Effective bond length of FRP stiffeners

International Journal of Non-Linear Mechanics 60 (2014) 46–57

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Effective bond length of FRP stiffeners$ Annalisa Franco a, Gianni Royer-Carfagni b,n a b

Department of Civil and Industrial Engineering, University of Pisa, Via Diotisalvi 2, I 56126 Pisa, Italy Department of Industrial Engineering, University of Parma, Parco Area delle Scienze 181/A, I 43100 Parma, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 5 September 2013 Received in revised form 17 December 2013 Accepted 18 December 2013 Available online 28 December 2013

The problem of an elastic bar bonded to an elastic half space and pulled at one end is considered to model the performance of FRP strips glued to concrete or masonry substrates. If the bond is perfect, stress singularities at both bar-extremities do appear. These can be removed by assuming cohesive contact forces à là Baranblatt that annihilate the stress intensity factor. We show that the presence of such cohesive zones is crucial to predict the experimentally measured effective bond length (EBL), i.e., the bond length beyond which no apparent increase of strength is attained. In particular, it is the cohesive zone at the loaded end of the stiffener, rather than that at the free end, that governs the phenomenon because the EBL coincides with the maximal length of such a zone. The proposed approach provides better estimates than formulas proposed in technical standards. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Fiber Reinforced Polymer (FRP) Stiffener Cohesive contact Effective bond length Debonding

A promising technique to strengthen concrete or masonry structures consists in gluing to them strips made of Fiber Reinforced Polymer (FRP). Experiments have provided a wealth of evidence that the most frequent failure mode for this arrangement is the debonding of the FRP from the substrate, triggered by high stress concentrations at the extremities of the stiffener. A mixedmode analysis [1–4], accounting for the normal stresses acting at the interface, is certainly the most accurate approach. However, considering the small thicknesses of the FRP strips, their bending strength can be neglected at least as a first order approximation, so that a pure mode II crack propagation can be assumed to describe the response of the bonded joint. The debonding process is certainly complex and different experimental setups are used for its characterization (an extensive list of references can be found in [5,6]). In any case, there are a few objective parameters that characterize the ultimate performance. One of these is certainly the effective-bond-length (EBL) of the stiffener, defined as the bond length beyond which no further increase of pull-out strength can be achieved. Knowledge of the EBL is necessary to properly design the reinforcement so as to assure the complete transfer of load to the substrate. To interpret the phenomenon of debonding, various shearanchorage-strength models have been proposed, for which a review can be found in [6]. In general, such models can be

classified into three categories: (i) empirical models based on the regression of test results [7]; (ii) engineering formulations based upon simplified assumptions and appropriate safety factors [8,6,9]; and (iii) fracture-mechanics-based models [10–12]. All these aim at defining the pull-out-force vs. end-displacement curves [13,14] when the bond length is varied, from which the EBL can be determined. To our knowledge, the major underlying assumption common to all the analytical approaches proposed so far consists in neglecting the elastic deformation of the substrate, so that the description of the entire phenomenon is deferred to the calibration of a proper shear-stress vs. slip interface constitutive law. But such approaches predict that the shear stress at the interface never reaches, but rather asymptotically approaches, the zero value. It is then difficult to objectively define the EBL, because the bond is active in the whole stiffener, whatever its length is. This is why many researchers have given an engineering interpretation of the EBL. For example, many models define the EBL as the bond length over which the resultant of the shear contact stress is at least 97% of the ultimate strength1 of an infinite stiffener [11,15–17]. According to other authors, the evaluation is purely experimental. Measuring the strain profile in the stiffener – usually employing resistance strain gages – the effective bond length is the length over which the strain decays from the maximum to the zero value [13,18–22]. Empirical formulas can thus be proposed on the basis of the experimental results. Both

$ This paper belongs to the special issue “4th Canadian Conference on Non-linear Solid Mechanics”. n Corresponding author. Tel.: þ 39 0521 906606; fax: þ 39 0521 905705. E-mail address: [email protected] (G. Royer-Carfagni).

1 Notice, in passing, that a characteristic coefficient that appears in the governing equations relying upon the rigid-substrate hypothesis [11] is tanh 2, and tanh 2 C0:97. Therefore, the limit value 97% is simply suggested by a mathematical formulation of the problem and is not justified on a physical basis.

1. Introduction

0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.12.003

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

definitions, however, carry intrinsic ambiguities. In the first case, the percentage is a priori defined, and the result is strongly dependent upon the assumed constitutive law at the interface. The second definition is affected by the experimental error and the sensitivity of the gages. Another approximation associated with the assumption of rigid substrate is that the slip, i.e., the relative displacement between stiffener and substrate, is theoretically and experimentally evaluated by simply integrating the axial strain in the stiffener [23–25]. A more precise calculation would require the evaluation of the strain in the substrate, which is far from being negligible especially at those zones, like the stiffener extremities, where stress concentrations do occur. In any case, if the substrate is rigid the slip is always non-zero whatever the bond length is, regardless of the assumed constitutive law for the interface. This paper continues and concludes a line of research by the authors where the effect of the elastic deformation of the substrate is assumed to play a significant role. The model problem now considered is that of an elastic stiffener in contact with the boundary of a semi-infinite plate, supposed in generalized plane stress. Problems of this kind in plane linear elasticity have been considered by various researchers [26–31], with the main purpose of evaluating the stress concentrations near the edges of the stiffener in relation with crack initiation and propagation in the substrate or along the interface. In [32], the authors have solved the problem when the stiffener is pulled at one extremity (loaded end) and the bond is perfect (no slip occurs). An extension of Irwin0 s formula has been obtained to correlate the mode II stress intensity factor with the release of elastic strain energy associated with the detachment of the stringer. Assuming a Griffith-like energetic competition, debonding is assumed to start and develop when the energy release rate equals the surface energy of detachment. With this model the ultimate strength can be correctly predicted, but the EBL was strongly underestimated. In fact, the stress singularity at the loaded end predicted by the theory of elasticity equilibrates the greatest majority, by far, of the applied load. In other words, the calculated shear stress at the interface shows a decay much more rapid than in the experimental measurements. To solve this inconsistency, in [14] a zone was supposed to exist in a neighborhood of the loaded end where cohesive forces à là Baranblatt may develop at the price of a relative slip between the adherents. Assuming a simple, step-wise, shear-stress vs. slip constitutive law at the interface, the length of the cohesive zone was evaluated by imposing that the stress intensity factor is null at the frontier with the perfectly adherent zone, in agreement with a procedure also followed in [33,34]. In this way, it was possible to demonstrate that the applied load is in practice equilibrated by the cohesive forces acting in the cohesive zone only. Therefore, the maximal length of the cohesive zone, compatible with the assumed constitutive interface law, provides a natural and physically motivated definition of the EBL. Experimental results confirm the analytical findings for various values of the bond length. One major question is still open at this point, which regards the possible effects of the second (physically inconsistent) singularity acting at the free end of the stiffener. This is still present in the model of [14]. The major contribution of this paper is consideration of a second cohesive zone, governed by the same interface law used for the loaded end of the stiffener, able to annihilate also the singularity at the free end. Of course, the analysis complicates of one order of magnitude with respect to [14], because the two cohesive zones are not independent, but they influence one another. The problem is solved using a Chebyshev expansion that provides a complicated system of equations for the unknown coefficients. The formulations of [32,14] become particular limit cases of this more general and complete approach.

