Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Stress analysis of masonry structures: Arches, walls and vaults

A. Baratta, I. Corbi & O. CorbiDepartment of Structural Engineering, University of Naples “Federico II”, Naples, Italy

ABSTRACT: In the paper, the authors present a synthetic overview of some results obtained by means of anumber of theoretical and experimental studies developed on some classical masonry typologies such as arches,panels and vaults. The proper implementation of the analyzed structural problem, specialized to the specific case,which derives from the extension of classical structural approaches to structures made of masonry material, isshown to provide a reliable approach to the problem itself, also in the case of some reinforcement, as demonstratedby experimental data which are in perfect agreement with numerical results.

1 INTRODUCTION

The basic assumption of no-tension masonry modelcoincides with the hypothesis that the tensile resistanceis null. Under this hypothesis, no-tension stress fieldsare selected by the body through the activation of anadditional strain field, the fractures (see Baratta, 1991,Baratta et al. 1981, Baratta & Toscano 1982, Bazant1996, Heyman 1966). The behavior in compressioncan be modeled in a number of different ways (elas-tic linear, elastic non-linear, elastic-plastic; isotropic,anisotropic; etc.), without altering substantially nei-ther the results nor the mathematical treatment of theproblem; some convenience exists for practical appli-cations in assuming a isotropic linearly elastic model,in order to keep limited the number of mechanicalparameters to be identified for masonry, since increas-ing the number of data causes increasing uncertainty inthe results. Because of these reasons, and being clearlyunderstood that there is no difficulty in introducingmore sophisticated models, it is convenient to set upthe fundamental theory on the basis of the assump-tion that the behavior in compression is indefinitelylinearly elastic.

Analysis of NRT (Not Resisting Tension) bod-ies proves that the stress, strain and displacementfields obey extreme principles of the basic energyfunctionals.

Therefore the behaviour of NRT solids under ordi-nary loading conditions can be investigated by meansof some extensions of basic energy approaches to NRTbodies (Baratta 1984, Baratta & O. Corbi 2005a, 2007,Baratta & al. 1981, Baratta & Toscano 1982).

In details, the solution of the NRT structural prob-lems can be referred to the two main variationalapproaches:

– the minimum principle of the Potential Energyfunctional;

– the minimum principle of the ComplementaryEnergy functional.

In the first case the displacements and the fracturesare assumed as independent variables; the solution dis-placement and fracture strain fields are found as theconstrained minimum point of the Potential Energyfunctional, under the constraint that the fracture fieldis positively semi-definite at any point.

The approach based on the minimization ofthe Complementary Energy functional assumes thestresses as independent variable. The complementaryapproach is widely adopted since the existence anduniqueness of the NRT solution are always assured interms of stress, if some conditions on the compatibil-ity of the loads are satisfied. The stress field can, then,be found as the constrained minimum of the Comple-mentary Energy functional, under the condition thatthe stress field is in equilibrium with the applied loadsand is compressive everywhere in the body.

The solution of both problems can be numericallypursued by means of Operational Research methods(see i.e. Rao 1978) suitably operating a discretizationof the analyzed NRT continuum (Baratta & I. Corbi2004, Baratta & O. Corbi 2003b, 2005a). One shouldnotice that discussion about existence of the solutionactually can be led back to some Limit Analysis of theconsidered NRT continua (Baratta & O. Corbi 2005a).

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Figure 1. (a) Masonry panel geometry, (b) with the applications of a light reinforcement by means of FRP strips.

In this regard, a special formulation of Limit Anal-ysis for No-tension structures has been performed,allowing the set up of theorems analogous to the basickinetic and static theorems of classical LimitAnalysis;thus, one can establish efficient procedures to assessstructural safety versus the collapse limit state (see e.g.Como & Grimaldi 1983) by specializing and applyingfundamental theorems of Limit Analysis to NRT con-tinua (Baratta 1991, Baratta & O. Corbi 2003a, 2005a,Bazant 1996, Como & Grimaldi 1983, Khludnev &Kovtinenko 2000). In details, the individuation of thecollapse (live) load multiplier for NRT continua canbe referred to the approaches relying on the two mainlimit analysis theorems:

– the static theorem;– the kinetic theorem.

