A simple model for performing nonlinear static and dynamic analyses of unreinforced and FRP-strengthened masonry arches

Accepted Manuscript A simple model for performing nonlinear static and dynamic analyses of unreinforced and FRP-strengthened masonry arches Barbara Pintucchi, Nicola Zani PII:

S0997-7538(16)30038-9

DOI:

10.1016/j.euromechsol.2016.03.013

Reference:

EJMSOL 3302

To appear in:

European Journal of Mechanics / A Solids

Received Date: 12 January 2015 Revised Date:

8 February 2016

Accepted Date: 20 March 2016

Please cite this article as: Pintucchi, B., Zani, N., A simple model for performing nonlinear static and dynamic analyses of unreinforced and FRP-strengthened masonry arches, European Journal of Mechanics / A Solids (2016), doi: 10.1016/j.euromechsol.2016.03.013. 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.

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A simple model for performing nonlinear static and dynamic analyses of unreinforced and FRP-strengthened masonry arches Barbara Pintucchia,* , Nicola Zania

Corresponding author. Tel. +39 0552756850; e-mail address [email protected] (B. Pintucchi).

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Department of Civil and Environmental Engineering (DICeA), University of Florence, Piazza Brunelleschi 6, 50121 Florence, Italy

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Abstract

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A numerical model for analysing unreinforced and fiber-reinforced masonry arches is presented. It is based on a constitutive equation formulated for one-dimensional masonry elements, assumed to be made of a nonlinear material with no resistance to tension and limited compressive strength. A softening behaviour under compression is accounted for, and a procedure is provided for irreversible damage occurring to the masonry. The reinforcement, which is instead assumed to have no resistance to compression, is applied on the inner and/or outer side of the arch. In order to capture any possible FRP debonding, the values of the shear and normal stresses at the FRP-masonry interface are evaluated. The model, implemented in the non-commercial computer code MADY, allows for short computational times in studying reinforced masonry arches with any geometry, restraints and loading conditions. Comparisons with some experimental results are carried out to verify the effectiveness and accuracy of the model’s predictions under static loads. Some dynamic analysis are also performed for a case study, and preliminary results are reported on the effectiveness of some FRP strengthening arrangements against earthquake actions.

1. Introduction

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Keywords: Masonry, Arches, Damage, Model, Strengthening, Composites, Debonding, Dynamic analysis

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In recent years, specialists have increasingly sought to address the problems inherent in strengthening masonry structures, in particular those with an arch shape. In fact, modern materials technology has provided new possibilities through light and removable retrofit techniques and fiber-reinforced polymer composites (FRP) have been adopted successfully to restore many masonry structures [1]. Wide-ranging research efforts have been devoted to formulating behavioral models and numerical procedures able to predict the structural response of reinforced, as well as unreinforced masonry arches [2], [3], [4]. Moreover, experimental research on a variety of FRP-strengthened masonry arches and vaults have proved very useful in clarifying their static behavior. Besides confirming the effectiveness of these techniques for increasing the ultimate load capacity of such structures, these investigations have helped to shed light on the mechanisms involved in the failure of strengthened masonry arches [5], [6], [7]. Unreinforced masonry arches fail primarily due to the formation of mechanisms. The first such mechanism is the classical four/five-hinge mechanism studied by Heyman [8], which is typical of arches subjected to a vertical pointwise load. Such a collapse mode is attributed to masonry’s low or null tensile strength. Cases of failure that actually involve masonry crushing are also possible, though unusual [9], [10]. Conversely, this failure mode is more 1

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likely to be exhibited by FRP-strengthened arches, as the reinforcement prevents or at least reduces the cracking of sections, and inhibits the formation of the typical hinge mechanisms of unreinforced structures. The experimental campaigns conducted over the last few decades have clearly highlighted such possible collapse mode in strengthened specimens, which often failed by crushing. Besides this mechanism, others failure modes related to sliding along the mortar joints and to FRP detachment have been revealed as well. Despite this wealth of fundamental knowledge acquired in the past, this field is still of great interest amongst researchers, as demonstrated by the large number of relevant works published recently [11], [12], [13], [14], [15]. However, in literature there are very few studies aimed at verifying the effectiveness of FRP-strengthening against dynamic actions. To our knowledge, the only research works dealing with the dynamic behavior of reinforced masonry arches are the experimental campaigns presented in [16], [17], [18] and [19]. So, even though the seismic performance of FRP-reinforced masonry arches does indeed appear to be promising [20], in-depth knowledge of their dynamic behavior is still lacking. In this paper, a simple model for masonry elements strengthened with externally bonded composite materials is presented. This model has evolved from previous studies devoted to defining a numerical model for performing non-linear static and dynamic analyses on masonry columns, arches, towers, and slender structures in general [21], [10], which was implemented in a FEM computational code, called MADY [22]. The latest version of the code now includes the new FRP-strengthened beam element and over all the masonry model described in the following. In defining the model, firstly, a constitutive equation has been developed for one-dimensional elements strengthened with bonded FRP composites. Masonry is a non-linear elastic material with no resistance to tension and limited compressive strength. The reinforcement, which is applied to the intrados and/or extrados of the beam, is instead assumed to have no resistance to compression. However, it has by now been well-established that modeling masonry without setting a bound to the inelastic strain leads to overestimating the load bearing capacity of masonry arches when collapse takes place by crushing [23]. Hence, the constitutive equation has been enhanced by introducing a nonlocal damage process of the material under compression. Essentially, the method consists of updating the values of Young’s modulus and the compressive strength of the generic section for representing the material post-peak softening behaviour. Obviously, the material thus obtained is no longer elastic, because the damage phenomenon is irreversible. Althought the study of the FRP debonding mechanism is beyond the scope of the present paper, premature FRP detachment may compromise the effectiveness of external FRP-strengthening systems, and need to be considered. Hence, the model allows to account for a limit to bond strength in order to identify when this phenomenon occurs. Then, a procedure -inserted into the numerical code- that reduces the composite performance can be selected and used for representing the FRP debonding. In summary, the model needs only few damage parameters to be calibrated; despite its simplicity, it is able to capture all possible failure modes, except for collapse resulting from sliding of stones due to the high shear stress. In this regard, however, experimental evidence shows that this mechanism occurs almost solely when the reinforcement is applied along the extrados of the arch. In such cases, therefore, the model should be used with caution. Thanks to the use of beam elements, performing numerical analyses with the MADY code is not time-consuming and the model furnishes predictions of the ultimate strength in accordance with experimental results for all the failure mechanisms mentioned. Information on dynamic behavior and on the efficacy of the strengthening intervention against seismic actions can also be obtained. In the following, the model and the procedure for accounting for damage due to crushing and debonding is first presented. For a detailed description of the proposed constitutive equation, the concepts of the state space and the processes as formulated in [24], [25] have been used. Then, comparisons of some numerical results obtained via the proposed models with experimental data available in literature are provided. A comparison with the results obtained via two other numerical models has also been carried out with reference to a simple example case. Lastly, some preliminary dynamic results are presented for a reference arch built on two piers. Various hypothetical strengthening systems have been considered involving the application of an externally bonded composite material strip - located either on the intrados or the extrados or either case including the piers. Then, dynamic analyses have been conducted in order to highlight the potentialities of the model for evaluating the effectiveness of different FRP arrangements on the arch’s seismic performance. 2

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Figure 1. Undamaged model. σ- diagram under uniaxial stress for a) masonry, b) reinforcement material; c) cross-section geometry.

