A Frame Element Model for the Nonlinear Analysis of FRP-Strengthened Masonry Panels Subjected to...

Preprint of the paper: Ernesto Grande, Maura Imbimbo, Alessandro Rasulo & Elio Sacco. A frame element model for the nonlinear analysis of FRP-strengthened masonry panels subjected to in-plane loads. Advances in Materials Science and Engineering. Volume 2013 (2013), Article ID 754162, 12 pages. http://dx.doi.org/10.1155/2013/754162

A frame element model for the nonlinear analysis of FRP-

strengthened masonry panels subjected to in-plane loads

Ernesto Grande, Maura Imbimbo, Alessandro Rasulo & Elio Sacco Department of Civil and Mechanical Engineering-DICeM, University of Cassino and Southern Lazio, via G. Di Biasio, 43, 0043-Cassino, (FR) - Italy

ABSTRACT: In this paper a frame element model for evaluating the nonlinear response of un-strengthened and

FRP-strengthened masonry panels subjected to in-plane vertical and lateral loads is presented. The

proposed model, based on some assumptions concerning the constitutive behaviour of masonry and

FRP material, considers the panel discretized in frame elements with geometrical and mechanical

properties derived on the basis of the different states characterizing the sectional behaviour. The

reliability of the proposed model is assessed by considering some experimental cases deduced from

the literature.

Keywords: Frame Models, Masonry Panels, FRP, Capacity Curve


1. INTRODUCTION

Masonry constructions certainly constitute an important part of the existing buildings in several

countries and, in many cases, of their cultural heritage [1]. Past and recent seismic events have

pointed out a significant level of vulnerability of such constructions and have posed the necessity to

develop adequate and effective retrofit and strengthening systems able to improve their seismic

resistant.

In the last decades, strengthening techniques based on the use of fiber-reinforced polymers (FRPs)

materials have been proposed. Among these, one of the most commonly used technique is

represented by FRP strips externally applied on the masonry surface.

The literature related to FRP strengthening systems have clearly demonstrated the capacity of such

systems to improve the structural performances of masonry structures [2-4]. Some of the major

drawbacks in modeling FRP-strengthened structures are related to the masonry response as well as

to the interaction mechanism between masonry and FRPs at the interface layer. Many studies have

specifically investigated on the complex interaction mechanisms between FRP and masonry

supports [5-7] and others have provided theoretical laws able to describe the local bond behavior of

FRPs applied on masonry supports and, consequently, suitable as a basis for deriving numerical

models [8-11]. On the other side, it is certainly important in the development of such systems to

have reliable numerical tools for the assessment of their performances and the design of the

strengthening systems itself.

In this context, different approaches are proposed in the current literature for modeling FRP-

systems applied on masonry supports, most of them developed within the finite element formulation

framework. In particular, many authors proposed modeling approaches based on the use of

nonlinear interface elements interposed between the support and the FRP-strengthening and

characterized by constitutive laws derived by tests or by theoretical considerations [12-15]. Other

https://www.researchgate.net/publication/228373574_Experimental_Bond_Behavior_of_FRP_Sheets_Glued_on_Brick_Masonry?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/257896094_Experimental_investigation_and_numerical_analysis_of_tuff-brick_listed_masonry_panels?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/234033165_Round_Robin_Test_For_Composite-To-Brick_Shear_Bond_Characterization?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/235003717_Experimental_tests_and_numerical_modeling_of_reinforced_masonry_arches?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/232411576_Bond_behaviour_of_CFRP_laminates_glued_on_clay_bricks_Experimental_and_numerical_study?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/264042477_Development_of_a_constitutive_model_for_the_masonry-FRP_interface_behaviour?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/235003729_A_Simple_Model_for_the_Bond_Behavior_of_Masonry_Elements_Strengthened_with_FRP?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/235003746_Finite_Element_Analysis_of_Masonry_Panels_Strengthened_with_FRPs?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/50848746_Bond_Behavior_of_Historical_Clay_Bricks_Strengthened_with_Steel_Reinforced_Polymers_SRP?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/222393645_In-plane_shear_performance_of_masonry_panels_strengthened_with_FRP?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==


approaches consider homogenization procedures for including the behavior of the FRP/support

interface into the model of the FRP-reinforcement or, directly, in the masonry model [16-20].