This means that, after defining the classes ofstatically admissible and kinetically sufficient loadpatterns, LimitAnalysis allows individuating the valueof the live load multiplier limiting the loading capac-ity of the body, i.e. evaluating the collapse live loadand/or the safety factor versus collapse. One shouldnote that in a NRT structure its own weight (the deadload) is an essential factor of stability, while collapsecan be produced by not-admissible additions of thevariable component of the load pattern.

Duality tools may also be successfully appliedin order to check the relationships between the twotheorems of Limit Analysis (Baratta & O. Corbi

2004). In the study of plane mono-dimensional struc-tures featuring a low degree of redundancy, theforce/stress approach appears the most convenientto be adopted if compared with the displace-ment/strain approach, whose number of governingvariables is higher and, moreover, increasing withthe order of discretization (Baratta & I. Corbi 2003,2004, 2006).

The following section reports some results show-ing how the proper implementation of the describedtheoretical approach for classical masonry structuraltypologies such as panels, arches and vaults, producesresults that are in a very good agreement with exper-imental data, demonstrating the overall reliability ofthe mentioned approach, for whose details one shouldrefer to cited references.

As shown in the following, results can also be suc-cessfully extended to the case of reinforcements withfiber-reinforced polymers (FRP).

2 TESTS ON PROTOTYPES OF MASONRYPANELS

2.1 Experimental investigation

This section reports some of the results of the wideexperimental campaign developed at the Laboratoryof Materials and Structural Testing of the Universityof Naples “Federico II” on masonry panels, which aresymmetrical, with a central hole covered by a steelarchitrave, and having upper part characterized by a

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concrete fascia lightly reinforced by steel (Baratta &I. Corbi 2003, 2004, 2006).

The geometry of the panels is shown in Figure 1.a.A laboratory prototype of masonry panel is referred to,made of tuff bricks (type “yellow tuff of Naples”, Italy)jointed to each other by a pozzolana mortar in order toconfer a light additional resistance to the masonry;the masonry itself is characterized by unit weightγ = 10300 Nm and Young modulus E = 5.5 GPa. Asregards to the loading condition, a varying force isapplied in the middle of the left side of the panel,in such a way to mitigate the proneness of the panelto sliding of bricks with respect to each other, andsome loading/unloading cycles are developed up tothe collapse condition.

Once reached the crisis, the panel is reinforcedby directly laminating on the masonry some FRPstrips according to the provision scheme shown in Fig-ure 1.b, at the same time with the impregnation ofthe fibers by means of a special bi-component epoxyresin, and a further experimental investigation is devel-oped on the reinforced structure by re-executing someloading/unloading cycles.

The adopted reinforcement, produced by FTS,is a BETONTEX system GV330 U-HT, made of12 K carbon fiber, jointed by an ultra light netof thermo-welded glass. The mechanical character-istics of the employed carbon fibers are: tensilelimit stress σfrp = 4.89 GPa, elastic modulus in trac-tion Efrp = 244 GPa, limit elongation σfrp = 2%. TheFRP strip is characterized by thickness of 0.177 mmand depth of 200 mm. The induced displacementsat some selected points [the transducers 1, 2, 3and 4 in Figure 1.a] of the panel both for the notreinforced and for the lightly reinforced panel arerecorded by a monitoring equipment consisting of: 4transducers, placed at different locations of the panelin order to record the absolute displacements, and 15strain-gauges, arranged in 3 blocks of 5 strain-gauges,in such a way that each block is devoted to recordthe related strain situation. In details two transduc-ers are located horizontally at two different heightson the panel right side (transducers 1 and 2), and twoare placed in correspondence of the opening, one inhorizontal position at the top of the left side of thehole (transducer 3) and the other one under the archi-trave, which is devoted exclusively to control the paneldeflection (transducer 4). The displacements s(mm)versus the varying force F(N) monitored by the trans-ducers during the experiment in the not-reinforcedand in the reinforced case with some horizontallyapplied C-FRP strips are shown in Figures 2.a–cand 2.d–f respectively, as regards to the first load-ing cycle. By the diagrams in Figure 2, which reportthe displacements s(mm) vs the varying force F(N)read by the transducers 1–3, some considerations canbeen made.

