Δxi k δ xi k + Δxi k i =1
)
h
a5 = ∑ Δxi k δ xi k
(5.48)
i =1
c1 = a4 + a5δλ1 c2 = a4 + a5δλ2 The load multiplier and the statically unknown parameters increments are updated setting: Δλ = Δλ k + δλ
(5.49)
Δx = Δx k + δ x = Δx k + δ x k + δλ δ xt k
The iteration process within each loading step is performed until the residual r = s is greater than a fixed tolerance.
5.7. Displacement formulation The displacement formulation approach has been developed considering both Model 1 and Model 2. In this section, only the Model 2 is described. The section AM of the masonry is specialized into: Ant where ε > 0 and, as consequence of equations (5.10), σ M = 0 τ M = 0 ; Ae
where
εy < ε ≤ 0
and,
as
consequence
σ M = αε 2 + βε τ M = GM γ ; Ap where ε ≤ ε y and σ M = σ y τ M = GM γ .
The resultants in the reinforced masonry are: 125
of
equations
(5.10),
⎧ ⎫ ⎪ ∫ σ y dA + ∫ (αε 2 + βε ) dA + N R ⎪ ⎪A p ⎪ Ae ⎪ ⎧N ⎫ ⎪ ⎧ cˆ ⎫ ⎪ ⎪ ⎪ * ⎪ * 2 c = ⎨ ⎬ = ⎨ M ⎬ = ⎨ ∫ y σ y dA + ∫ y (αε + βε ) dA + M R ⎬ ⎩T ⎭ ⎪ T ⎪ ⎪ Ap Ae ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ τ dA ⎪ A ∪∫ A M ⎪ ⎩e p ⎭
(5.50)
The axial resultant and bending moment are computed as:
N=
∫σ
dA +
y
Ap
∫ (αε
2
+ βε ) dA + N R
Ae
= σ y Ap +
∫ ⎡⎣⎢α (ε
+ y* χ ) + β (ε 0 + y* χ ) ⎤ dA + N R ⎦⎥ 2
0
Ae
(5.51)
= σ y Ap + A% ε 0 + S% χ + N R
M=
∫ yσ *
y
dA +
Ap
∫ y (αε *
2
+ βε ) dA + M R
Ae
= σ ySp +
∫y
Ae
*
⎡α ( ε + y * χ )2 + β ( ε + y* χ ) ⎤ dA + M 0 0 R ⎣⎢ ⎦⎥
(5.52)
= σ y S p + S%ε 0 + I% χ + M R
respectively, where A% =
∫ (αε
+α y* χ + β ) dA = (αε 0 + β ) A e +αχ Se
(5.53)
y* + 2α y*2 + β y* ) dA = (αε 0 + β ) Se + αχ I e
(5.54)
0
Ae
S% =
∫ (αε
0
Ae
126
I% =
∫ (α y
*3
)
χ + β y*2 + αε 0 y*2 ) dA = (αε 0 + β ) I e + αχ I
(5.55)
Ae
and AS = χT ( Ae + Ap ) is the shear area, χT is the shear correction factor, NR and MR are the stress resultants due to the reinforcement.
5.7.1. Kinematics The kinematics of the Timoshenko beam theory, schematically illustrated in Fig. 5.8, can be expressed as
u1 = 0 u2 = v
(5.56)
u3 = w + yϕ
Fig. 5.8: Timoshenko’s beam theory.
The strain field is given by
ε0 = w' χ =ϕ ' γ = v '+ ϕ 127
(5.57)
Note that the prime on a variable indicates its derivative with respect to z . The kinematics strain vector is introduced as ⎛d ⎜ ⎧ε 0 ⎫ ⎧ w ' ⎫ ⎜ dz ⎪ ⎪ ⎪ ⎪ d = ⎨χ ⎬ = ⎨ ϕ ' ⎬ = ⎜ 0 ⎜ ⎪ γ ⎪ ⎪v '+ ϕ ⎪ ⎜ ⎩ ⎭ ⎩ ⎭ ⎜ 0 ⎜ ⎝
0 0 d dz
⎞ 0 ⎟ ⎧ w⎫ ⎟ ⎧ w⎫ d ⎟⎪ ⎪ ⎪ ⎪ ⎨v ⎬ = L⎨v ⎬ ⎟ dz ⎪ ⎪ ⎪ϕ ⎪ ⎟ ⎩ϕ ⎭ ⎩ ⎭ 1 ⎟⎟ ⎠
(5.58)
5.7.2. Finite element implementation A discussion on displacement based or mixed formulation beam elements was presented by Crisfield [70], [71]. Reddy, [72]-[74], developed superconvergent (i.e. , yields exact nodal values) locking-free Timoshenko’s beam finite element based on the interdependent interpolation as well as assumed strain formulation. In the same paper, Reddy extended the procedure to develop a locking-free finite element for the third-order beam theory. A consistent beam finite element was proposed in reference [75]. The conventional finite element model of the Timoshenko’s beam is obtained by using Lagrange interpolation of v and ϕ . For example, linear interpolation of both v and ϕ is known to yield a finite element that exhibits locking. Reduced integration
of the shear stiffness alleviates this problem, but does not yield exact values of the displacements at the nodes without using a large number of elements. Here, we consider an alternative interpolation of the dependent variables that yields a lockingfree finite element. In particular, the transverse displacement v is approximated using the Hermite’s cubic interpolation, the rotation ϕ and the axial displacement w are approximated using Lagrange quadratic interpolation: 128
w = N1w w1 + N 2w w2 + N 3w w3 v = N1v v1 + N 2v v2 + N1θ θ1 + N 2θ θ 2
(5.59)
ϕ = N1ϕϕ1 + N 2ϕϕ2 + N 3ϕϕ3 where θ1 = −v '(0) and θ 2 = −v '( L) . Note that vi and θi are the transversal displacements and the slopes of the i − th node, respectively, with i = 1, 2 , and ϕi are the rotations of the cross-sections about the x axis, corresponding to the i − th node, with i = 1, 2, 3 , as schematically represented in Fig. 5.9.
ϕ3
x w1
v1
w2
z
v2
y
θ2 ϕ2
θ1 ϕ1 Fig. 5.9: Beam finite element.
In compact form, formulas (5.59) can be expressed as ⎧ w⎫ ⎪ ⎪ u = ⎨v ⎬ = NU ⎪ϕ ⎪ ⎩ ⎭
where
129
(5.60)
⎛ N1w ⎜ N=⎜ 0 ⎜ 0 ⎝
0 N1v 0
0 N1θ 0
0 0 N1ϕ
N 2w 0 0
0 N 2v 0
0 N 2θ 0
0 0 N 2ϕ
N 3w 0 0
0 0 0 ⎞ ⎟ 0 0 0 ⎟ 0 0 N 3ϕ ⎟⎠ (5.61)
U = {w1 v1 θ1 ϕ1
w2
v2 θ 2 ϕ 2
w3
0 0 ϕ3 }
T
with
v 1
N
N 2v θ
N1
N 2θ
1 N1w = ξ (ξ − 1) 2 1 N 2w = ξ (ξ + 1) 2 w N3 = 1 − ξ 2
(5.62)
(ξ + 2)(ξ − 1) 2 = 4 (ξ − 2)(ξ + 1) 2 =− 4 L(ξ + 1)(ξ − 1) 2 =− 8 L(ξ − 1)(ξ + 1) 2 =− 8
(5.63)
1 N1ϕ = ξ (ξ − 1) 2 1 N 2ϕ = ξ (ξ + 1) 2 ϕ N3 = 1 − ξ 2
(5.64)
L⎞ 2 ⎛ and ξ = ⎜ z − ⎟ , z ∈ [0, L] . 2⎠L ⎝ 130
The beam finite element was implemented in the code FEAP, developing an iterative numerical procedure able to solve the nonlinear arch problem. The proposed procedure is based on the secant stiffness method, and it allows to determine the solution of a nonlinear problem as solution of an opportune sequence of linear problems. A scheme of the numerical procedure is reported: 1) Initially the reagent section is all the geometrical section; 2) The nodal displacement are calculated; 3) At the generic cross section the deformations are noted; 4) Noted the deformations, the neutral and plasticity axes position can be valuated; 5) The new reagent section with its elastic and plastic part is defined; 6) Defined the reagent section the procedure comes back to step 2. The procedure iterates from step 2 to 6 until the residue, computed as the difference between the external forces and those determined by the state of deformation of the structure, is lower than a fixed tolerance.
