Finite element analysis of the out-of-plane behavior of FRP strengthened masonry panels

Accepted Manuscript Finite element analysis of the out-of-plane behavior of FRP strengthened masonry panels Alessia Monaco, Giovanni Minafò, Calogero Cucchiara, Jennifer D'Anna, Lidia La Mendola PII:

S1359-8368(16)32211-9

DOI:

10.1016/j.compositesb.2016.10.016

Reference:

JCOMB 4613

To appear in:

Composites Part B

Received Date: 10 August 2016 Revised Date:

5 October 2016

Accepted Date: 6 October 2016

Please cite this article as: Monaco A, Minafò G, Cucchiara C, D'Anna J, La Mendola L, Finite element analysis of the out-of-plane behavior of FRP strengthened masonry panels, Composites Part B (2016), doi: 10.1016/j.compositesb.2016.10.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT FINITE ELEMENT ANALYSIS OF THE OUT-OF-PLANE BEHAVIOR OF FRP STRENGTHENED MASONRY PANELS Alessia Monaco*, Giovanni Minafò, Calogero Cucchiara, Jennifer D’Anna, Lidia La Mendola

*

Corresponding author: [email protected]

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Abstract

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University of Palermo, Department of Civil, Environmental, Aerospace and Material Engineering (DICAM), Viale delle Scienze, 90128 Palermo - Italy

In the present study a numerical model is proposed for the response of out-of-plane

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loaded calcarenite masonry walls strengthened with vertical CFRP strips applied on the substrate by means of epoxy resin. A simplified structural scheme is considered consisting in a beam fixed at one end, subjected to constant axial load and out-of-plane lateral force monotonically increasing. Two different constraint conditions are taken

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into account: in the first one, the panel is assumed free to rotate at the top end while, in the second one, the rotation is restrained. Three-dimensional finite elements are used for the calcarenite parts and an equivalent constitutive law available in the literature is

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considered for the compressive behavior of the system ashlar-mortar. Conversely, shell elements are used for modeling the CFRP strips and linear elastic behavior is assumed

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for the composite while cohesive contact properties are introduced at the FRPcalcarenite interface. The model is validated using both experimental results available in the literature and simplified analytical formulations recently presented by the authors in a previous paper.

Keywords: masonry panels; FRP strengthening; out-of-plane behavior; finite element modeling; FRP-masonry interface.

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ACCEPTED MANUSCRIPT 1.

INTRODUCTION

In the last few decades, scientific interest in the rehabilitation of historical buildings had increasingly involved the use of innovative materials and techniques. In particular, the

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use of Fiber-Reinforced Polymers (FRP) for the strengthening of masonry structures is widely adopted, mainly because of the numerous advantages offered by this technique, such as easy installation and achievement of high mechanical performance.

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In the technical literature several topics are investigated in the field of the rehabilitation of masonries, including the evaluation of the global mechanical behavior of the

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strengthened structures [1-11], the analysis of local mechanisms concerning the bond behavior at the interface between FRP and substrate [12-19], the adoption of adequate numerical or analytical models for the simulation of the material properties and structural response of masonry elements [2,3,20-27] and so on. Among the several

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issues explored in the field of the restoration of ancient masonry structures, the problem of the out-of-plane response of masonry panels represents one of the most significant aspects to be investigated [20,28-31]. The fact is that flexural behavior of bi-

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dimensional elements loaded in the out-of-plane direction is one of the most common local failure mechanisms of masonry walls. This mechanism can be induced by several

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causes, for instance seismic forces, thrust of arches and vaults or also imperfections regarding the verticality of the panel itself as well as eccentricity induced by wind loads or dead/live loads. The mechanism can generally occur due to simple overturning around a cylindrical hinge on the base of the panel or vertical or horizontal flexure. In Fig. 1 the schemes of these failure mechanisms are shown together with pictures of real cases of masonry walls that collapsed after the earthquake in Abruzzo (Italy) in 2009 [32]. The first failure mode (Fig. 1a) is typical of panels that are not well anchored panels and have limited tensile strength. Besides the most traditional retrofitting 2

ACCEPTED MANUSCRIPT techniques, such as the adoption of steel rods, a very effective kind of intervention to repair a masonry wall that is collapsing according to this failure mode consists in the application of FRP sheets on the top of the panel refolded on the orthogonal walls. Conversely, the vertical flexure mechanisms (Fig. 1b) can occur even in masonry walls

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that are well constrained both on the top and on bottom, loaded by out-of-plane forces that cause the formation of three aligned hinges; the three-hinge horizontal flexure

mechanism (Fig. 1c) can also occur due to out-of-plane forces when the transverse walls

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are not able to ensure adequate confinement to allow the arch mechanism to be

activated. In this case, to prevent collapse due to horizontal flexure, FRP composites

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can be placed on the upper bound of the panel to enhance its flexural capacity, while, to avoid vertical flexure, vertical FRP strips can be used.

In the present work, attention is particularly focused on the mechanism caused by vertical flexure. About this local failure mechanism, many research works in the

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literature, as well as several case-studies, demonstrate the effectiveness of the application of FRP composites in the form of vertical strips placed and properly anchored on the panel to be strengthened, enhancing its flexural capacity, the

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compressions being absorbed by the masonry and the tensile stresses by the FRP reinforcement [4,5,20-22,28,29,33]. The study presented here is centered on the

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development of a numerical model aimed at simulating the response of out-of-plane loaded calcarenite masonry walls strengthened with vertical CFRP strips applied on the substrate by means of epoxy resin. In particular, a simplified structural scheme is considered which consists in a beam fixed at one end, subjected to constant axial load and out-of-plane lateral force monotonically increasing. Two models are generated which differ for the constraint condition at the top-end of the panel: in the first one, the panel is assumed free to rotate at the top end (model 1) while, in the second one, the rotation is restrained (model 2). In both models, three-dimensional finite elements are 3

ACCEPTED MANUSCRIPT used for the calcarenite parts and the system ashlar-mortar joint is modeled as a single element endowed with an equivalent constitutive law. In particular, the stress-strain relationship proposed by Sargin [34] and modified by Cavaleri et al. [23] is assumed. Conversely, a classical linear elastic behavior is assumed for the FRP composites [35].

interface by introducing cohesive interaction properties.

