paper anidis

8/3/2019 Paper Anidis

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Keywords: Irregular masonry, Limit analysis, Equivalent frame approach, FE method

ABSTRACTMasonry walls made of stones/bricks joined by mortar with poor mechanical properties, resulting in very irregularstructures, are very common in ancient buildings. In this case, piers and spandrels ultimate strength is not alwayswell predicted by simplified formulas suggested by codes of practice, which typically are tailored to regulartextures. In this framework, a two-step numerical model for the pushover analysis of in-plane loaded masonry wallsconstituted by the irregular assemblage of stones/bricks is proposed. In Step I, ultimate bending moment-shearforce strength domains of structural elements are derived by means of a heterogeneous upper bound FE limitanalysis and the results are stored in a database. In Step II, a frame model of entire walls is assembled, where piers

and spandrels are modeled as elasto-plastic Timoshenko beams. At each analysis step, the internal forces of each

structural element are compared to the failure loads stored in the database created in Step I. If the capacity isexceeded, suitable plastic hinges are introduced at the end of the structural elements. Furthermore, the resistance ofthe element is set to zero when a limit drift is exceeded. With the numerical tool developed, a real scale masonrywall arranged in irregular texture is analyzed in the inelastic range under increasing static loads. Results arecompared with those obtained by a very expensive plane-stress non linear heterogeneous approach, performedthrough the code STRAND and with those obtained by a standard approach available in common design practice,performed through the commercial code PC. E (AEDES 2010).

1 INTRODUCTION

Historical masonry buildings, especially in theSouthern Italy, are usually realized with irregularstones joined through mortar with poor mechanical

properties. For this reason, piers and spandrelsultimate strength cannot be predicted by simplifiedformulas suggested by codes of practice, as forinstance D.M. 2008, which are typically tailored toregular patterns. At the present stage, to propose acomprehensive numerical model able to give reliableinformation on the non linear behaviour of suchtypology of buildings, taking properly into account

the actual texture of the walls, seems a prohibitivechallenge. In this paper, a novel two step approachfor the pushover analysis of masonry walls with

irregular texture is presented and applied to a studycase. This model still uses for the global analysis ofthe masonry walls a simple equivalent frame model(Milani et al. 2009, Magenes and Della Fontana1998, Belmouden and Lestuzzi 2009) and is able, atthe same time, to estimate masonry macroscopicmechanical properties taking into account the actualmasonry texture. The final aim is to put at disposal to

practitioners a very simple instrument to be used inordinary design, but at the same time capable ofgiving realistic estimations of the load carryingcapacity of structural elements realized by irregularassemblages of stones with variable shape and size.The tool is based on a two step procedure. In the first

step (Step I), ultimate bending moment-shear forceresistances of piers and spandrels are derived bymeans of a heterogeneous upper bound FE limit

A Simple Model for Pushover Analysis of Masonry Walls with

Irregular Texture

Giuseppe Alfredo CundariPhD, Dipartimento di Meccanica e Materiali Universit Mediterranea di Reggio Calabria, Via Graziella,

Localit Feo di Vito, 89124 Reggio Calabria

Gabriele MilaniAssistant Professor, Dipartimento di Ingegneria Strutturale (DIS) Politecnico di Milano, Piazza Leonardoda Vinci 32, 20133 Milano

Adolfo SantiniFull Professor, Dipartimento di Meccanica e Materiali Universit Mediterranea di Reggio Calabria, Via

Graziella, Localit Feo di Vito, 89124 Reggio Calabria


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analysis (Sloan and Kleeman 1995) and the resultsare stored in a database. Piers and spandrels of anexisting oil-mill located in Calabria (Southern Italy)are meshed by means of triangular elementsfollowing the actual geometry of units and mortar

joints, on the base of a precise survey of the 2Dtexture of each structural element. A large databaseis collected, considering the behaviour of eachspandrel and pier belonging to the structure. A veryrefined discretization is used in order to take intoaccount as close as possible the effect of the irregulartexture and the presence of preferential planes ofweakness. A heterogeneous limit analysis is adopted