5.8. Post-computation of the shear stresses The computation of the stresses at the FRP-masonry interface can be very important as they are responsible for the decohesion of the reinforcement from the masonry. It can be remarked that, because of the heterogeneity of the masonry material, the stresses at the interface can present local concentrations. Thus, the normal and shear stresses at the interface, σ d and τ d respectively, can be computed as the sum of two quantities: the first ones σ T and τ T are evaluated enforcing the equilibrium condition of the FRP for a typical infinitesimal element of the arch, the second ones σ h and τ h correspond to the local normal and shear stress concentration due to the masonry material heterogeneity. 131
With reference to Fig. 5.10, the normal and the shear stresses at the extrados and intrados masonry-FRP interfaces can be computed as:
σ T− =
2 N R− b ( 2R − h )
τ T− =
2 N R+ σ =− b ( 2R + h )
2 dN R− b ( 2 R − h ) dθ
dN R+ 2 τ =− b ( 2 R + h ) dθ
+ T
(5.65)
+ T
with evident meaning of the symbols.
N R+ bτ T+
R+
h 2
N R−
bτ T−
+ bσ T− bσ T
N R+ + dN R+
dθ
N R− + dN R−
O
R−
h 2
Fig. 5.10: Normal and shear stress at the masonry-FRP interface.
Indeed, because of the heterogeneity of the masonry, also when the stress resultants in a reinforcement is constant, stresses can occur at the interface. In order to evaluate the stresses profile due to the material heterogeneity, a micromechanical analysis of the reinforced masonry is developed. In particular, it is assumed that the masonry is a periodic heterogeneous material. Because of the symmetry of the repetitive cell with respect to the plane orthogonal to the local beam axes (see Fig. 5.11), the study can be limited to a half of the cell, 132
considering an elastic interface joining the FRP reinforcement to the masonry governed by the relationship: ⎧τ h ⎫ ⎡ Kτ ⎨ ⎬=⎢ ⎩σ h ⎭ ⎣ 0
0 ⎤ ⎧ sτ I ⎫ ⎨ ⎬ Kσ ⎦⎥ ⎩ sσ I ⎭
(5.66)
and Kσ are the tangential and normal stiffness respectively, while sτ I
where Kτ
and sσ I are the relative displacements in the tangential and normal direction.
Block
Mortar
Block
Mortar
ym
yn FRP
FRP Half cell
Half cell
Unit cell
Fig. 5.11: Stresses acting on the mortar joint of the masonry unit cell.
The unit cell is subjected to the normal stresses derived from the structural analysis; in particular, the stresses computed from the structural analysis are applied on the mortar joint, as illustrated in Fig. 5.11. A two-dimensional finite element analysis is performed of the unit cell in the framework of plane stress analysis, evaluating the normal and the shear stress profiles at the interface. As the post-computation of the stress profiles at the masonry FRP interface should be performed for several sections of the arch and for different values of the loading level, a simple numerical strategy is developed. Let nj be the number of nodes of the section where the stresses are applied, nj pre-analyses are developed determining the relative displacements occurring between the FRP and the masonry due to a unit 133
force acting on a single node. The results of the analyses are organized in an influence matrix G, whose i-th column represents the results of the micromechanical analysis due to a unit force applied at the i-th node. Once a loading step is selected and a section of the arch is considered, the structural analysis allows to compute the normal stress profile in the mortar joint, which is considered as an external distributed load for the unit cell. The distributed load is transformed in equivalent nj nodal forces following the classical finite element procedure, defining the vector F. The vector of the relative displacements s I due to the actual distribution of the normal stresses acting on the mortar joint is determined by the matrix product:
s I = GF
(5.67)
Thus, the normal and shear stresses can be computed substituting the values of the relative displacements obtained by expression (5.67) into equation (5.66). The obtained stresses must be added to the quantities determined by formulas (5.65).
134
5.9. Numerical results In this section, the obtained numerical results are presented. The numerical results deal with the stress formulation and the displacement formulation. The reliability of the nonlinear elastic constitutive law (Model 1), compared with Model 2 is tested. The effectiveness of the numerical procedure is also experienced. Moreover the results obtained by the application of the kinematical theorem of the limit analysis are illustrated. Finally a comparison between numerical and experimental results is proposed.
5.9.1. Models and numerical procedures assessment The aim of this analysis is the assessment of the stress and displacement formulations. A round clamped-clamped arch is considered, so that it results three times statically undetermined; the arch is subjected to a vertical downward distributed load of intensity p=10 N/mm and to a horizontal distributed load q=1 N/mm amplified by the multiplier λ. The geometry and the mechanical properties of the adopted materials are reported in the following: Geometry o masonry: round arch with radius R=5000 mm and rectangular crosssection with dimensions b=300 mm and h=1000 mm; o FRP: the thickness is assumed t=0.17 mm, which corresponds to one layer of composite, while the width is taken bFRP=200 mm. Materials o masonry: the Young’s modulus is set Em=15000 MPa, which corresponds to rock blocks, while the Poisson ration is n=0.2 and the shear modulus is Gm=6250 MPa; the limit strength in compression is set sy=7.5 MPa;
135
o FRP: the Young modulus of the carbon-fiber is EFRP=400000 MPa.
In order to assess the effectiveness of the proposed masonry model and to verify the robustness of the numerical procedure based on the complementary energy approach within the dual version of the arc-length technique, displacement finite element analyses are carried out considering the no-tension elasto-plastic masonry model implemented in the code FEAP. Thus, the numerical results obtained by the stress formulation of the nonlinear elastic model are put in comparison with the ones carried out by the displacement approach, based an elasto-plastic open-ended model. The computations are performed setting nT=300 for the stress approach, while a mesh of 60 elements is considered for the displacement finite element formulation. Initially, the response of the un-reinforced arch is studied. In Fig. 5.12, the value of the multiplier λ of the distributed horizontal load is plotted versus the horizontal displacement vk computed at the key of the arch. The results are reported adopting the following acronyms:
NT NTP EC FEM
no-tension material with unlimited compressive strength; no-tension material with limited compressive strength; complementary energy approach; elasto-plastic displacement finite element formulation.
136
3.50 3.00 2.50 2.00
λ 1.50 1.00 NT EC NT FEM NTP EC NTP FEM
0.50 0.00 0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
vk [mm]
Fig. 5.12: Un-reinforced arch modeled considering a no-tension material with unlimited and limited compressive strength.
It can be emphasized that all the computed solutions are in very good agreement. In particular, it can be remarked that there is not significant differences when unlimited or limited compressive strength is considered; in fact, because of the no-tensile capacity of the material, the collapse of the arch occurs for the formation and opening of hinges located at the extrados and at the intrados of the arch. As a consequence, the limited strength in compression does not play a significant role in the overall behavior of the arch. Moreover, the stress approach demonstrates to be effective and robust in the developed computations. Then, the case of arch reinforced at the extrados is studied. As in the previous analyses, several computations are performed and the obtained results are put in comparison. In Fig. 5.13, the plot of the multiplier λ of the horizontal load versus the horizontal displacement vk of the key section is reported.
137
18.00 16.00 14.00 12.00 10.00
λ 8.00 6.00
NT EC
4.00
NT FEM 2.00
NTP EC NTP FEM
0.00 0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
vk [mm]
Fig. 5.13: Reinforced arch modeled considering a no-tension material with unlimited and limited compressive strength.
Once again, it can be remarked the ability of the stress approach to reproduce the nonlinear response of the reinforced arch: the results obtained adopting the nonlinear elastic model does not significantly differs from the ones obtained considering the elasto-plastic model, so that it can be claimed that the loading history does not influence in this case the response of the arch. The differences of the results obtained considering unlimited or limited strength in compression can be remarked. In fact, when the arch is reinforced at the extrados, the hinges at the intrados cannot occur as the crack openings at the extrados are not allowed and the limited strength in compression of the masonry material plays a fundamental role in the response of the arch. Moreover, looking at Fig. 5.12 and Fig. 5.13, it can be noted the effectiveness of the reinforcement in the overall behavior of the arch. Then, computations are performed considering the arch reinforced by 1, 2, 5, 10 and 15 FRP layers, subjected to a vertical distributed load of intensity p = 100 N/mm and to the increasing distributed horizontal load, simulating the effect of an earthquake 138
on the structure. In Fig. 5.14 the curves concerning the horizontal load multiplier versus the horizontal displacement of the key section are reported for the unreinforced and for the reinforced arch.
