Development of an interface numerical model for C-FRPs applied on flat and curved masonry pillars

Journal Pre-proofs Development of an interface numerical model for C-FRPs applied on flat and curved masonry pillars Gabriele Milani, Mario Fagone, Tommaso Rotunno, Ernesto Grande, Elisa Bertolesi PII: DOI: Reference:

S0263-8223(19)34451-4 https://doi.org/10.1016/j.compstruct.2020.112074 COST 112074

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Composite Structures

Received Date: Revised Date: Accepted Date:

22 November 2019 28 January 2020 14 February 2020

Please cite this article as: Milani, G., Fagone, M., Rotunno, T., Grande, E., Bertolesi, E., Development of an interface numerical model for C-FRPs applied on flat and curved masonry pillars, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct.2020.112074

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Revised paper, modifications highlighted in color YELLOW

Development of an interface numerical model for C-FRPs applied on flat and curved masonry pillars Gabriele MILANI*(1), Mario FAGONE(2), Tommaso ROTUNNO(3), Ernesto GRANDE(4), Elisa BERTOLESI(5) (1) (2) (3) (4) (5)

*

Department of Architecture, Built environment and Construction engineering (ABC), Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan (Italy) Dipartimento di Ingegneria Civile e Ambientale (DICEA), Università degli Studi di Firenze, Piazza Brunelleschi 6, 50121 Florence (Italy) Dipartimento di Architettura (DiDA), Università degli Studi di Firenze, Piazza Brunelleschi 6, 50121 Florence (Italy) Department of Sustainability Engineering, University Guglielmo Marconi, Via Plinio 44, 00193 Rome (Italy) ICITECH, Universitat Politècnica de Valencia, Camino de Vera s/n 46022, Valencia (Spain)

corresponding author, e-mail: [email protected]

Abstract A novel interface numerical model for the incremental analysis of the debonding phenomenon of Carbon Fiber Reinforced Polymer (C-FRP) reinforcements externally applied on flat and curved masonry pillars is 

presented. The interface tangential stress-slip behavior is suitably described by a C exponential function, that accounts for the ductility and residual strength variation due to the presence of interfacial normal stresses, according to a frictional-cohesive relationship. Such dependence is particularly important when dealing with C-FRP reinforcements applied to masonry curved structures (i.e. arches and vaults). The smooth interface relationship here adopted allows to deal with a boundary value problem for a system of second order differential equations, representing a standard delamination problem, without singularities. Consequently, an easy and robust numerical solution algorithm based on a standard finite differences approach can be adopted. The model is validated against some experimental and numerical results obtained previously by the authors and concerning shear-lap bond tests of flat and curved masonry pillars reinforced by C-FRP sheets. The obtained results underline an excellent robustness and reliability of the experimental global and local behavior. Keywords: masonry; arches and vaults; C-C-FRP reinforcement; non-linear Boundary Value Problem for ODEs; debonding; curved substrates.

1. Introduction Fiber Reinforced Polymer FRP reinforcements proved to be particularly effective in all those cases where masonry is unable to withstand moderate tensile stresses [1][2], as it typically occurs during earthquakes in the majority of existing/historical buildings. The debate on the application of FRP composites in general and C-FRP in particular for the rehabilitation and seismic upgrading of historical masonry structures or existing buildings in general is however still open [3]-[5], some authors raising doubts on the long term efficacy and cost of the intervention when compared with traditional techniques. The major drawback seems however related to the reversibility issue, which

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is nowadays considered a priority for any seismic upgrading with innovative materials. In order to be consistent with such conservation requirement, part of the scientific efforts have been recently channeled to alternative –appearing more reversible- innovative strengthening systems, such as Textile Reinforced Mortars TRMs [6]-[12]. Apart open issues related to reversibility, durability and vapor permeability of FRP strips, from a strictly structural point of view, the application of FRP on masonry walls and arches is certainly very interesting. The mechanical behavior is quite simple in principle, FRP absorbing tensile stresses that masonry is unable (or scarcely able) to withstand. The well-known inability of masonry to be performant in the tensile regime inspired a quite famous theoretical approach to study masonry, which is known in the specialized literature as No-Tension Material NTM model [13],[14]. NTM is particularly suited when dealing with the formation of flexural hinges in the masonry material rather than for the analysis of shear walls; in this regard, it is an approach that realistically predicts the actual behavior of masonry arches at failure [15],[16]. As matter of fact, arches fail for the formation of four flexural hinges and one of the most effective ways to drastically increase their load bearing capacity against horizontal loads is to externally retrofit (either at the intrados or extrados) by means of C-FRP strips [15]-[25]. C-FRP is therefore a quite effective reinforcement technique for masonry curved substrates, i.e. for arches and vaults in general. At a structural level, FRP counteracts the opening of a flexural hinge in NTMs and maximizes its efficacy when it is disposed in tensile region, on the opposite side with respect to the location of a hinge forming the mechanism [15][17]. Having precluded the formation of a flexural hinge in a specific position, the arch will fail either translating the position of the hinge outside the reinforced region, or using different modalities, for instance involving masonry sliding or crushing. In both cases, in agreement with limit analysis theorems, the failure multiplier will be higher than that without reinforcement [19]. The efficacy of FRP reinforcement is however guaranteed only if debonding from the glued surface does not occur prematurely [26]-[46]. Such mode of failure is particularly dangerous, because characterized by a sudden brittle propagation of damage inside the support, few millimeters under the glued surface of the reinforcement. For this reason, the study of the delamination phenomenon of FRP strips from masonry took particular importance in the last two decades and a variety of numerical, analytical and experimental works were presented in the technical literature [26]-[52], also related to durability issues [41]-[46] or in presence of local anchoring devices [47]-[52], all aimed at providing sufficient insight into the main features characterizing such phenomenon. Experimentally, several different apparatuses and testing protocols were conceived to study FRP debonding [37]-[40]: one of the most successful being certainly the single lap shear test [39][40]. At present, there is a superabundance of technical literature dealing with the experimental and