Among these, the approaches based on the schematization of masonry elements strengthened with

FRPs by using frame elements are of particular interest since they are in accordance with the

equivalent frame modeling approach useful in the design and implemented in many commercial

computer codes for the structural analysis of masonry structures [21].

The model herein proposed for the nonlinear analysis of masonry panels strengthened with FRPs,

belongs to the equivalent frame approach and aims at providing a simple and effective tool to be

used in the phase of assessment of masonry structures and, also, in the preliminary design of the

FRP strengthening systems.

2. THE PROPOSED MODEL

The proposed model is based on simple considerations concerning the structural response of

masonry panels subjected to in-plane loads.

Consider the masonry panel shown in Figure 1, subjected to a constant vertical load N acting along

the central axis of the panel and to a horizontal force V applied at the top of the panel. By

increasing the force V, the stress-strain state characterizing the behavior of the panel cross-sections

varies continuously along its height. In particular, assuming for the masonry material a no-tensile

behavior with an elastic perfectly-plastic response in compression, the following states can be

identified (Figure 1):

a) the whole section is in compression and the material behaves elastically ( max y );

b) the whole section is in compression and the material behaves plastically ( max y );

https://www.researchgate.net/publication/251669974_Limit_analysis_of_FRP_strengthened_masonry_arches_via_nonlinear_and_linear_programming?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/222530088_Homogenised_limit_analysis_of_masonry_walls_Part_I_Failure_surfaces?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/248502735_Load_carrying_capacity_of_2D_FRPstrengthened_masonry_structures?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==

https://www.researchgate.net/publication/234060878_Modelling_and_analysis_of_FRP-strengthened_masonry_panels?el=1_x_8&enrichId=rgreq-175fe2df-e2a7-490a-90df-5032b911dfee&enrichSource=Y292ZXJQYWdlOzI1ODI1NzgzODtBUzoxODgzMzY4MzY3MTg1OTJAMTQyMTkxNDQwOTIwMA==


c) only a portion x of the section is in compression and here the material behaves elastically

( max y );

d) only a portion x of the section is in compression and here the material behaves plastically

( max y );

being max the maximum value of the strain in compression and y is the strain value corresponding

to the elastic limit of the masonry material.

At each value of the force V, the masonry panel can be considered as composed, along its height, by

different homogenous zones each one characterized by one of the above states. The number of these

zones and their extension along the panel height depend on several parameters such as the

properties of masonry, the value of the applied loads and the geometry of the panel.

The model proposed herein consists of schematizing the panel as an assemblage of elements, with

flexure and shear deformability, representing the homogenous zones. An example of the model is

reported in Figure 2. In particular, Figure 2a shows the case of a single element while Figures 2b

and 2c the case of more elements, one element at the top with all the sections fully compressed and

the others where the sections are not entirely in compression.

The properties of each element - length and resisting cross-section – are, then, defined by the

properties of the corresponding homogenous zone. In particular, the cross-section area and the

moment of inertia of each element is assumed to be constant and, for each step of load, equal to the

average value of the cross-section area and the moment of inertia characterizing the resisting

sections located at the ends of the homogenous zone. Each element is, then, described by a stiffness

matrix given by:


tL

eIE

tL

IE

tL

eIE

tL

IEtL

IE

tL

IE

tL

IEL

AE

L

AEtL

eIE

tL

IE

symtL

IEL

AE

K

mmmm

mmm

mm

mm

m

m

element

)1(460

)12(260

120

6120

00

)1(460

.12

0

22

323

2

3

(1)

where:

L: length of the considered element;

A: average value of the cross-section area of the resisting sections located at the ends of the

element;

mE : Young’s modulus of masonry;

I: average value of the moment of inertia of the resisting sections;

2

3

ALG

IEe

m

m with G the shear modulus of the masonry and the shear factor of the resisting section;

et 41 .