80.00
15 FRP layers
70.00
10 FRP layers
60.00
5 FRP layers
50.00 2 FRP layers
λ 40.00
80
1 FRP layer 70
60
30.00 λ
50
20.00
30
Unreinforced
10.00
0.00 0.00
40
20
10
Compressive masonry failure Tensile FRP failure 50.00
100.00
150.00
0 0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
7000.00
8000.00
vk [m m ]
200.00
250.00
300.00
350.00
400.00
vk [mm]
Fig. 5.14: Collapse loading for the un-reinforced and reinforced arch.
In that figure, the diagram load multiplier vs horizontal displacement is reported considering two different scales for the displacements: in the main Figure the maximum displacement is set equal to 400 mm, while in the boxed image the maximum displacement is greater. Indeed, in both the considered scales, the reached displacements are much greater than the admissible ones obtained considering the limited compressive deformation of the masonry and the tensile strength of the FRP reinforcement. In fact, the boxed figure is reported only to evaluate the limit load multiplier in the case of un-reinforced and reinforced arch, λ≈13 and λ≈73 respectively, when no care is taken to the limited compressive deformation of the masonry and tensile strength of the FRP. In this case, the limit load for the un-reinforced and reinforced arch is due to the no-tensile response and to the plastic constitutive relationship of the 139
masonry, respectively. It can be remarked that the limit load for the reinforced arch does not depend on the number of FRP layers. Finally, it could be emphasized that the limit load is attained for very high values of the displacements, so that the classical assumption of small displacements and small deformations could be not valid anymore. As a consequence, from the developed numerical computations, it can be concluded that the classical limit analysis is not applicable to evaluate the maximum loading capability of reinforced arch. In the main scale of Fig. 5.14, the loading levels, with the corresponding values of the horizontal displacements, which induce the tensile failure of the FRP reinforcement and the reached limit compressive strain for the masonry, are reported with a square and a triangle, respectively. In particular, in the developed computations, the tensile failure of the FRP is set as σR,y=3500 MPa while the limit compressive strain for the masonry is taken εu=0.0035. When only 1 FRP layer is applied to reinforce the arch, the tensile failure of the reinforcement occurs before the compressive masonry collapse; increasing the number of FRP layers, a gradual inversion of the failure mechanism of the two materials can be noted. In correspondence of 15 FRP layers, the compressive collapse of masonry occurs before the FRP tensile failure. In particular, the following results are obtained: 1 FRP layer
λ≈23 masonry failure
λ≈15 FRP failure
2 FRP layer
λ≈29 masonry failure
λ≈21 FRP failure
5 FRP layer
λ≈42 masonry failure
λ≈35 FRP failure
10 FRP layer
λ≈49 masonry failure
λ≈48 FRP failure
15 FRP layer
λ≈62 masonry failure
λ≈71 FRP failure
It can be concluded that the reinforcement at the extrados of the arch is very effective from a structural viewpoint as its presence is able to enhance the performances of the arch with respect to horizontal loading.
140
5.9.2. Experimental surveys numerical results The numerical results are put in comparison with experimental results both available in literature and obtained by the experimental program realized at LAPS of Cassino.
5.9.2.1. Comparison 1 In this section the experimental evidences carried out by Briccoli Bati and Rovero [54] and numerical results obtained adopting the numerical procedures are illustrated. A reinforced masonry arch is studied and the geometry and mechanical properties of the adopted materials are: Geometry o masonry: arch with radius R=865 mm, rectangular cross-section b=100 mm and h=100 mm and clamped at 30o and 150o; the arch is composed assembling hollow clay masonry units of thickness 25 mm, joined by a mortar layer of 4 mm. o FRP: reinforcement applied at the whole intrados of the arch, with thickness t=0.17 mm, which corresponds to one layer of composite, and width bFRP=50 mm. Materials o masonry: the Young’s modulus of the hollow clay masonry units and of the mortar are 1785 MPa and 133 MPa, respectively, so that the overall modulus of the masonry is set Em=680 MPa, while the Poisson ration is n=0.2 and the shear modulus is Gm=283 MPa; o FRP: the Young modulus of the carbon-fiber is EFRP=230000 MPa.
The structure is subjected to an increasing concentrate force F applied at the key of the arch. The experimental test shows that the arch failure occurs for crushing of the masonry in compression. As consequence, the compressive strength of the masonry plays a fundamental role in the overall behavior of the arch.
141
In Reference [54], the compressive strength of masonry, strongly governed by the mortar strength, is evaluated testing some specimens and it results in the range between 7 and 8 MPa. It can be pointed out that this value is reasonably higher than the strength of the masonry constituting the arch, because of the possible imperfections occurring during the construction of the arch, mainly in the key. In particular, the thickness of the mortar bed between two blocks is not the same in whole experimental arch tested in Reference [54], and in correspondence of the key of the arch there is a great reduction of bed mortar. As a consequence, it is found that the arch tested in the laboratory presents some initial geometrical defects in the heterogeneities of the masonry, which induces localization phenomena that significantly reduced the compressive strength of the masonry material with respect to the value obtained on “perfect” specimens. Indeed, some difficulties arise in the evaluation of the limit compressive strength and the limit failure strain, which are necessary to study the arch and to understand its behavior. It can be reasonably assumed that the compressive strength in the masonry arch is reduced from 1/4 to 1/3 with respect to the strength deduced by the specimen tests. Thus, the masonry strength could be set in the range between 1.75 and 2.67 MPa In order to numerically reproduce the behavior of the arch, a parametric analysis is carried out considering different values of the limit compression strength σy in the range from 2.04 to 6.80 MPa. In Fig. 5.15, the comparison between the load-displacement curves obtained by experimental investigation and by numerical analyses is reported.
142
7.00 6.50 6.00 5.50 5.00 4.50 4.00
F [kN] 3.50 3.00 2.50
σy = 2.04 Mpa σy = 2.72 Mpa σy = 3.40 Mpa σy = 6.80 Mpa Experimental results
2.00 1.50 1.00 0.50 0.00 0.00
1.00
2.00
3.00
4.00
5.00
6.00
vk [mm]
Fig. 5.15: Comparison 1, reinforced arch subjected to a concentrate force.
For the different values of limit compression strength σy, it is computed the minimum strain επ/2 and the tensile stress in the reinforcement σR evaluated at the key section, i.e. for θ=π/2, when the experimental collapse load of the arch F=6.5 kN is reached. In particular, it results: επ/2=-0.02538
and σR=1387 MPa
when σy=2.04 MPa,
επ/2=-0.00784
and σR=1053 MPa
when σy=2.72 MPa,
επ/2=-0.00626
and σR=979 MPa
when σy=3.40 MPa,
επ/2=-0.00602
and σR=965 MPa
when σy=6.80 MPa.
It can be emphasized that, in any case, the FRP stress is lower than its failure strength assumed to be 3500 MPa. Looking at Fig. 5.15, the numerical results are in good agreement with the experimental ones when the strength of the masonry is set equal to 2.04 MPa, i.e. belonging to the range announced above taking into account the material imperfection.
143
As consequence of the above arguments, it can be concluded that the comparison between the numerical and experimental results can be considered satisfactory; in fact, it shows the effectiveness of the simple proposed model, which can be able to predict the collapse load. Of course, as the model does not consider any softening effect in compression, the post-critical behavior of the arch characterized by a quite brittle response once the maximum load is reached, cannot be numerically reproduced. As discussed in section 5.8, it can be very interesting to evaluate the normal and, mainly, the shear stresses at the interface between the FRP and the masonry, in order to predict the possibility of decohesion of the reinforcement. In Fig. 5.16, the shear stress profile evaluated close to the key section is reported when the external force is F=1 kN. In particular, as the key section is subjected to a concentrate force and, as consequence, to a very special stress state, the shear stresses are computed for a unit cell at a distance of about the section height from the key section.
