a frame element model for the nonlinear analysis of frp strengthen masonry

8/12/2019 A Frame Element Model for the Nonlinear Analysis of FRP Strengthen Masonry

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Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleA Frame Element Model for the Nonlinear Analysis ofFRP-Strengthened Masonry Panels Subjected to In-Plane Loads

Ernesto Grande, Maura Imbimbo, Alessandro Rasulo, and Elio Sacco

Department of Civil and Mechanical Engineering (DICeM), University of Cassino and Southern Lazio,Via G. Di Biasio , Cassino, Italy

Correspondence should be addressed to Ernesto Grande; [email protected]

Received May ; Accepted August

Academic Editor: Vladimir sukruk

Copyright Ernesto Grande et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A rame element model or evaluating the nonlinear response o unstrengthened and FRP-strengthened masonry panels subjectedto in-plane vertical and lateral loads is presented. Te proposed model, based on some assumptions concerning the constitutivebehaviour o masonry and FRP material, considers the panel discretized in rame elements with geometrical and mechanicalproperties derived on the basis o the different states characterizing the sectional behaviour. Te reliability o the proposed modelis assessed by considering some experimental cases deduced rom the literature.

1. Introduction

Masonry constructions certainly constitute an important parto the existing buildings in several countries and, in manycases, o their cultural heritage []. Past and recent seismicevents [,] have pointed out a signicant level o vulner-ability o such constructions and have posed the necessityto develop adequate and effective retrot and strengtheningsystems able to improve their seismic resistance.

In the last decades, strengthening techniques based onthe use o ber-reinorced polymers (FRPs) materials havebeen proposed. Among these, one o the most commonly

used technique is represented by FRP strips which can beexternally applied on both the concrete masonry structures[].

Te literature related to FRP-strengthening systems hasclearly demonstrated the capacity o such systems to improvethe structural perormances o masonry structures [].Some o the major drawbacks in modeling FRP-strengthenedstructures are certainly related to the interaction mechanismbetween masonry and FRPs at their interace layer. Manystudies have specically investigated the bond response oFRP applied on masonry supports [], and some o themhave provided theoretical laws or modeling such behaviorand developing numerical models [].

On the other side, some studies have ocused on thedenition o reliable numerical tools to be used in the assess-ment o the perormances o the strengthened constructionsand in the design o the strengthening systems themselves.Among these are the approaches developed within the niteelement ormulation ramework. In particular, many authorshave proposed modeling approaches based on the use ononlinear interace elements interposed between the supportand the FRP strengthening and characterized by constitu-tive laws derived by tests or by theoretical considerations[,]. Other approaches have considered homogenizationprocedures or including the behavior o the FRP/supportinteraceintothe model o the FRP-reinorcement or, directly,

inthe masonrymodel []. Inthis context, theapproachesbased on the schematization o masonry elements strength-ened with FRPs by using rame elements are o particularinterest since theyare in accordance with the equivalent frameapproach which is useul in the design and implemented inmany commercial computer codes or the structural analysiso masonry structures [].

Te model herein proposed or the nonlinear analysiso masonry panels strengthened with FRPs belongs to theequivalent rame approach and aims at providing a simpleand effective tool to be used in the phase o assessment omasonry structures and, also, in the preliminary design o theFRP-strengthening systems.


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Advances in Materials Science and Engineering

N

V

H

B

t

t

x

t

x

t

(a)

(b)

(c)

(d)

max y

max y

max > y

max > y

xy

F : States characterizing the behavior o the cross-sectionscomposing the panel.

2. The Proposed Model

Te proposed model is based on simple considerations con-cerning the structural response o masonry panels subjectedto in-plane loads.

Consider the masonry panel shown in Figure , subjectedto a constant vertical loadNacting along the central axis othe panel and to a horizontal orce Vapplied at the top othe panel. By increasing the orce V, the stress-strain state

characterizing the behavior o the panel cross-sections variescontinuously along its height. In particular, assuming orthe masonry material a no-tensile behavior with an elasticperectly-plastic response in compression, the ollowingstates can be identied (Figure ):

(a) the whole section is in compression, and the materialbehaves elastically (max );

(b) the whole section is in compression, and the materialbehaves plastically (

max> );

(c) only a portionxo the section is in compression, andhere the material behaves elastically (

max );

(d) only a portionxo the section is in compression, andhere the material behaves plastically (max> );

withmaxbeing the maximum value o the strain in compres-

sion andthe strain value corresponding to the elastic limito the masonry material.