Finally, the stiffness matrix of the panel Kpanel is obtained by assembling the stiffness matrices of

each element.

The model defined for the case of un-strengthened panels is, then, extended to the case of FRP-

strengthened panels. In this case, the following additional hypotheses have been introduced:

- a no-compression behavior for the FRP-strengthening system with a linear-elastic behavior in

tension until its failure or the de-cohesion occur;

- absence of slip phenomena between masonry and FRP-strengthening.


Considering the above hypotheses, it is evident that additional states, besides those a) to d)

previously defined, would occur when the strengthening system will be activated. Consequently, in

these cases additional homogeneous zone will characterize the behavior of the FRP-strengthened

panels.

The defined model is applied to evaluate the nonlinear response of the panel in terms of its capacity

curve. The procedure is articulated into two phases: a first phase aimed at analyzing the stress-strain

state and a second phase devoted to evaluate the lateral displacement exhibited by the panel during

the increments of the lateral force V.

The equations characterizing the first phase are equilibrium and constitutive equations and will be

reported in detail in the following sections for un-strengthened and strengthened panels. These

equations allow to evaluate the stress-strain state at the base section of the panel, where

deformations assume the maximum values, and at the sections where a change of the states occurs.

In the second phase, for each value of the applied force V it is possible to derive the corresponding

value of the lateral displacement D at the top of the panel.

The capacity curve of the panel, i.e. the curve in terms of the applied force V vs. the top

displacement D, can be finally derived. This curve is representative of the global response of the

panel and takes into account the process of formation of the homogenous zones characterizing the

structural response of the panel.

The procedure is analyzed in detail in the following sections for un-strengthened and strengthened

panels.


2.1 UN-STRENGTHENED PANELS

With reference to the panel shown in Figure 2, subjected to a vertical force N, constant and acting

along the central axis of the panel, and to a horizontal force V monotonically increased, the position

of the neutral axis, indicated by x, defines two possible states at the base section of the panel:

a. the whole section is in compression (x>B, Figure 2.a)

b. only a portion of the section is in compression (x<B, Figure 2.b and Figure 2.c).

In the first case, the panel is modeled through a single element (Figure 2.a) while, in the second

case the panel is modeled by two or more elements (Figure 2.b and Figure 2.c), one representing the

portion of the panel where the sections are fully compressed and the others representative of the

parts where the sections are not fully compressed (x<B). For both the cases, the attained maximum

strain max activated in the sections cannot exceed the ultimate value u which indicate the failure

condition for the panel. In both the cases, the maximum strain can assume values less or greater

than the yield strain y ; when max exceeds y , three elements are necessary for schematizing the

panel (Figure 2.c).

The set of equations used for deriving the stress-strain state of the panel sections and the selected

procedure for carrying out the capacity curve, have been implemented in Matlab [22]. The

flowchart shown in Figure 3 schematically represents the main steps of the procedure for the case of

un-strengthened panels. In particular, considering different positions of the neutral axis for the base

section of the panel, the corresponding horizontal force V and displacement D at the top of the

panel are derived by considering the following five steps of analysis:

1. assign the position of the neutral axis, x, which could be external or internal to the section (x≥B

or x<B respectively);

2. evaluate the maximum strain in the masonry, max , through the following set of equations:


Case of x≥B:

x

Bx

x

BxtBE

Ny

m

maxmin

maxmax ) (if

1

2

(2)

NxxBtEf

txxf

xBxx

smmk

smk

us

y

222

)(

) (if

min

maxyminmax

(3)

) (if 2

maxmax ym txE

N

(4)

) (if

2

1

maxymax umk

ymk

Nxtf

xtf

(5)

where mkf is the masonry strength, sx indicates the fiber where the elastic limit strain is attained.