0.60
τd τT τh
shear stress [MPa]
0.50
0.40
0.30
0.20
0.10
0.00 0.00
2.00
4.00
6.00
8.00
10.00
z* [mm]
Fig. 5.16: Shear stresses for F=1 kN.
144
12.00
14.00
In this Figure the decohesion shear stresses τ d and the single contributes τ T and τ h , due to the equilibrium condition of the FRP for a typical infinitesimal element of the arch and to the local shear stress concentration due to the masonry material heterogeneity, respectively, are reported. The curve of the decohesion shear stresses presents a maximum values in correspondence of the brick-mortar section because of the different consistency of the brick and the mortar and it shows that the contribute to the value of the shear stresses of the variation of the normal force into the reinforcement is less significant than the effect of the material heterogeneity. In Fig. 5.17, the comparison between the shear stresses obtained for different values of the applied force is illustrated. In particular, the considered force intensities are
F=1 kN, F=3 kN and F=5 kN; it is evident that increasing the value of the applied force, it corresponds a nonlinear increment of the shear stress.
2.00 1.80
F = 1 kN F = 3 kN F = 5 kN
1.60 1.40
shear stress [MPa]
1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
z* [mm]
Fig. 5.17: Shear stresses for different force intensities.
Moreover, for the considered case study, it can be noted that in correspondence of
F = 5 kN , the maximum shear stress reaches the value τ d ≈ 1.8 MPa in the brick. This value can be considered safe for the brick which, according to the experimental 145
investigations [54], is characterized by a limit strength in tension and in compression of 1.7 MPa and 17.39 MPa, respectively. On the other hand, experimental evidences demonstrated that the collapse of the arch occurred with no delamination effects. The decohesion of the FRP from the masonry support could be accounted for considering a suitable interface model a numerical procedure able to predict and reproduce the nonlinear phenomenon [12] and [16].
5.9.2.2. Comparison 2 In this section the numerical results are put in comparison with the experimental results obtained during the experimental campaign realized at LAPS of University of Cassino. The studied arch is schematically reported in Fig. 5.18.
146
5
6
7
8
11 12 13 14 9 10 15
16
17
18
4 3 2
19 20 21 22
1
23 O
Z
Y
Re Ri c f i
O
Z
Y
Fig. 5.18: Reinforced arch geometrical model.
In accordance with the introduced global system, the geometrical characteristics of the arch are:
147
R ext = 576.07 mm Rint = 456.07 mm R ext + Rint 2 ϑi = 8°, ϑ f = 172° RG =
(5.68)
ϑC = ϑ f − ϑi where R ext , Rint and RG are the external, internal and center-line radii respectively;
ϑi , ϑ f and ϑC are the initial, final and central angle, respectively. A very fine mesh is considered for the arch, which was subdivided in 100 linear finite elements. The mechanical properties of each masonry constituents have been reported in chapter 4. In the picture reported in Fig. 5.19 it can be noted the defects and the irregularities. In fact, a significant reduction of the size of the cross section of the arch is noted in the mortar.
Fig. 5.19: Mortar joints irregularities.
148
It was observed that the geometrical variation of mortar joints is comprised between 0.2 mm and 0.7 mm. Moreover, it was noted that the external part of the mortar was characterized by very low mechanical properties, as it was possible to damage the mortar by hands. Thus, the considered reagent masonry mortar section was reduced of 0.7 mm in height and width, with respect to the geometrical section. The size of homogenized reagent section is: bh = 70.5mm , hh = 106mm and lh = 241mm . The elastic modulus of homogenized masonry was determined assuming: 1) 2) 3) 4)
elastic-linear brick constitutive relationship: σ b = Ebε b ; elastic-linear mortar constitutive relationship: σ m = Emε m ; same stress for brick and mortar; elastic-linear masonry constitutive relationship: σ M = EM ε M .
Enforcing the equilibrium and congruence conditions:
σM = σb = σm
(5.69)
ε bbb + ε mbm = ε M (bb + bm )
(5.70)
Substituting the constitutive relationship into equation (5.69):
Ebε b = Emε m = EM ε M
(5.71)
which gives:
εb =
EM εM Eb
Substituting the expression (5.72) into equation (5.70) it results:
149
(5.72)
ε M ( bb + bm ) = ε bbb + ε mbm =
EM E ε M bb + M ε M bm Eb Em
(5.73)
Thus, the elastic modulus of masonry is:
EM =
( bb + bm )
(5.74)
⎛ bb bm ⎞ ⎜ + ⎟ ⎝ Eb Em ⎠
In this specific case, EM ≅ 8300 N mm 2 . This computed value for EM is greater than the true value of the elastic modulus of the masonry; in fact the mortar specimens realized and tested into the laboratories are characterized by mechanical properties significantly greater than those of the mortar joints. The shear modulus is calculated in classic way: GM =
EM ≅ 3400 N mm 2 . 2(1 + ν )
The arch was considered clamped at springs. It is subjected to the its weight and to an additional increasing force applied at section corresponding to the angle ϑ ≅ 70o . Initially the unreinforced arch is studied. The numerical results are obtained considering for masonry both Model 1 and Model 2. In Fig. 5.20 all the results are reported. It can be observed that there is not a substantial difference between the use of Model 1 and Model 2.
150
-800 -750 -700 -650 -600 -550 -500
F [N]
-450 -400 -350 -300
Experimental Arch 1 Experimental Arch 2 FEAP, model 1 FEAP, model 2 Safe theorem
-250 -200 -150 -100 -50 0.00 0
-0.20
-0.40
-0.60
-0.80
-1.00
-1.20
-1.40
-1.60
-1.80
vk [mm]
Fig. 5.20: Comparison 2, unreinforced arch load-displacement curve.
The numerical model approximates in a satisfactory manner the limit load of the masonry arch. The limit load of the arch was calculated also by applying of the kinematical theorem of the limit analysis, on the base of collapse mechanism characterized by the four hinges formation at extrados or at intrados. The hinges position was obtained by minimizing the lading factor. The position of the hinges is compared with the ones obtained from the results determined using the finite element approach. In Tab. 5.1 the hinges position is reported.
151
Element Node [number] [number] 39 20 40 41 87 44 88 89 143 72 144 145 199 100 200 201
Angle [°] [rad] 39.16 0.6831 40.80 78.52
Radius [mm]
39.9800
0.6974
51.6070
79.3400
1.3840
51.6070
125.2600
2.1851
51.6070
171.1800
2.9861
51.6070
0.7117 1.3697
80.16 124.44
1.3983 2.1708
126.08 170.36
2.1994 2.9718
172.00
Hinge angle [°] [rad]
3.0004
Tab. 5.1: Hinges position.
According to the hinges position, the arch was subdivided in three blocks, called block 1, block 2 and block 3. In Fig. 5.21 are schematically illustrated the position of hinges that coincides with the arch relative or absolute centers of rotation.
152
Fig. 5.21: Comparison 2, unreinforced arch kinematical mechanism.
Determined the position of arch relative and absolute centers, the vertical displacement components can be traced on the horizontal fundamental. The itself weight of each one block is: Block 1: P=187.82 N; Block 2: P=219.13 N; Block 3: P=219.13 N;
Applying the virtual work and considering that the external load must be equilibrated by the action of block 2 and 3, it has: Lve = − PIη I + PIIη II + PIIIη III = FLIMη LIM
(5.75)
Solving equation (5.75) the limit load is determined: Lve = 0 ⇒ FLIM = 591.87 N
153
(5.76)
In Fig. 5.20 it can observe that the limit load is not different from the limit load determined using the proposed finite element approach. Then, the reinforced arch was studied using the finite element formulation. The following further input data are considered: dimensions of FRP woven: height tFRP=0.17 mm and width LFRP=100 mm; mechanical properties of FRP: EFRP = 230000 N mm 2 ; homogenization coefficient (considering the perfect adhesion between
masonry and reinforcement): n0 =
EFRP ≅ 27 . EM
The kinematical mechanism of the unreinforced masonry arch is not possible for the reinforced arch. In fact, the presence of reinforcement does not allow the cracks opening on the reinforced side of the arch, consequently the hinges formation on the opposite side is prevented. In Fig. 5.22 the load-displacement curves for the reinforced arch are reported.