47

Effective material separation is supposed to start when the relative slip between the adherents exceeds a certain threshold. If the stiffener is sufficiently long, we show that there is maximal reachable length of the cohesive zone at the loaded end, which we will demonstrate to be coincident with the EBL. Comparison of the three approaches (no cohesive zone, single cohesive-zone and double cohesive-zone) shows that, modulo proper calibration, all of them can predict the ultimate pull-out load in agreement with experiments, but only the cohesive models can accurately evaluate the EBL. More in particular, the state of stress at the free end of the reinforcement has a very little influence. In other words, the singularity at the free end of the stiffener carries a negligible part of the applied load. Therefore, the single cohesive-zone model results to be very accurate, but it avoids the noteworthy computational complications of the double cohesive-zone approach. The cohesive models furnish values of the EBL in good agreement with experimental data recorded by the technical literature, evidencing the importance of cohesion forces in the analysis of a bonded joint. Comparisons with the formulas proposed by technical standards [35] evidence that such formulations, based upon the assumption of rigid substrate, tend in general to overestimate the EBL. To this respect, the proposed approach represents a substantial improvement.

2. Adhesion of an elastic stiffener to an elastic substrate The contact problem of an elastic stiffener of finite length bonded to the boundary of an elastic semi-infinite plate and pulled at one end by a coaxial load is governed by a singular integral equation involving the unknown tangential contact forces [36]. If no slippage occurs between stiffener and plate, the theory of elasticity predicts that interface shear forces have a singularity at both ends of the stiffener. In order to remove this physical inconsistency, two cohesive zones are introduced at both edges of the reinforcement. The length of these zones depends upon the applied load, and can be found from condition that interface forces are finite in the whole bond, according to the same rationale followed by Barenblatt in the theory of cohesive cracks [37]. In Section 2.1 the resulting system of singular integral equations is solved through a Chebyshev expansion, while Sections 2.2 and 2.3 recover the solutions of one cohesive zone [14] and no cohesive zone (perfect bond) [32] as limit cases. 2.1. Double-cohesive-zone (DCZ) model Consider an elastic stiffener of length l, thickness ts and constant width bs, bonded to the boundary of an elastic semiinfinite plate in generalized plane stress of width bp (Fig. 1). At one end, the stiffener is loaded by a coaxial concentrated force P. As indicated in Fig. 1, let c1 and c2 denote the length of the cohesive

Fig. 1. A finite stiffener bonded to the boundary of a semi-infinite plate with cohesive zones at both ends.

48

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

where

λ is the rigidity parameter, which reads

2E b l λ ¼ p p b; π Es As

ð2:5Þ

having defined lb ¼ l  c1  c2 ; a ¼ ðlb þ 2c1 Þ=lb ; b ¼ ðlb þ 2c2 Þ=lb :

ð2:6Þ

Solution to Eq. (2.4) is subjected to the equilibrium condition for the stiffener Z 1 2ðP  qc c1  qc c2 Þ qðtÞ dt ¼ : ð2:7Þ lb 1 Fig. 2. A finite stiffener bonded to the boundary of a semi-infinite plate. Free body diagram of a portion of the stiffener.

zones at the left-hand-side and at the right-hand-side extremities of the stringer, respectively. A reference system ðξ; ηÞ is introduced with the origin on the left-hand-side edge, so that the loaded-end cohesive zone is 0 r ξ r c1 and the free-end cohesive zone is l  c2 r ξ r l, while the perfectly bonded part is the interval c1 r ξ r l  c2 . With reference to the free-body diagram of Fig. 2, let qc ðξÞ be the (cohesive) tangential force per unit length acting over the length c1 and c2, while qðξÞ the contact tangential force per unit length in the bonded portion. The stiffener strain can be obtained through Hooke0 s law, from the equilibrium of that part of the stiffener comprised between the origin and a section ξ ¼ x, in the form ɛ s ðxÞ ¼

  Z c1 Z x Ns ðxÞ 1 ¼ P qc ðξ Þ dξ  qðξÞ dξ ; Es As Es As c1 0

ð2:1Þ

"Z 0

c1

qc ðξÞ dξ þ ξx

Z

l  c2 c1

qðξÞ dξ þ ξx

Z

l l  c2

# qc ðξ Þ dξ ; ξx ð2:2Þ

where Ep is the elastic modulus of the plate and bp its thickness. One obtains the singular integral equation that solves the problem by imposing that strains are equal over the perfectly bonded interval. In the simplest case one may assume that the cohesive forces are constant, i.e., qc ðξÞ ¼ const: ¼ qc . In general, it is agreed that the cohesive-forces vs. relative-slip constitutive equation can be represented by a trilinear law of the type later on represented in Fig. 4, but this assumption would lead to formidable complications. However, we will demonstrate that even the simplest (stepwise) constitutive law here adopted is capable of reproducing, with excellent agreement, the experimental results. The relationship between the trilinear law and the proposed stepwise approximation will be discussed in Section 3.1. Consequently, by equating (2.1) and (2.2) and introducing the dimensionless coordinate t in such a way that the completely bonded zone is the interval ½ 1; 1, that is t¼2

ðξ  c 1 Þ ðl  c1  c2 Þ ðt þ 1Þ þ c1 :  1⟺ξ ¼ ðl  c1  c2 Þ 2

one finds

  Z      t 0 þ 1   þ lnt 0  b þ qc ln t  1 t þ a 0

0

qðtÞ ¼

n 2Q pffiffiffiffiffiffiffiffiffiffiffiffi ∑ X s T s ðtÞ; 2s¼0 π lb 1  t

ð2:8Þ

where Ts(t) are the Chebyshev polynomials of the first kind [36], Xs are constants to be determined and, for simplicity of notation, we have set Q ¼ P  qc c1 qc c2 . Observe that there is a square-root singularity in the solution at both ends of the reinforcement, which is typical of most contact problems in linear elasticity theory. Following Bubnov0 s method [36], with a procedure similar to that of [32,14], substitution of (2.8) into conditions (2.7) and (2.4) allows one to obtain, with the orthogonality conditions for Chebyshev polynomials of the first kind (see Appendix of [14]), X 0 ¼ 1; and the system of linear equations

where Es is the elastic modulus of the stiffener and As its crosssectional area. Besides, on the boundary of the semi-infinite plate, the strain in the interval ½0; l due to the cohesive stress and to the tangential contact stress may be written as [36] 2 ɛ p ðxÞ ¼  π Ep bp