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The numerical study also accounts for the simulation of the FRP-masonry contact at the

The model is verified using the results of an experimental campaign by Accardi et al.

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available in the literature [21]. Furthermore, the finite element results are validated by means of a simplified analytical formulation proposed by the present authors in a

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previous paper [22]. The comparisons show a good agreement between the numerical prediction and the experimental and analytical reference data. Considerations on the failure mode of both reinforced and unreinforced masonry with two different restraint conditions on the top end are thus provided.

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In the following sections, the aims of the research and the finite element (FE) model are firstly presented; then, the reference experimental data and analytical formulation used for validation are briefly described. Finally, the FE results and the comparison with the

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reference data are presented and discussed.

AIMS AND APPLICATION FIELD

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2.

The numerical model presented in this paper is intended for the prediction of the overall flexural response of a masonry wall in terms of a curve representing the lateral force F vs. the lateral displacement δ. The main hypothesis consists in assuming the structural scheme of a beam of length L, fixed at one end, loaded with a horizontal force F and subjected to a constant axial load P and a bending moment M = FζL at the top end, ζ being a coefficient depending on the effective restraining level at the top. In particular,

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ACCEPTED MANUSCRIPT ζ=0 when the free end of the beam is allowed to rotate, while ζ=0.5 for a beam with fixed rotation at both ends. The structural scheme is represented in Fig. 2. These simplified assumptions are intended to be applicable when the real cases studied are those of masonry panels with inadequate restraints with the orthogonal walls or

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panels with significant width-to-height ratio. In such cases, the lateral constraints can be generally considered inefficacious or far enough for a unitary width strip of the wall to

3.

NUMERICAL MODEL

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3.1. GEOMETRY AND LOAD CONDITION

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be studied for evaluating the flexural response of the central part of the masonry wall.

The FE model was developed using the software Abaqus [36]. The model was tridimensional and took into account the material and geometrical nonlinearities and the stresses arising at the interface between FRP strips and masonry substrate. In the

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literature, different approaches have been proposed for modeling the system ashlarmortar joint, providing appropriate element types and constitutive behavior for each singular component or, conversely, adopting equivalent geometrical and mechanical

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properties able to reproduce the behavior of the system [20, 24-26]. In particular, La Mendola et al. [20] developed a two-dimensional FE model of the experimental tests by

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Accardi et al. [21] introducing an interface element type for reproducing the response of each mortar joint of the unreinforced specimen, adopting a normal stress-displacement interface law calibrated on the experimental results of the simulated test. Conversely, in the current case, the system ashlar-mortar joint was modeled using a continuum brick element with equivalent material properties. Thus the geometry of the specimen is realized as a single block whose geometry is represented in Fig. 3. The panel was 210 mm thick, 740 mm width and 2100 mm height. Two models were generated which differed for the constraint applied at the top end of the panel. In particular, in model 1 5

ACCEPTED MANUSCRIPT (Fig. 3a), the load condition and constraints replicated those used by Accardi et al. [21] in their experimental tests the top end of the wall was free to rotate (pinned end) and a lateral displacement δ was applied monotonically at the base of the wall, which was not allowed to rotate. Conversely, in model 2 (Fig. 3b), the top end of the panel was fixed

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so that the rotation was restrained and a bending moment arose. In both cases, a

constant axial load P=80 kN was applied. The reaction force F due to the impressed displacement was measured.

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Firstly, for both model 1 model and 2 the analysis of the unreinforced specimen was

to the masonry substrate (Fig. 3).

3.2. ELEMENT TYPES

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conducted; successively, four CFRP strips 50 mm wide and 0.13 mm thick were applied

The element type used for meshing the panel was a 8-node linear brick (C3D8R) with

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structured meshing technique. This 8-node linear brick was a first-order reducedintegration solid continuum element with a single Gauss integration point. Using a reduced-integration element makes it possible to reduce the running time in 3D models.

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Moreover, generally hexahedra have a better convergence rate than other solid

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elements, and they are not significantly affected by sensitivity to mesh orientation if regular meshes are used, as in the model presented here. Normally, brick elements with a good mesh provide accurate solutions at low computational cost. The mesh size used for the panel was 50 mm (Fig. 3c). Conversely, the element type used for meshing the FRP strips was a 4-node quadrilateral shell element (S4R) with reduced integration and large-strain formulation. S4R is a general-purpose conventional shell element. These element types allow transverse shear deformation but the transverse shear deformation becomes very small as the shell thickness decreases, as in the present case. The S4R element accounts for finite membrane strains and arbitrarily large rotations and, 6

ACCEPTED MANUSCRIPT therefore, is suitable for large-strain analysis. The mesh size used for CFRP reinforcement was 25 mm (Fig. 3c).