At each value o the orce V, the masonry panel canbe considered as composed, along its height, o different

homogenous zones with each one characterized by one o theabove states. Te number o these zones and their extensionalong the panel height depend on several parameters such asthe masonry properties, the value o the applied loads, andthe geometry o the panel.

Te model proposed herein consists o schematizingthe panel as an assemblage o elements, with exure andshear deormability, representing the homogenous zones. Anexample o the model is reported in Figure . In particular,Figure (a) shows the case o a single element while Figures(b) and(c) show the case o more elements, one element atthe top with all the sections ully compressed and the otherelements where the sections are not entirely in compression.

Te properties o each elementlength and resistingcross-sectionare dened by the properties o the corre-sponding homogenous zone. In particular, the cross-sectionarea and the moment o inertia o each element are assumedto be constant and, or each step o load, are equal to theaverage value o the cross-section area and the momento inertia characterizing the resisting sections located atthe ends o the homogenous zone. Each element is, then,described by a stiffness matrix derived by taking into accountalso the shear deormability introduced through the terms sandeand given by

element

=

0 123 sym.0 62

4 (1 + ) 0 0

0 123

62 0 123

0 6

2 2 (2 1)

0 6

2

4 (1 + )

, ()


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whereHis length o the considered element,Aaverage valueo the cross-section area o the resisting sections located at theends o the element,Iaverage value o the moment o inertiao the resisting sections, Youngs modulus o masonry, shear modulus o masonry; = 3/2 withthe shear actor o the resisting section and = 1 + 4.

Indeed, in the context o the equivalent rame modelingapproach, the shear deormability plays a crucial role inderiving the seismic response o masonry structures sincethe capacity curve obtained through a pushover analysis isdevoted to urnish both the peak load (resistance) o thestructure and the ailure mechanism affecting piers and span-drels, also a measure o the ultimate displacement capacitynecessary or evaluating the global ductility o the structure.

Finally, the stiffness matrix o the panelpanel

is obtained

by assembling the stiffness matrices o each element.

Te model dened or the case o unstrengthened panelsis, then, extended to the case o FRP-strengthened panels.In this case, the ollowing additional hypotheses have beenintroduced:

(i) a no-compression behavior or the FRP-strengthen-ing system with a linear-elastic behavior in tensionuntil its ailure or the decohesion occurs;

(ii) absence o slip phenomena between masonry andFRP-strengthening.

Considering the above hypotheses, it is evident that ad-ditional states, besides those (a) to (d) previously dened,would occur when the strengthening system is activated.Consequently, in these cases additional homogeneous zones

will characterize the behavior o the FRP-strengthened pan-els.

Te proposed model is applied to evaluate the nonlinearresponse o the panel in terms o its capacity curve. Teprocedure is articulated into two phases: the rst phase aimedat analyzing the stress-strain state, and the second phase isdevoted to evaluate the lateral displacement exhibited by thepanel during the increments o the lateral orceV.

Te equations characterizing the rst phase are equi-librium and constitutive equations and will be reportedin detail in the ollowing sections or unstrengthened andstrengthened panels. Tese equations allow evaluating thestress-strain state at thebase section o thepanel, where deor-

mations assume the maximum values, and at the sectionswhere a change o the states occurs.

In the second phase, or each value o the applied orceVit is possible to derive the corresponding value o the lateraldisplacementDat the top o the panel.

Te capacity curve o the panel, that is, the curve in termso the applied orce V versus the top displacement D, canbe nally derived. Tis curve is representative o the globalresponse o the panel and takes into account the processo ormation o the homogenous zones characterizing thestructural response o the panel.

Te procedure is analyzed in detail in the ollowingsections or unstrengthened and strengthened panels.

V

V V

VVN

V

H

B

t

t

N

N

H

H

Element1

Element1

Element1

Element2

Element2E

lement3

B

max

Bx

t

max

max > y

H1

H1

H2

y

x2

(a)

(b)

(c)

F : Unstrengthened panels: schemes o assemblage o rameelements.

.. Unstrengthened Panels. With reerence to the panelshown inFigure , subjected to a vertical orce, constantand acting along the central axis o the panel, and to ahorizontal orcemonotonically increasing, the position othe neutral axis, indicated by, denes two possible states atthe base section o the panel:

(a) the whole section is in compression ( > ,Figure (a));

(b) only a portion o the section is in compression ( < ,Figures(b) and(c)).