With reference to the panel’s reinforcement bymeans of the application of some C-FRP strips, themajor effect of the C-FRP intervention is the reductionof the stress in the masonry. In general lower displace-ments at the locations monitored by the transducers canbe recorded in the consolidated case with comparisonto the unconsolidated case.

To this regard, the pretty light type of reinforcementallows to read the influence of even a small provisionon the panel response, which, on the counterpart,cannot be expected to be macroscopic.

One should emphasize that the first objective of thisapplication is, then, to show the sensitivity of the NRTmodel even to small changes in the structural response,very differently from the elastic model, which, on thecontrary, for the specific case, is unable to detect anydifference in the behaviour of the wall. A number ofmore effective reinforcements have also been testedby the authors obviously resulting in more appreciableresults and a much higher performance (Baratta & I.Corbi 2006).

In the specific case, one can notice that, with refer-ence to the same load intensity [e.g. in correspondenceof the load value 3000 N in Figures 2.a–c], lower dis-placements can be recorded in case of FRP insertions.Moreover, the increase of the overall stiffness of thepanel results in a higher loading capacity with respectto the not-reinforced wall. In particular the trend ofeach curve, shows that it is closer to the x-axis (repre-senting the load variable), thus indicating an increasein the stiffness which is also related to an highercollapse value of the load.

2.2 Experimental/theoretical comparison

Actually the application of the general theory of NRTstructures to the considered case of the masonry panel,also in the presence of FRP reinforcements, can pro-duce numerical results which are in good agreementwith the results obtained by the above reported exper-imental campaign (Baratta & I. Corbi 2006). Thespecialization of the general problem to the case ofmasonry walls requires the definition of a discretemodel coupled to the real structural model, the set upof the energetic problem (in the case of masonry pan-els the potential energy approach is to be preferred)for the discrete problem, which, for masonry material,results in a Non Linear programming problem to besolved by means of Operational Research tools, and,finally, the search of the numerical solution of the setup OR problem by means of a suitably implementedcalculus code (Baratta & I. Corbi 2004, 2006).

Once followed the above described steps, thenumerical results can be compared to the ones comingout from the experimental investigation, for the finalvalidation of the theoretical set up.

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Figure 2. Comparison between the numerical (continuous line) and the experimental (dotted line) at the monitored positions1, 2, 3 for the not reinforced panel (a, b, c) and for the reinforced panel (d, e, f).

For the specific case one may compare the resultsrelevant to the first loading cycle with those relatedto experiments. As shown in Figures 2, the theoreticaldata (continuous lines) are in good agreement wit theexperimental ones (dotted lines) both as regards to thenot reinforced case (Figs 2.a–c) and to the consolidatedcase (Figs 2.d–f).

In the first unconsolidated case the masonryexhibits a behaviour which appears lightly stiffer thanin the theoretical model: this effect is maybe due tothe micro-fractures present at the first stage of thecomputational procedure, which are probably absentin the real behaviour of the masonry. The transducers2 and 3 show an overall pretty good agreement betweennumerical and experimental data even if also otherphenomena as sliding between bricks, micro-fractures,etc., should be taken into account, which cause the not

perfect agreement of the diagrams relevant to the firsttransducer 1 (Figs 2.a–c).