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numerical analysis of FRP-from-masonry debonding [26]-[46], especially in single lap shear tests. However, despite the fact that one of the most important applications of FRP reinforcement is the strengthening of masonry arches, technical literature dealing with the FRP debonding from curved masonry pillars is still quite limited [52]-[59]. Even the Italian Technical recommendations CNRDT200 [40] for FRP reinforcement on masonry and r.c. structures provide quite generic information on the role played by normal stresses acting at the FRP/masonry interface in the ultimate ductility and load carrying capacity. The present work follows a research stream by the authors focused on the experimental [52]-[54] and numerical analysis [51][55]-[59] of the debonding phenomenon occurring in curved masonry pillars reinforced by C-FRP strips, in presence [51][52] and absence [54]-[56] of anchor spikes. The experimental program relied in the analysis of pillars constituted by five common Italian clay bricks stacked along a circular arc and subjected to single lap shear. Two curvature radii were investigated, equal to 1500 mm and 3000 mm respectively, with reinforcement glued at the intrados and extrados Figure 1. Also, the flat configuration was investigated to provide a reference in a case where stresses normal to the interface are negligible. Two different numerical models were conceived to interpret experimental results, one based on a 3D FE discretization with damage spreading into the support [56], the other based on the C-FRP/masonry interface 1D FE modelling by means of normal and tangential coupled non-linear springs [55]. Whilst the first approach is useful at research level to obtain a better insight into the damage concentration inside the support, the second one is more oriented to utilization in practical applications, still however having the drawback to require dedicated FE software and experience in the utilization of FEs in presence of strong global softening phenomena. The present paper is aimed at presenting a simpler numerical approach which avoids a FE discretization of the interface and discusses a fast and robust interface numerical model for the incremental analysis of the debonding phenomenon of C-FRP strips in flat and curved masonry pillars. The core of the approach is the interface tangential stress-slip law assigned at the C FRP/masonry interface, which is a C exponential function accounting for the ductility and

residual strength variation due to the application of an interfacial normal stress, according to a frictional-cohesive relationship. Equilibrium equations along the tangential and horizontal direction, assuming that (i) all the non-linearity concentrates at the interface, (ii) C-FRP behaves as an elastic material and (iii) the substrate is stiff enough to disregard it deformations, allowing to describe CFRP debonding by means of a non-linear second order differential equation in the slip variable, with assigned boundary conditions on the C-FRP free and loaded edges. The choice of an infinitely

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differentiable interface relationship provides stability to finite differences algorithms, allowing to deal with a non-linear second order boundary value problem without singularities. The procedure does not require dedicated software and can be used by anyone having at disposal a standard finite differences solver for ordinary differential equations with assigned boundary conditions (ODEBVP, Ordinary Differential Equation- Boundary Value Problem solver [60]). The model is validated against experimental and numerical results obtained previously by the authors on previously mentioned C-FRP reinforced flat and curved masonry pillars, showing excellent robustness and predictivity of the experimental behavior, at a fraction of the time needed by the FE computations.

F

F

F

F

F

FRP

Lb

brick mortar

LCAI

LCBI

LFlat LCBE

LCAE

Hb R=1.5m

CAI LCAI= 330 mm

R=3m

CBI LCBI= 330 mm

R=

R=3m

CBE

Flat LFlat= 330 mm

LCBE= 354 mm

R=1.5m

CAE LCAE= 382 mm

Figure 1: The five families of experimentally tested C-FRP reinforced curved masonry pillars (Lb=250 mm, Hb=65 mm).