An approximate solution for (2.4) can be obtained by expanding the contact force q in terms of a series of Chebyshev polynomials2 [36,38,39], which are orthogonal in the interval ½  1; 1, i.e.,

ð2:3Þ

  Z t0 qðtÞ π 2 λ 2ðP  qc c1 Þ dt ¼   qðtÞ dt ; t  t l 8 0 b 1 1

Xj þ

λ n λ πλqc c2 ql ∑ a X s ¼  bj  c  c bd 4 s ¼ 1 js 4 4Q j π Q j;

for j ¼ 1; 2; …; n;

ð2:9Þ

here Z 1 U j  1 ðtÞU s  1 ðtÞð1  t 2 Þ dt; ajs ¼ 1=s 1 Z 1 pffiffiffiffiffiffiffiffiffiffiffiffi U j  1 ðtÞ 1  t 2 arccos t dt; bj ¼ Z

1 1

pffiffiffiffiffiffiffiffiffiffiffiffi 1  t 2 dt; 1   Z 1 pffiffiffiffiffiffiffiffiffiffiffiffi t þ 1    þ lnt  b dt; U j  1 ðtÞ 1  t 2 ln dj ¼   t þa t  1 1

cj ¼

U j  1 ðtÞ

Uj(t) being the Chebyshev polynomials of the second kind [36]. These expressions can be evaluated with the change of variable t ¼ cos φ, so that U j  1 ðtðφÞÞ ¼ sin jφ= sin φ. In conclusion, one finds 8 4j > < ajs ¼  for even j s; ½ðjþ sÞ2  1½ðj  sÞ2  1 > :a ¼0 for odd j s; js 8 2 > π > > b1 ¼ ; > > 4 > < 4j for even j; > bj ¼  2 > > ðj  1Þ2 > > > : bj ¼ 0 for odd j a1; 8 π < c1 ¼ ; 2 :c ¼0 for j ¼ 2; 3; …; n; j

1

ð2:4Þ

2 The main properties of Chebyshev polynomials are reported in the Appendix of [14].

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

and 8 ( ) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi π > 2 > 2  1  aÞ  lnja þ a2  1j  ½bð b2  1  bÞ  lnjb þ > d1 ¼ ½að a b  1 j ; > > 2 > > > > " # pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi > > j j þ 1 j  1 2 > ð a2  1  aÞ > < dj ¼ π 2ð  1Þ  ð a  1  aÞ þ 2 j2  1 jþ1 j1 > pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi " # > > 2 2 > j jþ1 > π 2ð  1Þ ð b  1  bÞ ð b  1  bÞj  1 > > þ ð  1Þj 2 ; þ  > > > 2 jþ1 j1 j 1 > > > > : for j ¼ 2; 3; …; n:

The parameters c1 and c2 add to the other n unknowns Xs, so that there are n þ2 unknowns for the n equations (2.9). Other two conditions need to be introduced, and these are accomplished by imposing that in ξ ¼ c1 ðt ¼  1Þ and in ξ ¼ l  c2 ðt ¼ 1Þ the shear stress must be finite, or, equivalently, that the mode II stress intensity factors KII are null. The resulting conditions become 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > qðξÞ 2π ðξ  c1 Þ ¼ 0; > < K II;load ¼ ξlim -c1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:10Þ > > : K II;free ¼ lim qðξÞ 2π ðξ  l þ c2 Þ ¼ 0;

49

Chebyshev polynomials of the second kind U j  1 , one finds 8 > < ajs ¼  > :a ¼0 js

4j ½ðj þsÞ2 1½ðj  sÞ2  1

8 > π2 > > ; > b1 ¼ > 4 > < 4j > bj ¼  2 > >  1Þ2 ðj > > > : bj ¼ 0

for even j s; for odd j s;

for even j; for odd j a1;

and 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi π > d1 ¼ ½1  a2 þ a a2  1  lnða þ a2  1Þ; > > 2" < # pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi j jþ1 2 ð a2  1  aÞj  1 > d ¼ π 2ð  1Þ  ð a  1  aÞ > þ j > 2 : 2 j 1 jþ1 j1

for j ¼ 2; 3; …; n:

ξ-l  c2

where the subscripts “load” and “free” refer to the loaded end and the free end of the bonded part of the stiffener, respectively. Substitution of the contact forces (2.8) into (2.10) gives the expressions 8 n 2Q > > K > ¼ pffiffiffiffiffiffiffiffiffiffi ∑ X s ð  1Þs ; > < II;load 2π l b s ¼ 0 ð2:11Þ n 2Q > > > K II;free ¼ pffiffiffiffiffiffiffiffiffiffi ∑ X s ; > : 2π lb s ¼ 0 which reduce to the conditions 8 n s > > > < ∑ X s ð  1Þ ¼ 0; s¼0

It is evident how the second term on the right-hand side of Eq. (2.9) disappears, because it was associated with the cohesive length c2. The expression of the coefficient dj results substantially simplified. In this case, the parameter c adds to the other n unknowns Xs, so that there are n þ1 unknowns for the n equations (2.9). The condition to be introduced is the annihilation of the

ð2:12Þ

n > > > : ∑ X s ¼ 0; s¼0

under the requirement that, of course, lb 4 0. This is the adaptation to the contact problem of the approach originally proposed by Barenblatt [37] to eliminate the stress singularity predicted by the elasticity theory in an opening crack, as a consequence of cohesive forces acting at its tip. Conditions (2.12) allow one to evaluate the length of the zones over which tangential slippage can occur at the interface, provided that the cohesive stress qc is known. 2.2. Single-cohesive-zone (SCZ) model The model of [14] can be considered as a limit case of the DCZ approach when the cohesive zone at the free end c2 is null. Setting for simplicity c1 ¼ c (Fig. 3(b)), this single cohesive zone (SCZ) model is governed by the set of algebraic equations Xj þ

λ n λ ql ∑ a X s ¼  b j  c b dj 4 s ¼ 1 js 4 πQ

for j ¼ 1; 2; …; n;

where lb ¼ l  c1 ¼ l  c, Q ¼ P  qc c1 ¼ P  qc c and expression of (2.5). The coefficients of (2.13) are Z 1 U j  1 ðtÞU s  1 ðtÞð1  t 2 Þ dt; ajs ¼ 1=s Z bj ¼ Z dj ¼

1

1 1 1 1

U j  1 ðtÞ

ð2:13Þ

λ has the same

pffiffiffiffiffiffiffiffiffiffiffiffi 1  t 2 arccos t dt;

U j  1 ðtÞ

pffiffiffiffiffiffiffiffiffiffiffiffi t þ 1 dt: 1  t 2 ln t þ a

From Eq. (2.6), one has b ¼1 and a ¼ ðl þ cÞ=ðl  cÞ. Therefore, with the change of variable t ¼ cos φ and the representation for the

Fig. 3. A finite stiffener bonded to a semi-infinite plate. (a) No-cohesive zone (NCZ) model; (b) single cohesive zone (SCZ) model; and (c) double cohesive zone (DCZ) model.