3. check the derived value of max in order to establish the number of elements composing the

panel;

4. evaluate the force V corresponding to the assigned position of the neutral axis through the

following set of equations which are dependent on the position of the neutral axis, the value of the

maximum strain and the characteristics of the elements composing the panel (number of elements,

length and properties of the cross section of each element).

Case of x≥B:

) (if

62

1

max

2

minmax

y

m

H

BtE

V

(6)


6

222

2

1

2

) (

min2

min1

maxy21

ssmmk

ssmmk

u

xBxxxBtEfY

xxBxxtEfY

H

YYV

(7)

Case of x<B:

V

BNHH

H

xBtxE

V y

m

6

) (if 322

1

1

max

max

(8)

tf

NB

V

NHH

V

BNHH

H

xxBtxxfx

xBtxf

V

mk

u

ssmk

ssmk

3

2

2

6

) (if 2

)(3

2

22

1

2

1

maxy

(9)

5. evaluate the top displacement of the panel corresponding to the force V calculated at the previous

step through the following relation:

VKpanel

1 (10)

being Kpanel the stiffness matrix of the panel obtained by assembling the stiffness matrices of each

element Kelement.

The process is then repeated starting from a new position of the neutral axis and it continues until

the ultimate value of the strain is attained.


It is important to emphasize that the different phases characterizing the behavior of the panel, and,

consequently, its schematization through frame elements, requires to operate in terms of increments

of load since the nonlinearity of the response leads to a variation of the stiffness matrix of the model

at each load increments.

2.2 FRP-STRENGTHENED PANELS

The model proposed for the case of the un-strengthened panels has been extended to the case of

masonry panels strengthened with FRP strips bonded on the external surface (Figure 4). The

equations used for deriving the capacity curve are those carried out for the case of the un-

strengthened panels with additional equations referring to the sections where the strengthening is

activated.

In Figure 5 is reported the flowchart schematizing the steps implemented in matlab for the case of

FRP-strengthened panels. It is possible to observe that the procedure differs with respect to the one

developed for the un-strengthened panels only when the position of the neutral axis leads to the

activation of the FRP-strips. In this case, the procedure provides the following steps:

1. assign the position of the neutral axis;

2. evaluate the maximum value of the strain in the masonry and the strengthening through the

following set of equations:

Case of B-d2≤x<B-d1:

NAExtE

xdBx

fffm

yf

11max

max1

1max

2

1

) (if

(11)


NAExxtfxtf

xxdBx

fffsmksmk

us

yf

11

maxy1

1max

2

1

) (if

(12)

Case of 0<x≤B-d2:

NAEAExtE

xdBxdBx

ffffffm

yff

2211max

max2

2

1

1max

2

1

) (if

(13)

2211

maxy2

2

1

1max

)(2

1N

) (if x

ffffffsmksmk

uff

AEAExxtfxtf

xdBxdB

(14)

where 1f , 2f are the tensile strains of the FRP strips at the base of the panel and 1fA , 2fA are the

corresponding cross-section areas.

3. check the evaluated maximum strain in the masonry and FRP derived from the previous set of

equations. In particular, the attainment of the ultimate deformation either in masonry or in FRP

leads to stop the procedure;

4. evaluate the force V by introducing a further set of equations. In Figure 6 and Figure 7 are

reported the schemes of the strengthened panels used for deriving the equations as follows.

Case of B-d2≤x<B-d1:


V

dBNHH

V

BNHH

H

dB

AExB

xtE

y

fffm

1

36

6

) (if 2322

1

V

12

1

max

11max

(15)

tf

AENx

V

dB

AExB

xtf

HH

V

dBNHH

V

BNHH

dB

AEY

xxBtxxfY

xxx

BxtfY

H

YYY

mk

fffy

fffy

ymk

fff

ssmk

sssmk

uy

2

1

2322

1

1

36

6

2

22)(

322

1

) (if V

13,1

112,1

3

12

1

13

2

1

max321

(16)