2. Brief overview of the experimentation carried out The experimentation carried out, already presented in [54] and here briefly recalled for the sake of clearness, was intended to study the role played by interfacial normal stresses in C-FRP reinforcement of masonry arches. Single lap shear tests were carried out on five different typologies of specimens, characterized by different radii of curvature R and reinforcement position (glued at the intrados or extrados). In particular two radii R equal to 1500 and 3000 mm were studied, see Figure 1, defining two sets of cases with high (labeled as CA specimens) and low (labeled as CB specimens) curvature, respectively. C-FRP was glued at the intrados (I) and extrados (E) in both curvature configurations, so to define in total four different cases (CBI, CBA, CAI, CAE). The flat configuration was also tested as reference case. Masonry specimens are constituted by five clay bricks of dimension 65x120x250 mm and four mortar joints. The authors performed several standard tests on the constituent materials to fully characterize their mechanical behavior in the elastic and inelastic range, which include compression

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tests, three-point bending tests and pull-off tests on reinforced bricks [54]. The reinforcement is a 100 mm wide C-FRP strip glued on the masonry curved surface by means of epoxy resin, whose mechanical properties are given by the producer. C-FRP is assumed linear elastic in the present simulations with a thickness tF equal to 0.165 mm and a Young modulus of 250 GPa. The reader is referred to [54] for further details on the experiments carried out and the preliminary mechanical characterizations, all details which are not reported here for the sake of conciseness. The test apparatus utilized to perform standard single lap shear tests is depicted in Figure 2. To properly prevent a rigid rotation of the whole specimen, the curved pillars were constrained by a steel plate placed on the upper edge and by a steel wedge disposed under the base the lower base (Figure 2). Tests were performed under displacement control –increased in the tests monotonically- applied by means of an actuator equipped with a fork and a steel cylinder. The final portion of the (dry) C-FRP was wrapped and glued to the steel cylinder. The steel fork was connected in series to a load cell (50 kN), applying the tensile load directly to the reinforcement. Five replicates for each typology were tested, with a total of 25 tests performed. The experimental load-slip curves obtained in the five different cases are shown in Figure 3. Such curves are the result of a “shifting back” procedure fully described in [54] and carried out postprocessing the global applied force-displacement curves by means of data provided by  transducers installed near the top glued edge of the fiber. Such post-processing is necessary to compare experimental data with any numerical prediction of the global load-slip curve, as for instance that obtained with the differential model of Figure 4. Four strain gauges for curved specimens (labeled in Figure 2 as SG01 SG02 SG03 SG04) and five for flat ones were disposed along the longitudinal direction of the fiber, to indirectly provide information of the evolution of tangential and normal stresses with slip. Data collected by straingauges allowed to experimentally provide interface



-slip relationships to compare with the

numerical model proposed, as done in this Paper in Figure 5 and Figure 6.

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Figure 2: Test setup and instrumentation. [54]

slip s [mm]

slip s [mm]

slip s [mm]

Figure 3: Load-slip (“shifted omega” displacement) diagrams obtained from the experimental tests.

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3. The mathematical model in brief: flat case The basic mathematical model here proposed takes its steps from the classic differential equations model adopted for flat brittle specimens reinforced with C-FRP, but here applied for the first time to curved masonry specimens, like those representing arches and vaults. Let us consider a generic flat masonry pillar reinforced with a C-FRP strip having thickness tF , as in Figure 4. Assuming that the C-FRP strips behave as elastic with sudden fragile rupture occurring at the ultimate strength, and that all the non-linearity concentrates at the interface between substrate and C-FRP reinforcement, then the following compatibility equations hold with the classic linear stress strain relationship for the fiber:

 F  EF F s  sF  sM  sF (1)

F 

ds dx

Where symbols have the following meaning: -

 F is C-FRP axial strain;

s , sM , sF are respectively, see Figure 4, interface slip, masonry (substrate) displacement

(considered negligible) and C-FRP displacement; and  F are elastic modulus and normal stress of C-FRP;

-

EF

-

x is the abscissa identifying point P, assuming the origin located on the free C-FRP edge.

By writing the equilibrium equation along the C-FRP longitudinal direction, see Figure 4, we obtain:

t F BF d F    s, x  BF dx  t F EF

d 2s dx

2

   s, x 

(2)

It is worth noting that Eq. ( 2 ), in order to account properly for the non-linearity of the delamination phenomenon at large applied forces near delamination, should mandatorily account for the non-linearity with softening of the tangential stress function   s, x  , which is the stress acting at the interface between reinforcement and substrate. In general, therefore, Eq. ( 2 ) is a nonlinear second order differential equation with known values at the boundary:

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t F EF

d 2s dx 2

   s, x 

(3)

Eq ( 3 ) is hereafter called for the sake of brevity 2ndONDEBV (2nd order ordinary non-linear differential equation with boundary values). Boundary values to assume are the following:

F

x 0

 EF

ds dx

0 x 0

ds dx

0 x 0

s x  L  s0

(4)

