Microsoft Word - 2232


 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

321 
 

Fracture and Structural Integrity: ten years of ‘Frattura ed Integrità Strutturale’ 

 
 
 
  
Numerical simulation of the de-bonding phenomenon of FRCM 
strengthening systems 
 
 
Ernesto Grande 
University Guglielmo Marconi, Department of Sustainability Engineering, via Plinio 44, 00193-Roma, Italy 
e.grande@unimarconi.it, http:// orcid.org/0000-0002-3651-1975 
 
Maura Imbimbo, Sonia Marfia 
University of Cassino and Southern Lazio, Dep. of Civil and Mechanical Engineering, via G. Di Biasio 43, Cassino, Italy 
mimbimbo@unicas.it, http://orcid.org/0000-0003-3163-3073 
marfia@unicas.it, http://orcid.org/0000-0002-2166-9788 
 
Elio Sacco 
University of  Naples Federico II, Department of Structures in Engineering and Architecture, Via Claudio 21, Naples. Italy 
Elio.sacco@unina.it http:// orcid.org/0000-0002-3948-4781  
 
 
 
ABSTRACT. Aim of the paper is to present a one dimensional simple model 
for the study of the bond behavior of Fabric Reinforced Cementitious Matrix 
(FRCM) strengthening systems externally applied to structural substrates. The 
equilibrium of an infinitesimal portion of the reinforcement and the mortar 
layers composing the strengthening systems allows to derive the governing 
equations. An analytical solution is determined solving the system of 
differential equations. In particular, in the first part of the paper a nonlinear 
shear-stress slip law characterized by a brittle post-peak behavior with a 
residual shear strength in the post peak phase is introduced for either the 
lower reinforcement-mortar interface (approach 1) or both the lower and the 
upper interface (approach 2). In the latter approach, a calibration of the shear 
strength of the upper interface is proposed in order to implicitly account for 
the effect of the damage of the mortar on the bond behavior. In the second 
part of the paper it is presented the solution of the problem in the case of 
softening behavior by approximating the shear-stress slip law throughout a 
step function. Comparisons with experimental data, available in literature, are 
presented in order to assess the reliability of the proposed approach. 
  
KEYWORDS. FRCM; De-bonding; Analytical model; Interface. 
 

 

 
 

Citation: Grande, E., Imbimbo, M., Marfia, 
S., Sacco, E., Numerical simulation of the de-
bonding phenomenon of FRCM 
strengthening systems, Frattura ed Integrità 
Strutturale, 47 (2019) 321-333. 
 
Received: 23.10.2018 
Accepted: 24.11.2018  
Published: 01.01.2019 
 
Copyright: © 2019 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided 
the original author and source are credited. 

 

 
 

http://www.gruppofrattura.it/VA/47/2232.mp4


 

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24                                                                  
 

322 
 

 
INTRODUCTION 
 

he reinforcement of existing structures has always been a relevant problem both in the technical and scientific civil 
engineering community. Lately, the study and design of new reinforcement materials is a challenging issue. In 
particular, fabric reinforced cementitious matrix (FRCM) is an emerging strengthening system obtained 