It is indeed because of these reasons that the numer-ical/experimental agreement is higher, almost perfect,in the reinforced case (Figs 2.d–f).

In this case, the sliding between bricks are reducedand do not influence the overall characteristics ofdeformability and stiffness of the masonry panel.

3 TESTS ON PROTOTYPES OF MASONRYARCHES

3.1 Experimental investigation

This section reports some of the results of the wideexperimental campaign developed at the Laboratoryof Materials and Structural Testing of the University of

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Figure 3. The portal arch model with the monitoring equip-ment: sketch of the monitoring equipment.

Naples “Federico II” on masonry arches, consolidatedor not by means of FRP strips (Baratta & O. Corbi2003b, 2005b).

The geometry of the portal arch (Figs 3 and 4)is symmetrical and is characterized by spanL = 1900 mm, rise f = 660 mm, arch thickness d =240 mm, piles thickness b = 385 mm, piles heighth = 1700 mm; the arch shape is a semi-ellipse. Thearch depth is 400 mm, whilst the two abutmentsare 480 mm deep. The masonry is characterized byunit weight γ = 12300 N · m−3 and Young modulusE = 5.5 GPa.

As mentioned, in the above, in the second stage ofthe experimental campaign one also considers someFRP continuous reinforcement applied on the archlength. In this case, the FRP reinforcements consistof continuous mono-directional FRP strips applied onthe extrados of the arcade.

The adopted reinforcement, produced by FTS, isa BETONTEX system GV330 U-HT, made of 12 Kcarbon fibre, jointed by an ultra light net of thermo-welded glass.

The mechanical characteristics of the employedfibres are: tensile limit stress σfrp = 4.89 GPa, elasticmodulus in traction Efrp = 244 GPa, limit elongationεfrp = 2%. The FRP strip is characterized by thicknessof 0.177 mm and depth of 100 mm.

After roughly preparing the masonry support inorder to render the application surface smoother, theFRP is directly laminated on the masonry, at the sametime with the impregnation of the fibres by means ofa special bi-component epoxy resin.

As regards the execution the tests, the structure issubject to its constant own weight and to a lumpedhorizontal force F, applied on the top right side ofthe right abutment in the rightward direction in theincreasing phase (Figs 3 and 4), which is transmitted

Figure 4. The portal arch model with the monitoring equip-ment: picture from laboratory tests.

by means of a loading equipment consisting of a loadcell placed on the right side of the portal arch.

This force is able to potentially produce collapse ofthe structure according to a mechanism that is typicalof earthquake failures of arch-portals (Fig. 6), and itis intended to represent a pseudo-seismic action, ableto yield a measure of the structure attitude to sustainearthquake shaking.

The monitoring stuff (Figs 3 and 4) consists of:

– 1 dial gauge G1, placed on the left side of theleft abutment, finalized to the monitoring of theabsolute displacement of the pile;

– 2 transducers T1 and T2, vertically placed on thefront side of the left abutment, finalized to the mon-itoring of the length variation of both edges of thepile;

– 1 inclinometer I1, placed on the top of the left abut-ment, finalized to the monitoring of the pile averagerotation;

– 1 extensometer E1, placed between the two abut-ments, finalized to the monitoring of the relativepiles’ displacement;

– 30 deformometer cells, placed on the front of thearch, finalized to the monitoring of the arcadedeformation.

For the un-reinforced structure (Baratta and Corbi,2003a,b) the critical condition is related to the activa-tion of a collapse mechanism composed by four hingesdistributed as follows:

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– 1 at the keystone on the extrados,– 2 at the reins on the intrados,– 1 at the bottom of the right pile on the extrados.

The collapse condition is reached at F∼80 N; thelow failure value of the force shows that, due to thechosen elliptical shape of the arch, the funicular linecompatible with the applied loads and admissible (i.e.interior to the arch profile) is already very close to theupper and lower bounds of the arch profile at the restcondition.