Case of 0<x≤B-d2:


V

dB

AExB

txf

HH

V

dBBNHH

V

BNHH

dB

AEY

dB

AEY

xBxtEY

H

YYY

fffy

ymk

fff

fff

m

y

13,1

3

22

1

2223

1112

max1

max321

2322

1

1

32

6

2

2

322

1

) (ifV

(17)


114,13

44442

4441

3214

113,1

3

12

1

2224

1113

2

1

max4321

2

22)(

3

2

22

1

2322

1

1

36

6

2

2

22)(

3

2

22

1

) (ifV

dB

AEZ

xxBxxtfZ

xx

BxtfZ

V

ZZZHH

V

dB

EAxB

xtf

HH

V

dBNHH

V

BNHH

dB

AEY

dB

AEY

xxBxxtfY

xx

BxtfY

H

YYYY

fff

ssmk

ssmk

fffy

ymk

fff

fff

ssmk

ssmk

uy

(18)

5. evaluate the top displacement of the panel corresponding to the force V calculated at the previous

step through the relation D Kpanel

1 V .

The process is then repeated starting from a new position of the neutral axis and it continues until

the ultimate value of the strain is attained.


3. NUMERICAL APPLICATIONS

The proposed model has been applied with reference to some cases of study deduced from the

literature and for which the capacity curves were deduced experimentally.

The main characteristics of the analyzed panels are summarized in Table 1 and they refer to the

following literature studies: Fantoni, 1981 [23], panels denoted as (a, b, c); Gianbanco et al., 1996

[24], panel denoted as panel (d); Giuffrè and Grimaldi, 1985 [25], panel denoted as panel (e);

Callerio, 1998 [26], panel denoted as panel (f); Marcari, 2005 [27], panel denoted (g). The panels

are characterized by different geometrical dimensions and mechanical properties of masonry,

important for assessing the sensibility of the model.

The reliability of the proposed model is analyzed, for each un-strengthened and strengthened panel,

by comparing the capacity curve obtained by the model itself with the experimental curve. Further

results have been also derived from the numerical analyses in order to better investigate the

structural response of the panels.

3.1 UN-STRENGTHENED PANELS

The capacity curves V-D deduced by using the proposed model developed for the un-strengthened

masonry panels, have been compared with the experimental curve derived from the related studies

[23, 24, 25, 26]. The comparisons are reported in Figure 8.a-f where the experimental results are

indicated by circular symbols and the numerical results by continuous lines. Figure 8 shows a good

agreement between the numerical and the experimental results in terms of the maximum load and,

in many cases, in terms of ultimate displacement.

The proposed model is, then, applied to examine different aspects of the panels behavior. In Figure

9 is analyzed the variation of the height H2, representing the amplitude of the plastic zone of the

panel where the maximum strain exceeds the yield strain value of the masonry, as a function of the


position of the neutral axis at the base section. From the plots of the Figure 9 it is possible to

observe a similar behavior for the panels (a), (c) and (f) which are characterized by a maximum

value of zone H1 about 5% of the whole height of the panel, and a portion of the base section where

max>y less than the 20% of its length B.

For the other panels, lower values of H1 are observed (less than the 5% of the height of the panel)

whilst a greater portion of the base section, about 30% of the base dimension, is characterized by

max>y.

In Figure 10 are reported the curves representing the variation of the length of the different zones

characterizing the behavior of the panel (b) and the panel (e) (H1 where max>y; H2 where max<y

and only a portion of the section is in compression; H3 where max<y and the whole section is in

compression), normalized with respect to the height of the panel and plotted as a function of the

position of the neutral axis at the base section.

The plots of the Figure 10 clearly show the process of formation of the various zones characterizing

the behavior of the panels when the external force V increases. It is interesting to observe that in

both cases H2=H3 when the neutral axis is located at the center of the section (x/B=0.5), and the

maximum value of the length H1 of the panel (b) is about 0.05 H (i.e. 120 mm) and about 0.1 H (i.e.