F

Which correspond respectively to the C-FRP free edge condition on the left (x=0) and the applied displacement s0 at the loaded right edge LF , where LF is the glued strip length. Assuming to solve the 2ndONDEBV Eq. ( 4 ) using a powerful but standard numerical solver like Matlab [60], then 2ndONDEBV Eq. ( 4 ) transforms into a system of first order equations:

dy  y1  s y2  1   dx  t E dy2    s, x  F F  dx  y2 x  0  0

(5)

y1 x  L  s0 F

Previously written 1st order ODE system may be solved numerically with a standard tool available in Matlab software [60]. In particular, a finite difference algorithm that implements the 3-stage Lobatto IIIa formula is adopted. In such approach, a collocation technique is used, and the collocation polynomial provides a C1-continuous solution that is fourth-order accurate uniformly in the interval of integration. Mesh selection and error control are based on the residual of the continuous solution. The collocation technique uses a mesh of points to divide the interval of integration into subintervals. The kernel determines a numerical solution by solving a global system of algebraic equations resulting from the boundary conditions and the collocation conditions imposed on all the subintervals. It then estimates the error of the numerical solution on each subinterval. If the solution does not satisfy user’s defined tolerance criteria, then the solver adapts the mesh and repeats the process. Typically, the solution depends on the selected non-linear s   behavior adopted at the C-FRPsubstrate interface: the most effective should be smooth and possibly infinitely differentiable. The

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relationship adopted in this paper, complying with such general requirements, will be discussed in the following Section.

Interface sliding x s(x)

sliding P''

FB Ft F

Equilibrium x dx P ( F+d

F )BFt F

P' (s,x)BF dx

F P x

BF tF

LF

substrate

Figure 4: Mathematic interface model proposed in case of C-FRP reinforcement on a flat surface. s   relationships to adopt at the C-FRP-substrate interface There are many different nonlinear slip-tangential stress ( s   ) relationships that can be adopted to

4. Effective

describe the behavior of the interface between C-FRP and substrate near delamination. According to consolidated literature, they are basically the following: -

Linear-suddenly fragile, as suggested for instance in [27][28]. Such approach is probably the most straightforward to carry out quasi analytical computations; in its simplicity, however, it sometimes shows some deviations from the experimental non-linear behavior of real reinforcing systems.

-

Bilinear (or multi-linear in general) behavior with finite ductility and softening, in agreement with indications provided by the Italian CNR DT 200 prescriptions [40] for FRP reinforcement. Such piecewise linear relationship of the interface is particularly suited in a standard discretization into FEs and has been already adopted for instance by Grande and co-workers in [51][55][57]. It is not suited however when ODEs equations must be solved numerically with finite differences.

-

Smooth exponential model with softening, similarly to that adopted in [30] by Fedele & Milani to study the same problem for flat cases.

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In the present paper the latter is adopted, because more convenient in the framework of the numerical solution of the BVP of Eq. ( 4 ), providing stability of the algorithm and fast convergence to the solution. In particular, a modification of the [30] interface relationship is assumed, with tangential stresses ruled by the following smooth exponential function:    s   1  2

2   s        s 2   s0   2 s (6)  s   M  * e   r 1  e  0   s0     Where  M is the maximum tangential stress,  r the residual tangential stress at infinite slip s ,  a





non-dimensional parameter ruling the amount of softening and s0 the slip at which the tangential stress is equal to  M   *  .

*

has more a mathematical meaning, but it can be physically defined as

the tangential stress necessary in order to enforce that the maximum stress  M occurs at In particular, the maximum of Eq. ( 6 ) occurs at the following slip value    s   1 

   s   1 

2

d  s  ds d  s  ds



 M  *

 s1 e

s    0 

2 

0

s* :

2



  M  *

 s  ss  0

2

e

2 

s    0 



r s s0 s0

0





s* .

 s   2 s e  0

  

2

2

 s 2         s s 1 2 s      M   * e 2     r   M  * e 2  e  0   0   s0 s0  s0   

r 





 r  

s* 

2





  *  4   M   e 2     











(7)

2

s0

2  M   * e 2

Substituting Eq. ( 7 ) into Eq. ( 6 ) it is possible to find numerically the value to assign to to obtain  M at



 M   M  *

*

in order

s* :

 ss

*

0

2     s*  

1  2   s0    e 

2    s*        2s    r 1  e  0    0    

(8)

Revised paper, modifications highlighted in color YELLOW *,

Making Eq. ( 8 ) explicit in

then variables

*

and

s*

can be found graphically intersecting the

following two curves:

M curve (1)  *   M 

2    s*        2s    r 1  e  0       2     s*  

1  s    0 

s* 2  e s0 curve (2)

* M 

(9)

r

   *  s 1   2 e  s s*  0      s 0   

The presence of a stress  n normal to the interface is assumed to affect both peak shear stress  M and s0 according to the following relationships: * M   M   n tan 

s0*

* M  s M 0

( 10 )

Residual tangential stress  r is again assumed ruled by a Mohr-Coulomb failure criterion, so that in Eq. ( 9 ), in case of a normal stress different from zero (  n  0 )  r is replaced by

 r*

according to the

following rule:

 r*   r   n tan   n  0  r*  0

n  0

( 11 )

As can be noted, in Eq. ( 10 )  M is ruled by a classic cohesive frictional law, whereas the slip s0 varies linearly with

* M

, in agreement with the actual behavior of the interface, which increases

ductility and fracture energy when subjected to compressive normal stress. Parameters constituting the slip-tangential stress relationship of Eq. ( 6 ) must be assumed consistently with experimental data available. Once such experimental data are at disposal, parameters of Eq. ( 6 ) can be evaluated either with a speedy “simplified” fitting or by the application of a rigorous non-linear least squares optimization.