because this is suitable for computing failure loads ofcomplex structures with a moderate computationaleffort. The limit analysis is carried out for the rangeof expected axial loads in the structural elements andthe expected relative rotations and displacements of

piers and spandrels two parameters that were foundto affect significantly the strength of unreinforcedmasonry. Appropriate static and kinematic boundaryconditions are imposed to account for the complexinteraction of internal forces and deformed shapes ofsingle elements. Once collected structural elementsstrengths at the meso-level, in the second step(macro-level), a equivalent frame model of themasonry wall is built. In this model, spandrels and

piers are modelled as elastic Timoshenko beamelements with concentrated rigid-plastic hinges at the

extremes. The strength of the piers and spandrels isdefined by the strength domains stored in thedatabase. At each analysis step, the bending momentand shear demands are compared to the respectivecapacities. If the capacity is exceeded, a plasticflexural hinge is introduced at one or both theextremes of the equivalent beam depending on thetype of achieved failure. In order to test the reliabilityof the approach proposed, a small two-bay two-storey wall is extracted from the structure andanalyzed as isolated panel when subjected toincreasing horizontal loads up to collapse. An

expensive 2D plane stress elasto-plasticheterogeneous approach (Strand 7 2004) is also

performed on the panel, to have at disposal areference global pushover curve, to compare to thatobtained with the model here presented. Pushovercurves obtained by using a commercial equivalentframe code (PC.E, Aedes 2010), where formulas ofthe Italian technical rules are implemented, are alsoreported. Moreover, it is found that the simpleapproach proposed seems generally more reliablewith respect to an equivalent frame software based

on formulas of the technical regulations, despite the

fact that a reasonable estimation of the failure loadmay be obtained also in this latter case.

2 DESCRIPTION OF THE REFERENCE

STUDY CASE

The building under study (see Figure 1) is an ancientoil-mill belonging to the family Nesci from the earlytwentieth century and located in the town of BovaMarina (RC), in the center of a flat area between theSan Pasquale river to the west, the Ionian coast to thesouth and the Agrillei hill to the east. This masonrystructure is composed of a main body withrectangular plan and two smaller bodies which giverise a overall planimetry in the shape of "C". Thedevelopment in elevation of the main body of the

building is two stories above ground, while the twoadvanced and symmetrical bodies have a single

storey. A basement 140 cm high and realized withlarge stiff and resistant irregular stones joinedthrough thick mortar with poor mechanical

properties is also present. This part of the perimeterwalls actually coincides with the continuation of thefoundation structure to above the ground surface.The remaining part of the walls is made of smallerstones, pieces of bricks and mortar with veryirregular texture, except for the presence of a fewhorizontal thin layers, equally stepped in vertical,obtained utilizing quite regular small clay bricks. A

reinforced concrete ring beam with rectangularsection (dimensions 40x80 cm), lightly reinforced (4 bars of diameter 10 mm at the corners) and withconcrete with relatively poor mechanical properties(assumed here C12/15) is also present between thefirst and second floor. Ground floor thickness isassumed equal to 80 cm, whereas the last floor wallsare assumed 50 cm thick. The slab of the first floor ismade by little brickwork vaults built with thin bricksand supported by iron beams with cross-section todouble "T". The supporting structures of the roofsare set up by timber trusses built with poor technical

implementation and currently very degraded. Thecoating of roofs is done with Marseillais tiles. Thefirst floor is made by small clay bricks vaultssupported by double "T" steel beams, whereas theroof is sustained by a truss-like timber structure,currently very degraded. For each pier and spandrel

belonging to the building under consideration,strength domains are numerically obtained in termsof ultimate bending moment ( uM ), ultimate shear( uV ) and compressive force (N).


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0 1 2 5 10

0 1 2 5 10

N

0 1 2 5 10

N

Figure 1. The ancient masonry oil-mill under study.