98 mm) for the panel (e). The different amplitude of the plastic zone affects the global response of

the panels (Figure 8), where it is evident a greater plastic deformation for the panel (e).

3.2. FRP-STRENGTHENED PANELS

The comparison between the experimental and the numerical capacity curve deduced for the

strengthened panel experimentally examined in [27] is shown in Figure 11. Also in this case the

developed model shows a good reliability in reproducing the global response of the panel. Indeed,

the numerical curve well approximates the pre-peak behavior and provides a good estimation of the


peak load. Nevertheless, it is evident a significant degradation characterizing the pre-peak

experimental behavior of the panel probably due to the gradual detachment of the strips, which is

not accounted by the proposed model. Moreover, differently from the un-strengthened panels, the

FRP-strengthened panel shown a significant softening in the pre-peak phase which emphasizes a

fragile response.

4. CONCLUSIONS

In this paper a simple approach for evaluating the nonlinear response of un-strengthened and FRP-

strengthened masonry panels subjected to in-plane vertical and lateral loads has been presented. The

model belongs to the equivalent frame approach and consists of schematizing the panel as an

assemblage of frame elements with flexure and shear deformability. The geometrical and

mechanical properties of the elements are derived on the basis of the different stress-strain states

characterizing the sectional behaviour.

The reliability of the proposed model is analysed by considering some experimental cases deduced

from the literature. The comparison in terms of the capacity curve obtained by the numerical and

the experimental study has provided a satisfactory agreement especially for the prediction of the

maximum load sustained by the panels and, in most of the cases, in terms of the ultimate

displacement. However, for some of the analysed un-strengthened panels and for the FRP-

strengthened panel the numerical values of the ultimate displacement resulted less than the

experimental values. These results, certainly due to the simplified hypothesis assumed for the

masonry and FRP and, in particular, for their interaction mechanism, have evidenced the restrictions

of the proposed model and could constitute the basis from which propose further development of

the model itself.


The model, as proposed herein and being based only on few parameters obtained by standard

experimental tests on masonry and FRP, aims at providing a simple tool to be used in the phase of

assessment of masonry structures and in the preliminary design of the FRP strengthening systems.

Moreover the model could be applied also to the case of building facades because it provides the

capacity curves of each wall bay constituted the facades.

LIST OF SYMBOLS

N: vertical load applied at the top of the panel along its central axis;

V: horizontal load applied at the top of the panel;

B: base of the panel;

t: thickness of the panel;

L: height of the panel;

1d : distance of the left FRP sheet from the panel edge;

2d : distance of the right FRP sheet from the panel edge;

mE : Young’s modulus of masonry;

mkf : masonry strength;

y : elastic limit strain of masonry;

u : ultimate strain of masonry;

max : maximum strain of masonry in compression;

1f : strain of the left FRP sheet at the base section;


2f : strain of the right FRP sheet at the base section;

3,1f : strain of the left FRP sheet at the section located at 3L ;

4,1f : strain of the left FRP sheet at the section located at 4L ;

4,..,1iLi : height of the panel’s portion with sections not fully compressed;

x : neutral axis of the section;

yx : neutral axis of the section when y max ;

sx : distance of the fiber where y is attained;

A: average value of the cross-section area of the resisting sections located at the ends of the

element;

I: average value of the moment of inertia of the resisting sections;

2

3

ALG

IEe

m

m with G the shear modulus of the masonry and the shear factor of the resisting section;

et 41 .

Af1, Af2: cross-section areas of the FRP strips

REFERENCES

[1] Grande E, Romano A. 2013. Experimental investigation and numerical analysis of tuff-brick

listed masonry panels. Materials and Structures - RILEM. 46(1):63-75.

[2] Marcari G, Manfredi G, Prota A, Pecce M. In-plane shear performance of masonry panels

strengthened with FRP. Composites Part B: Engineering 2007;38(7–8):887-901.