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Experimental slip-tangential stress data collected for three specimens of the flat case are depicted in Figure 5; it is immediately evident from the figure how the average maximum tangential stress is about 1.4 MPa in correspondence of a slip of 0.1 mm. At a fixed  and assuming  r =0, variables

are found by Eq. ( 9 ), so providing all the necessary data to plot Eq. ( 6 ).  parameter is

s*

and

*

then adjusted manually by a visual trial and error procedure in order to interactively best fit experimental data in all the range of slips experimentally investigated. A second rigorous procedure is to utilize a non-linear least squares solver. Mathematically, the best fitting is obtained minimizing the following objective function: n

min

 

  i



1 exp3   *   iexp 4   iexp5   i  M , , r , s0 ,   3  i  

2

( 12 )

where: -

n

is the total number of experimental points and i the i-th slip;

-

 iexp3 ,  iexp 4

-

 i is the numerical stress evaluated at the i-th slip.

and

 iexp5

are the experimental stresses for the three tests at the i-th slip;

It is worth mentioning that the independent variables to determine by the non-linear optimization are 5, namely

 M , *, r , s0 , 

, which results in a quite demanding numerical problem, affected by

convergence strongly dependent on the starting point selection and with a possible multiplicity of optimal solutions. Furthermore, experimental data must be preliminarily manipulated in order to obtain values for ,

 iexp 2

and

 iexp3

 iexp1

at the same slip-points. Such pre-processing phase can be managed by means of a

standard linear interpolation. When interpolation is not possible because data are outside the range experimentally registered, the average is done exclusively on those data physically available.

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Figure 5: Interface  -slip behavior assumed in the simulations with normal stress  n =0 obtained by the simplified approach, fitting experimentally determined  -slip values. In Figure 5, the tau-slip relationship adopted in the present simulations is reported, assuming as model parameters those summarized in Table 1 with the so-called simplified approach, i.e. that of Eq. ( 9 ). From Table 1 it is also evident that the results obtained with least squares optimization differ little from those determined manually, meaning that the procedure proposed in Eq. ( 9 ) for the simplified determination of parameters entering into Eq. ( 6 ) is preferable. Furthermore, it is worth mentioning that the smooth curve so-determined is very similar to the trilinear one assumed by Grande et al. in [55] to reproduce the same experimental data, represented in Figure 8 for the sake of comparison. In case of curved surfaces, the role played by the normal stress  n on tangential strength and ductility is taken into account through a Mohr-Coulomb failure criterion for strength and a linear amplification for ductility, according to Eq. ( 10 ). The additional parameter to determine is therefore only friction angle



, having assumed all the other 5 parameters known from the flat

case. Again and in the same way of the flat case, two strategies can be followed, namely a speedy fitting of the experimental with

* M

, s0 with

s0*



-slip curves (when  n  0 ) according to Eq. ( 9 ) and substituting  M

and  r with

 r* ,

or using non-linear least squares. In this latter case, the

optimization problem is less demanding and more stable, because the objective function depends only on one optimization variable, namely friction angle, and the range of research is very limited, spacing reasonably from 0° to 50°. Results obtained using the two alternative approaches are summarized in Table 1, where again the negligible difference is worth noting.



-slip curves

Revised paper, modifications highlighted in color YELLOW

numerically obtained with the simplified approach are compared with experimental data in Figure 6, for the 4 curvatures analyzed. As can be observed and analogously to the flat case, the fitting is quite satisfactory. In case of curved surfaces, the experimental normal stress  n varies during the test and is not constant along the longitudinal direction of the glued interface (Figure 7). As it will be shown later in the Paper,  n is proportional to the tensile stress of the fiber. Comparisons shown in Figure 6 refer therefore the peak experimental  n evaluated between the strain gauges. For the sake of completeness, Eq. ( 9 ) procedure is applied assuming different values of assigned normal stresses  n acting on the interface and results are summarized in Figure 8-a. Parameters assumed for the



determination of

* M

-slip curves are those reported in Table 1 and the graphical procedure for the and

s0*

is reported in Figure 8-b.  n is assumed varying between +0.4 MPa and

-0.4 MPa. As clearly visible the typical Mohr-Coulomb behavior of the interface is worth noting, with a reduction of ductility and peak strength for tractions and an increase in both ductility and peak strength for compressions. The same considerations hold for the residual strength, which progressively increases with the level of normal pre-compression of the interface. Figure 8 shows how the interface model proposed is capable of realistically describe the effect of the curvature of the surface. Table 1: Parameters adopted to characterize the interface behavior with the simplified approach and by non-linear least squares. M

r

s0





MPa

MPa

-

°

1.37

0.05

0.09

35

1.42

0.02

mm Simplified approach 0.028 Non linear least squares 0.031

0.088

35

Revised paper, modifications highlighted in color YELLOW

CAI

CAE

-a CBI

-b CBE

-c -d Figure 6: Interface  -slip behavior assumed in the simulations (obtained by the simplified approach) for the different curvatures analyzed and comparison with experimental data and Grande et al. [55] approach. –a: CAI. –b: CAE. –c: CBI. –d: CBE.