2. Constitutive equation

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2.1. Constitutive equation of the undamaged material By presuming that the sections remain plane and orthogonal to the line of the deformed axis, and accounting for axial stress alone, the deformation of a beam can be described in terms of two generalized strains only: extensional strain  and curvature of the axis κ. Under the aforementioned assumptions, the behavior of both masonry and the reinforcement material can be described as in Fig. 1a), where σc and c respectively denote the minimum admissible compressive stress and the strain at which such stress is first attained in the masonry. The reinforcement, which is applied to the intrados and/or extrados of the beam, is instead assumed to have no resistance to compression (Fig. 1b). Such modelling assumption is in accordance with the Italian guidelines [26], [27], and it seems suitable for FRP with a polymeric matrix. It might appear less suitable for other reinforcement - like those with a cementitious matrix - that could resists a certain level of compressive stress. Nevertheless, generalization to such a case would not involve any conceptual modifications, and hence is not addressed in the paper. A limited tensile strength can also be assumed for the FRP, though from a practical standpoint such a premise is less profitable. In order to render the model as general as possible, a distinction has been made between strips bonded to the intrados and the extrados, so that, for example, it can also be used to represent the common cases of reinforcement on the extrados or intrados alone. Hence, the geometrical and other mechanical properties will henceforth be specified in general terms, by means of the subscripts i and e, indicating FRP location. The amount of exterior and interior reinforcement in a given cross-section is taken into account, respectively, through the two parameters Ke and Ki : Ke = δe ne Ee , Ki = δi ni Ei (1) where, as shown in Fig. 1c), δe , δi represent the FRP thickness, ne = b f e /b, ni = b f i /b denote the width of the composites normalized to the section width b, and Ee and Ei are their Young’s moduli. As a consequence of the above kinematic assumptions and for the constitutive laws shown in Fig. 1, for a given generic rectangular cross section of dimensions b and 2h strengthened on both the intrados and extrados, the normal stress σ has a piecewise linear diagram. Let yn , yt , be the positions where σ reaches the values 0 and σc , respectively. Their values can be easily expressed as functions of  and κ, via the equations (yn ) =  + κyn = 0,

(yt ) =  + κyt = c

(2)

which are a consequence of the Euler-Bernoulli hypothesis. Consequently, the plane of generalized strains turns out to be naturally divided into 12 regions, and the constitutive relation - which associates the corresponding generalized stress, axial force N and bending moment M, to the generalized strain - can all be determined. 3

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To describe the constitutive law, it is convenient to introduce the following non-dimensionalized parameters: η=

 , c

χ=

κh , c

n=

N , bhEc

m=

M , bh2 Ec

k+ =

Ke , hE

k− =

Ki . hE

(3)

In this way, the partition in the plane (η, χ), shown in Fig. 2, is expressed by: E±1 = {±χ ≥ 0, η ∓ χ ≥ 0, ∓χ − η + 1 ≥ 0}

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where E is the masonry’s Young modulus in compression. Moreover, in the following the non-dimensionalised strain will be denoted as e = (η, χ) and the non-dimensionalised stored energy as φ(η, χ) . (4) φˆ = Ebhc2

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E±2 = {±χ − η ≥ 0, η ± χ ≥ 0, ∓χ − η + 1 ≥ 0}

E±3 = {χ ∓ η ≥ 0, ±χ − η + 1 ≥ 0, ±χ + η − 1 ≥ 0}

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E±4 = {±χ − η ≥ 0, ±χ + η − 1 ≥ 0} E±5 = {±χ ≥ 0, ∓χ − η ≥ 0}

E±6 = {±χ ≥ 0, ∓χ + η − 1 ≥ 0}

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and for each region the constitutive equations are:

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E±2 :

n = 2η, 2 m = χ, 32 2 ˆφ = 3η + χ , 3 (η ± χ)2 + k± (η ∓ χ), 2χ (η ± χ)2 (2χ ∓ η) ± k± (χ ∓ η), m=± 6χ2 (χ ± η)3 k± (η ∓ χ)2 φˆ = + , 6χ 2

E±3 :

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n=±

(η ∓ χ)2 − 2η ∓ 2χ + 1 , 2χ (χ ∓ η ± 1)2 (2χ ± η ∓ 1) , m= 6χ2 η3 ∓ 3η2 (χ ± 1) + 3η(χ ∓ 1)2 ∓ χ3 − 3χ2 − 1 1 φˆ = ∓ − , 6χ 2

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n=∓

E±4 :

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n = k− (±χ + η) ∓ k+ (χ ∓ η), m = k− (±χ + η) ± k+ (χ ∓ η), k+ (η ∓ χ)2 + k− (η + χ)2 φˆ = , 2 n = 2, m = 0, φˆ = 2η − 1,

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E±6 :

(6)

2χ ± 2η ∓ 1 + k± (η ∓ χ), 2χ 3η2 − 3η − 3χ2 + 1 m=∓ ± k± (χ ∓ η), 6χ2 3η2 + 3η(±2χ − 1) + 3χ2 + 1 k± (η ∓ χ)2 1 φˆ = ± + − , 6χ 2 2

n=

E±5 :

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E±1 :

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Figs. 3a) and b) show the non-dimensionalized energy φˆ as a function of (η, χ), respectively for the cases without and with reinforcement. It is worth noting that ∂φˆ = n, ∂η

∂φˆ = m. ∂χ

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2.2. Modelling damage for masonry crushing In order to account for damage processes, let us assume that masonry behaves as shown in Fig. 4a), where the softening behaviour after crushing is represented through a linear piecewise law. Thus, the normal stress σ over the cross-section of dimensions b ∗ 2h turns out to be

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Figure 2. Partition of the plane (η, χ).

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φ3 2 1 0 4

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η

3

2

2

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−2 −3

χ

0

−2

−1 −4

1

η

a)

−1 −4

−2 −3

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Figure 3. φˆ as a function of (η, χ), for the case a) without reinforcement (k± = 0), b) with reinforcement (for k± = 1).

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if if

(y) ≥ 0 c < (y) < 0

if

u ≤ (y) ≤ c

if

(y) < u

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  0       E(y)        (y)      − 1  σ(y) =  c      1 − β σ  c      µ − 1          0

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u ≥ 1 is the material ductility c and (y) =  + κy with y ∈ [−h, h]. It’s worth noting that the (limit) cases β = 1 and β = 0 correnspond, respectively, to the case of a stress-strain diagram with linear softening and to a diagram with a plateau before the limit value of the compressive deformability u . In the latter case, damage occurs only when u is attained. Looking at the plane (η, χ), the damage process regards the regions where σc is attained - i.e. regions E±3 , E±4 , E±6 in Fig 2. Moreover, denoting by yu the position where

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(yu ) =  + κyu = u

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where β ∈ [0, 1] denotes the slope of the softening branch between c and u , µ =

the conditions

−h ≤ yu = define the lines

(µ − η)h ≤ h, χ

χ = ±(η − µ )

(9) (10)

(11)

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that delimit different regions, where a progressive damage may occur (see Fig. 5). Referring by way of example + to E+4 ∪ E4 = E+4 , the  pattern in the section turns out to be the one shown in Fig. 4 b1) or b2) depending on whether ± (η, χ) belongs to region E±4 or E4 , to each of which corresponds, respectively, the stress pattern given in Fig. 4 c1) and c2). In both the cases shown in the figure, the actual stress distribution is represented herein by means of an equivalent section - which is always wholly reactive in compression (Fig. 4 d)) - for which the values of the material’s Young modulus and compressive strength, respectively E r and σrc , have been reduced. More specifically, the proposed damage model is defined by assuming that i) c and u are not modified by damage; ii) there exists a positive real valued damage function α, which for every generalized strain (η, χ) gives the maximum values that E r and σrc can attain. From these assumptions, it follows that υ=

E r σrc = ≤α E σc

(12)

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where E and σc refer to the undamaged material and υ will be called the damage parameter. Various possible damage function α can be chosen and used for reducing the initial mechanical properties of the material after crushing. Among the others, one choice may be to obtain the reduced parameters E r and σrc by assuming that the equivalent section has the same normal force as the damaged section. Of course, a function can also be obtained assuming the equivalence of the bending moment, in addition to the equivalence of the axial force. However, it leads to the violation of the (reasonable) assumption that c and u are not modified by damage. The damage function chosen has been described in detail below, where the expression of α have been provided for the different regions affected by damage. It is worth underlined that the equivalent section chosen serves only to define the damage function, as it will be clarified in Section 3. If damage occurrs in region E±3 (see Fig. 6) expressions for α are given by the equivalence   (y) Z yc Z ±h Z yc Z z   c − 1  r r  dy  (13) ± E b( + κy) dy ± bσc dy = ± Eb( + κy) dy ± bσc 1 − β µ − 1  ∓h yc ∓h yc 7

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Figure 4. (a) Masonry behavior assumed for modeling damage; (b) possible  pattern; (c) Reactive cross-section and actual σ pattern; (d) Equivalent section and σ pattern.