The experimental force-displacement diagram isreported in Figure 5.

After reaching the collapse condition, the portalarch is then unloaded in order to be prepared for thesubsequent experimental tests on FRP reinforcements.After completing the unloading process, the portal archis prepared for laboratory tests on FRP reinforcements,which are finalized to the evaluation of the benefitsinduced on the model response by the application ofcarbon fibre strips.

The reinforcement consists of a continuous FRPstrip bonded on the extrados of the arch. Since thecollapse mechanism of the not reinforced simple por-tal arch is characterized, as described in the above, bythe formation of two intrados hinges at the reins of thearch, corresponding to the fractures d4–d5and d12–d13at the extrados, the major effect of this interventionis supposed to be the prevention of these fractures,and, therefore, a wide increase in the model loadingcapacity.

The funicular line is now free to exceed the lowercontour of the portal arch cross section.

In this case the critical condition is related to theactivation of a collapse mechanism composed by fourhinges, distributed as follows:

– 1 at the top of the left pile on the intrados,– 1 at the keystone on the extrados,– 1 under the load cell on the intrados of the right pile

(where shear occurs),– 1 at the bottom of the right pile on the extrados.

The collapse is reached at F∼800 N with an increasein the loading capacity of the portal arch of approx-imately 10 times with respect to the unconsolidatedcase. The experimental force-displacement diagram isreported in Figure 7.

3.2 Experimental/theoretical comparison

Actually the application of the general theory of NRTstructures to the considered case of the masonry portalarch, also in the presence of FRP reinforcements, canproduce numerical results which are in good agree-ment with the results obtained by the above reportedexperimental campaign (Baratta & O. Corbi 2005a,2005b, 2007).

The specialization of the general problem to the caseof masonry arches requires the definition of a discretemodel coupled to the real structural model, the set upof the energetic problem (in the case of masonry archesthe complementary energy approach is to be preferred)for the discrete problem, which, for masonry mate-rial, results in a Non Linear programming problem(which in the specific case can be reduced to a LinearProgramming problem) to be solved by means of Oper-ational Research tools, and, finally, the search of thenumerical solution of the set up OR problem by meansof a suitably implemented calculus code (Baratta &O. Corbi 2003a, 2003b, 2005a).

Once followed the above described steps, thenumerical results can be compared to the ones comingout from the experimental investigation, for the finalvalidation of the theoretical set up.

Numerical investigation on the portal arch modelexperimentally tested results in the possibility ofappreciating the skill of the NRT model to capturethe major features of the structure behaviour. More-over also the correct modelling of the reinforcementand of its coupling with the main structure can beevaluated. Figure 5 reports the numerical/experimentalcomparison relevant to the right pile top displacementu (mm) versus the varying load F (N) for the consideredun-reinforced arch.

A very good agreement between the numerical andexperimental data can be observed. The calculus codeis demonstrated to be able to capture the behaviour ofthe portal arch following the whole loading path up tocollapse; Figure 6 depicts the collapse mechanism ofthe structure as it appears directly from the calculuscode, clearly due to the formation of four hinges: oneat the keystone on the extrados, two at the reins onthe intrados, one at the bottom of the right pile on theextrados.

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Figure 6. Unreinforced portal arch: picture of the collapsemechanism captured from the calculus code.

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Moreover one reports in Figure 7 the numeri-cal/experimental comparison relevant to the right piletop displacement u (mm) versus the varying load F(N) for the arch reinforced with an extrados FRPreinforcement.

Again a very good agreement can be observedbetween the numerical and experimental data. The cal-culus code is demonstrated to be able to capture thebehaviour of the portal arch ; Figure 8 depicts the col-lapse mechanism of the structure as it appears directlyfrom the calculus code, clearly due to the formation offour hinges: one at the top of the left pile on the intra-dos, one at the keystone on the extrados, one under theload cell on the intrados of the right pile, one at the bot-tom of the right pile on the extrados. Both numericaland experimental data agree in assessing at approxi-mately ten times the original value the increment of theloading capacity of the structure due to the extradosFRP reinforcement.