In the rst case, the panel is modeled through a singleelement (Figure (a)) while, in the second case, the panel ismodeled by two or more elements (Figures (b) and(c)),one representing the portion o the panel where the sectionsare ully compressed and the others representating the partswhere the sections are not ully compressed ( < ).For both cases, the attained maximum strain

max

activated


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Data

x B x < B

Unstrengthened

panelEqn. (2)

Eqn. (4) Eqn. (5) Eqn. (7) Eqn. (6)

Stop

Eqn. (3)

max max

max y max ymax> u max> u

V; D

y < max u y < max u

F : Flowchart o the procedure or unstrengthened panels.

in the sections cannot exceed the ultimate value whichindicates the ailure condition or the panel. In both cases,the maximum strain can assume values less or greater thanthe yield strain; whenmaxexceeds, three elements arenecessary or schematizing the panel (Figure (c)).

Te set o equations used or deriving the stress-strainstate o the panel sections and the selected procedure orcarrying out the capacity curve have been implemented inMALAB []. Te owchart shown inFigure schemati-cally represents the main steps o the procedure or the caseo unstrengthened panels. In particular, considering differentpositions o the neutral axis or the base section o the panel,

the corresponding horizontal orceVand displacementDatthe top o the panel are derived by considering the ollowingve steps o analysis.(1) Assign the position o the neutral axis, x, which couldbe external or internal to the section ( or < , resp.).(2) Evaluate the maximum strain in the masonry,max,through the ollowing set o equations.

Case of . Imax

,max

= 2 (1 +(( ) /)) ,

min= max , ()

i< max ,max =

min =

,

++ min2 2 2 + = . ()

Case of < .Imax

,max

= 2 , ()

V

N

H

Bd1

d2

t

F : Strengthening conguration o the panel.

i< max ,max

=(1/2) , ()whereis the masonry strength andindicates the berwhere the elastic limit strain is attained.(3) Check the derived value o

maxin order to establish

the number o elements composing the panel.(4) Evaluate the orce V corresponding to the assignedposition o the neutral axis through the ollowing set oequations which are dependent on the position o the neutralaxis, the value o the maximum strain, and the characteristicso the elements composing the panel (number o elements,length and properties o the cross-section o each element).

Case of .Imax

,

=(1/2) max min 2/6

, ()i< max ,

=1+ 2 ,

1= [+ min + 2 ] ,2= [12 min 2 2 +

2 + 6 ] .()

Case of < . Imax

, =(1/2) max (/2 /3) ,

1= 6, ()


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Data

Eqn. (8), (9) Eqn. (10), (11)

UnstrengthenedFRP-strengthened

panel

or or

Eqn. (15)Eqn. (12)Eqn. (13) Eqn. (14)

case

StopStop

x Bor

B d1 < x B

max> u max> u

y < max u

V; DV; D

y < max u

max

max

y max yfmax > fu

fi fu fi

fi

maxfi

fu

fi> fu

B d2 x < B d1

0 < x B d2

F : Flowchart o the procedure or FRP-strengthened panels.

i< max , =(1/2) /2 + 2/3

+ + /2 .1= 6,

2= 2 23 .

()

(5)Evaluate the top displacement D o the panel corre-sponding to the orce Vcalculated at the previous step byusing the stiffness matrix o the panel

panel.

Te procedure is then repeated starting rom a newposition o the neutral axis, andit continues until the ultimate

value o the strain is attained. Ten the capacity curveV-Dothe panel is derived.

Te capacity curve provides two pieces o inormation,the rst related to the maximum horizontal orce sustainedby the panel and the second related to its ductility whichis dened as the ratio between the ultimate displacement

and the displacement corresponding to the attainment o themasonry yield strain.

It is important to emphasize that the different phasescharacterizing the behavior o the panel and, consequently,its schematization through rame elements require to operatein terms o increments o load since the nonlinearity o theresponse leads to a variation o the stiffness matrix o themodel at each load increment.

.. FRP-Strengthened Panels. Te model proposed or thecase o the unstrengthened panels has been extended to thecase o masonry panels strengthened with FRP strips bondedon the external surace (Figure ). Te equations used or

deriving the capacity curve are those carried out or thecase o the unstrengthened panels with additional equationsreerring to the sections where the strengthening is activated.