3 STEP I: LIMIT ANALYSIS NUMERICAL

MODEL

All the numerical models adopted in this work,to perform the limit analysis, derive from anupper bound approach based on the kinematicdiscontinuous formulation originally presented bySloan and Kleeman (1995), which has already

been applied successfully to various masonryproblems (e.g., Milani et al. 2006) The formulation

is based on a triangular discretisation of the 2Ddomain and on the introduction of discontinuitiesof the velocity field along the edges of adjacenttriangles. At each node j, a horizontal velocity

j

xu and a vertical velocityj

yu are introduced. Theresulting velocity field within a triangularelement is linear whereas the strain rate field isconstant. Across the interfaces a linear velocity

jump is assumed and therefore for each interfacefour unknowns (

T

tntn uuuu ][2211

=u )are introduced representing the normal (

j

nu ) andtangential (

j

tu ) velocity jumps with respect tothe discontinuity direction evaluated at the nodes

1=j and 2=j of the interface. A full descriptionof the heterogeneous FE upper bound limit

analysis model used in this paper is given inCundari and Milani (2010) and Milani et al.(2009) and the reader is referred there for furtherdetails. Here, it is worth remembering that, froma numerical point of view, the evaluation ofthe ultimate load bearing capacity of both piersand spandrels can be evaluated solving asuitable linear programming problem, wherethe objective function is the total internal

power dissipated minus the power dissipated

by external loads not dependent on the loadmultiplier:

( ) ( )[ ]

=

+

0

0

bUA

uPbb

ass

I

ass

E

eqeq

assTass

I

Tin

assI

ass

E

Tin

ass

thatsuch

min 0,

(1)

wherein

assb andin

assI,b are the assembled right-handsides of the equalities, which determine thelinearised failure surface of the material of thecontinuum and of the interfaces, respectively, 0P is the vector of permanent loads,

][ass

I

assass

E

assuuU = is the vector of

global variables, which collects the vector of


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assembled nodal velocities (

assu ), the vector of

assembled element plastic multiplier rates (ass

E ),

the vector of assembled velocity jumps oninterfaces (

assu ), and the vector of assembled

interface plastic multiplier rates (ass

I );

eqA is the

overall constraints matrix and collects velocity boundary conditions, relations between velocity jumps on interfaces and elements velocities,constraints for plastic flow in velocitydiscontinuities, constraints for plastic flow incontinuum and normalization conditions.

3.1 Piers strength domainsWithin an equivalent frame approach context, ageneric pier can be schematically represented bya shear deformable beam with 4 dof represented

by nodal rotation ( 1 and 2 ) and displacementsperpendicular to the beam axis ( 1u and 2u ), seeFigure 2. Due to the very low axial deformabilityof the piers, we suppose that displacements along

beam axis are negligible. We consider that aninteraction between axial force N and bendingmoment uM , and axial force N and ultimate shear

uV occurs. A rigorous approach for piers wouldrequire the determination of the ultimate shear( uV ) and the ultimate bending moment ( uM )failure surfaces taking into account all the

possible combinations of the following


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a)

u1

u2

Limit Analysis

Model

u = u1 -u2

2= 0

1= 0

2= 0

1= 0

equivalentbeam

b)

1

2

P A

Q B B

A

H

1

2=1

2 (Q-B)

1 (P-A)

u

v

v (A)v (P)

E1

E2

c)

M1

M1Elastic Analysis

on the Timoshenko

beam element

V1

V2

Limit Analysis

Model

u1 = 0

u2 = 0

u2 = 0

u1 = 0

1

2

equivalentbeam

x

y

1= 1

2=1

1= 1

2=1

Figure 2. General procedure adopted for the determination of masonry piers strength domains. (a): shear limit analysisproblem; (b): additional equality constraints to impose for the flexural behavior; (c): bending limit analysis problem.

kinematic/static input variables:

1. the ratio 12/= between foot andhead rotations;

2. the ratio ( ) ( )221 / Huu = , where 1u and 2u are top and bottom horizontaldisplacements and H is the pier height;

3. the applied pre-compression N.

Nevertheless, this approach would require an

almost prohibitive computational effort, also inlight of the very refined discretizations adopted in

this paper to reproduce the actual microstructureof the structural elements. For this reason, wereasonably assume that ultimate shear andultimate bending moments are uncoupled, thusevaluating ultimate shear uV simply as a functionof vertical pre-compression N and keeping

012 == . Considering the importance of relativehead and foot rotations on the evaluation of the

ultimate bending moment, uM is computed as afunction of ratio and vertical pre-compression.