Figure 8. Portal arch with extrados reinforcement: pictureof the collapse mechanism captured from the calculus code.

Figure 9. Barrel vault with horizontal directrix.

4 PROTOTYPES OF MASONRY VAULTS

4.1 The problem of barrel vaults with indefinitelength

As regards to barrel vaults (Baratta & O. Corbi 2007),first of all, one should consider that, since the vaultgeometrically derives by the translation along a direc-trix of a generating arch curve, in this case, themeridian lines coincide with the generatrix in theirshapes; if one considers a rectilinear directrix, the vaultparallels are horizontal and rectilinear as well (Fig. 9).The surface of the shell of the mid-surface of the vaultmay be defined by the equation z = f(x). Because ofthe vault geometry, one has

where dsx and dsy denote the length of the sides ofthe generic vault element ABCD of area dA dx and

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dy the length of the corresponding sides on the ele-ment A′B′C′D′ projected in the xy-plane, and ϕ and θdenote the angles formed by the meridian sidesAB andDC of the element with the x-axis and by the parallelsides AD and BC with the y-axis, respectively. As con-cerns equilibrium, hypothesizing that the vault is in amembrane state of stress, a correspondence can be setbetween forces acting on the element ABCD (stressesNx, Ny, Nxy = Nyx and applied load for unit area, px,py, pz) and projected forces acting on the associatedelement A′B′C′D′ (Nx, Ny, Nxy = Nyx and px, py, pz)in the xy-plane (Baratta & O. Corbi 2007).

In absence of horizontal loads and if the verticalload is not dependent on “y”, as it happens when thevault is subject to only vertical loads due to the self-weight (i.e. pz = pz(x) ≥ 0), and assuming that the vaulthas an indefinite length in the direction y, equilibriummay be expressed in the form

which reduces the problem to the determination ofstress function ψ(y).

Assuming that the directrix curve of the vault isa circular arch (Fig. 10) of radius R, with constantthickness “s” and unit weight γ , and imposing suit-able constraint conditions, one yields the final solution(Baratta & O. Corbi 2007)

with

Figure 10. Cross section of a barrel vault with circular archgeneratrix.

where z0 and z1 are arbitrary ordinates, conditioned bythe fact that z(t) should be contained in the interior ofthe profile of the vault.

After this result, it is possible to calculate theinternal forces Nx ≤ 0, Ny = Nxy = 0 and Nx ≤ 0,Ny = Nxy = 0

It is also possible to realize that the equilibriumsolution allows the structure to behave as a sequenceof identical independent arches. From this result, onemay refer to the results reported in the previous sectionfor the portal arch model, reinforced or not with someFRP strips, whose analytical problem implementationhas been shown to give theoretical results in perfectagreement with the produced experimental data, alsoexhibiting very effective results in the reinforced case.

5 CONCLUSIONS

The paper reports some results proving the successfulapplication of a correct theoretical treatment, basedon the NRT material assumption, of structural prob-lems relevant to classical masonry typologies such asarches, walls and vaults.

The set up of the general energetic approach foranalyzing masonry structures under live loads, itsspecialization to the relevant discrete models, theimplementation of ad hoc built up calculus codes aredemonstrated to produce numerical results in verygood agreement with data produced by experimentalinvestigation.

One should emphasize that, differently from manymodels which require a number of parameters allow-ing a certain adaptation of the shape of the numericalcurve to the experimental one, the NRT model has thebig advantage that the only mechanical parameter tobe evaluated is the masonry elastic modulus. Since thetuning of the theoretical model is pretty simple, therewould be no possibility to force it to produce theo-retical results fitting with such a good agreement theexperimental data, because the tuning operation itselfcannot influence the shape of the numerical diagrambut only the displacements scale.