In Figure the owchart schematizing the steps othe procedure or the case o FRP-strengthened panels isreported. It is possible to observe that the procedure differswith respect to the one developed or the unstrengthenedpanels only when the position o the neutral axis leads to theactivation o the FRP strips. In Figuresandthe schemeso the strengthened panels used or deriving the equations arereported as ollows.

(1) Assign the positionxo the neutral axis.(2) Evaluate the maximum value o the strain in themasonry and the strengthening through the ollowing set oequations.

Case of 2 < 1. Imax ,max =

1 1 ,

12max 11= ,

()

i

< max

,

max = 1

1 =

,12 + 11= .

()

Case of0 < 2. Imax ,max =

1 1 =

2 2 ,

12max 11 22= ,

()


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V

V

N

t

H

B

t

N

H

Element

1

Element3

Element1

Element2

Element2

Eleme

nt3

Element4

B

max < y

max > y

H1H2

H1

H2

H3

y

x

x

s

f1

f1

y

f1,3 xy

d1 d2x

(a)

(b)

F : FRP-strengthened panels: schemes o assemblage o rameelements.

i< max ,max =

1 1 =

2 2 ,

=12 + + 11+ 22,()

where1and2are the tensile strains o the FRP strips atthe base o the panel and1and2are the correspondingcross-section areas.

(3)Check the evaluated maximum strain in the masonryand FRP derived rom the previous set o equations. Inparticular, the attainment o the ultimate deormation eitherin masonry or in FRP leads to stopping the procedure.(4)Evaluate the orce Vby introducing a urther set oequations.

Case of 2 < 1.Imax , =(1/2) max (/2 /3)

+1/2 1

,

V

V

N

t

H

B

N

H

Element1

Element3

Element1

Element2

Element4

Element2

Element3

Element4

Eleme

nt5

B

max < y

max > yy

H1H2

H1H2

H3

H3H4

x

x

t

s

f1,3

f1

f2

f1f2

f1,3

f1,4

xy

d1

d1

d2

d2

x

F : FRP-strengthened panels: schemes o assemblage o rameelements.

1= 6 2= 6+

131,

()

i

max

,

=1+ 2+ 3 ,1=12 [

2

3] ,2= 2

2 ,3= 2 1 ,1=

6 ,


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2= 6+13

1,

3= (1/2) /2 /3

,

++1,21/2 1 ,= 1,31(1/2) .

()

Case of0 < 2. Imax ,

=1+ 2+ 3

1=12max 232= 112 1 ,3= 222 21= 6 2=

2 2

3 1

3= (1/2) /2 /3+1,3/2 1()

i max ,

=1+ 2+ 3+ 4 ,1=1

2

2 +2

3 ,

2= 2 2 ,3= 112 1 ,4= 222 2 ,1= 6 ,2=

6+1

31

,

3= (1/2) /2 /3+1,31/2 1 ,

4= 1

+ 2

+ 3

,1=12 4

2 4+

243 ,2= 4 4 2

4 42 ,3= 1,412 1 .

()

(5)Evaluate the top displacement D o the panel corre-sponding to the orce Vcalculated at the previous step by

using the stiffness matrix o the panelpanel. Ten the capac-ity curve V-D o the strengthened panel is derived. Te capac-ity curve provides the maximum horizontal orce sustainedby the panel and its ductility dened as the ratio between theultimate displacement, corresponding to either the masonryailure or the FRP ailure/decohesion, and the displacementcorresponding to the attainment o the masonry yield strain.

Te process is then repeated starting rom a new positiono the neutral axis, and it continues until the ultimate value othe strain is attained.

3. Numerical Applications

Te proposed model has been applied with reerence to somecases o study deduced rom the literature and or which thecapacity curves were deduced experimentally.

Te main characteristics o the analyzed panels are sum-marized inable , and they reer to the ollowing literaturestudies: Fantoni, [], panels denoted by (a, b, and c);Giambanco et al. [], panel denoted by panel (d);Giuffre and Grimaldi, [], panel denoted by panel (e);Callerio, [], panel denoted by panel (); Marcari, [], panel denoted (g). Te panels are characterized by di-erent geometrical dimensions and mechanical properties omasonry, important or assessing the sensibility o the model.

Te reliability o the proposed model is analyzed, or

each unstrengthened and strengthened panel, by comparingthe capacity curve obtained by the model itsel with theexperimental curve. Further results have been also derivedrom the numerical analyses in order to better investigate thestructural response o the panels.