Figure 5. Plane (η, χ) with damage.

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Figure 6.  pattern for (η, χ) belonging to (a) E+3 , (b) E3 : Reactive cross-section, actual σ pattern and Equivalent σ pattern.

where z = ±h or z = yu depending on whether yu crosses the section or not, that is if the limit value u is attained or not. In view of Eqs. (3) and (9), Eq. (13), together with Eq. (12), yields the following expressions for α   β[(η ± χ)2 − 2η ∓ 2χ + 1] + (µ − 1)[(η ∓ χ)2 − 2η ∓ 2χ + 1]    if (η, χ) ∈ E±3   2 − 2η ∓ 2χ + 1]  (µ − 1)[(η ∓ χ) ±  α3 =  (14)  ±  (η ∓ χ)2 − 2µ + 1 + β(µ − 1)   if (η, χ) ∈ E   3 (η ∓ χ)2 − 2η ∓ 2χ + 1

yc

E b( + κy) dy ± r

± yn

Z

±h

bσrc

dy = ±

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Conversely, in regions E±4 (see Fig. 4), the following equation holds

yc

Z

yc

Eb( + κy) dy ±

Z

yn

z

yc

  (y)  − 1   c  dy bσc 1 − β µ − 1 

(15)

and once again accounting for Eqs. (12), (3) and (9), α takes on the expressions

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 β[(η ± χ)2 − 2η ∓ 2χ + 1] + (1 − µ )(2η ± 2χ − 1)       (1 − µ )(2η ± 2χ − 1) α±4 =   −β(µ − 1) + 2µ − 1     2η ± 2χ − 1

if

(η, χ) ∈ E±4

if

(η, χ) ∈ E4

±

(16)

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Lastly, where σc holds throughout the reactive section, i.e. for region E±6 , the further regions E±7 = {η > µ , 0 ≤ ±χ ≤ η − µ ≥ 0}

must be defined. Only the compressive strength is updated through α, which takes on the expression:  β(η − 1) + 1 − µ    if (η, χ) ∈ E±6    1 − µ ±  α6 =   ± (η ∓ χ − µ )[β(η ∓ χ + µ − 2) − 2(µ − 1)]    if (η, χ) ∈ E6  ∓4χ(1 − µ )

(17)

(18)

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1 0.8 0.6

α

0.4

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2

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η

−4

χ

Figure 7. α as a function of (η, χ), for µ = 2 and β = 0.5.

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2.3. Admissible domain in the plane (n, m) Fig. 8 shows the admissible domain in the half-plane (n, m+ ), under the assumption that reinforcement is present on the extrados, i.e. k+ > 0. The domain in the half-plane Ω− will be qualitatively similar to that of Ω+ in the case of reinforcement also on the intrados, i.e. for k− > 0, and symmetric with respect to the n-axis only for k+ = k− . In the presence of reinforcement, the domain is limited by the half straight line r1 with equation (r1 ) n ≤ 2,

m = −n + 2;

(19)

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and it presents the following partition, which is the image of the one shown in Fig. 2. The linear elastic region Ω+1 is identified by the segments r2 and r3 with equations (r2 )

0 ≤ n ≤ 1,

(r3 )

1 ≤ n ≤ 2,

n , 3 2−n m= , 3 m=

while the curves δ, β and the half-line r4 with equations p (k+ − n + 3) (k+ )2 + 2k+ (2 − n) + n2 − (k+ )2 + k+ (2n − 5) − n2 (δ) n ≤ 1, m = ; 3 (2 − n)(2n − 1) (β) 1 ≤ n ≤ 2, m = ; 3 (r4 ) n ≤ 0, m = −n;

(20)

(21)

delimit the remaining regions Ω+i with i = 2, ..5, as shown in Fig. 8. Lastly, region E+6 is mapped onto point (2, 0). When damage is introduced, the partition of regions Ω+3 , Ω+4 is provided by the two curves Γ1 and Γ2 with equations p (n − 2)( (2 − n)(4µ − n − 2) − 3µ + n + 1) (Γ1 ) n p ≤ n ≤ 2 m = ; 3(µ − 1) 10

(22)

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Figure 8. Partition in the plane (n,m) for m >0 and k+ > 0.

2[3(µ k+ (n − 4) + 2k+ − n2 )Λ + 3µ2 (k+ )2 n − 6µ k+ (k+ + n2 − 4n) + 4(k+ )2 − 12k+ n + 3n3 ] 3(Λ + µ k+ − n)2 (23) p where Λ = µ2 (k+ )2 + 2µ k+ (4 − n) − 4k+ + n2 and n p is the abscissa of the point (Γ2 ) n ≤ n p

m=−

2µ − 1 3µ − 2 ) , µ 3µ2

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P≡(

(24)

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It’s worth noting that for µ = 1, P coincides with point A, the curves Γ1 and Γ2 coincide respectively with r3 and δ, and regions Ω+3 , Ω+4 die out. Fig. 9 refers to the special case of a section without reinforcement, i.e. for k± = 0. In such case, as is widely known, the admissible domain turns out to be delimited by a parabola γ, with equation: (γ)

n m = n(1 − ) 2

0 ≤ n ≤ 2,

(25)

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to which the line r1 is tangent. Regions Ω+1 and Ω+3 -where the FRP does no work- does not change with respect to the general case, while for k+ = 0, the curve δ with Eq. (21)1 becomes the parabola (δ)

0 ≤ n ≤ 1,

m=

3n − 2n2 . 3

(26)

Lastly, the partition of regions Ω+3 , Ω+4 due to the damage model is identified by the curves Γ1 and Γ2 , the latter becoming (Γ2 ) n ≤ n p

m=n−

n2 n2 − , 2 6(2µ − 1)2

(27)

as obtained from Eq. (23) as k+ → 0. Also in this case, when µ = 1, P coincides with A, and Γ1 , Γ2 coincide with r3 and δ, respectively.

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Figure 9. Partition in the plane (n,m) with m >0: special case of k± = 0.

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2.4. State space The previous section presented a procedure for reducing the mechanical parameters for damaged sections. Now the evolution of the mechanical parameters of the section for a generic strain process is described. The damage process can be described in rough terms as follows: if a section is about to evolve into a state with a degree of damage greater than its current state - i.e the new values of the mechanical parameters are lower - then these new mechanical parameter values are assumed for the section; conversely, if a section is instead about to evolve into a state with a degree of damage that would be lower - i.e with equal or greater values of the mechanical parameters -then the mechanical parameters of the section are not changed. In order to express this in a rigorous fashion, to furnish a precise evolution equation for the mechanical parameters during the damage process and to check the thermodynamic consistency of the constitutive equation proposed, the concepts of states and processes have been used. Of the differing versions of the state-space formalism that have been ˇ developed by many authors, here the one proposed by Silhav´ y [24], [28], [25] and [29] has been adopted here. For our needs, only the case of infinitesimal isothermal deformations has been examined. Formally, a set Σ, the state space, and another set, Π, the set of processes are associated to each section of the beam. Each ς ∈ Σ represents a possible state of any given section of the beam. Π is a set of functions π : [0, dπ ] → Σ, defined on the real interval [0, dπ ] (with dπ > 0), that take their values in the state space. Each element π ∈ Π is called a process and the interval [0, dπ ] is interpreted as the time-interval during which the process takes place; dπ is called the duration of the process, and πi := π(0) and π f := π(dπ ) denote the states where the process π starts and ends, respectively. It is required that the process set Π satisfy the following two properties 1) and 2). 1) If processes π1 and π2 are such that π1 ends where π2 starts, i.e., π1f = πi2 , then Π also contains the composition of π1 with π2 , i.e. the process    if τ ∈ [0, dπ1 ], π1 (τ) π1 ∗ π2 :=  (28)  π2 (τ − dπ1 ) if τ ∈ [dπ1 , dπ1 + dπ2 ] of duration (dπ1 + dπ2 ). 2) For each π ∈ Π and for every pair τ1 , τ2 with 0 ≤ τ1 < τ2 ≤ dπ , Π contains the [τ1 ,τ2 ]-segment of π, i.e. the process π[τ1 ,τ2 ] (τ) := π(τ + τ1 ), τ ∈ [0, τ2 − τ1 ], (29) of duration (τ2 − τ1 ), which starts at π(τ1 ) and ends at π(τ2 ). Let us consider the damage function α : R2 → [0, 1], 12