3.3. MATERIALS AND CONTACT PROPERTIES

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In the technical literature, several models are available for reproducing the damaged

behavior of masonry panels (among others see [25-27]). The approach used in this work for the modeling of masonry was the Concrete Damaged Plasticity (CDP) model

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available in ABAQUS [36]. The CDP model is a plasticity-based continuum damage

model in which two principal failure mechanisms are assumed, i.e. tensile cracking and

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compressive crushing, and the evolution of the failure surface is controlled by both compressive and tensile hardening variables. The CDP model is generally suitable for modeling quasi-brittle materials such as masonry or concrete. Generally, there are few limitations to the applicability of the CDP model, one of these being that a regime of

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low confining pressures is required for both monotonic and cyclic loading conditions; moreover, the inelastic behavior is defined through the combination of isotropic damaged elasticity and isotropic tensile and compressive plasticity. However, for the

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mechanical modeling in the present case, all these assumptions were allowed and the CDP model proves to be accurately representative of the constitutive behavior of the

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masonry. Actually, it is worth remarking that the CDP model is ruled by several parameters that should be derived from several specific tests on the material such as uniaxial compression tests, uniaxial tension tests and a sufficient number of triaxial tests also for accounting for the triaxial effects. However, in our case, the only available data were those obtained from uniaxial compression tests on masonry panels (see Ref. [21]) and, therefore, the behavior of the material was modeled exploiting the existing results in the technical literature. In particular, the material properties assumed in this paper corresponded to those used by Accardi et al. [21] in their experimental tests. The 7

ACCEPTED MANUSCRIPT parameters of the CDP model were calibrated reproducing by means of finite element method the experimental tests on the specimens of masonry panels available in Ref. [21] concerning uniaxial compression tests on six specimens constituted by three layers of block of calcarenite stone and mortar joints. The results of these analyses are not

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reported in the paper for brevity’s sake.

Specifically, the constitutive law assumed in this model for the compressive behavior of masonry was the stress-strain relationship proposed by Sargin [34] and successively

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modified by Cavaleri et al. [23]:

(1)

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Aε% + ( D − 1) ε% 2 σ% = 1 + ( A − 2 ) ε% + Dε% 2

in which σ% = σ / σ 0 and ε% = ε / ε 0 represent the stress and strain normalized with respect to the corresponding peak values; A = Ei / E0 is the ratio between tangent and secant elastic modulus; and D is a coefficient that influence the descending branch,

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giving more ductile sub-horizontal post-peak response for high value of D. Eq. (1) was implemented within the CDP model assuming the following values:

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σ 0 = 4 MPa ; ε0 = 1.3 mm /m ; A=2.8 and D=1.5 which were previously obtained in Ref. [21] fitting the results of the uniaxial compression tests on the specimens of

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masonry panels. Furthermore, the CDP model assumes that the compressive response of material is characterized by damaged plasticity, i.e. when the specimen is unloaded from any point on the strain-softening branch of the stress-strain curve the unloading response is characterized by a certain loss of initial stiffness. Therefore the following expression is assumed for the unloading branch:

(

σ = (1 − d ) E0 ε − ε% pl

)

(2)

in which d represents a damage variable aimed at describing the degradation of the elastic stiffness and ε% pl is the plastic strain. 8

ACCEPTED MANUSCRIPT Finally, under uniaxial tension the stress-strain response follows a linear elastic relationship with reduced elastic modulus equal to 2000 MPa until a threshold is reached for failure stress. Specifically, in the absence of experimental tensile tests on a masonry panel specimen constituted by the assembly of calcarenite blocks and mortar

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joints, which are generally aimed at identifying the effective tensile stress-strain

behavior, the tensile strength value to be implemented in the CDP model was calibrated exploiting some results available in the literature dealing with the non-linear behavior of

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masonry under tension [37]. In particular, Ref. [37] provides an overview of

deformation controlled tensile and flexural tests in the literature carried out for

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establishing the behavior of masonry systems prior to and beyond the maximum load under tension. Based on those results, a precautionary value for the tensile strength is adopted in the model assuming σ t = 0.16 MPa . Beyond the failure stress, the onset of micro-cracking in the material is represented macroscopically with a softening stress-

Gf = 0.008 N/mm .

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strain response, which was defined in terms of fracture energy assuming

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For the FRP reinforcement a linear elastic material behavior was adopted [35], assuming an elastic modulus Ef equal to 230000 MPa and a debonding stress ftb= Ef εdu= 624.29

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MPa, the debonding strain being εdu= 2.71 mm/m. Finally, the definition of the interface behavior between FRP strips and masonry substrate was described by means of a cohesive formulation primarily intended for situation in which the interface thickness is negligibly small. The elastic part of the interface can then be written as: tn   Knn Kns Knt  δ n      t = ts  =  Kns K ss K st  δ s  = Kδ t   K    t   nt K st Ktt  δt 

(3)

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ACCEPTED MANUSCRIPT where K is the elastic stiffness matrix, t is the nominal traction stress vector, tn, ts and tt are the normal and the two tangential local directions, respectively, and δn, δs and δt the corresponding separations. When the normal and tangential stiffness components are uncoupled, the extra-diagonal terms of the matrix K are equal to zero, as set in the

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current case. The damage evolution in the normal direction is defined assuming Knn = 0, meaning that the normal separations immediately produce the failure of the normal

contact. Conversely, the tangential cohesive constraints are enforced by establishing

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firstly an initiation criterion, and secondly a damage evolution criterion. The tangential direction considered here was s, i.e. along the longitudinal direction of the FRP strip.