Revised paper, modifications highlighted in color YELLOW

CAI

CAE

-a CBI

-b CBE

-c -d Figure 7: Experimental normal stresses for the four different specimens analyzed. –a: CAI. –b: CAE. –c: CBI. –d: CBE.

Revised paper, modifications highlighted in color YELLOW

-a

-b Figure 8: Interface  -slip behavior assumed in the simulations at different levels of normal stress  n (-a) and graphical determination of s* and  * in the different cases (-b).

Revised paper, modifications highlighted in color YELLOW

5. Formulation of the delamination problem in presence of curvature of the glued surfaces The original contribution of this paper is to provide both a rigorous mathematical formulation in case of C-FRP applied on masonry curved surfaces and a numerical procedure independent from an ad-hoc FE implementation. Certainly, such problem is one of the most important, because C-FRP strips are mainly applied to vaults and arches to increase their load carrying capacity, especially when subjected to horizontal static loads mimicking the action of an earthquake. In such a case, assuming to formulate the problem for a curved pillar with constant curvature radius R, equilibrium equations along the tangential and normal direction on a point P, see also Figure 9, read as follows:

t F BF d F    s, x, n  BF Rd 2t F BF  F

d   n BF Rd 2

( 13 )

Where  is defined in Figure 9 and all the other symbols have been already introduced. Assuming again an elastic behavior for the fiber(  F

t F EF

 EF  F  EF

1 ds R d

), Eq. ( 13 ) modifies as follows:

1 d 2s

   s , x,  n  ( a ) R 2 d 2 t n  F F (b) R

Eq. ( 14 )(b), substituted in Eq. ( 14 )(a) assuming again that

( 14 )

 F  EF  F  EF

1 ds R d

provides the

following differential relationship: t F EF d 2 s R

2

t E ds      s, x, F 2F  d R d  

( 15 )

2

Eq. ( 15 ) is a non-linear second order differential equation. The mechanical problem obtained, assuming the same boundary conditions imposed for the flat case and with ODE-BVP:

x  Rd ,

is the following

Revised paper, modifications highlighted in color YELLOW

t F EF ds dx

d 2s

t F EF ds     s , x , 0  R dx  dx 2  0

( 16 )

x 0

s x  L  s0 F

The numerical procedure adopted to solve ODE-BVP problem ( 16 ) is identical to that used for the flat case and is not repeated here for the sake of conciseness. Force resultants on P

R

C F(x)

+12d

n(x)

-12d F(x)+d F

P x

(x)

dx

R n(x)

FRP

P

tF

F

BF

(x) substrate LF +12d

Interface sliding x s(x)

sliding P''

n BFdx

FtFBF

P'

P BFdx

-12d ( F+d F)tFBF

Force resultants on P dx

x

Figure 9: Interface behavior in case of curved specimens.

6. Numerical results In the present Section, the numerical results obtained in the flat and curved cases are compared with both experimental evidences and a previous model developed by the authors and based on a FE discretization with springs, see [55]. Results are critically commented in light of their capability to fit experimental data, in particular considering the role played by the stresses arising at the interface along the normal direction. In Figure 10, the global force-displacement curve obtained with the proposed model (black curve) for the flat case is compared with both experimental data and the already mentioned previous numerical model by the authors [55], based –as already said- on a FE discretization with springs.

Revised paper, modifications highlighted in color YELLOW

As it is possible to notice, there are some few discrepancies between numerical results and experimental data, which however can be explained with the irregular trend exhibited by the experimental force displacement curves, which show odd ultimate ductility without reaching a clear plateau of the ultimate load. This is particularly evident comparing the experimental results obtained in the flat case with those relevant for the curved substrates, see Figure 3. Flat configuration shows indeed a ductility much lower than CBI and CAI specimens, whereas an intermediate behavior between intrados and extrados reinforcement was expected. As a matter of fact, see Figure 3-a and –c, a synoptic comparison between the force-displacement results obtained in all the curved cases (CAI, CAE, CBI and CBE) shows that the ultimate ductility monotonically increases passing from reinforcement at the intrados to reinforcement at the extrados. However, in the flat case, the ultimate ductility observed is unexpectedly lower than those found for reinforcement at the intrados, without a clear theoretical explanation, exception made for a faulty preparation of the specimens. Such irregularity obviously reflects on the quality of the numerical results found. As a matter of fact, whilst the numerical peak load lays inside the experimental fuse, the ductility is overestimated. Furthermore, experimental curves seem not to reach a clear plateau before global failure, continuing to increase up to collapse, again showing some irregularities when data are compared with those found for the curved cases (Figure 3). On the other hand, the comparison with the approach reported in [55] confirms the reliability of the procedure adopted in the present paper. The almost perfect agreement with [55] is not surprising, because the differences between the models stand exclusively on the complexity of the tangential stress-slip relationship adopted, see also Figure 5.