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   0     α= 1     αi

e ∈ A0 , e ∈ A1 , e ∈ E±i ,

if if if

13

(30) i = 3, 4, 6.

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where e = (η, χ), αi have been defined in Eqs.(14)-(18), A0 = E−7 ∪ E+7 = {e : η > µ , χ ∈ [µ − η, η − µ ]},

(31)

A1 = {e : η ≤ 1, χ ∈ [η − 1, 1 − η]}

(32)

and

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is the union of the sets E±i , i = 1, 2, 5. It can be verified that α is a function symmetric with respect to the axis χ = 0 and continuous in R2 − {(µ , 0)}. As already stated, υ := E r /E (33)

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is the non-dimensionalised Young’s modulus of the section, i.e. the ratio between the current and original value of the Young’s modulus. The state of any given generic section of the beam is completely specified by the (non-dimensionalised) strain e, the parameter υ and two flags, f − and f + , which can only take the value 0 or 1, which indicate the presence of an active FRP at the section’s intrados and the extrados, respectively. Because the current treatment deals with a specified section of the beam, f − and f + are fixed and are therefore not included amongst the state variables. In the next Section it will be shown how to modify the formalism of states and processes in order to take damage by debonding into account, as well. Eqs.(12)-(18) imply that, for each strain e, υ ≤ α(e), and can thus furnish the following definition of the state space

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Σ := {ς = (e, υ) ∈ R2 \A0 × (0, α(e)]}.

(34)

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Therefore, Σ can be identified as the subset of R3 delimited by R2 \A0 -not included- and the graph of function α -included- (see Fig.7). For each section of the beam, by knowing the corresponding state ς, t := (υn, υm)

(35)

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can be evaluated from Eq. (6). Then the normal force and bending moment N = bhc υEn,

M = bh2 c υEm

(36)

±

and

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can be computed by means of Eqs. (3). Of course, k in the constitutive equation (6) must be understood as equal to zero when the corresponding strip is absent in that section. Let us denote by b e : Σ → R2 , bt : Σ → R2 ,

ς 7→ b e(ς) = (η(ς), χ(ς)),

(37)

ς 7→ bt(ς) = υ(ς)(n(ς), m(ς)),

(38)

b υ : Σ → (0, 1],

ς 7→ b υ(ς) = E r (ς)/E

(39)

the mappings that at each state ς deliver the corresponding (non-dimensionalised) strain, stress and Young’s modulus of the section, respectively. Moreover, for each process π ∈ Π, let us define b eπ = b e ◦ π,

bt π = bt ◦ π, 13

b υπ = b υ ◦ π,

(40)

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so that b eπ , bt π and b υπ describe the evolution of the corresponding functions during the process π. These are called, strain path, stress path and Young’s modulus path, respectively. Here, for each process π ∈ Π, let us assume that b eπ , bt π and b υπ are Lipschitz continuous functions on [0, dπ ], with a right derivative at each τ ∈ [0, dπ ). Moreover, e˙ π , ˙t π and υ˙ π denote the time (right) derivative of b eπ , bt π and b υπ . The set Π of the processes is made up of the mappings π : [0, dπ ] → Σ such that, given b eπ (τ),    0 if b υπ (τ) < α(b eπ (τ)),     (41) υ˙ π (τ) :=  0 if b υπ (τ) = α(b eπ (τ)) and α(b ˙ eπ (τ)) ≥ 0,     α˙ π (τ) if b υπ (τ) = α(b eπ (τ)) and α(b ˙ eπ (τ)) < 0.

πi1 = πi2 ,

dπ1 = dπ2 ,

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It is a simple matter to verify that the set of processes Π satisfies conditions (28) and (29). We call the quadruplet (Σ, Π,b e,bt) a material section of the beam. Note that any two processes π1 and π2 that begin in the same state, have the same duration and determine the same strain path, coincide i.e. b eπ1 = b eπ2

⇒

π1 = π2 ;

(42)

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in other words, each process is entirely determined by the corresponding strain path [29],[30]. From Eq. (41) it follows that given two states ς1 and ς2 , there exists a process π with πi = ς1 and π f = ς2 if and only if b υπ (0) − b υπ (dπ ) ≥ α(b eπ (0)) − α(b eπ (dπ )).

(43)

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From this it follows that the material section does not satisfy the condition of perfect accessibility, that is, it is not true that for each pair of states ς1 and ς2 there exists at least one process π with πi = ς1 and π f = ς2 . On the other hand, from Eq. (43) with b υπ (0) = 1 and α(b eπ (0)) = 1, it can be deduced that, for every e = (η, χ) ∈ R2 \A0 there i exists a process π such that π = (0, 1) and π f = (e, α(e)). A fortiori there exists a process π such that πi = (0, 1) and π f = (e, υ) with υ < α(e). In other words, any state can be reached by a process starting in ς0 = (0, 1), which is called a base state. For each π ∈ Π, Z dπ

bt π (τ) · e˙ π (τ)dτ

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w(π) :=

(44)

0

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is the non-dimensionalised work (per unit volume in the reference configuration) done by the exterior on the material section during process π; the corresponding (density of) work W(π) is given by W(π) = bhc2 υEw(π). We shall now verify that for each state ς ∈ Σ, there is a real constant θ(ς) with the property that w(π) ≥ θ(ς)

(45)

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for each π ∈ Π, such that πi = ς0 , π f = ς. This is a version of the second law of thermodynamics, which is appropriate to our context [31]. Indeed, let ς be an arbitrary state and π be a process such that πi = ς0 and π f = ς. From Eqs. (44), (38) and (7) it follows that bt π (τ) = b υπ (τ)∇b φ(eπ (τ)),

(46)

and thus, integrating by parts, Z

w(π) := 0

dπ

bt π (τ) · e˙ π (τ)dτ =

Z

0

dπ

b υπ (τ)∇b φ(eπ (τ)) · e˙ π (τ)dτ =

Z 0

dπ

b υπ (τ)φ˙ π (τ)dτ = [b υπ (τ)b φπ (τ)]d0π −

Z 0

dπ

υ˙ π (τ)b φπ (τ)dτ (47)

so that w(π) ≥ b υ(ς)b φ(ς) ≥ 0 14

(48)

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as b φπ (0) = 0, and for every τ ∈ [0, dπ ] υ˙ π (τ) ≤ 0 by Eq. (41), υπ (τ) > 0 by Eq. (33) and b φπ (τ) ≥ 0. Thus, inequality (45) is verified with θ(ς) = b υ(ς)b φ(ς). By defining the function b ψ : Σ → R, b c ψ := υ φ,

0

dπ

0

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the inequality Z dπ Z b w(π) = υπ (τ)φ˙ π (τ)dτ =

(49)

! Z dπ   d  d  b b υπ (τ)b φπ (τ) − υ˙ π (τ)b φπ (τ) dτ ≥ υπ (τ)b φπ (τ) dτ = b ψ(π f ) − b ψ(πi ) (50) dτ dτ 0

t · e˙ − ψ˙ = −υ˙ b φ≥0

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follows. That is, b ψ satisfies the internal dissipation inequality (50) and is thus a (non-dimensionalised) free energy function for our material section [31], [30]. Moreover, from Eqs. (35), (41) and (49) the Clausis-Duhem inequality (51)

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can be obtained, where −υ˙ b φ is the (non-dimensionalized) local dissipation. By way of example, an explicit computation of the work done and the dissipated energy during a rectilinear strain process is described in the Appendix.