embedding a grid of the carbon, glass or aramid reinforcement in an inorganic matrix. In general, the matrix is applied as a 
double layer incorporating the reinforcement. Nowadays, FRCM systems are used in the current practice to reinforce 
concrete and masonry structures. 
Some experimental investigations [1-7] and theoretical/numerical studies ([2, 3, 8-16] on FRCM strengthening systems are 
available in the recent literature. They testify the efficacy and advantages of FRCM systems together with the need to 
investigate aspects specifically characterizing the bond behavior of this new family of strengthening systems. 
The experimental investigations are mainly shear-lap tests that analyze the local bond behavior of FRCMs. From the 
experimental evidence different failure mechanisms can occur, such as a cohesive failure of the substrate, de-bonding at 
the reinforcement/substrate interface, de-bonding at the reinforcement/matrix interface, sliding of the reinforcement, 
tensile failure of the reinforcement in the un-bonded portion and tensile failure of the reinforcement within the mortar.  
The above mechanism occurrence depends on the characteristics of the strengthening system as well as of the support, 
such as the mechanical properties of the materials, the thickness of the mortar layers and the configuration of the 
reinforcement. These mechanisms particularly underline the role of additional phenomena to be necessarily considered for 
the study and the development of theoretical models/design formulas specific for FRCMs. 
Regarding the theoretical and numerical studies, the approaches available in literature show particular interest to the 
derivation of simple laws able to simulate the de-bonding phenomenon and, moreover, models able to account for 
additional phenomena specific of FRCMs. 
In [2] the shear stress-slip law at the interface level was obtained throughout a procedure applied to steel and carbon 
FRCM strengthening systems externally embedded on masonry supports. The procedure was carried out by directly 
considering the experimental data and in particular the strain gauge measurements. A procedure for the derivation of the 
shear stress-slip law for FRCMs based on the experimental data was also proposed in [3]. In particular, the authors firstly 
estimated the fracture energy by using the well-known formula derived by the theory of linear fracture mechanics and, 
subsequently, they performed numerical FE analyses to detect the optimal values of the parameters of the cohesive shear 
stress-slip law.  
In[4] the bond behavior of FRCM-to-concrete was analytically examined by using a general approach applied to the case 
of FRP-materials. The results emerged from this study particularly emphasized the role of the pronounced descending 
branch of the calibrated laws in leading to large values of the effective anchorage length. In addition, lower values of bond 
shear stresses on the concrete surface with respect to those typically characterizing FRP strengthening systems were 
observed. A similar approach was also presented in [3] for the case of carbon-FRCM materials externally applied on 
masonry supports. 
In [13] two approaches for numerically studying the bond behavior of masonry specimens strengthened with FRCMs 
were proposed. The first one, consisted of an analytical-numerical approach specifically accounting for the interaction 
between the reinforcement and the mortar; the second approach consisted of a full 3D-FEM non-linear approach 
obtained as an extension of the procedure originally adopted in [13] and in [15]. 
A recent study presented in [9,16] was mainly devoted to investigate the influence of the upper mortar layer on the bond 
behavior of FRCM-strengthening systems applied on structural supports. In particular, the authors carried out a 
theoretical modeling approach based on the solution of a system of differential equations obtained by introducing 
equilibrium considerations. From the study emerged interesting aspects concerning the role of the upper mortar layer on 
the debonding process of FRCMs. Among these, it was observed that increasing the applied load after the occurrence of 
the de-bonding between the reinforcement and the upper interface it does not lead to further increases of the peak value 
of normal stresses of the upper mortar. On the other hand, after the occurrence of the first crack at the upper mortar, 
only the peak value of slips at the lower interface continues to increase whilst the peak value of slips at the upper interface 
does not significantly increase.  
In [10] it was proposed a simple approach for the study of the bond behavior of FRCM applied to concrete supports able 
to enable the use of a common interface modeling strategy by implicitly introducing the effect of the damage of the matrix 
into the shear behavior of the reinforcement/mortar interface layer.  
In this paper a one dimensional simple model, based on the one presented in [9] and in [16], is proposed for the study of 
the bond behavior of FRCM strengthening systems externally applied to masonry substrates. The model is mainly 

T 



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

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characterized by the derivation of the explicit solution of a system of differential equations obtained by considering the 
equilibrium of an infinitesimal portion of the reinforcement and the mortar layers composing the strengthening systems. 
In order to model the slip between the reinforcement and the upper and lower mortar layers, two approaches are 
considered. The first approach (denoted in the following approach 1), considers a nonlinear behavior of the lower 
reinforcement/mortar interface only, by considering a shear stress-slip constitutive law characterized by a linear brittle 
behavior with a residual strength in the post-peak phase. On the other hand, the approach 2 assumes a nonlinear behavior 
for both the lower and the upper reinforcement/mortar interface, still considering a shear stress-slip constitutive law 
characterized by a linear fragile behavior with a residual strength in the post-peak phase. Moreover, in the latter approach, 
a calibration of the shear strength of the upper interface is proposed in order to implicitly account for the effect of the 
damage of the mortar on the contribution of this component of the strengthening system.  
In addition to these approaches, in the second part of the paper is presented the analytical solution in case of a shear 
stress-slip law characterized by a linear softening behavior in the post-peak phase. In particular, a step function 
approximating the law together with the procedure carried out from the approach 2 are used in order to derive the 
analytical solution. 
The proposed approaches are validated in the paper by considering experimental results derived from the literature. 
Moreover, the results are also compared with the ones obtained by the model recently proposed by [9,16], where, 
differently from the proposed approaches, the damage of the upper mortar was explicitly introduced in the model by 
assuming a nonlinear behavior in terms of normal stress-strain for the upper mortar layer. Although this assumption 
allows to account for the phenomena generally observed, it leads to a computational effort significantly greater than the 
one characterized the two approaches proposed in this paper. 
 