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The separations in the second tangential direction t were assumed to be negligible. Therefore the damage was assumed to begin when the condition ts ts0 = 1 was reached, ts0 being the threshold value of the shear stress along the s direction. Once the initiation

criterion was achieved, an evolution law describes the rate at which the cohesive

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stiffness was degraded by means of the definition of the fracture energy Gf dissipated during delamination. The elastic and post-elastic behavior of the cohesive delamination model is represented in Fig. 4. In the figure, the symbol δ s represent the separation

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0

0 f corresponding to the threshold value ts while δ s corresponds to the ultimate value of

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separation at complete failure. The model parameters are calibrated on the basis of experimental results obtained by Accardi et al. [38], assuming the following 0 values: ts = 0.5MPa , Kss=8.33N/mm3, Gf=0.1028 N/mm.

4.

REFERENCE STUDIES USED FOR VALIDATION

The model was validated using the results of an experimental campaign of out-of-plane lateral loading of masonry panels carried out by Accardi et al. [21] and then the FE results were also compared with the iterative analytical procedure recently presented by 10

ACCEPTED MANUSCRIPT the authors in Cucchiara et al. [22] for theoretical prediction of the lateral loaddeflection curves of out-of-plane loaded masonry walls.

4.1. EXPERIMENTAL TESTS BY ACCARDI ET AL. [21]

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The tests carried out by Accardi et al. [21] (also described in Ref.[33]) refer to masonry walls, of dimensions 740 x 210 x 2100 mm realized with calcarenite ashlars and mortar joints having elastic modulus equal to 9000 MPa and 2800 MPa, respectively. The

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average compressive strength of the masonry was σ0=4 MPa with a corresponding peak strain ε0=1.3 mm/m. Specimens reinforced with two, four and six CFRP strips were

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realized. In the present study only the results of the specimen strengthened with four strips were used for validation. The specimen indicated as FRP-W4 was reinforced with four CFRP strips 50 mm wide and 0.13 mm thick using the wet-lay-up technique (Fig. 5a). The mechanical properties of CFRP strips provided by the producer were: elastic

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modulus Ef=230000 MPa, tensile strength ft=3450 MPa and ultimate strain εfu= 1.5%. The strips were applied by means of epoxy resins with tensile strength equal to 30 MPa. In addition, an unreinforced specimen was manufactured for comparison (specimen

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URM). The load condition and the constraints are reported in Fig. 5b: a constant axial

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load P of 80 kN was applied to the panel by means of a hydraulic jack; then monotonically increasing displacements were transmitted at the bottom by a screw jack. During the test, the specimen was supported at the base by a trolley connected to the screw jack and at the top it was constrained by an unmovable cylindrical hinge. The tests were carried out by applying first the vertical load and then a monotonically increasing history of displacements at the base. The resulting load-deflection curves for the unreinforced and reinforced specimens are depicted in Fig. 5c. As expected, the initial stiffness was not influenced by the presence of reinforcement, while the following branch of the curves of strengthened walls was governed by the performance 11

ACCEPTED MANUSCRIPT exhibited by the CFRP strips after joint opening. In particular, it was observed that the higher amount of bonded CFRP strip allowed the ultimate wall strength to increase but the initial stiffness of the walls did not change until mortar bed joint opening occurred.

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4.2. ANALYTICAL MODELS BY CUCCHIARA ET AL. [22]

The authors recently presented a simplified iterative procedure based on an analytical approach for the assessment of the flexural behavior of out-of-plane loaded masonry

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walls. The model was based on a simplified structural scheme of cantilever beam of

length L (Fig. 6a) in which the mechanical non-linearities were introduced by means of

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moment-curvature relationships where the ultimate debonding strain of FRP was taken into account and the second order effects were considered adopting an iterative calculation procedure. In particular, a continuum discretization of the wall was assumed, considering n parts of length ∆ with control points in the middle section of each part.

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For each control point of coordinate x, the first and second order moments M I and MII were thus defined as follows:

M I = FjL ( ζ − 1) + Fj x

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MII = −P ( wn − w )

(4a) (4b)

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In Eqs. (4) Fj is the lateral force, P is the constant axial load, ζ is a coefficient depending on the effective restraining level at the top of the beam, and wn and w are respectively the displacement at the top of the beam and the displacement of the considered control point. The effective displacement is calculated adopting Mohr’s analogy and assuming the curvature as constant in each part of the beam (Fig. 6b). In Mohr’s model the fictitious force T* on each portion of the beam is calculated as:

T* = φ⋅∆

(5)

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ACCEPTED MANUSCRIPT φ being the curvature at the control point, which is a function of the bending moment expressed by Eqs. (4). Finally, the displacement of each control point is expressed as: i

w i = −∑ Tk* ( x i − x k )

(6)

k =1

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The solution is achieved with an iterative procedure, considering the geometrical features and the axial load P as known and constant during the analysis and calculating the moment-curvature diagram preliminary for an assumed axial load P. The main

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phases of the iterative procedure are the following: 1) a value is assigned to the lateral force Fj at the j-th step; 2) the first-order moments are calculated by Eq. (4a) and the

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second-order moments are assumed to be zero at the first iteration; 3) the corresponding curvatures and the fictitious loads T*(1) are calculated by means of Eq. (5); 4) the lateral (1)

displacements at the first iteration, w , are therefore calculated with Eq. (6); 5) the displacements w(1) obtained are subsequently adopted to evaluate the second-order

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(1) (1) moments at the successive iteration M (2) ) ; 6) the updated value of the II = − P ( w n − w

total moment is calculated as M (2) = M I + M (2) II and points 3) and 4) are repeated,

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calculating the new values of the lateral displacements w(2) . The final values are obtained when the balanced difference between the deflection at the k+1-th and k-th

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iteration is less than a fixed tolerance equal to 0.01. A displacement value at each control point is calculated for a fixed lateral force Fj. The overall force-displacement curve is therefore obtained by assigning a new value to the lateral force Fj+1 and repeating the abovementioned procedure.

5.