-55000 -50000 -45000 -40000
F [N]
-35000 -30000 -25000 -20000 Experimental reinforced arch
-15000
FEAP, model 1
Compressive masonry failure
-10000
FEAP, model 2 -5000 -1 0
-3
-5
-7
-9
-11
-13
-15
-17
-19
-21
-23
-25
-27
-29
-31
-33
vk [mm]
Fig. 5.22: Comparison 2, reinforced arch load-displacement curve.
154
-35
The numerical results agree very well with experimental curve, in particular when the Model 2 for the masonry material is adopted. The collapse of the reinforced arch determined during the laboratory tests occurred because of a shear mechanism; in fact, the sliding of the brick under the applied force occurred, leading to the crushing of masonry. This type of mechanisms is not accounted for in the proposed model, so that the limit collapse displacement cannot be numerically evaluated.
155
6. MULTISCALE APPROACHES
6.1. Introduction The aim of this chapter is to introduce multilevel strategies. Generally, we talk of multilevel strategies when we have a global “ macroscopic” problem, associated to a coarse solution, and a local “microscopic” problem, associated to a finer mesh in a limited zone of interest. In literature there are three families of multilevel approach: the methods based on the homogenization, these based on super-position and these based on decomposition of domain. The reference problem is a problem of classic mechanics: the quasi-static study of a little perturbation on the body Ω , subjected to an imposed displacement field u d on part ∂Ω1 of its surface, to a surface forces field Fd on the complementary part ∂Ω 2 = ∂Ω \ ∂Ω1 and to a volume force field called f d , as illustrated in Fig. 6.1.
Fd
∂Ω 2
fd
∂Ω1
Ω ud
Fig. 6.1: Multiscale approaches, reference problem.
156
We suppose the material is linear elastic, except explicit mention; K is the Hook’s tensor, u is the displacement field, σ is the Cauchy stress field and ε is the deformation field; thus the reference problem can be written as: Kinematics admissibility: u = u d on ∂1Ω ε=
1 ∇u + ∇uT ) in Ω ( 2
(6.1)
Static admissibility:
divσ + f d = 0 in Ω
σ n = Fd on ∂ 2 Ω
(6.2)
Constitutive relationship: σ = Kε in Ω
(6.3)
6.2. Methods based on the homogenization The most famous multiscale methods are based on the homogenization theory. The first works on this technique were analytic or semi-analytic studies on the macroscopic behavior structures from some “effective” medium quantities [77] [80]. These methods could not enable to analyze the local effects. Then the microscopic level was introduced by the “Unit cell methods” [81] - [83], in order to obtain a local solution into a Representative Volume Element. Finally, the periodic media theory [84] - [87], based on the asymptotic analysis has proposed a really 157
multiscale approach: it enables to obtain a local solution from one macroscale problem and one microscale problem.
6.2.1. Theory of homogenization for periodic media The homogenization technique is applied when a problem can be schematized by a repeated unit cell; indeed the fundamental hypothesis is the periodicity. This repeated unit cell is called RVE, Representative Volume Element and has shown in Fig. 6.2a and Fig. 6.2b; in particular Fig. 6.2a shows the generic body that can be schematized by its repeated part and Fig. 6.2b shows the RVE.
Fd
Y2
X2
Y1
ud
X1
RVE
Y3
X3 a)
b)
Fig. 6.2: Homogenization technique.
Two representative scales were introduced: a macroscale with the position vector Xi (i = 1, 2, 3) defined on the body Ω and a microscale with the position vector Yi (i = 1, 2, 3) defined on the RVE; the smallness parameter Z = X / Y put in comparison the two scales. The homogenization methods are based on the following assumptions: 158
Periodicity; Solution is periodic in statistic sense; The macroscopic fields are constant in the RVE.
Obviously these assumptions are not verified in the neighbourhood of boundaries and when the heterogeneities of the material are not small enough with respect to the dimensions of the macrostructure. The displacement solution is u = u(x, y ) . The idea is to develop the solution into asymptotic form as: uiε (x, y ) = uio (x, y ) + ε ui1 (x, y ) + ε 2 ui2 (x, y ) L
(6.4)
Analogously the asymptotic form of the stress field is: σ ijε (x, y ) = ε −1σ ij−1 (x, y ) + σ ijo (x, y ) + εσ ij1 (x, y ) + ε 2σ ij2 (x, y ) + L
(6.5)
Once injected inside the equilibrium and constitutive equations, these developments lead to a succession of problem at different orders. In this way the macroscopic field uio and the microscopic field ui1 can be determined.
6.3. Methods based on the super-position The homogenization approaches permit to pass from the macrolevel to the microlevel defining the macroscale problem and analyzing it at the microscale. The methods based on the superposition do a different thing: they superpose a microscopic enrichment into the interest zone for the solution of the macroscopic problem.
159
6.3.1. Variational multiscale method This method was initially proposed by Hughes [88]: all the elements problem are not soluble numerically, as Hughes said. The microscopic effects that are not “soluble” can not be represented by finite elements size superior to the microstructure size. Hughes has proposed a superposition principle that permit to consider the effects of the small cell at macroscopic level. Solving the local problem, the small elements effects are condensed to the macroscopic level, obtaining a quasi-exact solution for the macroscopic problem. The solution of the problem is decomposed into a macroscopic and microscopic part, called u M and u m respectively:
u = uM + um
(6.6)
The choice for the approximation of u m is very important; a good solution is to utilize the Green’s function.
6.4. Methods based on the domain decomposition When the microstructure is analyzed two cases can occur: the analysis of only an interest zone with a fine mesh, it is the case of a local-global analysis for which a natural separation between a coarse and fine zone occurs. The other case occurs when the fine mesh is for all the structure; it happens when the structure is strongly heterogeneous. Then the direct solution is very complex and it is necessary to apply domain decomposition by the subdivision in sub-structures of the initial structure. The presence of sub-structures permits the resolution of small interface problem. The decomposition domain methods are subdivided in three great families: the primal approaches (Balancing Domain Decomposition Method, BDDM, [89]), the dual approaches (Finite Element Tearing Interconnecting, FETI, [90]) and the mixed 160
theory (based on the Lagrange’s algorithm [91] or on the LATIN [92]. In every case the solution is based on the application of an iterative procedure. In order to obtain a quickly numerical solution the problem of the propagation of a global information must be solved. This implies the grew up of a coarse problem to verify the partial transmission condition into the sub-structures. In the first case (primal approaches) a force condition is imposed, for the dual approaches the condition is imposed on the partial verification of displacements, in the mixed approaches the conditions are imposed both forces and displacements.
6.4.1. Primal approach Considering the reference problem illustrated in section 6.1, let us consider a partition of the domain Ω in two substructures Ω(1) and Ω(2) , as illustrated in fig. 6.3.
Γ
Ω (2)
Ω(1)
Fig. 6.3: Two subdomain decomposition.