 
ACCOUNTED MODEL AND APPROACHES 
 

he model here considered for the study of the bond behavior of FRCM systems externally applied on masonry or 
concrete supports is based on the work in [9, 16]. Indeed, considering the scheme shown in Fig. 1, the main 
components characterizing the model are: a cohesive support, a lower mortar layer, a lower interface, the 

strengthening, an upper interface and an upper mortar layer. Introducing a reference axis x in the direction of the 
reinforcement system and fixing the origin in correspondence of the unloaded section, the equilibrium of forces 
characterizing an infinitesimal portion of the reinforcement and the upper mortar layer (see Fig. 1) leads to the following 
system of differential equations governing the problem of the bond behavior: 
 

   

 

0

0

p e e i i
p p p

e
e e ec

p c p

d
b t s s b

dx

d
b t s b

dx


 




      

  

        (1) 

 

where p  and 
e
c  are the normal stresses in the reinforcement and in the upper mortar, respectively; pt  and 

e
ct  are the 

thicknesses of the reinforcement and the upper mortar, respectively; i  and e  are the shear stresses at lower and upper 

interfaces, respectively, both depending on the corresponding slips is  and es ; pb  is the width of the reinforcement. 

Introducing the following hypotheses: 
- the support and the lower mortar layer are assumed to be rigid; 
- the (lower and upper) mortar/reinforcement interfaces are modeled as zero-thickness elements with only shear 
deformability; 
- the upper mortar layer and the reinforcement are assumed deformable only axially. 

it is possible to write the displacements of both the reinforcement and the upper mortar layer (namely pu  and 
e
cu  , 

respectively) as functions of the slip of the lower and upper interfaces: 
 

i
p

e i e
c

u s

u s s



 
           (2) 

T 



 

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24                                                                  
 

324 
 

 

 
 

Figure 1: Schematic of an infinitesimal portion of the strengthening system and the upper mortar component used for performing the 
equilibrium of the involved forces. 
 
Considering a linear-elastic behavior for both the reinforcement and the mortar: 
 

i
p

p p p

e i e
e c
c c c

du ds
E E

dx dx

du ds ds
E E

dx dx dx





 

 
   

 

         (3) 

 
the system of differential Eqns. (1) becomes: 
 

   

 

2

12

2 2

22 2

0

0

i
e e i i

i e
e e

d s
K s s

dx

d s d s
K s

dx dx

 



      

      

         (4) 

 

where 1K  and 2K  are two constants equal to: 
 

1 2
1 1

, 
e

p p c c

K K
E t E t

            (5) 

 
Considering the system (4), the explicit solution is here derived by introducing different shear stress-slip laws 
characterizing the behavior of the reinforcement/mortar interface. 
 
Approach 1: nonlinear behavior of the lower interface 
A preliminary approach is based on the assumption of a linear-fragile behavior with a residual shear strength in the post-
peak stage only for the lower interface: 
 

1( )     

( )       otherwise

i i i i i

i i i
res

s G s s s

s



 

  



         (6) 

 

where ires  is the residual value of the shear strength in the post-peak stage, and 
iG  is the shear stiffness of the lower 

interface in the pre-peak stage. 

Differently, a linear-elastic behavior is assumed for the upper interface ( )e e e es G s  , where eG  is the shear stiffness of 
the upper interface.  

P
p pd p

c cd c

e
e
isupport

lower interface

strengthening

upper interface

upper mortar

lower mortar

x



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

325 
 

On the basis of these assumptions it is evident that, after the attainment of the slip threshold value at the lower interface, 
two different parts characterize the behavior of the specimen: part “1” where the upper mortar and the interfaces are both 
in the pre-peak stage and part “2” where the upper mortar and the upper interface are both in the pre-peak stage while the 
lower interface is de-bonded for a length a, representing an unknown of the problem. Consequently, four differential 
equations govern the problem.  
The first two equations are derived by considering the equilibrium involving an infinitesimal portion of the strengthening 
system in the part “1”: 
 

2
1

3 1 12

2 2
1 1

4 12 2

0

0

0

i
e i

i e
e

d s
K s s

dx
x L a

d s d s
K s

dx dx




     
  

      

       (7) 

 

where: 3 1 4 2, ,  
i

e e
e

G
K K G K K G

G
   . 