As a point of fact, the sensitivity of the mod-elling to material assumptions reduces to the inverseproportionality between the material elastic modulusand displacements, without any influence on the loadcapacity and on the evolution of displacements withthe loads.

Actually the extension to the case of some rein-forcement directly applied on the masonry can also bestudied by properly modeling the reinforcement itselfand its connection with the masonry.

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The theoretical/numerical agreement, which is anoriginal result also for the case of masonry con-structions with FRP reinforcements, demonstrates thatthe overall approach is reliable for the treatmentof masonry constructions also in the presence ofconsolidation interventions.

REFERENCES

Baratta, A. 1984. Il materiale Non Reagente a Trazione comemodello per il calcolo delle tensioni nelle pareti murarie.J. Restauro, 75: 53–77.

Baratta,A. 1991. Statics and reliability of masonry structures.In General PrinciplesApplications in Mechanics of Solidsand Structures. CISM, Udine: 205–235.

Baratta, A. & Corbi, I. 2003. Investigation of FRP consol-idated masonry panels. Proc. 9th Int. Conf. Civil andStructural Engineering Computing. Egmond Aan Zee,Netherlands.

Baratta, A. & Corbi, I. 2004. Iterative procedure in No-Tension 2D problems: theoretical solution and exper-imental applications”, In G.C.Sih & L.Nobile (eds),Restoration, Recycling and Rejuvenation Technology forEngineering and Architecture Application, Aracne Ed:67–75, Bologna.

Baratta,A. & Corbi, I. 2006. On the reinforcement of masonrywalls by means of FRP provisions. In A. Mirmiran &A. Nanni (eds) Composites in Civil Engineering., Miami,2006.

Baratta, A. & Corbi, O. 2003a. Limit Analysis of No Tensionbodies and non-linear programming. Proc. 9th Int. Conf.

Civil and Structural Engineering Computing. EgmondAan Zee, Netherlands.

Baratta, A. & Corbi, O. 2003b. The No Tension model for theanalysis of masonry-like structures strengthened by FiberReinforced Polymers. J. Masonry Intnl. 16 (3): 89–98.

Baratta,A. & Corbi, O. 2004.Applicability of DualityTheoryto L.A. Problems with Undefined Flow Law. Proc. 1stInt. Conf. RRTEA Restoration, Recycling and RejuvationTechnology for Engineering, Cesena.

Baratta, A. & Corbi, O. 2005a. On variational approaches inNRT continua. J. Solids and Structures. 42: 5307–5321.

Baratta, A. & Corbi, O. 2005b. Fibre reinforced compositesin civil engineering. Experimental validation of C-Fibremasonry retrofit. J. Masonry Intnl. 18(3): 115–124.

Baratta, A. & Corbi, O. 2007. Towards a new theory formasonry vaults assessment. Proc. 7Th Int. Masonry Conf.London.

Baratta, A., Vigo M. & Voiello G. 1981. Calcolo di archi inmateriale non resistente a trazione mediante il principiodel minimo lavoro complementare. Proc. 1st Nat. Conf.ASS.I.R.C.CO. Verona.

Bazant, Z.P. 1996. Analysis of work-of fracture method formeasuring fracture energy of concrete. J. EngineeringMechanics ASCE 122 (2): 138–144.

Como M. & Grimaldi A. 1983. A unilateral model for LimitAnalysis of masonry walls. In Unilateral Problems inStructural Analysis, Ravello: 25–46.

Heyman, J. 1966.The stone skeleton. J. Solids and Structures,2: 269–279.

Khludnev, A.M. & Kovtunenko, V.A.. 2000. Analysis ofcracks in solids. Applied Mechanics Review, 53 (10).

Rao, S.S., Optimization: Theory and applications, Wiley &Sons, New Delhi, 1978.

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