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On the other hand, within a limit analysisframework, velocity field has to be consideredinstead of displacement and rotation fields. It istherefore necessary to evaluate piers resistance asa function of rotations rates ratio 12/ = . Inthis way, the limit analysis procedure proposed

furnishes, at fixed pre-compression, two values ofthe ultimate shear (depending if the pier is loadedfrom west to east or vice versa) and a series of

bending moments associated each one to a singlevalue of the non-dimensional parameter . In theframework of a pushover analysis, at the end ofthe iteration i , the axial force ( )iN acting on the

beam as well as the rotations ( )i1 and ( )i2 at the beam extremes are known. The coefficient

( ) ( ) ( )iii 12 / = is defined as the ratio of therotations at beam extremes at the iteration i and

is kept also equal to rotation rates ratio. Given( )i and ( )iN , corresponding uM and uV values are

computed from the collected database andcompared to the actual bending moment andshear, to determine if the piers strength has beenexceeded. Differently from a standard limitanalysis problem, additional kinematic boundaryconstraints must be imposed on piers toreproduce, in the collapse mechanism, a fixedratio of rotations (rates) 12/= of the lower( 2 ) and upper ( 1 )boundary (see Figure 2)For spandrels, the same considerations may berepeated, provided that such constraints areapplied to the left and right boundaries. In aheterogeneous FE limit analysis, the kinematicconstraint on is taken into account by imposing aconstraint on the rotational velocities of top and

bottom edges ( 1 and 2 ). A full descriptionof the additional constraints to impose to the FElimit analysis model can be found in Milani et al.(2009) and they are omitted here for the sake ofconciseness Here it is worth noting that mayrange between Inf and +Inf, since it is possiblethat 12 > . In order to cover all the possibilities

that can be encountered (at least theoretically) atstructural level and considering that in limitanalysis represents a ratio between rotationsrates, it may be useful to limit the range ofvariability of between -1 and 1 and tointroduce a further non-dimensional variable '

defined as 21/' =

ranging between -1 and 1.By means of the alternative imposition ofboundary conditions related to and ' , all the possibilities that can be encountered in practicemay be covered. In Figure 3 and Figure 4, fewresults of a massive numerical analysis campaignconducted using a heterogeneous FE limitanalysis software (Cundari and Milani 2010)are summarized. In particular, in Figure 3, twodeformed shapes at collapse of one ground floor

pier subjected to shear at increasing pre-

compression and the corresponding ultimate shear-vertical pre-compression curve are represented. Therole played by vertical pre-compression in thechange of the failure mechanism is particularlyevident. The presence of irregular cracks whichzigzag between rigid stones suggests once againthat simplified formulas provided by technicalnorms may be not reliable and well suited for thistypology of masonry. Figure 4 shows somedeformed shapes of the same pier subjected to

bending moment. In these simulations, ismaintained equal to zero, whereas the vertical

pre-compression is increased starting from zeroand ending to the 80% of mortar compressivestrength. The corresponding vuM curves arealso reported in Figure (here v indicates verticalstress acting at the top of the pier). As expected, amoderate vertical pre-compression increasesconsiderably the ultimate bending resistance ofthe pier, whereas for v exceeding 50% of thecompressive strength of the joint the load

bearing capacity of the. structural elementdecreases, which seems again in agreement withexperimental evidences.

Figure 3. Pier 1, deformed shape at collapse corresponding to Vu at two increasing levels of pre-compression and


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corresponding shear strength as a function of vertical pre-compression.

Figure 4. Pier 1, Mu (clockwise) strength domain at different pre-compression levels and corresponding deformedshapes at collapse.