RESULTS

The numerical results are given, firstly, in terms of lateral load vs. displacement curves and, secondly, the failure modes observed from the FE models are described for model 13

ACCEPTED MANUSCRIPT 1 (upper hinge) and 2 (upper fixed end) for cases of both unreinforced masonry (URM) and masonry reinforced with four CFRP strips (FRP-W4).

5.1. LOAD-DISPLACEMENT CURVES

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In Fig. 7 the lateral force F vs. lateral displacement δ is represented for the URM and FRP-W4 specimens. The numerical curves refer to model 1 (upper hinge) and are compared with the corresponding experimental ones given by Accardi et al. [21],

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showing quite good agreement. The maximum numerical loads are 2.68 kN for URM

and 3.81 kN for FRP-W4. In the case of the strengthened panel (Fig. 7b), it is observed

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that, after the initial elastic phase, the FE model provides a slight overestimation of the masonry stiffness until achievement of the peak of the curve where, conversely, the numerical curve accurately predicts the experimental behavior. It might be worth mentioning that a similar trend of the FE load-displacement curves of both specimens

et al. [20].

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URM and FRP-W4 was found in the two-dimensional model presented by La Mendola

Validation of the FE results of model 1 was also conducted through comparison with

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the iterative analytical procedure by Cucchiara et al. [22]: in Fig. 8 the FE curves are

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compared with the analytical and experimental ones up to lateral displacement values of about 12 mm, in both cases showing good agreement, especially in the initial phase up to load values of about 1 kN. Subsequently the FE result provides slight overestimation of the system stiffness with respect to the analytical prediction, particularly for the FRPW4 specimen, as previously seen in Fig. 7b. The FE results concerning model 2 (upper fixed end) are given in Fig. 9 for unreinforced and reinforced specimens. As in model 1, both unreinforced and reinforced specimens exhibited the same initial elastic stiffness, which was not affected by the presence of the strengthening strips. As expected, the different boundary condition on 14

ACCEPTED MANUSCRIPT the top of the panel in model 2 provided initial stiffness of the specimens that was four times higher than in model 1. The maximum lateral force value was 11.55 kN for the unreinforced panel and increased up to 13.78 kN in the case of the CFRP strengthened specimen, exhibiting an increase of 19.4%. After the peak, both specimens experienced

stopped.

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5.2. NUMERICAL FAILURE MODES

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softening behavior up to lateral displacement of 50 mm, at which point the analysis was

5.2.1. MODEL 1

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The progressive evolution of the post-elastic response of the URM specimen with top hinge (model 1) is depicted in Fig. 10. In particular, in the first column, the analysis step corresponding to the initial phase beyond the elastic behavior is represented. In this step the displacement at the base of the panel umax was 5.73 mm, the maximum value of

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tensile stress was σt=0.154 MPa in the descending branch of the tensile stress-strain curve and the corresponding plastic strain εt was equal to 1.27e-4 mm/mm. The last row of the column shows the tensorial representation of the maximum principal strains, thus

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indicating the direction of the cracks at the base of the panel. The same output variables

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are shown in the second and third columns of Fig. 10 for the steps corresponding respectively to the peak and the end of the analysis, i.e. for umax=17.26 mm (peak of the F-δ curve) and umax=29.51 mm (end of the analysis). It is possible to observe the increase in the volume of the masonry panel where the material response reached the tensile strength and worked in the descending branch, producing cracks with increasing length and width. The main fractures are at the base of the panel in the transversal direction orthogonal to the maximum principal strains. Similarly, Figs. 11 and 12 describe the failure mode of specimen FRP-W4 hinged on the top end. In the transition between elastic and post-elastic response (Fig. 11), the 15

ACCEPTED MANUSCRIPT maximum principal stresses are concentrated on the bottom of the specimen (Fig. 11a), as already observed for the unreinforced panel. However, in this case, the presence of CFRP strips makes is possible to absorb a significant amount of traction as shown by the color mapping of the tensile state in the FRP reinforcement in Fig. 11b. The CFRP

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strips thus also made it possible to contain the transversal cracks on the base of the

panel as observed in Fig. 11c. In the post-elastic range, the collapse mode observed

involved failure of the interface through sliding and separation of those nodes that were

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previously in contact. Indeed, in Fig. 12a the nodes of the mesh experiencing contact

opening are represented according to a color code from “blue” (no contact opening)” to

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“red” (contact opening). The corresponding contact pressures are depicted in Fig. 12b, where the blue color indicates that no contact pressures were present and thus the nodes were separated. Therefore from Fig. 12b the debonding length of CFRP strip can be identified. Similarly, in Figs. 12c and 12d the relative tangential motions and the

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associated frictional shear stresses are represented. The strips mainly affected by debonding of the interface are the two external ones. The numerical failure mode

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observed proves to be in accordance with the experimental evidence [21,33].

5.2.2. MODEL 2

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The analysis of the failure mode was also conducted for the unreinforced and reinforced specimens with fixed top-end (model 2). From Fig. 13 the deformed shape of the URM specimen (Fig. 13a) can be observed together with the maximum principal stresses in the masonry (Fig. 13b): as expected, they are concentrated on the top and bottom of the panel where the material is in tension. At the maximum tensile stresses, transversal cracks develop orthogonally to the maximum principal strains, which are represented in Figs. 13c and 13d. When the masonry is strengthened by CFRP strips (specimen FRP-W4, model2), the 16

ACCEPTED MANUSCRIPT main transversal cracks develop on the top and bottom of the panel, as in the URM specimen, as observed in Fig. 14a. In the transition between elastic and post-elastic response, the CFRP strips on the base of the panel start experiencing a relative tangential motion (Fig. 14b) with respect to the substrate, producing frictional shear

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stresses (Fig. 14c), meaning that failure of the interface is occurring through sliding. At the end of the analysis, local contact openings also affect the bottom of the panel in line with one external strip, producing null contact pressure values (see Fig. 15a). On the top

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of the specimen, interface failure produces local contact openings (Fig. 15b) and

diffused sliding (Figs. 15c and 15d), which mainly affect the nodes of the mesh of the

6.