.. Unstrengthened Panels. Te capacity curves V-D deducedby using the proposed model developed or the unstrength-ened masonry panels have been compared with the experi-mental curve derived rom the related studies []. Tecomparisons are reported in Figures (a)() where theexperimental results are indicated by circular symbols andthe numerical results by continuous lines. Figure shows


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2.5

2

1.5

0.5

1

00 2 4 6 8 10

105

V(N

)

D(mm)

(a)

2.5

2

1.5

0.5

1

00 1 3 52 4 6

105

V(N

)

D(mm)

(b)

2.5

2

1.5

0.5

1

0

105

V(N)

0 2 4 6 8 10

D(mm)

(c)

104

0 10 20 30 40

D(mm)

V(N)

0

1

3

5

2

4

7

6

(d)

104

V(N)

D(mm)

10

8

6

4

2

0

0 0.5 1 1.5

(e)

104

V(N)

D(mm)0 5 10 15

0

1

3

5

2

4

7

6

8

()

F : Comparisons between numerical and experimental capacity curves o unstrengthened panels.

a good agreementbetween the numerical and the experimen-tal results in terms o the maximum load and, in many cases,in terms o ultimate displacement.

Te proposed model is, then, applied to examine differentaspects o the panels behavior. In Figure the variation o

the heightH2, is analyzed representing the amplitude o theplastic zone o the panel where the maximum strain exceedsthe yield strain value o the masonry, as a unction o theposition o the neutral axis at the base section. From the plotsoFigure it is possible to observe a similar behavior or


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10

8

6

4

2

010

(a)

(c) (f)

(d)

(b)

(e)

20 30 40 50

x/B(%)

H 1/H(%

)

F : Variation o the dimensions o the plastic zone or theunstrengthened panels.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

010.90.80.70.60.50.40.30.20.1

H3

H2

H1

Whole

compressed

sectionelasticbehavior

Whole

(b)

(e) compressed

sectionelasticbehavior

Partialized

sectionelastic

Height of the

plastic zone

Height of the

plastic zone

x/B

10.90.80.70.60.50.40.30.2

x/B

H i/H

1

0.9

0.8

0.7

0.60.5

0.4

0.3

0.2

0.1

0

H i/H

H3

H2

H1

behavior

Partializedsectionelastic

behavior

F : Variation o the dimensions o the zones characterizingthe behavior o the panel.

: Characteristics o the literature panels.

Re.(mm)(mm)(mm)(MPa)mk(MPa)(kN)Unstrengthened panels

[] .

[] .

[] . [] .

[] .

[] .

FRP-strengthened panel

[] .

panels (a), (c), and( ) which are characterized by a maximumheight1 o the plastic zone less than % o the whole heighto the panel and by a maximum base o the plastic zone lessthan % o its lengthB.

For the other panels (b) and (d), although the height

1o the plastic zone is less than % o the height o the panel, agreater portion o the base, about % o the base dimension,characterized the plastic zone.

InFigure the curves representing the variation o theheight o the different zones characterizing the behavior opanel (b) and panel (e) are reported: zone with height equalto1, where max> ; zone with height equalto2 , wheremax

< , and only a portion o the section is in compression;zone with height equal to3, wheremax< , and thewhole section is in compression. In the gure the heightis normalized with respect to the heighto the panel and isplotted as a unction o the position o the neutral axis at the

base section.Te plots oFigure clearly show the process o orma-tion o the various zones characterizing the behavior o thepanels when the external orceV increases. It is interestingto observe that in both cases2= 3 when the neutralaxis is located at the center o the section (/ = 0.5), andthe maximum value o the height1 o the panel (b) isabout .H(i.e., mm) and about .H(i.e., mm) orpanel (e). Te different amplitudes o the plastic zone affectsthe global response o the panels (Figure ), where a greaterplastic deormation or the panel (e) is evident.

.. FRP-Strengthened Panels. Te comparison between the

experimental and the numerical capacity curves deduced orthe strengthened panel experimentally examined in [] isshown inFigure . Also in this case the developed modelshows a good reliability in reproducing the global responseo the panel. Indeed, the numerical curve well approxi-mates the prepeak behavior and provides a good estimationo the peak load. Nevertheless, a signicant degradationcharacterizing the pre-peak experimental behavior o thepanel probably due to the gradual detachment o the strips,which is not accounted by the proposed model is evident.Moreover, differently rom the unstrengthened panels, theFRP-strengthened panel shows a signicant sofening in theprepeak phase which emphasizes a ragile response.