[3] Konthesingha KMC, Masia MJ, Petersen RB, Mojsilovic N, Simundic G, Page AW. Static

cyclic in-plane shear response of damaged masonry walls retrofitted with NSM FRP strips –


An experimental evaluation. Engineering Structures, In Press, Corrected Proof, Available

online 14 December 2012.

[4] Cancelliere I, Imbimbo M, Sacco E. Experimental tests and numerical modeling of

reinforced masonry arches. Engineering Structures 2010;32(3):776-792.

[5] Grande E, Imbimbo M, Sacco E. 2011a. Bond behaviour of CFRP laminates glued on clay

bricks: Experimental and numerical study. Composites Part B: Engineering, 42(2):330-340.

[6] Grande E, Imbimbo M, Sacco E. 2011b. Bond Behavior of Historical Clay Bricks

Strengthened with Steel Reinforced Polymers (SRP). Materials, 4(3):585-600.

[7] Valluzzi MR, Oliveira DV, Caratelli A, Castori G, Corradi M, de Felice G, et al. 2012.

Round Robin Test For Composite-To-Brick Shear Bond Characterization. Materials and

Structures - RILEM. 45(12):1761-1791.

[8] Panizza M, Garbin E, Valluzzi MR, Modena C. (2008) Bond behaviour of CFRP and GFRP

laminates on brick masonry. In: Proc. of VI Int. Conf. on Structural Analysis of Historical

Constructions, Bath (UK), July 2-4 2008, D’Ayala & Fodde eds.: 763-770.

[9] Oliveira D, Basilio I, Lourenço P.B. (2011) Experimental bond behavior of FRP sheets

glued on brick masonry. J. of Composites for Construction, ASCE 15(1), 32-41.

[10] Grande E, Imbimbo M, Sacco E. 2011c. Simple Model for the Bond Behavior of Masonry

Elements Strengthened with FRP. Journal of Composites for Construction. 15 (3):354-363.

[11] Malena M, de Felice G. 2013. Externally bonded composites on a curved masonry substrate:

experimental results and analytical formulation. Submitted for publication to Composite

Structures.

[12] Failla A, Cottone A, Gianbanco G. Numerical modeling of masonry structures reinforced by

FRP plate/sheets. In: Lourenço, Roca, editors. Structural analysis of historical constructions

– Modena. London: Taylor and Francis Group; 2005. p. 857–66.


[13] Maruccio C, Lourenço PB, Oliveira DV. Development of a constitutive model for the

masonry–FRP interface behaviour. In: Proc of the 8th world congress on computational

mechanics, Venice; 2008.

[14] Rahman A, Ueda T. Numerical simulation of FRP retrofitted masonry wall by 3D finite

element analysis. In: IABSE symposium Bangkok 2009. Sustainable infrastructure.

Environment friendly, safe and resource efficient, Bangkok, Thailand; 2009. p. 96.

[15] Grande E, Imbimbo M, Sacco E. 2013. Finite element analysis of masonry panels

strengthened with FRPs. Composites: Part B, 45:1296-1309.

[16] Grande E, Milani G, Sacco E. 2008. Modelling and analysis of FRP-strengthened masonry

panels. Eng Struct 2008;30(7):1842–60.

[17] Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls. Part I:

failure surfaces. Comp Struct 2006;84:166–80.

[18] Ascione L, Feo L, Fraternali L. Load carrying capacity of 2D FRP/strengthened masonry

structures. Compos B Eng 2005;36(8):619–26.

[19] Caporale A, Feo L, Luciano R. Limit analysis of FRP strengthened masonry arches via

nonlinear and linear programming. Compos B Eng 2012;43(2):439–46.

[20] Grande E, Imbimbo M, Sacco E. A beam finite element for the nonlinear analysis of

masonry buildings strengthened with FRP. Int. Journal of Architectural Heritage:

Conservation, Analysis and Restoration 2011;5(6):693–716.

[21] Magenes G, Della Fontana A. Simplified non-linear seismic analysis of masonry buildings.