The other two equations are obtained through the equilibrium involving an infinitesimal portion of the strengthening 
system of the part “2”: 
 

2
2

3 22

2 2
2 2

4 22 2

0

0

i
e

i e
e

d s
K s

dx
L a x L

d s d s
K s

dx dx




     
  

      

       (8) 

 

where 
i
res

eG


  . 

The system of differential Eqns. (7) and (8) has an analytical solution that depends on eight constants of integration 
determined by introducing suitable boundary conditions. In particular, the following conditions are indeed enforced: 
 

 
   
   
   

       
 

1

1 2

1 2

1 2

1 2 1 2

1 1

0 0

0 0 0

P

e e
c c

e e
c c

e e
p p

i i e e

i

L

L a L a

L a L a

s L a s L a s L a s L a

s L a s



 

 

 



 

  

  

     

 

          (9) 

 
The solution is graphically reported in Fig. 2 by considering a length value of the part “2” equal to a=50 mm, a residual 
value of shear strength equal to zero and the data reported in Tab. 1. 
 
Approach 2: nonlinear behavior of both the interfaces 
As shown in [11,17], the damage of the upper mortar generally occurs before the slipping of the reinforcement/mortar 
interfaces by particularly influencing the shear stress transfer mechanism. 
This phenomenon is here simple introduced by considering an elastic-fragile behavior also for the upper interface and 
assuming for this component of the strengthening system a bond strength equal to the shear stress corresponding to the 
attainment of the tensile strength of the upper mortar layer. In other words, the effect of the damage of the upper mortar 
is implicitly introduced into the behavior of the upper interface. 



 

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326 
 

a) b) c) 
 

Figure 2: Approach 1: a) shear stress developing at the interfaces; b) slip of the interfaces; c) normal stresses at the upper mortar layer. 
 

 symbol [unit] value 
Young’s modulus of the 

reinforcement 
Ep [MPa] 206000 

Young’s modulus of the mortar Ec [MPa] 7000 
equivalent thickness of the 

reinforcement 
tp [mm] 0.054 

thickness of the mortar tc [mm] 4 
width of the reinforcement bp [mm] 60 

width of the mortar bc [mm] 60
bond length L [mm] 1000 

 

Table 1: Data accounted for numerical analyses. 
 
Considering this assumption, the system of equations governing the problem has to account for the development of three 
possible parts characterizing the behavior of the specimen: part “1”: 0<x<L-(a+b), where both the interfaces are in the 
pre-peak stage and normal stresses in the upper mortar are lower than the tensile strength; part “2”: L-(a+b)<x<L-b, 
where the upper interface is in the post-peak stage (the length of this zone is equal to a); part “3”: L-b<x<L, where both 
the interfaces are in the post-peak stage (the length of this zone is equal to b). In particular, while the equations 
characterizing the part “1” are the eqns. (7), the equations characterizing the part “2” are: 
 

2
2

5 22

2 2
2 2

22 2

0

( )

0

i
i

i e
e
res

d s
K s

dx
L a b x L b

d s d s
K

dx dx






     

    
      

      (10) 

 
and the equations characterizing the part 3 are: 
 

2
3

12

2 2
3 3

22 2

0

0

i
e i
res res

i e
e
res

d s
K

dx
L b x L

d s d s
K

dx dx

 




     

        

       (11) 

 

where: 
e
res

iG


  , 5 1

iK K G , a is the length of the part 2, b is the length of the part 3, eres  and 
i
res  are the residual shear 

strength values are of the upper and lower interfaces respectively. 
The whole system of differential Eqns. (7), (10) and (11) has an analytical solution that depends on twelve constants of 
integration determined by introducing suitable boundary conditions similar to the ones introduced for the approach 1. 
The solution is graphically reported in Fig. 3 by considering a length value of the part “2” equal to a=50 mm, a length of 



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

327 
 

the part “3” equal to b=50 mm, a shear strength of the lower mortar equal to 0.9 MPa, a shear strength of the upper 
mortar equal to 0.45 MPa, a residual value of shear strength equal to zero for both the interfaces and the data reported in 
Tab. 1. 
 

a) b) c) 
 

Figure 3: Approach 2: a) shear stress developing at the interfaces; b) slip of the interfaces; c) normal stresses at the upper mortar layer. 
 