161
The interface between the two substructures is defined as Γ = ∂Ω(1) ∩ ∂Ω(2) and the equations governing the reference problem can be rewritten on the restrictions Ω(1) and Ω(2) of Ω : ⎧ divσ ( s ) + f ( s ) = 0 in Ω( s ) ⎪ (s) (s) (s) (s) ⎪ σ = a : ε(u ) in Ω ⎪ s = 1 o 2 ⎨ 2ε = ∇u ( s ) + ∇u ( s )T in Ω ( s ) ⎪σ ( s ) • n ( s ) = F ( s ) on ∂Ω ∩ ∂Ω( s ) d 1 ⎪ (s) (s) ⎪⎩ u = u d on ∂Ω 2 ∩ ∂Ω( s )
(6.7)
The interface connection conditions are the continuity of displacements:
u (1) = u (2)
on Γ
(6.8)
and the stresses equilibrium:
σ (1)n (1) + σ (2)n (2) = 0 on Γ
(6.9)
Of course the system constituted by equations (6.7), (6.8) and (6.9) is exactly the same as the system represented by equations (6.1), (6.2) and (6.3). In reality, a structure can be decomposed in N subdomains denoted Ω ( s ) . In this case three interfaces
are
defined:
the
interface
between
two
subdomains
Γ (i , j ) = Γ ( j ,i ) = ∂Ω( i ) ∩ ∂Ω( j ) , the complete interface of one subdomain (local
interface) Γ ( s ) = U j Γ ( s , j ) and the geometric interface at the complete structure scale (global interface) Γ = U s Γ ( s ) . Defined the interfaces, the reference problem can be discretized in order to obtain a classical finite element solution: 162
Ku = f
(6.10)
In order to rewrite equations (6.10) in a domain decomposed context, to introduce
λ ( s ) , the reaction imposed by neighbouring subdomains on subdomain (s). This reaction is defined in the whole subdomain, but it assumes non-zero value only on its interface. For every subdomain a local equilibrium condition is defined as: K ( s )u ( s ) = f ( s ) + λ ( s )
(6.11)
On the interface, a global equilibrium condition and a global condition of continuity of displacements are defined as:
∑A
t λ (s) = 0
(6.12)
t u(s) = 0
(6.13)
(s) (s)
s
∑A
(s) (s)
s
where t ( s ) and A( s ) are the local trace operator (restriction from Ω ( s ) to Γ ( s ) which permits to cast data from a complete subdomain to its interface) and the assembly operator (it is a strictly Boolean operator) respectively. The aim of this method is to write the interface problem in terms of one unique unknown: the interface displacement field ub. the problem can be solved introducing the primal Schur complement S p( s ) . The basic idea for every subdomain is to condense its behavior on its interface. Let consider the local equilibrium of a subdomain under interface loading:
K ( s )u ( s ) = λ ( s ) = t ( s ) λ b( s )
163
(6.14)
In order to separate the internal and boundary degree of freedom system (6.14) assumes the following form: ⎡ K ii( s ) ⎢ (s) ⎣ K bi
K ib( s ) ⎤ ⎧ui( s ) ⎫ ⎧ 0 ⎫ ⎨ (s) ⎬ = ⎨ (s) ⎬ (s) ⎥ K bb ⎦ ⎩ub ⎭ ⎩λb ⎭
(6.15)
from the first line it is obtained: ui( s ) = − K ii( s ) −1 K ib( s )ub( s )
(6.16)
Then the Gauss elimination of ui( s ) furnishes the primal Schur complement S p( s ) :
(K
(s) bb
− K bi( s ) K ii( s ) −1 K ib( s ) ) ub( s ) = S p( s )ub( s ) = λb( s )
(6.17)
Moreover if the subdomain is also loaded on internal degree of freedom the condensation of the equilibrium at the interfaces gives:
(s)
K u
(s)
= f
(s)
⇒
S p( s )ub( s ) = bp( s ) bp( s ) = f b( s ) − K bi( s ) K ii( s ) −1 f i ( s )
(6.18)
Using the primal Schur complement, the primal formulation of the interface problem assumes the form: S p ub = ( AS p◊ AT )ub = Ab ◊p = b p
Where the superscript
◊
(6.19)
denotes the row-block repetition of local vectors and the
diagonal-block repetition of matrices and S p = ∑ s A( s ) S p( s ) A
( s )T
Schur complement of the decomposed structure. 164
is the global primal
Primal approach comes with an efficient preconditioner, called the Neuman-Neuman preconditioner, enriched by a coarse problem associated to the well posedness of local problems with imposed forces on the interfaces. This preconditioner and its associated coarse problem is very similar to realizing a dual step as described in next section.
6.4.2. FETI method This method searches by an iterative procedure a field of forces that is continuous a priori at the interfaces to guarantee the continuity of displacements at convergence,
in other words the interface problem is formulated in terms of one unique unknown interface stresses field. The field of stresses must verify the equilibrium constrain of every sub-domain with the respect of external loads because of the local problem is well posed. Thus an iterative procedure is necessary; it is based on the conjugated gradient and its projection is associated to the equilibrium constraint of each subdomain. This projection can be considered as a macroscopic coarse problem that guarantees the rigid motions continuity at every iteration into the interface. For this reason the FETI method is considered as a multiscale strategy of calculus. But this macroscopic problem might be too poor because it is only associated to the rigid motions. In order to exceed this reef, the method was improved by the FETI 2 [93]: some additional constraints on the displacement field are introduced to guarantee the continuity at the interface middle. The projection becomes an enhanced macroscopic problem. A more recent version is called FETI-DP (Dual-Primal FETI Method [94]). The performances obtained with FETI method are very remarkable, but they are connected to the appropriate chosen of the pre-conditioner for the algorithm of resolution. This chosen depends essentially from the analyzed problem and it is possible because of the generality of the method. A classic chosen is that to consider as pre-conditioner a lumped or Dirichlet. For the heterogeneous structures the 165
Reference is [95]; for preconditioning and enhanced macroscopic problems based on the reuse of Krylov’s subspaces in a multiresolution context, References [96] and [97].
6.4.3. Mixed method: the micro-macro approach The principle of the mixed method is to rewrite the interface conditions in terms of a new local interface unknown, which is a linear combination of interfaces stresses and displacements. Mixed methods give a mechanical behavior to the interface and in the case of perfect interfaces it can be mechanically interpreted as the insertion of springs to connect the subdomains. The method is characterized by a micro-macro approach, i.e. [98] - [100]. The mixed formulation is indicated for the analysis of structures that are strongly heterogeneous, i.e. [101] and [102], and for the study of the nonlinearities, i.e. [103] and [104]. The characteristics of the method are the subdivision in sub-structures, the introduction of the multiscale effect only at interface level, where the stresses and the displacements are split into “macro” contributions and “micro” complements, and the satisfaction of the transmission conditions a priori by the interfaces macro stresses. So, the initial structure is subdivided in sub-structures subjected to the actions of the neighboring, as illustrated in Fig. 6.4.
166
FE F E’ ΩE
ΩE’
WE W E’ ΓEE’
Fig. 6.4: Sub-domains decomposition.
Consequently to the micro and macro separation, the force and displacement field are expressed in function of the micro and macro part: F = FEm + FEM
(6.20)
W = WEm + WEM
The micro and macro quantities must satisfy the splitting of the virtual work:
∫
Γ EE '
F • Wd Γ =
∫
FEm • WEm d Γ +
Γ EE '
∫
FEM • WEM d Γ
(6.21)
Γ EE '
At each interface the macro projector is defined as: F M = Π ΓMEE ' (F) F m = Π ΓmEE ' (F) W M = Π MΓ ( W)
(6.22)
EE '
W = Π Γ (W) m
m
EE '
The projector can be chosen in order to extract the linear part from the interface quantity. Finally the interface forces must satisfy the transmission conditions a priori
167
(in the case of perfect interfaces, macro-displacement can also be made continuous a
priori). To solve this problem the LATIN method [92] is applied. The LATIN method is an iterative resolution technique which takes into account the whole time interval studied. At each iteration, an homogenized macroproblem, defined over the whole time-space domain, is solved as well as a set of independent microproblems which are linear evolution problems defined within each substructure or at boundaries between the substructures. The LATIN method, illustrated in Fig. 6.5, is based on the idea of dealing the difficulties separately, splitting the equations in two subsets: ⎧Static and kinematic admissibility: ⎪ (σ , F ), (ε , W ) ∀M ∈ Ω ⎪ E E E E E Ad ⎨ Macro equilibrium: ⎪ ⎪⎩ FEM + FEM' = 0
(6.23)
⎧ Dissipation law of each substructure Γ⎨ ⎩The behavior and equilibrium at interfaces
(6.24)
Ad is the space of the global linear equations defined, while G is the space of the local nonlinear equations.
168
sn+1/2 E+ Esref.
sn+1
sn
Fig. 6.5: LATIN scheme for one iteration.
The solution of the problem is obtained by an iterative scheme. Each iteration consists of a local step and a linear step: the method permits to find a solution that verifies alternatively the equations of the first and second set of equations and at last it converges towards the solution sref.
6.5. Numerical results In this simple example a clamped beam, schematically illustrated in Fig. 6.6, subjected to an horizontal pressure was analyzed. Two tests were performed: the first one with the use of a single material and the second one with two different materials. The results obtained with the application of FETI method and of the mixed method were put in comparison. The beam properties are reported in the following: Geometrical characteristics: o Brick: rectangular cross section, b=100 mm and h=100 mm; o Mortar: rectangular cross section, b=4 mm and h=100 mm; o External load: horizontal pressure P=10 MPa.