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2.5. Modeling damage by debonding The performance of the bond between the FRP layer and the substrate plays a key role in the effectiveness of external FRP-strengthening systems. Hence, a large amount of research has been devoted to investigating and modeling FRP debonding phenomena [32], [33]. Such studies have helped to shed light on interfacial bonds and on the initiation and propagation of the debonding mechanism. Moreover, various models have been developed and used to represent in detail the behavior of the interface between the the FRP layer and the beam substrate throughout the whole range of response, starting from the linear elastic range up to the complete debonding [34], [35], [36]. Most of the studies focused on masonry plane substrates, where FRP debonding is primarily governed by shear stress and likely affects the terminal part (bond zone) of the composite layer. Nevertheless, in the last years attention has also been paid to debonding mechanism in the case of a curved beam substrate [37], [38], [39]. Indeed, experimental tests performed so far have clearly evidenced that FRP debonding in curve structures, i.e. reinforced arches, is even more critical. In particular, if reinforcement is applied to the intrados, normal (peeling) tensile stresses arise together with the shear stresses, making easy FRP debonding to occurr [40]. Some refined models have been proposed also for these cases, based on cohesive interfacial laws for shear and normal stress and mixed-mode fracture criteria [41], [42], [43]. The study of interfacial issues is not the aim of the present paper. In addition, the kinematic hypothesis of the model does not allow us to account for the possible slip of the FRP with respect to the substrate. Nevertheless, many of the aforementioned papers suggested some code-like formulas which provide a good estimation or an upper bound of the bond strength during the debonding process (debonding load) [44], [45] which can be used in the proposed model as a limit value of the axial force in the generic cross-section. When this limit is reached, the performance of the FPR in the cross-section can be considerably reduced with a numerical procedure implemented in the code. As an alternative, monitoring the possible onset of the earliest debonding phenomena can be conducted by estimating the values of the FRP-masonry interaction stresses [46], [40]. Indeed, normal stress and shear stress, here respectively referred to as σn and σt , can be easily obtained through equilibrium considerations, once the axial force N f = Ehn f with n f = k± ( ∓ κh) (52) in the reinforcement has been determined. In fact, the reinforcement may be interpreted simply as a membrane subjected to axial force alone, in equilibrium with the distributed tangential and radial load, p and q, which result from both the arch’s actions on the composite and any eventual external loads. As the bending moment and shear force in the FRP are null, the equilibrium equations are: q=−

Nf dN f ,p=− ds rf 15

(53)

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where s is the natural parameter and r f is the radius of curvature. Therefore σt = −

Nf 1 dN f , σn = − b f ds bf rf

(54)

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where b f is the width of the composite layer. In the following analyses, the latter approach has been adopted in case of FRP bonded on the intrados of the arch. As debonding generally coincides with failure of a masonry layer below the FRP-masonry interface, rather than failure of FRP’s adhesive power, the following criterion seems to be appropriate [46]: r σ2n σn + + σ2t = fb . (55) 2 4

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though different criteria can also be selected. Of course, it requires knowing the limiting value, f b , of the interfacial stress fb , associated to debonding [40]. Once fb reaches f b , the numerical procedure for reducing the composite material’s properties of a cross-section has been used. Conversely, for FRP-strengthening placed on the arch’s extrados, debonding has not been allowed, as the composite layer - though debonded - could still work significantly thanks to the curvature of the arch. Now it is described in rough terms how to extend the concepts of states and processes to the case in which damage by debonding is taken into account. Because the occurrence of such damage during a process may lead the FRP in a certain section to stop existing, as already stated, the flags f − and f + in the state of each section of the beam need to be included. These flags can only take the values 0 and 1 and indicate the presence of an active reinforcement, at the intrados and the extrados of the section, respectively. Thus, in place of Eq. (34), the set of states Σ := {ς = (e, υ, f − , f + ) ∈ R2 \A0 × (0, α(e)] × {0, 1} × {0, 1}}

(56)

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is defined; accordingly, Π is the set of functions π : [0, dπ ] → Σ that satisfies the[33] properties defined in Eqs. (28) and (29). The functions b e, bt, b υ, b f −, b f + , which are defined on Σ, assign (non-dimensionalized) values of the strain, stress, Young’s modulus and flags, respectively, to each state. Let B be the reference configuration of the beam, which is defined as a planar curve with arc length s. Any deformation process e : B × [0, d] → R2 defines, for each section s, a strain path e(s, ·) that, in view of Eq. (41), induces a process π(s, ·) of duration dπ = d, which can be interpreted as a ’global process’ of the beam, π : B × [0, dπ ] → Σ,

(s, τ) 7→ π(s, τ).

(57)

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Accordingly, for each global process of the beam π, the functions eπ = b e ◦ π, t π = bt ◦ π, υπ = b υ ◦ π, fπ− = b f − ◦ π, fπ+ = b f + ◦ π,

(58)

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which are defined on B × [0, dπ ], describe the (global) evolution of the strain, stress, Young’s modulus and flags, respectively. For each deformation process eπ , the evolution of υπ is furnished once again by Eq. (41), where α and υ must now be considered functions of both, s and τ. For the evolution of fπ± the following rules are set down. Let fb = fb (σn , σt ) as defined by Eq. (55), where σn and σt are functions of s and τ. Moreover, let s0 ∈ B be any section of the beam and τ0 ∈ [0, dπ ) any time. (i) If fπ± (s0 , τ0 ) = 0, then fπ± (s0 , τ) = 0 for every τ > τ0 ; (ii) if fπ± (s0 , τ0 ) = 1 and fb (s0 , τ0 ) < f b , then there exists  > 0 such that fπ± (s0 , τ) = 1, for each τ ∈ [τ, τ + ); (iii) if fπ± (s0 , τ0 ) = 1 and fb (s0 , τ0 ) ≥ f b , then fπ± (s0 , τ) = 0 for every τ > τ0 . 3. Finite element procedure and nonlocal damage integration A finite element approach has been followed for solving the static/dynamic equilibrium problems. In particular, arches are discretized into ’flat’ elements and: i) three degrees of freedom - axial and transverse displacements plus rotation - have been assigned to each node; ii) conforming elements and Hermite shape functions have been selected 16

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to guarantee the continuity of both transverse displacements and rotations; iii) linear shape functions have been used for axial displacements; vi) standard numerical techniques, based on the Newton-Raphson iterative method have been used to solve the nonlinear system resulting from the structure’s discretisation; the converge check is performed on the residual loads; v) for dynamic analyses, the Newmark method has been used to integrate the equations of motion, and the effects of viscous damping are taken into account by means of a constant damping matrix C, according to the Rayleigh method. The numerical FE method is not described in details since already presented in [21], [10], [47] with reference to other types of ’beam’ elements. Herein, the FRP-strengthened masonry beam element has been implemented into the code MADY [22]. The procedure to account for the irreversible damage under compression has also been introduced. In this regard, however, strain softening may causes mesh sensitivity and convergence to incorrect solutions when the finite elements become of vanishing size [48], [49]. To overcome this limitation, a standard nonlocal averaging approach has been used. Going into details, in its local formulation, the procedure consists of updating - at each load step in the static analysis and at each time step in the dynamic analysis - the current values of Young’s modulus and compressive strength (i.e. E r and σrc given in Section 2.2) at each Gauss point according to Eq. (41). To this regard, it’s worth noting that the numerical procedure guarantees, when the convergence is achieved with the new values of the mechanical parameters, the equilibrium of both the axial force and the bending moment with the loads, with whichever α function is used [50]. For the nonlocal formulation, in evaluating the variation υ˙ of the damage parameter from Eq. (41), the function α is determined by means of the nonlocal generalized strain e¯ = (η, ¯ χ) ¯ with Z Z η(z) ¯ = w∞ (z)(z − ζ)η(ζ) dζ, χ(z) ¯ = w∞ (z)(z − ζ)χ(ζ) dζ (59) R