Numerical applications for the approach 1 and the approach 2 
In order to assess the capability of the proposed model in providing a reliable prediction of the bond behavior of FRCM 
strengthening systems, some case studies derived from the current literature are considered ([4]). The case studies, here 
considered, consist of single lap shear tests of concrete blocks strengthened by a bidirectional unbalanced PBO fiber net 
with two mortar layers. In particular, in the present research the specimens are considered characterized by a bond length 
equal to 450 mm and two different values of the reinforcement width: bp=60 mm and bp=80 mm. These tests are of 
particular relevance since the experimental outcomes showed the damage of the upper mortar layer of tested specimens 
before the slipping at the interface level. 
Regarding the application of the approach 1, a shear stress-slip law for the lower interface characterized by a shear 
strength equal to 0.9 MPa and a slip threshold equal to 1.2 mm is considered (see [4]). On the other hand, for the 
approach 2 a shear strength value of the upper interface corresponding to the attainment of the tensile strength of the 
upper mortar layer (fct=3.5 MPa) is considered. For both the approaches, a null value of the residual shear strength is 
assumed for both the interfaces. 
The obtained results are shown in Fig. 4 in terms of applied load P versus the slip of the lower interface at the loaded 
section. In the same figure the envelop of the experimental curves (grey region) and the curve carried out in [4] are also 
reported. 
 

a) b) 
 

Figure 4: Comparison with experimental tests in terms of applied force vs. slip curves. 
 

0

2

4

6

8

10

0 2 4 6 8 10 12

Grande et al. 2018

approach 1

approach 2

slip si(x=L) [mm]

a
p
p
li
e
d
lo
a
d
P
 [
k
N
]

D’Antino et al. 2015 – bp=60mm

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Grande et al.
2018
approach 1

approach 2

slip si(x=L) [mm]

a
p
p
li
e
d
lo
a
d
P
 [
k
N
]

D’Antino et al. 2015 – bp=80mm



 

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328 
 

From the plots clearly emerges the importance of introducing in the model the influence of the damage of mortar on the 
contribution of the upper interface. Indeed, while the curves deduced from the approach 1 overestimate the experimental 
peak load, the curves derived by using the approach 2 provide a good approximation of the experimental outcomes. 
 
 
APPROXIMATE SOLUTION IN CASE OF SOFTENING BEHAVIOR  
 

n the previous section a solution of the de-bonding problem has been presented by introducing simplified shear 
stress-slip laws for the interfaces assumed with a perfect brittle post-peak behavior. This has allowed to obtain the 
analytical solution of the problem and to easily analyze different damage stages. 

The majority of studies available in the current literature are instead based on the use of constitutive laws where the 
softening governs the post- peak phase of the interface (cohesive models). The use of this type of laws at the interface 
level generally does not lead to computational problems when a Finite Element approach, or other types of numerical 
techniques, are used. On the contrary, it can present some drawbacks for the derivation of an analytical solution, 
particularly in the case of FRCM systems where there are two interfaces interacting among them in the de-bonding 
process.  
For this reason, the analytical solution is here derived by approximating the linear descending post-peak branch of the 
shear stress-slip law throughout a step function, i.e. a piecewise constant function (Fig. 5), and following the approach 
presented in the first part of the paper. Moreover, taking into account the ‘approach 2’ presented in the first part of the 
paper, the softening behavior is only introduced for the lower interface, whilst a linear-brittle behavior is accounted for 
the upper interface by opportunely calibrating the corresponding bond strength. 
 

 
 

Figure 5: Scheme used for approximating the softening branch of the shear stress-slip law. 
 