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two external strips.

CONCLUSIONS

In this study a three-dimensional finite element model is presented for the response of

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out-of-plane loaded calcarenite masonry walls strengthened with vertical CFRP strips. A simplified structural scheme is considered consisting in a beam fixed at one end, subjected to constant axial load and monotonically increasing out-of-plane lateral force.

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Two different constraint conditions are taken into account: in the first one, the panel is assumed free to rotate at the top-end while, in the second one, the rotation is restrained.

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The ashlar-mortar joint system is modeled using a continuum brick element with equivalent material properties, while the CFRP strips are modeled by means of shell elements. The FRP-substrate interface response is taken into account introducing a specific surface-to-surface contact property endowed with cohesive constitutive behavior, calibrated exploiting experimental results available in the literature [38]. The model is validated using both experimental results available in the literature [21] and a simplified analytical iterative procedure recently presented by the authors in a previous paper [22]. Comparison between the results shows good agreement. The 17

ACCEPTED MANUSCRIPT analysis of the failure mode is conducted for both unreinforced (URM) and reinforced (FRP-W4) specimens with top end free to rotate (model 1) and fixed top end (model 2). The FE outputs for unreinforced panels show that, when the top end is free to rotate, the masonry material achieves tensile strength and works in the descending branch,

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producing cracks with increasing length and width, mainly at the base of the panel. Conversely, when the top end of the URM specimen is prevented from rotated, as

expected, the main fractures also involve the top surface of the panel. The strengthened

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FRP-W4 specimen shows more contained transversal cracks in the masonry material in both models 1 and 2 due to the CFRP strips that absorb a significant amount of traction.

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In both models, failure of the FRP-substrate interface occurs, evidencing relative tangential motions with local contact openings on the bottom of the panel for models 1 and also on the top of the panel for model 2, especially in the two external FRP strips. The numerical failure modes of the interface are in accordance with the experimental

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evidence described in Refs. [21,33].

The FE model presented in the paper is appropriate for several possible future fields of application. First of all, it could be used for generating a parametric study on the

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influence of different types of FRP composites and the number of layers on the effectiveness of the strengthening system. Similarly, for given geometry, boundary

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conditions and applied loads, the optimization of the strengthening system with respect to the masonry mechanical properties could be analyzed. Furthermore, it is known that, during the last decade, several studies in the literature have been addressed to evaluation of effective confinement pressure, ultimate compressive stress and strain, and the overall trend of the stress-strain curve of confined masonry in compression, sometimes producing contrasting results due to the different key variables involved in each model. In this field, the FE model proposed in the paper could certainly be employed as a useful numerical support for formulation of appropriate analytical models aimed at 18

ACCEPTED MANUSCRIPT predicting the stress-strain curve of FRP confined masonry in compression, allowing the generation of numerous case studies to be used for validation.

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Comparison of Different Approaches through a Case Study. Engineering Structures 2002; 24: 1057-1069.

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FRP systems for strengthening existing structures. Rome (Italy). The Italian National

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Research Council, Advisory Committee on Technical Recommendations for Construction, 2013.

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arches via nonlinear and linear programming. Composites: Part B 2012; 43: 439-446. 7.

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surface of unreinforced and FRP-/FRCM-reinforced masonry domes. Composites: Part B 2015; 83: 297-305.

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Bertolesi E, Milani G, Fedele R. Fast and reliable non-linear heterogeneous FE

approach for the analysis of FRP-reinforced masonry arches. Composites: Part B 2016; 88: 189-200. 10.

Feo L, Luciano R, Misseri G, Rovero L. Irregular stone masonries: Analysis and

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strengthening with glass fibre reinforced composites. Composites: Part B 2016; 92: 8493. 11.

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Experimental investigation on masonry arches strengthened with PBO-FRCM

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masonry pillars during delamination tests. Composites: Part B 2012; 43: 1999-2011. 13.

Carrara P, Ferretti D, Freddi F. Debonding behavior of ancient masonry elements

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strengthened with CFRP sheets. Composites: Part B 2013; 45: 800-810. Ghiassi B, Oliveira DV, Lourenco PB, Marcari G. Numerical study of the role of

mortar joints in the bond behavior of FRP-strengthened masonry. Composites: Part B

D’Ambrisi A, Feo L, Focacci F. Experimental and analytical investigation on

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2013; 46: 21-30.

bond between Carbon-FRCMmaterials and masonry. Composites: Part B 2013; 46: 1520. 16.

Ceroni F, Ferracuti B, Pecce M, Savoia M. Assessment of a bond strength model

for FRP reinforcement externally bonded over masonry blocks. Composites: Part B 2014; 61: 147-161. 17.

Mazzotti C, Murgo FS. Numerical and experimental study of GFRP-masonry

interfacebehavior: Bond evolution and role of the mortar layers. Composites: Part B 20

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Mazzotti C, Ferracuti B, Bellini A. Experimental bond tests on masonry panels

strengthened by FRP. Composites: Part B 2015; 80: 223-237. Maljaee H, Ghiassi B, Lourenco PB, Oliveira DV. Moisture-induced degradation

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of interfacial bond in FRP-strengthened masonry. Composites: Part B 2016; 87: 47-58. 20. La Mendola L, Accardi M, Cucchiara C, Licata V. Nonlinear FE analysis of out-of-

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plane behaviour of masonry walls with and without CFRP reinforcement. Construction and Building Material 2014; 54: 190-196.