169
Mechanical properties: o Brick: E=1785 MPa; n=0.2; o Mortar: E=1785 MPa; n=0.2.
P L Fig. 6.6: Beam scheme.
The mesh is reported in Fig. 6.7, while in Fig. 6.8 it is remarked the single sub-domain considered.
Fig. 6.7: Beam mesh.
Fig. 6.8: Beam sub-domain.
Two analysis are effected, with the application of the FETI method and the mixed method. In the following figures are reported the displacement and stress fields; the results are the same for the two methods.
170
Fig. 6.9: Case 1, displacement field for FETI.
Fig. 6.10: Case 1, displacement field for the mixed method.
Fig. 6.11: Case 1, stress field for FETI.
Fig. 6.12: Case 1, stress field for the mixed method.
171
The same clamped beam, schematically illustrated in Fig. 6.6, subjected to the same horizontal pressure was re-analyzed considering different mechanical properties for the two materials: Geometrical characteristics: o Brick: rectangular cross section with b=100 mm; h=100 mm; o Mortar: rectangular cross section with b=4 mm; h=100 mm; o External load: horizontal pressure P=10 MPa. Mechanical properties: o Brick: E=1785 MPa; n=0.2; o Mortar: E=113 MPa; n=0.2.
In this case it is evident that the second material is effectively more deformable and at interface some local effects are visible, as illustrated in the following.
Fig. 6.13: Case 2, displacement field for FETI.
Fig. 6.14: Case 2, stress field for FETI.
172
Also in this case the analysis was conduced by the mixed method and the results are perfectly the same, as illustrated in Fig. 6.15 and Fig. 6.16.
Fig. 6.15: Case 2, displacement field for the mixed method.
Fig. 6.16: Case 2, stress field for the mixed method.
Moreover these results were put in comparison with the classical theoretical results and with the results obtained by FEAP using the implemented three nodes finite element; in particular in Tab. 6.1 and in Tab. 6.2 are reported the axial displacement values for the first and second case, respectively. FETI w [mm]
≈5.58
MIXED FEAP THEORIC ≈5.88
≈5.83
≈5.83
Tab. 6.1: Case 1 results.
FETI w [mm]
≈11.16
MIXED FEAP THEORIC ≈9.07
≈12.53
Tab. 6.2: Case 2 results.
All the results are in good accordance. 173
≈12.53
The conclusion after the study of this simple example is the possibility of applying these methods to analyze reinforced masonry arches, reducing the solution of a great problem into the solution of a set of simplest problems. In fact, once defined the mesh, with a manual or automatic mesh generator, the subdivision in subdomains is immediate and the study of the single sub-structure could become simple enough, in terms both of numerical procedures and computational resource requirement. It must be emphasized that the domain decomposition methods are adapted to parallel processing, consisting of independent tasks having their own data that can be allocated to the various processors of the system. The DDM offer a framework where different design services can provide the models of their own parts of a structure, each assessed independently, and they can evaluate the behavior of the complete structure just setting specific behavior at interfaces. From an implementation point of view, often programming DDM can be added to existing solvers as an upper level of current code.
174
CONCLUSIONS The research activity presented in this thesis work has been focused on the experimental and numerical analysis of masonry arches strengthened with fiber reinforced plastic (FRP) materials. The experimental program regarded different tests performed on both masonry constituents (bricks and mortar) and structures (unstrengthened and FRPstrengthened arches). From the performed tests, important aspects concerning the effect of the FRP reinforcement on the structural response of masonry arches was observed. In fact, comparing the behavior of unstrengthened and FRP-strengthened arches, it was observed that the application of FRP at extrados surface of arches produces an increase both in terms of load-bearing capacity (strength) and in terms of ultimate displacement (ductility). These effects are related to the collapse mechanism. In fact, while in the case of unreinforced arches the collapse is due to the formation of the classic four hinges, the FRP-strengthening prevents the crack opening at the extrados, i.e. the presence of hinges at intrados, and leads to a collapse mechanism characterized by shear and crushing failure of masonry. On the basis of the experimental observations and in order to understand further aspects concerning the nonlinear response of FRP-strengthened masonry arches, in the second part of the thesis numerical analyses have been developed. In particular, two models have been considered for the masonry material; both the models assume the masonry material characterized by no-tension behavior and limited compressive strength. For reduced values of the compressive strain, the first one considers a linear stress strain relationship, while the second one considers a quadratic relationship. In order to solve the nonlinear unreinforced and reinforced masonry arch problem, a stress formulation, based on the complementary energy, and a displacement formulation, based on the implementation of a three nodes finite element into the FEAP code, have been developed. 175
Moreover, as the delamination phenomenon between the FRP and the masonry support can play an important role in terms of FRP-strengthening contribution, an effective procedure, based on a simplified approach of the multiscale method for the evaluation of the normal and tangential stresses at the interface has been developed. Moreover, in the context of the multiscale approaches, the domain decomposition methods are analyzed. Numerical applications based on the use of the proposed models have been developed with reference to the performed experimental tests. The comparison between the numerical and the experimental results demonstrated the ability of the proposed models to reproduce the global and local response of unstrengthened and FRP-strengthened arches. In particular, while in the case of the unstrengthened arches the two proposed models give the same results, in the case of the FRPstrengthened arch substantial differences occur between the two considered models. In fact, in this case the second model gives the best results both in terms of pre and post-peak behavior.
176
APPENDIX: RELUIS SCHEDE
MODELLO PER LA DESCRIZIONE SINTETICA DI PROVE SPERIMENTALI STRUTTURE/ELEMENTI STRUTTURALI IN MURATURA
SPERIMENTAZIONE DI ARCHI IN MURATURA CON E SENZA RINFORZO IN FRP
CANCELLIERE ILARIA, RICAMATO MARIA, SACCO ELIO
177
ISTRUZIONI Il modello è diviso in quattro sezioni: 1) Dati generali della prova sperimentale (G) 2) Descrizione elemento/ struttura testata (D) 3) Proprietà dei materiali (M) 4) Risultati della prova (R) La compilazione delle prime due sezioni richiede l’inserimento di figure con fotografie e/o disegni che si ritengono utili alla comprensione del setup di prova. La sezione materiali è basata su tabelle da compilare con i risultati disponibili per mattoni, malta e muratura. Cliccando sulle relative tabelle si apre una finestra di Excel in cui sono disponibili tutti i comandi dell’applicazione. In alternativa è possibile copiare nello stesso spazio una qualsiasi tabella dai contenuti analoghi a quella già predisposta. La sezione finale non ha un formato prestabilito e la sua compilazione è lasciata agli autori della prova. E’ possibile inserire in ciascuna sezione tutte le pagine necessarie alla descrizione della prova e dei risultati. La numerazione delle pagine ha il formato: codice sezione/N. totale di pagine della sezione – numero progressivo di pagina
178
DATI GENERALI DELLA PROVA SPERIMENTALE Arco • Oggetto della Prova • Autori
Portale
Volta
Cupola
Pannello Colonna
Altro
X Cancelliere Ilaria, Ricamato Maria, Sacco Elio
• Data di Esecuzione
Luglio - Settembre 2007
• Sede Laboratorio
Laboratorio di Progettazione Strutturale LaPS, Università di Cassino
• Riferimento Bibliografico
Cancelliere I. “Analisi numerica e sperimentale di archi in muratura rinforzati con FRP”, Tesi di Laurea specialistica in Ingegneria Civile, Università di Cassino (FR), Ottobre 2007 Ricamato M. “Numerical and experimental analysis of masonry arches strengthened with FRP materials”, Tesi di Dottorato, Università di Cassino, Novembre 2007
Messa in opera dell’arco
Esecuzione della prova sperimentale sull’arco non rinforzato
Esecuzione della prova sperimentale sull’arco rinforzato
179
DESCRIZIONE SETUP DI PROVA Geometria e Vincoli
Re Ri Rb Fi Ff Fc
8 172 164
576.07 [mm] 456.07 [mm] 516.07 [mm] [°] 0.1396 [°] 3.0004 [°] 2.8609
[rad] [rad] [rad]
Vincoli L’arco è stato fissato alla base tramite elementi di contrasto per la realizzazione di una condizione di incastro.