Revised paper, modifications highlighted in color YELLOW

Figure 10: Flat specimen, global load-displacement curves. Comparison between present model and Grande et al. (2018) approach and present model and experimental data. In Figure 11, the evolution at different values of the applied displacement s0 of interface slip (subfigure –a), interface tangential stress (subfigure –b) and fiber tensile stress (subfigure –c) is represented. First of all, the stability of the algorithm is worth noting, with an evident excellent robustness even for full delamination applied displacements, i.e. in that range where global softening is particularly severe. This is certainly an advantage of the numerical procedure adopted, especially when compared with a standard FE approach, where it is rather difficult to follow such kind of softening, as clearly shown by Figure 10, where Grande et al. [55] curve halts at about 1.1 mm of displacement, losing convergence where the global stiffness matrix becomes almost singular.

Revised paper, modifications highlighted in color YELLOW

Increasing imposed displacement s0

-a

Increasing imposed displacement s0

-b

Increasing imposed displacement s0

-c

Figure 11: Flat specimen. –a: abscissa x-slip diagram –b: abscissa xtangential stress at the C-FRP substrate interface. –c: abscissa x- normal stress in C-FRP reinforcement.

Revised paper, modifications highlighted in color YELLOW

Global force-displacement curves obtained for the four different curved cases discussed (CAI, CAE, CBI and CBE) are reported in Figure 12, comparing again the present numerical results both with Grande et al. model [55] and with experimental data. From an overall analysis of the global results obtained, the following aspects are worth noting: -

For intrados reinforcement, the effect of the positive tensile stresses is to decrease progressively the tangential load bearing capacity passing from low curvatures to high curvatures.

-

In the tensile range of normal stresses, i.e. for CAI and CBI specimens, the frictional behavior assumed (Mohr-Coulomb relation between tangential and normal stresses) is not visible, consistently both with what found by Grande et al. [55] and experimental evidences.

-

For compressive normal stresses, i.e for CAE and CBE specimens, an evident increase of the ultimate ductility and ultimate load carrying capacity is worth noting. Again, the agreement with results reported in [55] is excellent, as well rather satisfactory the comparison with experimental data [54], considering the unavoidable scatter of these latter results. In both cases, the effect induced by friction is clear, with the global curves that can be fairly approximated by a bilinear relationship with second branch exhibiting a slope very similar to the experimental one.

From Figure 13 to Figure 16, the evolutions of slip s (subfigure –a), tangential stress  (subfigure – b), C-FRP axial stress  F (subfigure –c) and interface normal stress  n (subfigure –d) at different values of applied displacement s0 at the loaded edge are depicted. Figure 13 refers to CAI specimens, Figure 14 to CAE, Figure 15 to CBI and Figure 16 to CBE, respectively. Analyzing the response provided by the model for specimens with high curvature (Figure 13 and Figure 14), it is worth noting that the values of the tensile and compressive stresses  n at the interface for CAI and CAE reach values of about +0.095MPa and -0.12 MPa respectively. For CAI at peak delamination force,  n remains roughly constantly equal to the maximum (in absolute value) for about 65% of the bonded length, on the opposite side of the free edge. For CAE, a similar behavior is observed, exception made that  n exhibits in the same region a linear increase, which justifies (via the assumed Mohr-Coulomb relationship for the  -slip interface law) the very different global force-displacement curves observed for reinforcement at the intrados and extrados (compare Figure 12-a and –b). A very similar behavior –but attenuated- can be seen for low curvature specimens (CBI and CBE, see Figure 15 and Figure 16), with absolute maximum values of  n roughly equal to one half of those observed for specimens with high curvature. Such outcome

Revised paper, modifications highlighted in color YELLOW

is not surprising, because  n is inversely proportional to curvature radius R and directly proportional to  F , see Eq. ( 14 )(b). Since  F exhibits variations lower than 10% passing from high to low curvatures (compare for instance Figure 14-c and Figure 16-c), the major role is played by R, from which the values of  n found passing from CAI/CAE to CBI/CBE. As intuitively clear, the smaller values of  n justify the less evident differences in the global response. The distribution of normal stresses is systematically the same, a remark that helps in providing simplified formulas for the analytical determination of the debonding strength in case of curved masonry specimens, directly applicable in practice for a fast design of the reinforcement. Finally from Figure 17 to Figure 20, a comparison between numerical predictions and the experimentally determined average tangential (subfigure -a) and normal (subfigure -b) interface stresses between the four strain-gauges of Figure 2 is provided. Figure 17 refers to CAI specimens, Figure 18 to CAE, Figure 19 to CBI and Figure 20 to CBE respectively. As far as the experimental values are concerned, stresses are computed in post processing assuming the fiber elastic, using equilibrium Eq. ( 14 ) and the experimental strains provided by each strain gauge during the test. Generally, considering the obvious scatters and inaccuracies exhibited by experimental data (in some cases data are missing because the gauge did not work properly, the sample was not instrumented or stopped to register data at large slip values near the total debonding), the proposed numerical model proved to be predictive of the local behavior during the entire debonding process, giving a useful insight into the expected distribution of both tangential and normal stresses in each phase of the test. The good agreement with experimental data is confirmed also in the flat case, as visible in Figure 21, where the numerical  -slip curves are again compared with post-processed measures provided by strain gauges. Such capability to locally reproduce the state of stress arising at the interface, both in the flat and curved configurations, may be potentially useful in light of proposing simplified distributions of stresses along the longitudinal direction of the glued surface, with the final aim of providing more reliable closed form formulas at the ultimate and serviceability state to use in design practice.