R

(60)

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where the weight function w∞ (z) is the bell-shaped function   z2 15    (1 − 2 )2 if − L ≤ z ≤ L w∞ (z) =  16L L   0 elsewhere

EP

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and 2L is the characteristic length of the material. The same nonlocal approach has been chosen for the procedure used to account for FRP debonding. MADY enables now analysing both unreinforced and reinforced arches, with any geometry and restraints, under any static and dynamic loading conditions. Despite the nonlinear behaviour due to cracking, crushing and debonding - in the latter two cases even with (possible) abrupt change in materials properties, the model does not suffer of any significant problem of convergence. 4. Behavioral example and validation

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In this Section, some results are presented as a simple example case that is provided for illustrative purposes only. The structure (see Fig. 10 a)) is an unreinforced 1 m-long, square cross-sectional masonry flat arch, 0.1m on a side. It is perfectly constrained at both ends. The following values have been assumed for the mechanical parameters: Young’s modulus E = 1000 MPa, compressive strength σc = -1.0 MPa, and limit value of the compressive deformability u = - 0.002 (i.e. µ = 2). For the material density, γ = 1900 kg/m3 has been assumed. Firstly, some comparisons between the results obtained via the local and nonlocal damage procedure presented in Section 3 have been provided. All the numerical analyses have been conducted through MADY by applying a monotonically increasing vertical displacement at the midpoint of the beam of Fig. 10 a). For the damage parameter β, the median value β = 0.5 has been selected. Moreover, for the nonlocal approach, the characteristics length has been assumed equal to the cross-sectional height (2L= 0.1m). On the left, Fig. 10 b) shows the curves load versus displacement at the beam’s midpoint obtained with the local procedure by varying the finite element discretization, specifically assuming a number of 40, 80 and 100 finite elements (FE) respectively. The figure highlights the difference of the three curves and the difficulties in convergence as the size of the finite elements decreases. The results obtained for the same three discretizations by means of the nonlocal intagration are plotted on the right graph of Fig. 10 b). 17

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b)

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Figure 10. a) Skectch of the example case; b) Load-displacement curves obtained via a local and nonlocal procedure with various FE discretizations.

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This graph shows results that are not significantly dependent on the mesh size; it also highlights that difficulties in convergence for small FE have been overcome thanks to the nonlocal numerical technique. In the following analyses of this Section, the flat arch is always discretized into 80 finite elements and the characteristics length 2L is assumed equal to 0.1m. To highlight the effects of the proposed masonry damage model, Figs. 11 show the results obtained again by applying a monotonically increasing vertical displacement at the midpoint of the beam. Specifically, Fig. 11 a) shows the curves of load versus displacement obtained with the proposed damage model, for three different values of the parameter β = 0, 0.5, 1. The response predicted through the same model but neglecting damage under compression is also reported for comparison. Fig. 11 b) shows the corrensponding graphs of the bending moment as a function of the curvature for the beam mid-section. To validate the proposed model, a comparison with the numerical results obtained via two other different models have been carried out, with reference again to the foregoing example case. In particular, these comparisons are provided in order to highlight that the idea of spreading damage all over the cross-section -when just some fibers reach their limit state- does not undermine the global results predicted by the proposed model. Going into details, the first model used for comparison is the nonlinear (force-based) beam model with a fiber section available within the open-source OpenSees software [51]. The uniaxial material Concrete01 has been chosen as stress-strain law of each fiber, as it is the most similar to the constitutive law assumed in the proposed model. Namely, by adopting β = 1 in Eq. 8 and by setting in the OpenSees model the concrete crushing strength -i.e. the strength for  = u - equal to zero, the shape of the two constitutive law becomes quite similar. A difference still 18

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a)

b)

Figure 11. Monotonic loading: a) Load versus displacement and b) Bending moment versus curvature at the mid-section of the beam, for three different values of the parameter β.

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remains in the first branch of the two graphs (i.e. for c <  < 0), as the law of the OpenSees code is nonlinear. The comparison has been carried out by setting the same values of the Young’s modulus (E = 1000 MPa), of the compressive strength σc = -1.0 MPa and the ductiliy µ . In particular, the curves (solid-line) shown in Figs. 12 are those obtained for µ = 3, which is the lowest value for which the OpenSees analysis has reached the numerical convergence. Similar results have been obtained for higher values of µ . The response without softening behaviour obtained through the two models is also depicted in Figs. 12 (dashed-line curves). The slight difference between the slope of the curves shown in Figs. 12 appears to be strictly related to the above-mentioned difference in the constitutive laws used, as confirmed also by the trend of results obtained without softening. When the softening branch is achieved, the results predicted by the two models are similar, expecially in the moment-curvature response, thus suggesting that accounting for a fiber-by-fiber damaging does not induce significant variation. The second model is a 2D masonry-like model with damage, developed by the same authors: it is an extension of the classical masonry-like (no-tension) model which accounts also for a (nonlocal) damage process in compression [52]. Using a four nodes element in the plane stress framework, this 2D model has been implemented into the MADY code and used here for comparison. The load-displacement curve obtained with the same values of the mechanical parameters is depicted in Fig.13 and compared with the results provided by the beam model (already given in Fig. 11 a). Unsurprisingly, the beam model predicts a stiffer behaviour with respect to the 2D model; the trend of the response affected by damage is, however, in good agreement. Lastly, Fig. 14 shows the results obtained for a cyclic process conducted by applying a vertical displacement at the midpoint of the beam and by assuming β = 0.5; in particular as in Figs. 11, a) and b) show the vertical load versus displacement and the bending moment versus the curvature, respectively. It is worth noting the stiffness and strength degradation due to damaging, and the perfect pinching of the diagram - due to the nature of the constitutive model, which does not provide for residual deformations. Fig. 14c) shows the work done during the process W (solid line) and the free energy function Ψ (dashed line), whose corresponding non-dimensionalized density are defined in Eq. (44) and (49), respectively. The graph highlights the dissipation due to damage (see Eq.(50)): W1 and W2 gives, respectively, the energy that has been dissipated during the cicles 1 and 2. 5. Comparison with experimental data In order to check the proposed model’s ability to effectively predict the ultimate load and failure modes of reinforced and unreinforced arches, some comparisons have been carried out with reference to arches whose behavioral 19

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a)

b)

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Figure 12. Comparison with OpenSees results: a) Load versus displacement and b) Bending moment versus curvature at the mid-section of the beam.

Figure 13. Comparison with results obtained through a 2D masonry-like model with damage.

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80

0.6

Bending moment [Nm]

0

−0.2

−0.4

20 0

−20 −40

−2

−1.5

−1 −0.5 0 0.5 Vertical displacement [mm]

1

1.5

2

−60 −0.8

a)

−0.6

−0.4

−0.2 0 Curvature [m−1]

0.2

0.4

0.6

b)

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−2.5

40

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Vertical load [kN]

0.2

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60

0.4

c)

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Figure 14. Cyclic loading: a) Load versus displacement, b) Bending moment versus curvature at the mid-section of the beam, c) Work W and free energy function Ψ versus displacement.