4 STRUCTURAL IMPLEMENTATION ON

A SINGLE WALL IN-PLANE LOADED

In order to test the reliability of the simpletwo-step frame model proposed at a structurallevel, a benchmark medium size masonry wall

built in irregular texture, as depicted in Figure 5,is analyzed when subjected to a standard pushoveranalysis. A computationally very expensive plateheterogeneous mesh is built using a commercialelastic-plastic software, namely Strand 7 (2004),to have at disposal a reference global pushovercurve, to compare to that obtained with thesimple inexpensive frame model presented. Over200000 elements (see Figure 5) have beenused to discretize the wall, assuming for theconstituent materials the hypothesis of perfect

plasticity. For joints, a classic Mohr-Coulombfailure criterion has been adopted, obeying anassociated flow rule, with friction angle andcohesion equal to 37 and 1.4ft (ft = 0.02 MPa),respectively. Mortar Young modulus is assumedequal to 1300 MPa, shear modulus equal to 220MPa and Poisson ratio equal to 0.25, in

agreement with experimental data collected in theoil-mill (Cundari and Milani 2010). Irregular

blocks and bricks are modelled with linear elastic

elements (Young Modulus: 15000 MPa, Poissonratio: 0.20). It is authors' opinion that only anexpensive 2D heterogeneous approach mayrepresent a reliable reference model to test the

performance of a simplified frame approach.However, the non linear analysis required morethan 8 days to run, using a PC equipped with 5Gb RAM and a Pentium Dual Core 2.10 GHz,within a Windows 7 64 bit OS. Obviously, thelong time needed to complete such a non-linearsimulation precludes totally the usage at

professional level by any practitioner and analternative suitable approach is needed. With theaim of comparing results also with standardapproaches available in common design practice,the commercial code PC. E (Aedes 2010) has

been also utilized. Within this software, themasonry walls are modeled by means of anequivalent frame approach, where the practiceformulas of Italian regulations (D.M. 2008) areutilized to evaluate piers and spandrels ultimateresistance. When dealing with the Italian rules,reference is made to new or existing buildings


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Pier Spandrel RC Ring Beam Rigid Offset

290 150 605 150 290

1485

Spandrel 1

Spandrel 2

Spandrel 3

Spandrel 4

Pier 1

Pier 2

Pier 3

Pier 4

Pier 1

Pier 2

50

80

Wall thickness

roof level

1st floor level

390

300

80

250

6060

175

65

Spandrel 1

Spandrel 2

Spandrel 3

Spandrel 4

140

Section geometry (cmxcm)

Pier 1

Spandrel 1

Pier 2

Pier 3Pier 4

Spandrel 2

Spandrel 3

Spandrel 4

290x80

290x50

605x80605x50

80x80

40x80

85x50

120x50

Figure 5. Geometry of the wall realized with irregular texture and its discretization into a frame model and a heterogeneous FEapproach

made with regularly assembled masonry, raisingdoubts on their reliability for an ancient buildingwith very irregular masonry texture. Due to theimpossibility to set ad hoc masonry mechanical

properties in the commercial code, two differentmodels (hereafter labelled as I and II are tested).The first is a frame where cohesion for masonry

is kept equal to tfc 4.1= , whereas for the secondcohesion is evaluated through a plane stress

condition imposition (with known values offriction angle and tensile strength), which givesan indirect evaluation of cohesion as

MPa0.02=c . Due to the impossibility to set properly all input data, it is expected that astandard software is unable to provide good

predictions of the non linear behaviour under

horizontal static loads of walls assembled inirregular texture. The structural example under


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consideration is a two storey two bay wall,with approximate dimensions equal to 14.85 x6.90 m (length x height). Small openingsregularly distributed are present both at theground and first floor, making piers

particularly squat. A basement 140 cm high andrealized with large stiff and resistant irregularstones joined through thick mortar (modelledwith plate elements) with poor mechanical

properties is also present. This is a typicalsituation for existing buildings, where the firstfloor level was placed in elevation with respect tothe ground. The remaining part of the wall is builtwith smaller and less stiff irregular stones, again

bonded with thick mortar joints with irregulargeometry, except for the presence of a fewhorizontal thin layers, equally stepped in vertical,obtained utilizing quite regular small clay bricks.