Also in this case, the system of equations governing the problem depends on the status of the two interfaces (see Fig. 6). 
Indeed, increasing the applied load P, the first phase is certainly characterized by the pre-peak stage of both the interfaces 
(Fig. 6.a). In this case, the system of the two equations governing the problem is: 
 

   

 

2

12

2 2

22 2

0

0

i
e e i i

i e
e e

d s
K s s

dx

d s d s
K s

dx dx

 



      

      

         (12) 

 
where: 
 

( )     

( )

i i i i

e e e e

s G s

s G s








           (13) 

 

s1 s2 s3 s4

1
2
3

4



s

I 



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

329 
 

a) b) 
 

c) 
 

Figure 6: Scheme used for approximating the softening branch of the shear stress-slip law: a) phase 1 *1 1;
i es s s s  ; b) phase 2 zone 

1: *1 1;
i es s s s  ; zone 2 - zone 3: *1 1;

i es s s s  ; c) phase 3 zone 1: *1 1;
i es s s s  ; zone 2 - zone 3: *1 1;

i es s s s  ; zone 4 - 

zone 5: *1 1;
i es s s s  . 

 
The analytical solution of this system depends on four constants of integration determined by introducing suitable 
boundary conditions. In particular, considering the attainment of the slip s1 at the lower interface (generally, the lower 
interface exhibits higher values of shear stresses than the upper interface), the following conditions are indeed enforced: 
 

 
 
 
  1

0 0

0 0

0

p

e
c

e
c

i

L

s L s















           (14) 

 

The subsequent phase is then characterized by the upper interface in the pre-peak stage and the lower interface in the 
post-peak stage (Fig. 6.b). This phase ends when the upper interface attains the slip s1 at the loaded end. The occurrence 
of this condition identifies the number of steps approximating the descending branch of the shear stress-slip law of the 
lower interface involved in this phase. The number of equations governing the problem depends on the number of the 
zones in which the systems is divided and, thus, on the number of steps considered in the approximate descending branch 
of the interface constitutive law. 
For instance, considering the case of the Fig. 6.b, it is supposed that there are three zones characterizing the behavior of 
the lower interface, which include two steps approximating the descending branch. Consequently, the differential 
equations composing the system are the following: 
- zone 1: 
 

2
1

1 1 12

2 2
1 1

2 12 2

0

        0 - ( )

0

i
e e i i

i e
e e

d s
K G s G s

dx
x L a b

d s d s
K G s

dx dx


     

  
      

          (15) 

 

- zone 2: 
 

2
2

1 2 22

2 2
2 2

2 22 2

0

        - ( ) -

0

i
e e

i e
e e

d s
K G s

dx
L a b x L b

d s d s
K G s

dx dx




     
  

      

          (16) 

upper interface

lowe interface

(1) (2) (3)

(1) (2) (3) (4) (5)



 

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24                                                                  
 

330 
 

- zone 3: 
 

2
3

1 3 32

2 2
3 3

2 32 2

0

        -

0

i
e e

i e
e e

d s
K G s

dx
L b x L

d s d s
K G s

dx dx




     
       

       (17) 

 

where 2  and 3  are the shear stress values identified by the steps approximating the descending branch in the phase 2. 
In this case, it is necessary to introduce twelve boundary conditions. Four boundary conditions concern the two end 
sections of the specimen where it is stated that: the stress is zero at the free end of the reinforcement; the stresses are zero 

at two ends of the upper mortar layer; the upper interface attains the slip *1s  at the loaded end, i.e. the slip corresponding 
to the attainment of the tensile strength of the upper mortar layer, i.e.: 
 

 
 
 

 

,1

,1

,3

*
3 1

0 0

0 0

0

p

e
c

e
c

e

L

s L s















           (18) 

 
The other boundary conditions concern the continuity of normal stresses in the mortar layer and in the reinforcement, the 
continuity of the slip at the lower and upper layers at the section: 
 

           
       

           
       

1 2 1 2

2 3 2 3

1 2 1 2

1 2 1 2

e e e e
c c p p

e e e e
c c p p

i i e e

i i e e

L a b L a b L a b L a b

L b L b L b L b

s L a b s L a b s L a b s L a b

s L b s L b s L b s L b

   

   

         

     

         

     

    (19) 

 
Moreover, due to the introduction of the additional unknowns a and b, i.e. the length of the zone 2 and the zone 3 
respectively, it is necessary to introduce the following two additional boundary conditions: 
 

 
 

2 2

3 3

i

i

s L b s

s L s

 