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A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

mode II stress intensity factor KII in ξ ¼ c ðt ¼  1Þ, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K II;load ¼ limqðξÞ 2π ðξ  cÞ ¼ 0; ξ-c

ð2:14Þ

which reduces, after substitution of the contact stress (2.8) into (2.14) and simplification, to the first condition of Eq. (2.12) under the condition that, of course, lb ¼ l c 4 0. 2.3. No-cohesive-zone (NCZ) model When no cohesive zone exists at the interface between stiffener and substrate, i.e., c1 ¼ c2 ¼ 0, we have (Fig. 3(a)) Xj þ

λ

λ

n

∑ a X s ¼  bj 4 s ¼ 1 js 4

for j ¼ 1; 2; …; n

ð2:15Þ

which is the solution of an elastic stiffener bonded to the boundary of an elastic semi-infinite plate [36,32]. Hereinafter, this will be referred to as the no-cohesive zone (NCZ) model. In this case the coefficients cj and dj of Eqs. (2.13) and (2.15) disappear, because there is no more dependence by the cohesive zone size, while the coefficients ajs and bj have the same expression of Sections 2.1 and 2.2. In this case, the rigidity parameter λ involves the entire bond length, i.e., lb ¼ l, and reads

λ¼

2 E p bp l

π Es As

:

Fig. 4. Typical experimentally measured shear-stress vs. slip constitutive law at the interface. Trilinear and step-wise approximation.

the substrate whenever the release of elastic energy becomes equal to the work consumed to fracture the interface. Assume that the stress is constant on the width bs of the interface. Remarkably, as discussed at length in [32], the energy release rate is associated with the stress intensity factor KII by an expression à là Irwin of the type

ð2:16Þ

The solution of (2.15) presents singularities at both ends of the reinforcement, which are typical of most contact problems in the linear theory of elasticity.

3. Theoretical prediction of the contact shear stress The coefficients of the system of equations given by (2.9) and (2.12) depend upon the elastic properties of the materials in contact. Once they are assigned, the model furnishes the shear contact stress at the interface between the stiffener and the substrate. Results obtainable with the three approaches DCZ, SCZ and NCZ of Section 2 are now compared one another and with experimental results. Although the model applies to every pair of materials, reference will now be made to the paradigmatic case of Fiber-Reinforced-Polymer (FRP) stiffeners bonded to concrete substrates. 3.1. Constitutive law for the cohesive interface Any adhesive junction is characterized by an interface constitutive law, correlating the shear bond-stress τ with the relative slip s of the two adherents through the adhesive. In general, the τ  s curve is evaluated by measuring experimentally the strain in the stiffener and the substrate, as done e.g. in [13]. A typical trend is of the type represented in Fig. 4: after a pseudo-linear branch up to the peak stress, a strain-softening phase follows that ends when the zero-stress level, associated with complete debonding, is reached. As suggested in recent technical standards [35], the τ  s interface law may be approximated by a trilateral (Fig. 4), formed by a linearly ascending branch up to peak stress τf, followed by a linear softening phase approaching s ¼ sf where τ ¼ 0 and, finally, a zerostress plateau. The fracture energy per unit-surface is Gf ¼ 12τ f sf and, in general, such a value is made to coincide with the integral of the τ vs. s experimental curve. This equivalence allows one to evaluate the limit slip sf once the peak load τf is known. In the simplest NCZ approach of Section 2.3, the only relevant parameter is the material fracture energy. Debonding is regulated by Griffith energetic balance, i.e., the stiffener detaches from

bs Gf ¼

K 2II : 2Ep bp

ð3:1Þ

Consequently, the debonding condition for the NCZ model is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K II Z bs τf sf Ep bp . The models SCZ and DCZ for cohesive debonding of Sections 2.2 and 2.1 have been derived under the hypothesis that the cohesive force per-unit-length qc ¼ τc bs is constant. To comply with this simplification, an equivalence may be established between the triangular and a step-wise constitutive law by imposing the same slip limit sf and the same delamination fracture energy Gf. This is obviously achieved when τc ¼ 12τf . We will show that, despite this simplification, the obtainable results are in excellent agreement with experimental measurements. 3.2. Interfacial shear stress The interfacial shear stress associated with the NCZ, SCZ and DCZ models is now compared in an example that uses the material data from the experiments of [23]. Results for a bond length of l¼ 150 mm are represented in Fig. 5, which shows the normalized interfacial shear force distribution q=qc at various stages of loading. Results are shown for increasing complexity of the models, i.e., from the NCZ model to the DCZ model. (i) The NCZ model. It is evident in Fig. 5(a) the presence of singularities at both ends of the reinforcement. The stress rapidly diminishes going towards the free end of the stiffener: it is almost null for most part of the bond length, except for a very small zone near the free end where another singularity occurs. The results that can be obtained with this model have been discussed at length in [32]. (ii) The SCZ model. For each value of the applied load, the length c of the cohesive zone can be calculated with the equations of Section 2.2. In the normalized interfacial-force graph of Fig. 5(b) it is evident that at the loaded end the shear distribution tends to the maximum allowable stress qc, i.e., qðξÞ lim ¼ 1: ξ-c qc

ð3:2Þ

This means that the shear stress at the frontier between the cohesive and the perfectly bonded zones is continuous. At the free edge of the stiffener, the solution still presents the singularity

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

51

increasing the load, the cohesive portion is the one that gives by far the most important contribution (P cohes =P C 1Þ. As the load is increased, the cohesive zone reaches a maximal length cu after which debonding starts and, as shown in [14], the cohesive zone translates towards the free end of the stiffener maintaining its length practically unaltered. This phase is associated with a plateau in the force vs. displacement diagram. During this process, the greatest majority of the load applied to the stiffener is always carried by the cohesive forces qc in the cohesive zone [14]. (iii) The DCZ model. Once the value of the cohesive lengths associated with a given load and the respective constants Xs is evaluated, the interface force per unit length q can be calculated with the expression (2.8). Results are represented in Fig. 5 (c) where, in particular, the values of the cohesive lengths c1 and c2, as defined in Fig. 3(c), have been indicated near the curve corresponding to each load. Also for this case, at the frontier between the bonded part and the cohesive portions the stress results to be continuous, i.e., lim

qðξÞ

ξ-c1 qc

Fig. 5. Interfacial shear force distribution for different values of the applied load. Same materials of [23], with initial bond length l ¼ 150 mm. (a) No-cohesive zone (NCZ) model; (b) single cohesive zone (SCZ) model; and (c) double cohesive zone (DCZ) model.