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r = 91.5cm y

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P

b = 10 cm h = 10 cm

Figure 15. Analised structure.

bfi

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Θ = 30˚ x

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response up to collapse have recently been determined through experimental tests [53]. The results presented in the following are also useful as an example of the procedure used to represent FRP debonding. The structure is a flattened circular arch with clamped springings and mean radius r = 91.5 cm, a uniform rectangular cross-section of dimensions b = h = 10 cm, and springing angle Θ equal to 30◦ (Fig. 15). It is subjected to the pointwise load P, which is increased until collapse occurs. The experimental trials presented in [53] comprise several specimens reinforced with CFRP strips, bonded to both the extrados and the intrados in various configurations. Referring specifically to these latter cases, three variants of the FRP-strengthened arch have been considered by varying the reinforcement fiber width b f , from 1.25 cm to 5 cm. To summarise the experimental outcomes briefly, Table 1 shows the collapse load values reported for each of the considered cases, which also exhibited different failure modes. The unreinforced arch undergoes the classical five-hinge collapse mechanism, ascribable to masonry’s low tensile strength. In the other cases, either FRP debonding or compressive masonry failure occur, depending on the reinforcement width. More precisely, the case with the smallest amount of FRP (width = 1.25 cm) is characterized by a premature detachment, followed by a failure mode similar to the unreinforced arch; conversely, for the most highly reinfoced arch (width = 5 cm), debonding does not occur, and the collapse mode is governed primarily by material crushing at the crown. Lastly, in the median case (width = 2.5 cm), progressive debonding takes place, accompanied by the onset of masonry crushing at the crown.

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FRP width b f (cm) Unreinforced 1.25 2.5 5

Collapse Load (N) 450 3560 4520 6580

Table 1. Collapse loads for the unreinforced and reinforced arches obtained in [53].

In performing the numerical analysis, the mechanical properties assumed for masonry and composite materials are those suggested in [53]. Namely, masonry is assumed to have a Young’s modulus Ee = 1550 MPa, a compressive strength σc = -8.6 MPa, and a material density ρ=1800 kg/m 3 . For FRP composite, thickness δi is 0.08 mm, and the Young’s modulus is equal to Ei = 23000 MPa. This value has been changed with respect to the value suggested in [54] on the basis of the homogenization tecnique proposed in [55] to better fit the experimental results. All the analysed arches are discretized into 120 beam elements, and the characteristics length 2L for the nonlocal integration has been assumed equal to 0.1m. All the numerical simulations have been conducted by applying a monotonically increasing vertical displacement at the midpoint of the arches, with a step that becomes smaller approaching the collapse (up to 10− 3 mm). A refinement of the load steps does not modify the results and the numerical convergence. 22

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3.5 3

unreinforced arch FRP 1.25 cm FRP 2.5 cm FRP 5 cm

2 1.5 1 0.5

b)

1

2 3 4 Vertical displacement [mm]

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a)

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Vertical load [kN]

2.5

5

6

Figure 16. Load-displacement diagrams a) recorded during the laboratory tests (from [53]); b) obtained with MADY.

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For the unreinforced arch, when the value F = 450N is reached, the model evidences the five-hinge collapse mechanism observed in the specimen, thus highlighting the good agreement between the model’s prediction and the experimental results. The comparisons made with the experimental results on the strengthened arches have instead been used firstly to identify preliminary, but reasonable, values for the two model parameters governing the phenomena of masonry softening and FRP debonding: masonry’s inelastic limit strain u and a limit value f b of the stress at the FRPmasonry interface that leads to strips detachment. Naturally, the issue of assuming appropriate values for the material parameters - in particular those influencing the damage process - is an open one, which requires comparisons with a substantial amount of experimental data, and which is beyond the scope of the present paper. Going more into details, the results obtained for the specimen where only material crushing was observed are used herein to limit the maximum allowable ductility. Tentative values of u have been assigned until the collapse obtained numerically is achieved for both an applied load and an exhibited ductility approaching the respective value observed experimentally. In the same way, f b has been obtained by comparison with the laboratory results on the arch with reinforcement’s width equal to 1.25 cm, which exhibits only the composite detachment. In this manner, the values u = -6.5 10−3 and f b = 1.2 105 N/m2 have been obtained. Then, the case which experimentally reveals the interaction between FRP debonding and masonry crushing at collapse, is used as a further case to verify the reliability of the model calibrated on the two previous cases. Figures 16(a) and (b) show for the three analysed arches the graphs of the force applied at the crown as a function of the vertical displacement at the same point obtained by MADY, compared with the laboratory results. The same graph is also reported for the unreinforced arch; in this case, the maximum vertical displacement corresponds to the 90% of the collapse load. As shown by the figure, the numerical results are acceptably consistent with the experimental data in terms of force and deformation up to collapse. The collapse mode exhibited in each case are also consistent with the experimental evidence. Limiting together u and fb in the first two arch cases does not change the predicted mechanism of failure. For the case with the largest amount of FRP, collapse occurs by progressive damaging of the masonry in compression, while the composite layer does not, at any point in the arch, reach the limit of detachment. For the case with the smallest amount of FRP, fb reaches the limit value at the crown section, and debonding of the fiber occurs when there is no damage to the masonry; then collapse occurs without masonry damage, as shown by the experimental test. Lastly, both phenomena of masonry damage and FRP detachment are present at collapse for the median case (width = 2.5 cm), consistently with the experimental evidence. To highlight these findings, Fig. 17 shows the graphs of two response parameters - the first one referring to masonry damage, the second to FRP debonding - as a function of the arch’s curvilinear abscissa s, at some successive load steps closely approaching collapse. More precisely, the first graph shows parameter υ - as a function of s - which provides information on whether and where damage occurs; it also reveals how much the mechanical parameters have been reduced (i.e. the degree of damage). In the second 23

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Figure 17. Arch case with the median amount of FRP (width = 2.5 cm): υ and fb as functions of s, at three load steps approaching collapse.

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graph - which represents fb versus s - the comparison of the interfacial stress ( fb ) with the limit value f b - i.e. the straight dashed line in the graphs - indicates whether the FRP detaches and, thus, how much its performance has been degraded. Step A (red line) is the loading step at which the first masonry crushing occurs (at the crown): the minimum of υ is equal to 0.93, whereas the maximum of fb is smaller than f b . Successively, at Step B, FRP debonging also occurs at the crown, as long as crushing progressively increases. It’s worth noting that the strip ’detachment’ at the keystone leads to a considerable reduction of the FRP performance all around the crown, than can in some way represents the spread of debonding. 6. Dynamic behaviour: a case study

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The proposed model has been used to perform the dynamic analysis of a case study subjected to the action of a real earthquake. The aim is to show the potentialities of the model for investigating the effectiveness of FRP-retrofitting to improve the arch’s seismic performance. The analysed structure is a simple but realistic masonry arch built on piers. Its main geometrical characteristics are as follows: span = 3.7 m, rise = 1.4 m, springing angle = 16◦ , thickness = 0.35 m, width = 0.4 m, for the arch; thickness = 0.75 m, width = 0.4 m, height = 3.5m for the piers (see Fig. 18a)). The following values are used for the mechanical properties: ρ=1900 kg/m3 , E = 3 GPa, σc = -4 MPa, u = -2.0 10−3 (so that µ is equal to 1.5) and β = 0, and a value of 4% for the viscous damping coefficient [56]. The arch has been discretized into 308 elements (309 nodes), while the characteristic length for the nonlocal damage procedure has been set equal to 0.35m. Moreover, in all the analyses, firstly, it has been loaded by its own weight. The line of trust obtained is shown in Fig. 18b). As earthquake input, the strong ground motion shown in Fig. 19 recorded during the 1978 Tabas, Iran, earthquake has been used. The event has a magnitude of 7.4, a duration of 63.40s and a PGA of 0.925g [57]. Firstly, the dynamic analysis has been conducted for the unstrengthened arch. Fig. 20 shows the time-histories of the displacements in the horizontal (x) and vertical (y) directions at the arch’s crown. Diagrams are represented up to t = 14s; at this time-step, damage to crushing is dramatically extended and the numerical procedure stops converging. In order to help in analysing the damage process, some time steps are also marked in the same diagrams with vertical lines. 24

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Figure 18. Geometry of the analysed structural scheme (a); Line of trust under the self-weight (b).