A reinforced concrete ring beam withrectangular section (dimensions 40 x 80 cm),lightly reinforced (4 bars of diameter 10 mm atthe corners) and with concrete with relatively

poor mechanical properties (assumed hereC12/15) is also present between the first andsecond floor. The presence of the ring beam

justifies the choice of an equivalent frameconstituted by three separate spandrels at the firstfloor level. Indeed, it is reasonable to assume thatthe r.c. beam and masonry disposed below and on

top of the ring beam undergo independentdeformations. Ground floor thickness is assumedequal to 80 cm, whereas the last floor walls areassumed 50 cm thick. Vertical loads are mainlyconstituted by masonry self weight, which isassumed equal to 2000 Kg/m3. Horizontal loadincreased until failure is applied following a firstmode distribution. While in the equivalent frameapproach, horizontal static seismic action issimulated with concentrated forces on nodes

belong to the first and second level, in theheterogeneous 2D model, loads are applied at

each element centroid, which seems reasonableconsidering that the vertical loads are mainly dueto masonry self weight. In Figure 6, the pushovercurve (load direction: west-east) obtainednumerically with the model proposed is depicted

and compared with those obtained by using thecommercial codes. Additionally, the proposedmodel is slightly modified in order to reduceductility of the elements proposed by the Italiantechnical rules (D.M. 2008). In this case, indeed,where piers are particularly squat and with verylow deformability (thanks also to their thickness),it seems not reasonable to link the deformationcapacity with the height of the elements. It isauthors opinion that this assumption probablywould overestimate the ductility of the structure.Here, an alternative approach is proposed,limiting the ultimate deformation capacity of theelements to be equal to 1.5 times the elastic limitdeformation, both in shear and flexion, inagreement with the classic POR approach (D.M.1981). From an overall analysis of pushovercurves results, it can be stated that pushover

curves provided by the present approach are veryclose to that obtained with the heterogeneousexpensive 2D Strand mesh, meaning that thenumerical procedure presented seems quitereliable.

0 2 4 6 8 100

500

1000

1500

Roof displacement [mm]

Baseshear[kN]

A

Present numerical model: u

uD.M.2008

Present numerical model: u=1.5*

e

2D elasto-plastic

1D commercial code I

1D commercial code II

Figure 6. Comparison between pushover curve obtained withthe present frame model, the expensive 2D heterogeneousapproach and the commercial equivalent frame software.


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Present model shear failure

Present model

Commercial code

Diagonal cracking shear failure

Sliding shear failure Flexural failure

Figure 7. Deformed shapes near the collapse obtained by the proposed frame approach, the commercial frame software and theexpensive 2D heterogeneous approach.

The extra resistance of the proposed model withdeformation capacity from the D.M. 2008 slightlyoverestimate the maximum base shear reached bythe structure, whereas the approach with themodified deformation capacity seems moreconservative and therefore preferable in design

phase. From an overall analysis of the resultsobtained, the following key issues may behighlighted:

The peak total shear at the base provided by the commercial frame software where

practice formulas of Italian technical rulesare implemented is lower with respect tothe present approach and the reference2D solution, despite the fact that model Iis capable of give acceptable predictionsof the peak base shear. As expected,formulas provided by the Italian regulationsdifficultly can be adapted to a situationwhere texture is strongly irregular andwhere cracked zones, which zigzaginside thick mortar interspersed between


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stones, cannot be predicted easily withsimplified approaches.

The failure mechanism provided by theframe model approach involves piers ofthe first floor in shear and all thespandrels (see Figure 7), which typically

fail in bending for rotation ratios at theextremes near 1. As expected, the centralground floor pier, which is very squat,fails for the formation of shear hinges. Itscontribution to the global non linearresponse of the wall is predominant withrespect to the other deformable elements.The deformed shape of the 2Dheterogeneous model seems similar to that

provided by the frame approach.

The role played by the spandrels in thenumerical model is fundamentals at lowlevels of the external load and lowdeformations. Due to their small dimensionsand small strength if compared to those ofthe piers, they fail very early during thedeformation process, resulting in a prematuredecrease in the global stiffness of the

pushover curve and, when their ultimatedrift is exceeded, in moderate reductionsof the load bearing capacity of the wall.