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21. Accardi M, Cucchiara C, Failla A, La Mendola L. CFRP flexural strengthening of masonry walls: experimental and analytical approach. In: Proceedings of FRPRCS-8 Conference. Patras, Greece, July, 2007.

22. Cucchiara C, La Mendola L, Minafò G, and Monaco A. Flexural behaviour of

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calcarenite masonry walls reinforced with FRP sheets. In: Proceedings of ANIDIS 2015 Conference - Earthquake Engineering in Italy. L’Aquila, Italy, September, 2015. 23. Cavaleri L, Failla A, La Mendola L, Papia M. Experimental and analytical response

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of masonry elements under eccentric vertical loads. Eng Struct 2005; 27(8): 1175-1184.

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24. Lotfi HR, Shing PB. Interface Model Applied to Fracture of Masonry Structures. Jr. of Structural Engineering 1994; 120(1): 63-80. 25. Gambarotta L, Lagomarsino S. Damage models for the seismic response of brick masonry shear walls. Part I: the mortar joint model and its applications. Earthq Eng Struct Dyn 1997; 26(4):423-39. 26. Gambarotta L, Lagomarsino S. Damage models for the seismic response of brick masonry shear walls. Part II: the continuum model and its applications. Earthq Eng Struct Dyn 1997; 26(4):441-62. 21

ACCEPTED MANUSCRIPT 27. Lourenco PB. Anisotropic Softening Model for Masonry Plates and Shells. Jr. of Structural Engineering 2000; 126(9): 1008-1016. 28. Gilstrap JM, Dolan CW. Out-Of-Plane Bending Of FRP-Reinforced Masonry

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Walls. Composites Science and Technology 1998; 58:1277-1284. 29. Hamed E, Rabinovitch O. Failure characteristics of FRP-strengthened masonry walls under out-of-plane loads. Engineering Structures 2010; 32: 2134-2145.

Trung T, Liman A. Out-of-plane behaviour of hollow concrete block masonry

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walls unstrengthened and strengthened with CFRP composite. Composites: Part B 2014;

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67: 527-542.

Maddaloni G, Di Ludovico M, Balsamo A, Prota A. Out-of-plane experimental

behaviour of T-shaped full scale masonry wall strengthened with composite connections. Composites: Part B 2016; 93: 328-343.

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32. Milano L, Mannella A, Morisi C, Martinelli A. Schede illustrative dei principali meccanismi di collasso locali negli edifici esistenti in muratura e dei relativi modelli cinematici di analisi - Allegato alle Linee guida per riparazione e rafforzamento di

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elementi strutturali, tamponature e partizioni a cura di Dolce, M., Manfredi, G. 2011;

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Dipartimento della Protezione Civile – Reluis – Doppiavoce Edizioni (Italian). 33. La Mendola L, Failla A, Cucchiara C, Accardi M. Debonding phenomena in CFRP strengthened calcarenite masonry walls and vaults. Advances in Structural Engineering 2009; 12: 745-760. Special Issue. 34. Sargin M. Stress-Strain Relationship for Concrete and the Analysis of Structural Concrete Sections. In: Study 4, Solid Mechanics Division, University of Waterloo, Waterloo, Canada, 1971. 35. Täljsten B. Strengthening of concrete prisms using the plate-bonding technique. 22

ACCEPTED MANUSCRIPT International Journal of Fracture 1996; 82: 253-266. 36. Abaqus 6.13 Theory Manual. Dassault Systèmes Simulia Corp, 2013. 37. Van der Pluijm R. Non-linear Behavior of Masonry under Tension. Heron Journal

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of Delft University of Technology 1997; 42(1):25-54.

38. Accardi M, Cucchiara C, La Mendola L. Bond behavior between CFRP strips and

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calcarenite stone. In: Proceedings of FraMCos-6 Conference. Catania, Italy, June, 2007.

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ACCEPTED MANUSCRIPT Figure captions Figure 1. Local failure mechanism: a) simple overturning; b) vertical flexure; c) horizontal flexure (Ref. [32]).

Figure 2. Structural scheme for: a) general case; b) ζ=0; c) ζ=0.5.

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Figure 3. Geometry, load conditions and constraints for a) model 1; b) model 2 and c) mesh of the elements.

Figure 4. Traction-separation response of FRP-calcarenite interface.

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Figure 5. Reference experimental tests by Accardi et al. [21]: test setup; b) scheme of

FRP-W4.

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the load condition and constraints; b) load-displacement curves of specimens URM and

Figure 6. Reference model by Cucchiara et al. [22]: a) geometry and load path of the cantilever beam; b) Mohr’s scheme.

Figure 7. FE vs. experimental results for model 1: a) unreinforced masonry (URM); b)

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reinforced panel (FRP-W4).

Figure 8. Validation of model 1 against analytical model by Cucchiara et al. [22]: a) URM; b) FRP-W4.

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Figure 9. Load-displacement curves of model 2.

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Figure 10. Phases of numerical failure mode for specimen URM according to model 1. Figure 11. Transition between elastic and post-elastic response for specimen FRP-W4 according to model 1: a) maximum principal stresses; b) Von Mises stresses in CRFP strips; c) direction of maximum principal strains.

Figure 12. Failure modes for specimen FRP-W4 according to model 1: a) contact opening; b) contact pressures; c) relative tangential motion; d) frictional shear stresses.