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Figure 19. Input ground motion used in the analysis (from Tabas earthquake).

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Going into more detail, Fig.21 shows the time-histories of the damage parameter υ for the left and right piers and haunches and for the crown of the arch. At t = 0, υ is equal to 1, i.e. the undamaged state, in all the graphs. As shown by the figure, the damage starts at the base of the left pier at time t = 10s and grows in intensity until t = 11.1s. At the same time, it gradually extends up the pier for a certain length from the base. This is shown by Fig. 22 a) which is the diagram of υ along the central line (red line) of the arch-piers structure at t = 11.1s. In other words, it gives the value of υ for each cross-section of the structure at this given time. From this moment on, damage on the left pier does not augment, due to the reversal phase of the input ground motion (see Fig. 20); it develops, instead, on the right pier (from t = 11.15s until t = 12.17s), as shown in Fig.21 and Fig. 22b). The structure, now strongly damaged, is rapidly approaching collapse: crushing damage also affects the left haunch (see Fig.21 and the υ-graph for t = 12.48s in Fig. 22c)). Damage then occurs also at the right haunch and at the crown (see Fig.21 and Fig. 22d)). The seismic responses of the structure strengthened in various ways have then been evaluated and compared with that of the unstrengthened arch. In all the strengthened cases, the reinforcement has been assumed to have a Young’s modulus of 31.5 GPa, a thickness of 1.610−3 m and width of 0.2 m. FRP debonding has been accounted for at the intrados of the arch, and the criterion given in Eq. 55 has been used. Four different FRP reinforcement configurations have been considered (see Fig. 23): FRP located along the intrados of the arch (in the following referred to as Case i); FRP applied along the intrados of the arch and on the interior side of the piers (Case ip); FRP applied only along the extrados of the arch (Case e) and, lastly, FRP mounted externally on the piers and along the extrados of the arch (Case ep). In order to simulate the strengthening of an existing structure, the self-weigth has first been applied to the unstrengthened arch; then, the model has been modified by introducing the FRP-strengthening system and the dynamic analysis has been performed. The results obtained for all the considered cases are provided by Fig. 24 and 25, that show the x- and ydisplacements time-histories at the crown and at the left abutment, respectively. By comparing the results to each other and to those of the unstrenghtened case, which are also reported in the 25

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Figure 20. Unstrengthened arch: time-histories of the x-displacement and y-displacement at the arch’s crown (plus or minus signs are consistent with the reference system depicted in Fig.18a).

Figure 21. Unstrengthened arch: time-histories of the damage parameter υ at the piers, haunches and at the crown of the arch.

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Figure 22. Unstrengthened arch: diagram of the damage parameter υ along the central line (red line) of the arch-piers system at progressive time steps (from t=10.82s to t=15.32s).

Figure 23. Strengthening configurations.

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Figure 24. Time-histories of the x-displacement and y-displacement at the arch’s crown.

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graphs, it appears that: 1. FRP strengthening is effective in improving the arch seismic performance avoiding the consistent damage occurring in the unstrengthened structure. This is true for all the different reinforcement configurations considered. FRP composite bonded on the arch provides a restrain to cracks opening during the dynamic action that limits the increase of arch’s deformation, as evidenced by the time-histories of the y-displacement at the crown and the xdisplacement at the abutment. The overall structural damage is consistently reduced. Fig. 26a) and b) show the damage parameter υ observed at the end of the seismic action on Case i and Case e, respectively: crushing slightly affects only the right pier. When the FRP is also bounded to the piers (i.e. Case ip and Case ep), the damage is completely avoided. 2. FRP bebonding does not occur in any of the cases considered. Further analyses -whose results are not shown for the sake of brevity- conducted by reducing the width of the strip (from 20 cm up to 5 cm), exhibit only a slight increase of the interfacial stress. Even in these cases, however, debonding does not occur. Such a result suggests that a small amount of FRP is sufficient to improve the seismic performance of the arch. 3. In terms of dynamic behaviour, no significant differences appear between the cases of placying the FRP composite at the intrados or at the extrados. The histories of displacement are almost similar as well as the damage occurred (Figs. 24, 25, 26). Dynamic analyses have been performed only for a case study under a single earthquake. It is clear that these are preliminary results that should be improved and extended by means of a wider and more systematic study. 7. Conclusions A numerical model for studying FRP-strengthened masonry arches has been developed and implemented in the finite element procedure of the non-commercial code MADY, which enables performing both static and dynamic analyses with low computational costs. The model is formulated for beam elements made of an inelastic material with no resistance to tension and bounded compressive strength; the deformability of the material under compression is also limited, and an irreversible damage 28

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Figure 25. Time-histories of the x-displacement and y-displacement at the arch’s abutment.

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Figure 26. Strengthened arch: diagram of the damage parameter υ along the central line (red line) of the arch-piers system occurred under the seismic action: a) Case i; b) Case e.

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process is accounted for. In order to capture FRP debonding, the values of the tangential and normal interactions between the masonry and the composite can be monitored and several criteria can be used for limiting the FRP-masonry interfacial stress. Then, a procedure that reduces the FRP performance can be selected to model the detachment. The proposed model is simple yet effective. It can furnish valuable information for more mindful practices in the design and assessment of strengthening interventions for masonry arches, especially with regard to their effectiveness against seismic actions. The numerical simulation of experimental tests under static loads shows a fair agreement between the obtained results. A preliminary investigation on the seismic behaviour of a case study shows that FRP strengthening is effective in improving the arch seismic performance avoiding the consistent damage occurring in the unstrengthened structure. Of course, the performed analyses should be extended to cover arches with different mechanical and geometrical properties and different FRP configurations, under a significantly large number of input ground motions.

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By way of example of a process, let us consider a strain path b e that starts at point e0 = 0, coincides with the straight line χ = η until reaching point e1 = (µ , µ ) and then returns back up to point e0 along the same line (see Fig. 5). We put τ = χ = η and denote by eµ = ( 21 µ , 12 µ ) the point of intersection of the line χ = η with the line χ = µ − η. Moreover, let us denote by π1 the process that corresponds to the first part of the strain path, i.e. for 0 ≤ τ ≤ µ , and by π2 the process that corresponds to the strain path back from e1 to e0 . From Eqs. (41) and (14) it can be deduced that    for τ ∈ [0, 12 µ ],  1 b υπ1 (τ) = α(b eπ1 (τ)) =  (A.1) 2µ − 1   for τ ∈ [ 21 µ , µ ]  4τ − 1 and then ! 2µ − 1 f i π1 = π2 = e1 , . (A.2) 4µ − 1 From Eqs. (48) and (6)3,4 it follows that ! Z µ 4τ − 1 3τ − 1 b b w(π1 ) = φ(b eµ ) + υπ (τ) dτ + 1 2τ 6τ2 2 µ (A.3) 16(2µ − 1) 1 1 2 + µ + = (2µ − 1) ln − . 6 2µ − 1 6µ 3 Moreover, from Eq. (41) it can be deduced that, during the process π2 , b υπ2 is constant, b υπ2 (τ) = b υπ1 (µ ) = α (e1 ) =

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A FE beam model for analysing unreinforced and fiber-reinforced masonry arches is presented. Masonry is an inelastic material with no resistance to tension and bounded compressive strength. Masonry deformability is limited and an irreversible damage process is accounted for. A procedure is provided for capturing FRP debonding. Dynamic analyses are performed on a case study to evaluate the effects (and benefits) of FRP-retrofitting in improving seismic performances.

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