Revised paper, modifications highlighted in color YELLOW

-a

-b

-c

-d

Figure 12: Comparison between present model and Grande et al. (2018) approach and present model and experimental data. –a: CAI. –b: CAE. –c: CBI. –d: CBE.

Revised paper, modifications highlighted in color YELLOW

Increasing imposed displacement s0

-a

-b Increasing imposed displacement s0

-c -d Figure 13: CAI. –a: abscissa x-slip diagram –b: abscissa x- tangential stress at the C-FRP substrate interface. –b: abscissa x- tangential stress at the C-FRP substrate interface.

Revised paper, modifications highlighted in color YELLOW Increasing imposed displacement s0

-a

-b Increasing imposed displacement s0

-c -d Figure 14: CAE. –a: abscissa x-slip diagram –b: abscissa x- tangential stress at the C-FRP substrate interface. –b: abscissa x- tangential stress at the C-FRP substrate interface.

Revised paper, modifications highlighted in color YELLOW

Increasing imposed displacement s0

-a

-b Increasing imposed displacement s0

-c -d Figure 15: CBI. –a: abscissa x-slip diagram –b: abscissa x- tangential stress at the C-FRP substrate interface. –b: abscissa x- tangential stress at the C-FRP substrate interface.

Revised paper, modifications highlighted in color YELLOW

-a

-b

-c -d Figure 16: CBE. –a: abscissa x-slip diagram –b: abscissa x- tangential stress at the C-FRP substrate interface. –b: abscissa x- tangential stress at the C-FRP substrate interface.

Revised paper, modifications highlighted in color YELLOW

-a -b Figure 17: CAI specimen, tangential (-a) and normal (-b) interface stresses. Comparison between numerical prediction and experimental data provided by strain gauges SG01-04.

Revised paper, modifications highlighted in color YELLOW

-a -b Figure 18: CAE specimen, tangential (-a) and normal (-b) interface stresses. Comparison between numerical prediction and experimental data provided by strain gauges SG01-04.

Revised paper, modifications highlighted in color YELLOW

-a -b Figure 19: CBI specimen, tangential (-a) and normal (-b) interface stresses. Comparison between numerical prediction and experimental data provided by strain gauges SG01-04.

Revised paper, modifications highlighted in color YELLOW

-a -b Figure 20: CBE specimen, tangential (-a) and normal (-b) interface stresses. Comparison between numerical prediction and experimental data provided by strain gauges SG01-04.

Revised paper, modifications highlighted in color YELLOW

Figure 21: Flat specimen, tangential interface stresses. Comparison between numerical prediction and experimental data provided by strain gauges SG01-04.

7. Conclusions The paper discussed a new simple interface numerical model suitable to predict the experimental behavior (in terms of global force-slip curves and local evolution of interface stress) in standard single lap shear tests carried out on flat and curved masonry pillars reinforced with C-FRP. The  interface tangential stress-slip relationship adopted was a C exponential function, accounting for

the main feature exhibited by a composite delaminating from fragile curved surfaces, namely the cohesive-frictional behavior which affects peak load, ductility and residual strength. Such dependence is crucial when dealing with the C-FRP reinforcement of a masonry arch, despite the literature available giving information on such problem is still very limited. The smooth interface relationship adopted allowed to deal with a boundary value problem for a system of second order differential equations ODE-BVP, without singularities and which can be therefore solved using a standard commercial numerical solution algorithm based on a standard finite differences. The model has been extensively validated against existing experimental data available for C-FRP reinforced curved and flat masonry pillar, showing always an excellent predictivity. The main advantage of the procedure proposed is the robustness and velocity of the algorithm, especially in presence of strong global softening, i.e. after full delamination, and the total independence from ad-hoc conceived Finite Element implementation. The generality of the tangential-stress slip law adopted, with parameters set on experimental data available either through

Revised paper, modifications highlighted in color YELLOW

classic non-linear least squares or a simplified direct procedure comprehensively discussed in the paper, represent one of the further major outcomes of the research presented. The implementation of parametric relationships to predict the entire delamination process by means of simplified formulas/equations is indeed straightforward by means of the approach proposed, can be tackled starting for the actual evolution of the slip function point by point along the horizontal direction of the fiber, and is one of the future developments of the procedure. The same model will be extended in case of C-FRP strips glued on the same specimens in presence of anchor spikes. From a mathematical point of view, the anchor spike may be regarded as a particular portion of the interface with modified tangential and normal behavior, having set the corresponding parameters according to ad-hoc experimentation carried out on the anchoring system. The prediction of the ultimate load bearing capacity, particularly important at the ultimate limit state, could be handled by means of efficient 3D kinematic limit analysis approaches with preassigned failure mechanisms.

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