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V q

1480

1570

400 250 120 mm

Tuff unit:

(a)

2.5

2

1.5

1

0.5

00 5 10 15 20

Panel (g)

25

V(N)

D(mm)

105

(b)

F : (a) Dimensions and strips arrangement o the considered FRP-strengthened panel []. (b) Comparison between numerical andexperimental capacity curves o the FRP-strengthened panel.

4. Conclusions

In this paper a simple approach or evaluating the nonlinearresponse o unstrengthened and FRP-strengthened masonrypanels subjected to in-plane vertical and lateral loads hasbeen presented. Te model belongs to the equivalent rameapproach and consists o schematizing the panel as an assem-blage o rame elements with exure and shear deormability.Te geometrical and mechanical properties o the elements

are derived on the basis o the different stress-strain statescharacterizing the sectional behaviour.

Te reliability o the proposed model is analysed byconsidering some experimental cases deduced rom the liter-ature. Te comparison in terms o the capacity curve obtainedby the numerical and the experimental studies has provideda satisactory agreement especially or the prediction o themaximum load sustained by the panels and, in most o thecases, in terms o the ultimate displacement. However, orsome o the analysed unstrengthened panels and or theFRP-strengthened panel, the numerical values o the ultimatedisplacement were less than the experimental values. Teseresults, certainly due to the simplied hypothesis assumed orthe masonry and FRP and, in particular, or their interactionmechanism, have evidenced the restrictions o the proposedmodel and could constitute the basis rom which urtherdevelopment o the model itsel is proposed.

Te model, as proposed herein and being based only onew parameters obtained by standard experimental tests onmasonry and FRP, aims at providing a simple tool to beused in the phase o assessment o masonry structures andin the preliminary design o the FRP-strengthening systems.Moreover the model could be applied also to the case obuilding acades because it provides the capacity curves oeach wall bay constituting the acades.

Symbols

N: Vertical load applied at the top o thepanel along its central axis

V: Horizontal load applied at the top o thepanel

D: Horizontal displacement at the top o the panel

B: Width o the panelt: Tickness o the panelH: Height o the panel1: Distance o the lef FRP sheet rom the

panel edge2: Distance o the right FRP sheet rom thepanel edge1,2: Cross-section areas o the FRP strips( = 1, . . . , 4): Height o the panels portion withsections not ully compressed( = 1, . . . , 3): Height o the homogenous zones o thepanel: Youngs modulus o masonry : Shear modulus o masonry

: Strength o masonry

: Elastic limit strain o masonry : Ultimate strain o masonry max

: Maximum strain o masonry incompression

min: Minimum strain o masonry in

compression: Ultimate strain o FRP1: Strain o the lef FRP sheet at the basesection2: Strain o the right FRP sheet at the basesection1,3: Strain o the lef FRP sheet at the sectionlocated at

3


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1,4: Strain o the lef FRP sheet at the sectionlocated at4 : Neutral axis o the section

x: Neutral axis o the section whenmax=: Distance o the ber whenis attainedelement: Stiffness matrix o the single elementpanel: Stiffness matrix o the panel

A: Averagevalueo the cross-section areaothe resisting sections located at the endso the element

I: Average value o the moment o inertiao the resisting sections: Shear actor o the resisting section.

References

[] E. Grande and A. Romano, Experimental investigation andnumerical analysis o tuff-brick listed masonry panels,Mate-

rials and Structures, vol. , no. -, pp. , .[] L. Decanini, A. deSortis, A. Goretti, R. Langenbach,F. Mollaioli,

and A. Rasulo, Perormance o masonry buildings during the Molise, Italy, earthquake, Earthquake Spectra, vol. ,supplement , pp. SS, .

[] D. F. DAyala and S. Paganoni, Assessment and analysis odamage in LAquila historic city centre afer th April ,Bulletin of Earthquake Engineering, vol. , no. , pp. , .

[] A. Khalia, W. J. Gold, A. Nanni, and M. I. Abdel Aziz,Contribution o externally bondedFRP to shear capacityo RCexural members,Journal of Composites for Construction, vol., no. , pp. , .

[] E. Grande, M. Imbimbo, and A. Rasulo, Effect o transversesteel onthe response o RC beams strengthenedin shearby FRP:experimental study,Journalof Composites for Construction, vol., no. , pp. , .

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a frame element model for the nonlinear analysis of frp strengthen masonry

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