predicted by the theory of elasticity. Debonding starts when the relative slip between stiffener and substrate exceeds the limit value sf. In this particular example, the applied load is always lower than the debonding limit. The post-critical analysis has been performed in [14], to which the reader is referred to for details. What is important to notice for this case is that most of the applied load is equilibrated by the tangential force acting in this cohesive portion; in particular, the stress singularity at the free end does not play a significant role in the equilibrium of the stiffener. To illustrate, Fig. 6 shows the load fraction carried by the cohesive part (P cohes =P) and by the remaining part of the bond length (P bond =P) as a function of the applied load P. In the same picture the value of the cohesive zone length c corresponding to each load-level is indicated at the top border. It is clear that

¼ 1;

lim

qðξÞ

ξ-l  c2 qc

¼ 1:

ð3:3Þ

In any case, the length of the cohesive zone at the free end of the stiffener is much smaller than that at the loaded end. Comparing the values of c1 with the corresponding values of c for the SCZ model, also highlighted in Fig. 5(b), it is clear that at the loaded cohesive zone the SCZ and the DCZ model give in practice identical results. The shear stress profile at the interface does not appreciably change if the singularity at the free end is removed, apart of course in a neighborhood of the free end. In any case, that part of the applied load that is equilibrated by the second singularity at the free end is not significant. To make this clearer, Fig. 7 represents the fraction of the axial load equilibrated by that portion of bond length laying on the lefthand side of the generic abscissa ξ. The results obtained with the three approaches for P ¼15 kN are juxtaposed: the NCZ model (continuous line), the SCZ model (dashed line) and the DCZ model (dash-dotted line). For what the NCZ model is concerned, notice that a bonded length of 20 mm is sufficient to balance 97% of the axial load: in rough terms, most of the load is balanced by the singularity at the loaded edge. Instead, the SCZ and the DCZ curve evidence that a bond length higher than  120 mm is necessary to balance the relevant part of the applied load. The curves obtained with both the SCZ and DCZ models almost overlap, confirming that the part of load carried out by the singularity at the free end is negligible. Fig. 7(b) shows a magnification of Fig. 7(a) in a neighborhood of the free end of the bond length. It is again evident that the main discrepancy between the dashed curve (SCZ model) and the dash-dotted curve (DCZ model) is in a very small part of the bond length, and that the difference between the results obtainable with the two approaches is not substantial. In conclusion, the NCZ model predicts a rapid decay of the interface shear stress because most of the applied load is carried in a small neighborhood of the stress singularity at the loaded end of the stiffener. In the cohesive interface models, most of the load is carried in the yielded portion of the bond length in proximity of the loaded end. The SCZ model still predicts a singularity at the free end of the stiffeners, but this does not furnish a significant contribution. Indeed, the length of the cohesive portion in proximity of the loaded end of the stiffeners, which is the most important, remains substantially the same in both the SCZ and the DCZ model. Consequently, if one is mainly interested in the engineering evaluation of the mechanism of adhesion, reference could be made to the SCZ model, which requires a computational effort much lower than the DCZ model.

52

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57 c [mm] 6.19

1

14.18

22.43

30.80

39.22

47.69

56.18

5

6

7

64.69

73.22

81.76

90.31

98.87

8

9

10

11

12

107.44 116.02 124.62

0.9 0.8

DEBONDING LIMIT

Pbond/P, Pcohes/P

0.7 0.6 Cohesive portion Bonded portion

0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

13

14

15

P [kN]

Fig. 6. Load fraction balanced by the interface stresses acting on the cohesive portion and on the perfectly bonded portion as a function of the applied load. Results from the SCZ model (bond length l ¼150 mm and mechanical parameters of [23]).

1

1 0.99

0.6

P/Pmax

P/Pmax

0.8

0.4 NCZ model SCZ model DCZ model

0.2

0.98

0

0.96 0

20

40

60

80

100

120

140

NCZ model SCZ model DCZ model

0.97

160

125

130

135

140

145

150

ξ [mm]

ξ [mm]

Fig. 7. Fraction of the axial load balanced by the interfacial shear force acting in the portion 0 r x r ξ. Mechanical parameters of [23] (l ¼ 150 mm and P max ¼ 15 kN). (a) Comparison between the NCZ, the SCZ and the DCZ model and (b) details of the portion comprised between ξ ¼ 125 mm and ξ ¼ 150 mm.

4. Effective bond length: comparison with experiments There is a general agreement that the adhesion strength (in pure mode II) of a stiffener on a substrate is characterized by an intrinsic length usually referred to as the effective bond length (EBL). This can be defined as the length necessary to transfer the load from the stiffener to the substrate. In fact, it has been experimentally verified that increasing the bond length beyond such a limit does not lead to any increase of load carrying capacity, confirming that only part of the bond is active. For this reason, the determination of this limit is of fundamental importance in the characterization of the joint performance.3 The aim of this section is to assess the capability of the three considered models to capture, besides the ultimate load, the value of the EBL. Such a value can be experimentally determined from pull-out tests on stiffeners with different bond lengths: by definition, the EBL is the bond length beyond which the ultimate load remains almost constant. Comparisons will be made between the analytical outputs and the results from relevant experimental campaigns recorded in the technical literature.

3 The response of a joint in terms of load–displacement curve strongly depends upon the bond length l. “Short” stiffeners, for which l o EBL, show a post-peak softening while “long” stiffeners, whenever l 4 EBL, are characterized by a plateau, usually followed by a snapback phase [13,23,14]. Therefore, the definition of a proper effective bond length makes possible the correct distinction between “short” and “long” stiffeners.

4.1. Assessment of the constitutive properties of the interface from experiments Several experimental results for FRP reinforced concrete will be considered now. With reference to [13,23,40,25,41,15], Table 1 reports the specimen properties and the parameters τf and sf that are associated with the trilinear constitute interface-law τ  s of Fig. 4. Recall that, following the equivalence established in Section 3.1, in the cohesive models here considered a step-wise approximation of the interface-law will be used, with τc ¼ τf =2. Since most of the times the values of τf and sf are not explicitly provided in the technical references, it is necessary to describe how they can be derived from generic experimental results. When only the characteristic compressive strength of concrete fck is known [42], one can evaluate τf through an expression borrowed from technical standards [35] of the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffi

τf ¼ 0:64κ b f ck f ctm ;

ð4:1Þ

qffiffiffiffiffiffiffiffiffiffiffiffi 3 where f ctm ¼ 0:30 ðf ck Þ2 , with fck expressed in MPa, is the value of the paverage tensile strength of concrete [42], while ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ b ¼ ð2  bs =bp Þ=ð1 þ bs =400 ½mmÞ Z 1. The values in Table 1 that have been obtained with this procedure have been evidenced by an asterisk. If the fracture energy per unit area Gf is known, then one readily has sf ¼ 2Gf =τf . When Gf is not explicitly given, it can be

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

53

Table 1 Mechanical properties of materials used in experimental campaigns and parameters of the interface law. Testa

Concreteb

Ali Ahmad et al. [13] Carrara et al. [23]n Chajes et al. [40]n Mazzotti et al. [25] Taljsten [41]n Yuan et al. [15]

Interface lawc

FRP

Ep (MPa)

tp (mm)

bp (mm)

Es (MPa)

ts (mm)

bs (mm)

τf (MPa)

sf (mm)

33 230 28 700 34 411 30 700 35 000 28 600

125 90 152.4 200 200 150

125 150 152.4 150 200 150

230 000 168 500 108 478 195 200 170 000 256 000

0.167 1.3 1.016 1.2 1.25 0.165

46 50 25.4 50 50 25

7.07 7.71 8.78 9.14 9.04 7.20

0.230 0.150 0.234 0.0971 0.154 0.160

Experimental tests for which the interface-law parameters are not explicitly provided are evidenced by an asterisk (n). When the literature provides the cylindrical strength fck only, then, as suggested in technical standards [42], Ep is calculated through Ep ¼ 22 000 ðf cm =10Þ0:3 MPa, being f cm ¼ f ck þ 8 MPa, fck in MPa. c When the literature does not provide the value for the peak stress τf, then expression (4.1) from the Italian Standard [35] has been used. a

b

approximated through the simple expression Gf ¼

ðP max;e Þ 2

2

2bs Es t s

;

ð4:2Þ

where P max;e is the experimentally measured peak load4 and ts is the thickness of the stiffener. Such an expression, also suggested by standards [35], neglects the elasticity of the substrate [43] and results quite accurate for FRP-reinforced concrete. However, for the proper evaluation of the shear interface forces, the elasticity of the substrate cannot in general be neglected. Let us then discuss the results that can be obtained with the various formulations just presented. For what Gf is concerned, the values calculated in the experiment by evaluating the work consumed during the fracture process (integration of the load vs. displacement curve) may differ from the values obtained through (4.2), but the discrepancy is in general very small. This parameter is the only one that needs to be considered in the NCZ approach: the ultimate load can be calculated through the evaluation of the stress intensity factor as per (3.1). If one calculates from the experimentally determined value of the ultimate load the debonding surface energy through (4.2), which derives from a Griffith-like energetic balance where the elasticity of the substrate is neglected, and afterwards re-calculate the ultimate load through (3.1), which considers the elasticity of the substrate, the results that are obtained are in practice the same. This confirms that, at least for concrete, the elasticity of the substrate does not give a substantial contribution for what the evaluation of Gf is concerned. In general we have found that evaluating Gf through (4.2) provides a slightly better approximation than the integration of the experimental load vs. displacement curve (when this is provided in the technical reference), which is usually subjected to measurement errors. The parameters τf and sf are of importance for the SCZ and the DCZ model. In order to understand how they may affect the results, reference is made to the tests of [13,40], where the sophisticated experimental apparatus allowed a precise measurement of the constitutive interface law. Fig. 8(a) refers to the tests of [13] and shows Pu as a function of the virgin bond length l. The points indicated with dots refer to data obtained by the same authors in [44] with very accurate numerical experiments that took into account the exact, experimentally measured, interface-law. The graph drawn with a continuous line on the left-hand-side picture refers to the results obtainable with the SCZ model by considering τf ¼ 5:03 MPa, sf ¼ 0.23 mm, i.e., the average peak stress and the fracture slip limit of the interface-law that was experimentally measured in [13]. 4 The maximum axial loads derived from experiments are recorded later on in Table 2.

Since this graph does not exactly match with experiments, we attempted at varying the fracture slip sf while keeping unchanged the maximum shear stress τf. By considering the average experimentally measured value Gf ¼ 0.735 MPa mm [13], one obtains sf ¼ 2Gf =τf ¼ 0:292 mm. The corresponding graph, which is indicated by the dashed curve on the left-hand-side graph of Fig. 8(a), still underestimates the ultimate load. But it is also possible to evaluate the fracture energy from (4.2): taking P max;e ¼ 11:5 kN as the average experimental value of [13], one obtains Gf ¼0.812 MPa mm and, leaving unchanged τf ¼ 5:03 MPa, the value sf ¼0.3235 mm. The curve obtained in this way is the one indicated by a dotted line on the left-hand side of Fig. 8(a). This shows excellent results for what the evaluation of the maximum load is concerned, but results are still inaccurate for short bond lengths (l o 100 mmÞ. Because of this discrepancy, a further elaboration has been made assuming that now sf ¼ 0.23 mm is fixed and by changing τf. The graphs on the right-hand side picture of Fig. 8(a) show, respectively, again the curve obtained with the experimentally measured values τf ¼ 5:03 MPa and sf ¼0.23 mm (continuous line); the curve corresponding to Gf ¼0.735 MPa mm and τf ¼ 2Gf =sf ¼ 6:39 MPa (dashed line); the curve associated with Gf ¼ 0.812 MPa mm from Eq. (4.2) and the corresponding τf ¼ 7:07 MPa (dotted line). It is clear that it is the dotted line that approximates the best experiments. A similar procedure has been followed for the experimental data of Chajes et al. [40], where the authors did not directly provide the parameters of the interface law. At first, an attempt has been made to use the method by Ferracuti et al. [24], who proposed a procedure to derive a non-linear mode II interface law starting from experimental data. With a calibration procedure, they obtained τf ¼ 6:64 MPa and sf ¼0.475 mm for the experiments of [40]. The corresponding curve, which is shown with a continuous line in Fig. 8(b), is not accurate. In a second attempt, the expression (4.2) for Gf has been calculated using the maximum experimental load of [40]: setting τf ¼ 8:78 MPa as per (4.1), one finds sf ¼ 0.234 mm. The results, drawn with a dashed line in the same picture, show a very good agreement with the experimental data. In conclusion, the best approximations are usually achieved when Gf is estimated through (4.2) from the maximum load obtained in pull-out experiments. This quantity defines the product τ f  sf . The best way to find the relative values of these parameters is through a calibration of the model on the basis of simple experimental campaigns, where the pull-out load is measured for various values of the bond length (short and long stiffeners). This approaches by-passes all the technical difficulties of a sophisticated experimental apparatus that always becomes necessary to evaluate the constitutive interface-law and, moreover, all the uncertainties of such a complicated measure.

54

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

12 Critical Load, Pu [kN]

Critical Load, Pu [kN]

12 10 τ =5.03 MPa, s =0.323 mm

8

τ =5.03 MPa, s =0.292 mm τ =5.03 MPa, s =0.230 mm

6

Numerical Results Results with original data Results with s from experimental G

4 2

Results with s from evaluated G

10 τ =7.07 MPa, s =0.230 mm τ =6.39 MPa, s =0.230 mm τ =5.03 MPa, s =0.230 mm

8 6

Numerical Results Results with original data Results with τ from experimental G

4 2

Results with τ from evaluated G

0

0 0

50

100

150

200

250

300

0

50

100

150

200

250

300

Bond Length, l [mm]

Bond Length, l [mm]

Critical Load, Pu [kN]

15

10

τ =6.64 MPa, s =0.475 mm τ =8.78 MPa, s =0.234 mm Experimental Results Results with data from Ferracuti

5

Results with τ from CNR-DT200 and s from evaluated G

0

0

50

100 150 200 Bond Length, l [mm]

250

300

Fig. 8. Maximum applied load as a function of the bond length. Influence of the parameters that define the interface law. (a) Tests of Ali-Ahmad et al. [13] and (b) tests of Chajes et al. [40].

The values of τf and sf that are indicated in Table 1 have obtained following this procedure.

The values of the critical load as a function of the bond length l obtained for the mechanical parameters of [13,23,40,25,41,15] are indicated by a continuous line in Fig. 9 as a function of the bond length. Each graph is compared with the experimental data, here indicated by dots,5 and with the results from the SCZ model (dashed line) and the DCZ model (dotted line), whose derivation is done in the sequel. For what the NCZ model is concerned, notice that the values of the bond length beyond which there is no substantial increase of the ultimate load, i.e., the EBL, are of the order of few millimeters. This is due to the rapid decrease of the interfacial shear stress

beyond the singularity, but the result is not corroborated by experiments. In other words, the NCZ model underestimates the EBL. The values of the ultimate load calculated with the mechanical parameters of the experimental campaigns of [13,23,40,25,41,15] are summarized in Table 2, together with the results of the SCZ model, the DCZ model and experimental data recorded in the literature. More precisely, in the “experimental data” columns, the mean experimental value on the peak load has been indicated with P max;e , while the values of the EBL evaluated from the experimental data6 have been referred to as le;e . (ii) The SCZ model. The results of the SCZ model are shown in Fig. 9 by dashed lines. Comparison with the experimental data [13,23,40,25,41,15] evidences the good agreement with the prediction of the model for what the ultimate load is concerned. Notice that Pu increases with the bond length l until the limit of the EBL, which is also well captured by the model. The values of the ultimate load and of the EBL so calculated are also summarized in Table 2. As largely discussed in [14], since the greatest part by far of the applied load is equilibrated by the cohesive shear forces acting in the yielded portion of the adhesive, in this model the EBL may be associated with the maximal length cu of the cohesive zone attained in long stiffeners. Increasing the bond length beyond this limit does not increase the load bearing capacity of the joint. The value of cu, calculated through the model, is also evidenced in Fig. 9 with a circular marker and denoted by cu;SCZ . It matches very well with the limit bond length according to the standard definition. The EBL could thus be evaluated through a strain-driven pullout test on long stiffeners. Measuring the relative displacement of the loaded end, debonding starts when the relative slip of the

5 In the case of [13,15], the dots refer to very careful numerical experiments that consider the full constitutive interface-law obtained through sophisticated experimental apparatus.

6 The effective bond length is here defined as that limit of the bond length beyond which no apparent increase of ultimate load is experimentally observed.

4.2. Results from the various models The results obtainable with the NCZ, SCZ and DCZ models are now compared with the experiments of [13,23,40,25,41,15], using the material parameters of Table 1. (i) The NCZ model. As in Section 3.1, let Gf represent the fracture energy per unit surface and Gf bs the fracture energy per unit length of the stiffener. Then, from the expression (3.1) for the strain energy release rate associated with an infinitesimal crack growth, one finds [32]  n 2 K 2II P2 ∑ X s ð  1Þs ; Gf bs ¼ ¼ ð4:3Þ 2Ep bp π Ep bp l s ¼ 0 so that the critical value Pu of P reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π Ep bp l P u ¼ Gf bs n : ½∑s ¼ 0 X s ð 1Þs 2

ð4:4Þ

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

55

10

Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c -P (SCZ Model)

8 6

u

u

cu-P u (DCZ Model)

4 2

c

c

=69.02 mm

Critical load, Pu [kN]

Critical load, Pu [kN]

12 15 Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c P (SCZ Model)

10

u

5

=69.029 mm

c

0

-

u

cu-P u (DCZ Model) =125.4 mm

=125.45 mm

c

0 0

50

100

150

200

0

50

100

150

200

250

300

Bond Length, l [mm]

Bond Length, l [mm]

14

25

10

6

Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c P (SCZ Model)

4

cu-P u (DCZ Model)

8

u

2

c

u

=106.319 mm

c

=106.3087 mm

-

Critical load, Pu [kN]

Critical load, Pu [kN]

12 20 Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c -P (SCZ Model)

15 10

u

c

=92.48 mm c

=93.07 mm

0

0 0

50

100

150

0

200

50

100

Bond Length, l [mm]

35

6

30

5

25

Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c P (SCZ Model)

20 15

-u cu-P u (DCZ Model) u

10 5

c

=115.29 mm c

3

100

150

200

250

u

2

0 50

200

250

300

350

400

Analytical results (NCZ model) Analytical results (SCZ model) Analytical results (DCZ model) Experimental results c -P (SCZ Model)

4

300

350

400

u

cu-P u (DCZ Model)

1

=115.269 mm

0 0

150

Bond Length, l [mm]

Critical load, Pu [kN]

Critical load, Pu [kN]

u

cu-P u (DCZ Model)

5

c

0

=60.05 mm c

50

=60.05 mm

100

150

200

250

Bond Length, l [mm]

Bond Length, l [mm]

Fig. 9. Ultimate load Pu as a function of the initial bond length l. Predictions of the NCZ, SCZ and DCZ models and comparisons with experimental results. (a) Tests of AliAhmad et al. [13]; (b) tests of Carrara et al. [23]; (c) tests of Chajes et al. [40]; (d) tests of Mazzotti et al. [25]; (e) tests of Taljsten [41]; and (f) tests of Yuan et al. [15].

Table 2 Results from NCZ, SCZ and DCZ models and comparison with the values predicted by Italian standards [35] and experimental or numerical tests. Test

NCZ model P u (kN)

Ali Ahmad et al. [13] Carrara et al. [23] Chajes et al. [40] Mazzotti et al. [25] Taljsten [41] Yuan et al. [15]

11.51 15.10 12.10 22.80 27.22 5.52

SCZ model le (mm) – – – – – –

DCZ model

CNR-DT200

Experiments

P u;SCZ (kN)

c u;SCZ (mm)

P u;DCZ (kN)

c u;DCZ (mm)

P max (kN)

le (mm)

P max;e (kN)

le;e (mm)

11.488 15.095 12.09 22.79 27.18 5.49

69.02 125.40 106.30 92.48 115.29 60.05

11.491 15.097 12.095 22.78 27.19 5.49

69.03 125.45 106.32 93.02 115.27 60.05

11.50 15.09 12.09 22.79 27.20 5.51

80.08 185.00 129.29 175.33 175.48 91.89

11.50 15.11 12.09 22.65 29.83 5.53

70–90 120–150  100.00  100.00 100–150  60.00

reference point reaches the fracture slip sf predicted by the interfacial constitutive law (see Fig. 4). At this point, the maximum load that can be carried by the FRP stringer is attained. The maximal cohesive zone cu at the beginning of the debonding gives a physical characterization of the EBL. Recall that, as demonstrated in [14], the value of the ultimate cohesive length cu does not change as debonding proceeds, but simply the cohesive zone moves towards the free end of the stiffener, leaving its length unaltered, so that the ultimate load Pu remains almost constant. This confirms that increasing the bond

length over the EBL limit does not increase the anchorage strength of the joint. However, it certainly improves the joint ductility! (iii) The DCZ model. There are noteworthy analogies with the SCZ model. A strain driven test can be conducted in order to measure the relative slip of the loaded end of the stiffener for each stage of loading. When the relative slip of the reference point reaches the fracture slip sf provided by the interface constitutive law (see Fig. 4), debonding starts and correspondingly the maximum value of the cohesive zone length c1 is attained, while the cohesive length c2 undergoes inappreciable changes.

56

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

The graph showing the ultimate load Pu as a function of l is drawn in Fig. 9 with a dotted line. The limit length of the cohesive zone c1 is here indicated with cu;DCZ and evidenced by a triangular marker. Notice that in all the considered cases, the graphs of the SCZ model and of the DCZ model overlap in practice, giving almost the same value of ultimate load and effective bond length, confirming that the influence of the singularity at the free end of the stiffener in the SCZ model is negligible to this respect. The numerical values of the outputs are reported in Table 2, where the accuracy of both cohesive models SCZ and DCZ is even more evident. The main conclusion from this discussion is that the SCZ model is the most convenient engineering approach for the characterization of the joint response, since it involves a reasonable computational effort if compared to that required by the DCZ model. Finally, it may be useful to compare the values of the EBL just obtained with those obtainable with formulas suggested by technical standards. To this respect, the recent Italian instructions CNR-DT200 [35], which appear to be one of the most modern references, suggest to take EBL ¼ le , with sffiffiffiffiffiffiffiffiffiffiffi Es t s ; ð4:5Þ le ¼ 2f ctm where fctm is the mean tensile strength of concrete [42]. The main underlying assumption for (4.5) is a trilinear shear-stress vs. slip model, of the type represented in Fig. 4, together with the hypothesis of rigid substrate. The standard also suggests to evaluate the ultimate load Pmax through an energetic balance, leading to an expression of the same type of (4.2). Using the data of Table 1, the results from the CNR-DT200 are also reported in Table 2. Notice that the cohesive models, which are not based upon an energetic balance but simply rely upon the calculation of the state of stress with the classical theory of elasticity, give values which are in excellent agreement with the standards for what the ultimate load is concerned. On the other hand, the expression (4.5) seems to excessively overestimate the EBL with respect to the experimental data, which are instead very well captured by the proposed cohesive models.

5. Conclusions The contact problem between an elastic stiffener and an elastic half-space has been considered to assess the interfacial conditions of detachment in pure mode II of the two adherents. Contrarily the traditional approaches that neglect the important role played by elastic deformation of the substrate, here this aspect has been emphasized. This paper extends and develops previous work by the authors. In [32] the adherents were supposed in perfect contact; in this nocohesive-zone (NCZ) approach, stress singularities are predicted at both extremities of the stiffener. In [14], a single-cohesive-zone (SCZ) was introduced at the loaded-end of the stiffener, allowing for relative slip under constant cohesive forces so as to annihilate the corresponding stress singularity. Here, we have considered a more complicated problem that accounts for a second cohesive region at the free end of the stiffener, so as to mitigate also the second singularity. The solutions of this double-cohesive-zone (DCZ) problem, obtained through a Chebyshev expansion, prescribes a continuous interfacial shear stress that never exceeds the cohesive limit. The potentialities of the three approaches have been discussed. In the NCZ model, the interfacial shear stress shows an extremely rapid decrease from the maximum concentration near the loaded end. The SCZ approach, just assuming a very simple step-wise

interface law, predicts the formation of a cohesive zone that produces a more gradual decay of the contact stress in agreement with experimental results. The DCZ model prescribes two cohesive zones at the edges of the reinforcement, but it has been shown that the zone at the free end does not play a significant role. The stress distribution practically coincides with that of the SCZ model, apart from a very small neighborhood of the free end where the singularity is present. A method has also been proposed to calibrate the parameters that determine the interface shear vs. slip constitutive law of the cohesive models on the basis of simple experimental campaigns. The interface fracture energy Gf can be estimated from the maximum pull-out forces through simple formulas proposed in technical standards [35]. Then, the cohesive parameters can be conveniently calibrated from a series of elementary pull-out tests on specimens with a sufficiently wide range of bond lengths. This is done by requiring the equivalence with the expected value of Gf and the best fitting with the experimental results, trying to capture in particular the limit value of the bond length beyond which no further increase of the pull-out load can be obtained. Indeed, such a limit value is usually referred to as the effective bond length (EBL) of the reinforcement. The NCZ model underestimates by far the experimentally measured EBL, because the shear stress at the interface decays too rapidly. The SCZ and the DCZ approach both give excellent predictions of the EBL, because their shear stress distribution is almost the same except in a small neighborhood of the free-end. As explained in detail in [14], the SCZ approach allows a complete description of the post-critical response of bonded joints, after delamination has started. A maximal length cu of the cohesive zone is reached when the relative slip at the loaded end reaches the fracture limit sf, representing a key parameter of the model. Debonding initiates at this stage at a critical value Pu of the applied load. The length cu does not change appreciably but simply translates as delamination propagates along the interface, until it reaches the opposite free end. Since the resultant of the cohesive forces only is sufficient to equilibrate almost the whole applied load P, this remains almost constant and equal to Pu during the delamination process. Therefore, the length cu gives a physical characterization of the EBL, i.e., the length necessary to transfer the load from the stiffener to the substrate. Obviously, increasing the bond length beyond its effective limit does not increase the load bearing capacity, although it increases the ductility of the reinforcement. The length of the ultimate cohesive zone cu predicted by the DCZ models practically coincides with that of the SCZ model because the length of the second cohesive zone is usually very small, as small is the resultant of the shear stress at the freeend singularity in the SCZ model. The ultimate load Pu obtained through the three models matches very well not only with experimental results, but also with the relevant formulas proposed in technical standards [35]. For what the effective bond length is concerned, the NCZ is not accurate, but both the SCZ and DCZ models give predictions in good agreement with relevant tests recorded in the literature, here considered for the sake of comparison. On the other hand, it must be observed that the formulas suggested by standards [35] give excessively overestimated values, that in some cases are about twice the experimental results. To this respect, the SCZ and DCZ approaches seem to be an improvement of what proposed so far. In conclusion the SCZ model, which considers only one cohesive zone and the simplest stepwise interface constitutive law, is able to predict correct values of the critical pull-out load as well as of the EBL. The DCZ model is physically more accurate, but gives in practice identical results, though at a price of much more complicated calculations. In an engineering approach, thus the SCZ formulation appears to be the best compromise.

A. Franco, G. Royer-Carfagni / International Journal of Non-Linear Mechanics 60 (2014) 46–57

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