24

ACCEPTED MANUSCRIPT Figure 13. FE output of model 2 for specimen URM: a) deformed shape at the peak (scaling factor x20); b) principal tensile stresses; c) direction of maximum principal strains on the top; d) direction of maximum principal strains on the base.

Figure 14. Transition between elastic and post-elastic response for specimen FRP-W4

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according to model 2: a) maximum principal strains; b) relative tangential motion; c) frictional shear stresses.

Figure 15. Failure mode for specimen FRP-W4 according to model 2: a) contact

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the top; d) frictional shear stresses on the top.

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pressures on the bottom; b) contact opening on the top; c) relative tangential motion on

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a)

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b)

c)

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Figure 1. Local failure mechanism: a) simple overturning; b) vertical flexure; c) horizontal flexure (Ref. [32]).

1

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M=0.5 FL P

P

P

F; δ

F; δ

L

L

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L

F; δ

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M=FζL

b)

a)

c)

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Figure 2. Structural scheme for: a) general case; b) ζ=0; c) ζ=0.5.

1

ACCEPTED MANUSCRIPT axial load

axial load top fixed end

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top hinge

lateral displacement on the base

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FRP-W4

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URM

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a)

lateral displacement on the base

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CFRP

c) Figure 3. Geometry, load conditions and constraints for a) model 1; b) model 2 and c) mesh of the elements.

1

ACCEPTED MANUSCRIPT traction

ts

0

δs

0

δs

0

f

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Gf

Kss

separation

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Figure 4. Traction-separation response of FRP-calcarenite interface.

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ACCEPTED MANUSCRIPT axial load P

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lateral force F displacement

b)

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a) 4.00 3.50

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3.00

F [kN]

2.50 2.00 1.50 1.00

URM (exp)

0.50

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FRP-W4 (exp)

0.00 0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

δ [mm]

c)

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Figure 5. Reference experimental tests by Accardi et al. [21]: test setup; b) scheme of

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the load condition and constraints; b) load-displacement curves of specimens URM and FRP-W4.

1

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Mj=FζL P Fj

i

i

2

T2* T1*

T i*

2

1

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1

Tn*

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n

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n

FICTICIOUS MODEL

a)

b)

Figure 6. Reference model by Cucchiara et al. [22]: a) geometry and load path of the

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cantilever beam; b) Mohr’s scheme.

1

ACCEPTED MANUSCRIPT 4.00 3.50 3.00

2.00 1.50 1.00

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F [kN]

2.50

model 1 - URM (fem) 0.50

URM (exp)

0.00 0.00

5.00

10.00

15.00

20.00

25.00

a)

35.00

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4.00 3.50 3.00 2.50

F [kN]

30.00

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δ [mm]

2.00 1.50 1.00

0.00 0.00

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model 1 - URM (fem)

0.50

5.00

10.00

15.00

FRP-W4 (exp) 20.00

25.00

30.00

35.00

δ [mm]

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b)

Figure 7. FE vs. experimental results for model 1: a) unreinforced masonry (URM); b)

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reinforced panel (FRP-W4).

1

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3.00

2.50

2.50

2.00

2.00

F [kN]

1.50

1.50

1.00

1.00

0.50

0.50

0.00 0.00

4.00

8.00

0.00 0.00

model 1-FRP W4 (fem) FRP-W4 (exp) FRP-W4 (analytical)

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F [kN]

3.00

model 1-URM (fem) URM (exp) URM (analytical)

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3.50

3.50

12.00

4.00

8.00

12.00

δ [mm]

δ [mm]

b)

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Figure 8. Validation of model 1 against analytical model by Cucchiara et al. [22]: a)

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URM; b) FRP-W4.

1

ACCEPTED MANUSCRIPT 14.00 12.00

8.00 6.00 4.00

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F [kN]

10.00

model 2 - URM (fem) 2.00

model 2 - FRP-W4 (fem) 0.00 0.00

10.00

20.00

30.00

40.00

50.00

60.00

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δ [mm]

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Figure 9. Load-displacement curves of model 2.

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δmax= 17.26 mm; scale factor x30

σt,max= 0.154 MPa

σt,max = 0.152 MPa

δmax= 29.51 mm; scale factor x20

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σt,max= 0.147 MPa

εt,max = 1.32e-4

εt,max = 1.38e-4

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εt,max = 1.27e-4

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δmax= 5.73 mm; scale factor x50

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εt,max = 1.27e-4

εt,max = 1.32e-4

εt,max = 1.38e-4

Figure 10. Phases of numerical failure mode for specimen URM according to model 1.

1

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a)

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b)

c)

Figure 11. Transition between elastic and post-elastic response for specimen FRP-W4

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according to model 1: a) maximum principal stresses; b) Von Mises stresses in CRFP

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strips; c) direction of maximum principal strains.

1

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b)

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a)

c)

d)

Figure 12. Failure modes for specimen FRP-W4 according to model 1: a) contact

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opening; b) contact pressures; c) relative tangential motion; d) frictional shear stresses.

1

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b)

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a)

c)

d)

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Figure 13. FE output of model 2 for specimen URM: a) deformed shape at the peak (scaling factor x20); b) principal tensile stresses; c) direction of maximum principal

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strains on the top; d) direction of maximum principal strains on the base.

1

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b)

a)

c)

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Figure 14. Transition between elastic and post-elastic response for specimen FRP-W4 according to model 2: a) maximum principal strains; b) relative tangential motion; c)

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frictional shear stresses.

1

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b)

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c)

d)

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Figure 15. Failure mode for specimen FRP-W4 according to model 2: a) contact pressures on the bottom; b) contact opening on the top; c) relative tangential motion on

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the top; d) frictional shear stresses on the top.

1