           (20) 

 
where s2 and s3 are the slips identified by the two steps along the softening branch. 
The subsequent phase (Fig. 6.c) is characterized by both the interfaces in the post-peak stage. An increase of the number 
of equations is still due to the steps discretizing the descending branch of the shear stress-slip law of the lower interface. 
Moreover, some of the zones are also characterized by a null value of the shear stress at the upper interface. Indeed, 
considering the example shown in Fig. 6.c, the set of equations characterized by this condition are the ones related to the 
last two zones: 
- zone 4: 
 

 

2
4

1 42

2 2
4 4
2 2

0

        -

0

i

i e

d s
K

dx
L c d x L d

d s d s

dx dx




 


        

       (21) 

 



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

331 
 

- zone 5: 
 

2
5

1 52

2 2
5 5
2 2

0

        -

0

i

i e

d s
K

dx
L d x L

d s d s

dx dx




 


      

        (22) 

 
The boundary conditions are the same of those accounted for the previous phases and, also in this case, it is necessary to 
introduce additional boundary conditions for deriving the length of the different zones. 
 
Numerical applications for the case with softening 
Considering the approach based on the step function approximation, numerical applications are performed with reference 
to the case studies presented in the first part of the paper ([4]). 
In particular, considering the shear stress-slip law proposed in [4], a bi-linear law has been derived (Fig. 7.a). Subsequently, 
this law has been approximated throughout a step function and assumed for the behavior of the lower interface (Fig. 7.b). 
Regarding the upper interface, a linear brittle shear stress-slip law has been considered by assuming a bond strength value 
equal to 0.9 MPa, which corresponds to the attainment of the tensile strength of the upper mortar layer (see the approach 
2). 
 

a) b) 
 

Figure 7: Shear stress-slip law accounted for the lower interface in the case of the approach based on the step function approximation. 
 

a) b) 
 

Figure 8: Results in terms of applied load vs. slip: comparison between the approach 2 and the case with softening. 
 

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

sh
e
a
r 
st
re
ss
 [
M
P
a
]

slip [mm]

bi‐linear

step‐function

0

2

4

6

8

0 1 2 3 4

a
p
p
li
e
d
 l
o
a
d
 P
 [
k
N
]

slip si(x=L) [mm]

D'Antino et al., 2015 ‐
bp=60mm

approach 2

softening

0

2

4

6

8

10

0 1 2 3 4

a
p
p
li
e
d
 l
o
a
d
 P
 [
k
N
]

slip si(x=L) [mm]

D'Antino et al., 2015 ‐
bp=80mm

approach 2

softening



 

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24                                                                  
 

332 
 

Then, considering the equations at the basis of the problem and the boundary conditions, both presented in the previous 
section, the analytical solution has been derived and presented in Fig. 8 in terms of applied load vs. the slip. In the same 
plot it is also reported the curve previously obtained by applying the approach 2 and then considering a linear-brittle 
behavior also for the lower interface. By examining the curves of the plot it emerges a similar value in terms of peak load 
and post-peak behavior. This outcome is due to the fact that the law assumed for the approach 2 and the one with 
softening are characterized by the same value of fracture energy: f=0.54 N/mm. On the other hand, the greater initial 
stiffness and the smoother post-peak branch of the case with softening are strictly related to the shape of the shear stress-
slip laws. 
In Fig. 9, it is reported for the case with bp=60 mm the distributions along the bond length of shear stresses at the 
interfaces and normal stresses at the upper mortar layer for the load step corresponding to the attainment of the slip s1 at 
the lower interface. The plots underline the length of the four zones (a1=10mm; a2=28.5mm; a3=54mm; a4=25mm) 
characterizing the behavior of the lower interface. Moreover, it also emerges that: the maximum value of normal stresses 
in the upper mortar layer remains equal to the tensile strength because of the assumption of a reduced bond strength for 
the upper mortar; after the zone a1, since shear stresses in the upper interface are equal to zero, also the normal stresses at 
the upper mortar layer are zero. 
 

a) b) c) 
 

Figure 9: Results in terms of distribution along the bond length of a) slips, b) shear stresses, c) normal stresses at the step 
corresponding to the attainment of the slip value si=sf=1.2 mm at the lower interface: case with softening. 
 