Note: Il carico è stato applicato mediante un martinetto idraulico disposto in posizione eccentrica. Tra l’estradosso dell’arco e il martinetto è stata posizionata la cella di carico. Sono stati utilizzati tre potenziometri e due comparatori: un potenziometro e un comparatore in direzione verticale in corrispondenza del martinetto; un potenziometro ed un comparatore in direzione verticale in chiave dell’arco; un potenziometro in direzione orizzontale posizionato in chiave.
180
PROPRIETA’ DEI MATERIALI MATTONI Descrizione (Tipo, Marca, Forno d'origine)
Dimensioni nell'elemento strutturale
Prova di Resistenza Resistenza misurata Norma di riferimento compressione D.M.20/11/1987
N. Prove
Dimensione campione Media 6 55x55x55 38,5 Mpa
Risultati sui singoli campioni Mattoni pieni in laterizio denominati "di Salerno". Cava e fornace ubicate nel comune di Salerno (Italia).
250x120x55 mm
3
Numero campione 1 2 3 4 5 6
181
Resistenza 41,5 35,7 40,06 34,7 36,6 41,7
NOTE rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra rottura a clessidra
Scarto 7,47
PROPRIETA’ DEI MATERIALI MALTA Descrizione
Spessore dei letti di malta nell'elemento strutturale
Prova di Resistenza Resistenza misurata flessione flessione compressione compressione
Norma di riferimento UNI-EN 196/1 UNI-EN 196/1 D.M.20/11/1987 D.M.20/11/1987
Numero campione 1 2 3
Resistenza 2,36 2,62 2,59
1 2 3
2,69 2,66 2,27
1 2 3 4 5 6
9,147 9,4475 8,1937 8,4806 9,008 8,22
1 2 3 4 5 6
9,83 9,85 8,355 8,87 8,53 8,75
N. Prove 3 3 6 6
Dimensione campione 40x40x160 40x40x160 40x40x80 40x40x80
Risultati sui singoli campioni
Malta bastarda: pozzolana, calce aerea, cemento pozzolanico (Duracem 32.5R) ed acqua
il valore medio misurato sulla struttura messa in opera è pari a 15 mm.
182
NOTE
Media 2,53 2,54 8,75 9,03
Scarto
PROPRIETA’ DEI MATERIALI MURATURA Prova di Resistenza Resistenza misurata (trazione, compressione, taglio)
Tipo di prova (compressione diagonale, compressione, etc..)
Norma di riferimento
N. Prove
Dimensione campione
Risultati sui singoli campioni Numero campione
NOTE (modalità di rottura, etc..)
Resistenza
Descrizione della prova (modalità di applicazione del carico, diagrammi carico spostamento, setup di prova, figure del campione, etc..)
183
Media
Scarto
PROPRIETA’ DEI MATERIALI FIBRE Descrizione: proprietà
unità di misura
Spessore (lamina)
0.17 mm
larghezza
100 mm
lunghezza
162 mm
Metodo di prova normativa di riferimento
Geometria della sezione (barre, cavi) mm2
Area nominale (barre, cavi) Perimetro nominale (barre, cavi)
mm
Colore
nero
densità
Contenuto in fibra
fibra
1.80 g/cm3
matrice
1.31 g/cm3
in peso
%
in volume
%
ISO 1183-1:2004 (E)
ISO 11667:1997 (E) ISO 11357-2:1999(E) (DSC) ISO11359-2:1999(E) (TMA)
Temperatura di transizione vetrosa della resina (Tg) Temperatura limite massima di utilizzo
°C
Conducibilità elettrica
S/m
Modulo di elasticità normale a trazione
230GPa
ISO 527-4,5:1997(E)
3900 MPa
ISO 527-4,5:1997(E)
Resistenza a trazione (valore caratteristico)
MPa
ISO 527-4,5:1997(E)
Deformazione a rottura a trazione
1.5 %
ISO 527-4,5:1997(E)
Modulo di elasticità normale a compressione (barre)
GPa
ISO 14126:1999(E)
Resistenza a compressione (barre) (valore medio)
MPa
ISO 14126:1999(E)
Resistenza a compressione (barre) (valore caratteristico)
MPa
ISO 14126:1999(E)
%
ISO 14126:1999(E)
Resistenza a trazione (valore medio)
Deformazione a rottura per compressione (barre) Resistenza a creep
ISO 899-1:2003(E)
Rilassamento (barre, cavi) Aderenza: tensione tangenziale (barre, cavi)
Prova di pull-out
184
Note
PROPRIETA’ DEI MATERIALI RESINA Descrizione resina: (nome commerciale, mono o bicomponente, pasta o liquida, tipologia di utilizzo ed ogni altra informazione generale ritenuta utile)
caratteristiche della resina non miscelata unità di misura
proprietà colore viscosità a 25°
comp. A
comp. B
miscela
bianco
grigio
grigio
note
ISO 2555:1989(E) ISO 3219:1993(E)
Pa s
indice di tissotropia densità rapporto di miscelazione condizioni di stoccaggio (contenitore siggillato)
metodo di prova
ASTM D2196-99
g/cm3 4:1 %
in volume in peso tempo
1.31
ISO 1675:1985(E)
mesi
temperatura
°C
caratteristiche della resina miscelata condizioni di miscelazione: condizioni di applicazione: proprietà tempo di lavorabilità (a 35°) a 5°C tempo di gelo a 20°C a 35°C temperatura minima di applicazione tempo picco esotermico temperatura tempo di completa a 5°C reticolazione (full a 20°C core) a 35°C
unità di misura 30 min min
metodo di prova normativa di riferimento
note
ISO 10364:1993(E) ISO 9396:1997 (E) ISO 2535:2001 (E) ISO 15040:1999 (E)
10°C min °C
ISO 12114:1997 (E)
min
ISO 12114:1997 (E)
proprietà della resina reticolata condizioni di stoccaggio: precauzioni d'uso e sicurezza: proprietà
unità di temperatura misura di prova
valore
metodo di prova normativa di riferimento
stagionato stagionato 5gg. a 22°C 1 ora a 70°C ritiro volumetrico coefficiente di dilatazioe termica
ISO 12114:1997 (E)
10-6 °C-1
ISO 11359-2:199 (E)
temperatura di transizione vetrosa, Tg
°C
ISO 11357-2:1999 (E) (DSC) ISO11359-2:1999(E) (TMA) ASTM E 1640 (DMA)
modulo di elasticità normale a trazione resistenza a trazione
Gpa Mpa
ISO 527:1993 (E) ISO 527:1993 (E)
185
RISULTATI DELLA PROVA F(v) in corrispondenza della forza -700.00 Ciclo I Ciclo II Ciclo III
-600.00
-500.00
F [N]
-400.00
-300.00
-200.00
-100.00 0.50
0.00 -0.50 0.00
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
-4.00
-4.50
-5.00
-5.50
-6.00
-6.50
-7.00
100.00
v [mm]
Cinematismo di collasso e curva forza - spostamento (relativa ai tre cicli di carico e scarico effettuati) per l’arco non rinforzato.
Immagini relative alle fasi di prova dell’arco rinforzato. F(v) in corrispondenza della forza -60000 Ciclo I Ciclo II Ciclo III
-50000
-40000
F [N]
-30000
-20000
-10000 6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23
0
10000
v [mm]
Curva forza - spostamento (relativa ai tre cicli di carico e scarico effettuati) per l’arco rinforzato. Note
186
NOTATIONS
The following notation were used throughout the text:
Vectors and tensors Quantities
Scalar Vector Second order tensor Third order tensor
a a A && A
Inner product Vectorial product Dyadic product Gradient operator
• × ⊗ Δ
Operators
187
Matrices and columns Quantities Scalar
a
Column
⎧a1 ⎫ ⎪ ⎪ a=⎨M⎬ ⎪a ⎪ ⎩ i⎭
Line
a = {a1 L ai }
Matrix
A = ⎡⎣ Aij ⎤⎦
Operators
Matrix product
AB
Transposition
AT
Inversion
A −1
Any notation which has not been explicitly defined in this section will be explained at its first point of use.
188
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