5 CONCLUSIONS

A simple model for the pushover analysis ofancient masonry walls in-plane loaded andconstituted by the assemblage of irregular stonesand mortar with poor mechanical properties has

been presented. The approach is based in thepreliminary determination of piers and spandrelsstrength domains through a heterogeneous FElimit analysis package and in the successiveimplementation of the strength domains collectedin a database in an equivalent frame model

program. The novelty of the method relies on thecomputation of the strength domains for piers andspandrels assembled in irregular texture by meansof limit analysis and their implementation into aframe analysis software. Due to the geometricaldimension of the piers, the very irregular textureof the walls and the small dimension of thestones/blocks if compared to the overall width ofthe piers, a considerable number of heterogeneouslimit analysis problems with several optimizationvariables have been solved. In the first step,

isolated masonry piers and spandrels have beenanalysed. In both cases, shear and bendingstrength have been uncoupled, assuming that theultimate shear depend exclusively on vertical pre-

compression and that the ultimate bending is afunction of the axial load applied to the spandreland the relative end rotations of the beam, whichcharacterize the type of loading to which the

beam is subjected. For spandrels it is assumedthat the deformation of the large zones where the

beam and pier axes cross, does not significantlyaffect the capacity of the spandrel. Thisassumption is also corroborated by numericalresults reported in Milani et al. (2009). Whencarrying out the limit analysis, two mainassumptions were necessary:

1. it was assumed that the ratio of rotationalvelocities applied in limit analysis can beset equal to the ratio of the end rotationsof the structural element, which is relevantfor the frame analysis;

2. it was postulated that the ratio of endmoments 12/MM= applied for theevaluation of ultimate bending moments,which are both increased by the plasticmultiplier during the limit analysis, can

be estimated from the moment distributionof an elastic Timoshenko beam subjectedto end rotations with 12/= .

Limit analysis results obtained both for piers andspandrels have been then stored in a database,which has been used to analyze by means of anon commercial software a mid-size masonry

wall (two bays, two stories) arranged in irregulartexture and loaded with an equivalent staticseismic load increased until the failure of wall.Structural pushover curves obtained with the

proposed model have been finally compared withthose provided by commercial codes and adiscussion on the differences found has beenreported in order to evaluate the limitations andthe capabilities of simplified formulas provided

by the Italian technical rules. It is found thatparticular care should be used by practitioners in

the utilization of formulas tailored for regulartexture in case of ancient masonry structureshaving irregular texture and mortar with poormechanical properties.

REFERENCES

Aedes 2010. Users' Manual.

Belmouden, Y., Lestuzzi, P., 2009. An equivalent frame

model for seismic analysis of masonry and reinforcedconcrete buildings, Construction and buildingmaterials, 23 (1), 40-53.

Cundari, G.A., Milani, G, 2010. Homogenized andheterogeneous limit analysis model for the pushover

analysis of masonry walls with irregular texture, under

review.


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D.M. 2/7/81, 1981. Normativa per le riparazioni ed il

rafforzamento degli edifici danneggiati dal sisma nelle

regioni Basilicata, Campania e Puglia.

D.M. Infrastrutture e Trasporti, 2008. Norme tecniche per lecostruzioni, Italian National Norms on Constructions.

Magenes, G., Della Fontana, A., 1998. Simplified non-

linear seismic analysis of masonry buildings, Proc. ofthe British Masonry Society, 8, 190-195.

Milani, G., Loureno, P. B., Tralli, A., 2006. Homogenised

limit analysis of masonry walls Part I: failure surfaces,Computers and Structures, 84, 166-180.

Milani, G., Beyer, K., Dazio, A., 2009. Upper bound limit

analysis of meso-mechanical spandrel models for the

pushover analysis of 2D masonry frames,Engineering

Structures, 31 (11), 2696-2710.Sloan, S.W., Kleeman, P.W., 1995. Upper bound limit

analysis using discontinuous velocity fields, ComputerMethods in Applied Mechanics and Engineering, 127(1-4), 293-314.

Strand 7.2. (2004). Theoretical Manual. Available from: Sydney, Australia.

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