Proc Brit Masonry Soc 1998;8:190–5.

[22] Mathworks, Inc., 2002 Natick, Massachusetts. Matlab refer-ence manual, Version 6.5.

[23] Fantoni L., 1981 Rapporto sulle prove della capacità portante di murature in pietrame

iniettate, CRAD: Centro di ricerca applicata e documentazione, Udine.


[24] Giambanco G., Rizzo S., Spallino R., 1996, Il modello di inter-faccia a doppia asperità per

l’analisi delle strutture discontinue, Atti del convegno nazionale: La meccanica delle

murature tra teoria e progetto, Messina.

[25] Giuffrè A., Grimaldi A., 1985, Studi italiani sulla meccanica delle murature, Atti del

Convegno: Stato dell’arte in Italia sulla meccanica delle Murature, Roma.

[26] Callerio A., 1998, Comportamento sismico delle murature ne-gli edifici monumentali: un

metodo di analisi basato sulla meccanica del danneggiamento, Doctoral Dissertation,

University of Naples

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Thesis, University of Naples.


Figures

Figure 1. States characterizing the behavior of the cross-sections composing the panel


elem

ent 1

elem

ent 2

elem

ent 1

V

V

H1

x2

max

B

H

t

NV

VN

max

t

H

B

H2

x

y

B

H

t

NV

H1

max>y

elem

ent 2

elem

ent 1

V

elem

ent 3

a)

b)

c) Figure 2. Un-strengthened panels: schemes considered for deriving the equations


Figure 3. Flowchart of the procedure for un-strengthened panels


d2d1

B

t

NV

H

Figure 4. Strengthening configuration of the panel


Figure 5. Flowchart of the procedure for FRP-strengthened panels


f1

HH1

H2

max<y

x

VN

t

B

f1

N

fram

e 4

H3y

B

t

V

x

max>y

H2

H1

H fram

e 3

fram

e 1 fram

e 2

fram

e 3

fram

e 1

fram

e 2

y

xs

f1,3

d1d2

xy

a

b

Figure 6. Strengthened panels: schemes considered for deriving the equations


d2d1

B

t

NV

x

max<y

H3

H1

H

f1

f1,3

f2

H2

H2

f2

f1,4

f1

HH1

H4

max>y

x

VN

t

Bd1d2

y

xs

xyf1,3

H3

elem

ent 3

elem

ent 1

elem

ent 2

elem

ent 4

elem

ent 3

elem

ent 1

elem

ent 2

elem

ent 5

elem

ent 4

a

b

Figure 7. Strengthened panels: schemes considered for deriving the equations


PANEL (a)

PANEL (b)

a) b)

PANEL (c)

PANEL (d)

c) d)

PANEL (e)

PANEL (f)

e) f)

Figure 8. Comparisons between numerical and experimental capacity curves of un-strengthened panels


(a)

(b)

(c)

(d)

(e)

(f)

Figure 9. Variation of the height H2 of the plastic zone for the un-strengthened panels


Figure 10. Variation of the dimensions of the zones characterizing the behavior of the panel cross-sections.


V q

1480

1570

PANEL (g)

a) b)

Figure 11. a) Dimensions and strips arrangement of the considered FRP-strengthened panel [27]; b) Comparison

between numerical and experimental capacity curves of the FRP-strengthened panel.


Table 1. Characteristics of the accounted panels for numerical analyses.

rif B (mm)

H (mm)

t (mm)

Em (MPa)

fmk

(MPa)N

(kN)Un-strengthened panels

[1] 1250 1820 500 1118 6.0 343[2] 1200 2400 500 726 4.5 311[3] 1250 1850 500 1290 3.69 358[4] 980 980 290 4880 3.4 242[5] 1000 2000 250 [6] 1250 1830 500 1125 6.0 355

FRP-strengthened panels [7] 1480 1570 530 700 1.1 300

A Frame Element Model for the Nonlinear Analysis of FRP-Strengthened Masonry Panels Subjected to...

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