 
CONCLUSIVE REMARKS 
 

n this paper a one-dimensional simplified model for studying the bond behavior of FRCM strengthening systems 
externally applied to masonry structures is proposed. The model is based on the study of an infinitesimal portion of 
the strengthening system composed by the reinforcement and the mortar layers, computing the explicit solution of a 

system of equilibrium differential equations. Interfaces are introduced between the reinforcement and the upper and lower 
mortar layers to model the possible slip phenomena. A nonlinear shear-stress slip laws, characterized by a brittle fracture 
with a residual strength in the post-peak stage, is adopted for the interfaces to derive the two approaches presented in the 
first part of the paper. These approaches differ only for the behavior of the upper interface: in the first one the upper 
interface is characterized by a linear behavior, while, in the second one by the proposed nonlinear response. Both the 
approaches have been applied to two case studies available in the literature. 
From the results, approach 2 is able to better describe the experimental results, both in terms of peak load and post peak 
behavior, with respect to approach 1. In approach 2 the value of the peak shear stress in the upper interface is calibrated 
in order to take into account the damage occurring in the upper layer of mortar. The results are also compared with the 
ones obtained by the model proposed by [9,16]. 
From a computational point of view, the presented model results simpler than the one proposed in [9,16], as it doesn’t 
model directly the damage mechanism in the upper layer of mortar but it is able to take it into account by suitable setting 
the peak shear stress in the constitutive law of the upper interface. 
In the second part of the paper it has been presented the analytical solution in the case of softening behavior of the lower 
interface. In particular, a step function has been introduced in order to approximate the softening branch. Also in this case 
the approach has been applied to the same case studies of literature. From the obtained results it is emerged a response 
similar to the one obtained in the approach 2 due to the important role played by the fracture energy, parameter common 
to the two models.  

I 



 

                                                               E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 
 

333 
 

 
REFERENCES 
 
[1] D’Ambra, C., Lignola, G.P., Prota, A., Sacco, E., Fabbrocino, F. (2018). Experimental performance of FRCM retrofit 

on out-of-plane behaviour of clay brick walls, Compos. Part B Eng., 148, pp. 198-206.  
DOI: 10.1016/j.compositesb.2018.04.062. 

[2] D’Ambrisi, A., Feo, L., Focacci, F. (2012). Bond-slip relations for PBO-FRCM materials externally bonded to 
concrete, Compos. Part B Eng., 43(8), pp. 2938-2949. DOI: 10.1016/j.compositesb.2012.06.002. 

[3] D’Ambrisi, A., Feo, L., Focacci, F. (2013). Experimental analysis on bond between PBO-FRCM strengthening 
materials and concrete, Compos. Part B Eng., 44, pp. 524-532. DOI: 10.1016/j.compositesb.2012.03.011. 

[4] D’Antino, T., Sneed, L.H., Carloni, C., Pellegrino, C. (2015). Influence of the substrate characteristics on the bond 
behavior of PBO FRCM-concrete joints, Constr. Build. Mater., 101(1), pp. 838-850.  
DOI: 10.1016/j.conbuildmat.2015.10.045. 

[5] de Felice, G., De Santis, S., Garmendia, L., Ghiassi, B., Larrinaga, P., Lourenço, P.B., Oliveira, D. V., Paolacci, F., 
Papanicolaou, C.G. (2014). Mortar-based systems for externally bonded strengthening of masonry, Mater. Struct. 
Constr., 47(2), pp. 2021-2037. DOI: 10.1617/s11527-014-0360-1. 

[6] Grande, E., Imbimbo, M., Sacco, E. (2015). Investigation on the bond behavior of clay bricks reinforced with SRP 
and SRG strengthening systems, Mater. Struct. Constr., 48(11), pp. 3755-3770. DOI: 10.1617/s11527-014-0437-x. 

[7] Marcari, G., Basili, M., Vestroni, F. (2017). Experimental investigation of tuff masonry panels reinforced with surface 
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[9] Grande, E., Imbimbo, M., Sacco, E. (2017). Local bond behavior of frcm strengthening systems: Some considerations 
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[11] Carbone, I. (2010). Delaminazione di compositi a matrice cementizia su supporti murari. Università degli Studi Roma 
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[13] Carozzi, F.G., Milani, G., Poggi, C. (2014). Mechanical properties and numerical modeling of Fabric Reinforced 
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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /UKR <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>
    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice