Ultimate behaviour of FRP wrapped sections under axial force and bending: Influence of stress–strain confinement model

Composites: Part B 54 (2013) 85–96

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Ultimate behaviour of FRP wrapped sections under axial force and bending: Influence of stress–strain confinement model Rosario Montuori ⇑, Vincenzo Piluso, Aldo Tisi University of Salerno, Department of Civil Engineering Fisciano (SA), Italy

a r t i c l e

i n f o

Article history: Received 23 October 2012 Accepted 15 April 2013 Available online 28 April 2013 Keywords: A. Carbon fibre C. Analytical modelling C. Computational modelling

a b s t r a c t In the present work, the influence of the adopted confined concrete constitutive law, among those available in the technical literature, on the flexural strength and curvature ductility of reinforced concrete sections strengthened by FRP (fibre reinforced polymer) wrapping is investigated. An important issue to be underlined is that the stress–strain relationship of confined concrete depends not only on the number of layers and on the type of FRP used for wrapping, but also on the size and the shape of the section. By using the main constitutive laws proposed in the technical literature to model the confined concrete behaviour, the moment–curvature diagrams have been evaluated for a significant number of study cases by means of a specifically developed computer program based on a refined fibre model. The results show that even if the different constitutive laws exhibit large differences in the resulting stress–strain behaviour, they lead to negligible differences in terms of flexural resistance, but to very significant differences in terms of curvature ductility. Therefore, the accurate evaluation of the ultimate strain seems of paramount importance compared to the whole stress–strain curve. In addition, the influence of pre-existing loads acting on the structure at the time of the strengthening intervention has been investigated showing that it affects the knee region of the moment–curvature relationship, while the ultimate flexural resistance remains almost unaffected. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In last years, among the different technological solutions for the retrofitting and the strengthening of reinforced concrete structural elements [17–23], the technique of confinement by means of FRP has gained more and more interest of researchers and practicians. Aiming to the retrofitting and strengthening of existing buildings, it is evident the need to provide designers with valid calculation models accounting for all the parameters affecting the ultimate behaviour of reinforced concrete members strengthened by means of FRP wraps or sheets. In particular, in case of reinforced concrete columns confined by FRP wraps, because of the availability of a lot of different stress–strain constitutive laws, proposed in the technical literature, the influence due to the possibility of using different material models needs to be clarified. In addition, it should be noted that for a given stress–strain constitutive law, the effectiveness of confinement is strongly affected not only by the characteristics and the amount of layers of the used FRP material, but also by the shape of the section. In particular, square and rectangular sections show a performance improvement, both in terms of strength and ductility, less than that occurring in the case of circular sec⇑ Corresponding author. Tel.: +39 089 963421; fax: +39 089 968764. E-mail address: [email protected] (R. Montuori). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.04.059

tions. This is due because in circular sections the whole area is laterally confined, while in case of square and rectangular sections, according to the well-known ‘‘arch effect’’, only a part of the section area is effectively confined [1,2]. In order to investigate the influence of the confined concrete stress–strain constitutive law on the ultimate flexural strength and ductility of reinforced concrete sections of columns confined by means of FRP wraps, a computer program, in Visual Basic programming language, able to work with a wide set of constitutive laws [2–12] and dealing with reinforced concrete sections having different geometry has been developed. This computer code allows the computation of the moment– curvature diagram, for a given value of the axial load, for all the constitutive laws whose main features and drawbacks have been already discussed in a previous work [1]. The developed fibre model accounts for the following issues to be faced for an accurate evaluation of the ultimate resistance of strengthened columns: the deformations resulting from the loads acting on the original preexisting section; the different behaviour of effectively confined concrete with respect to the unconfined one; the variation of the effectively confined concrete area as a consequence of the strengthening intervention; the variation of the r–e law for the effectively confined concrete considering the difference between the concrete effectively confined by FRP wraps only, the concrete

R. Montuori et al. / Composites: Part B 54 (2013) 85–96

effectively confined by existing hoops only and the concrete effectively confined by both FRP wraps and existing hoops; the possibility of buckling of longitudinal bars. Therefore, with reference to several sections representative of members belonging to existing buildings typically designed to withstand vertical loads only, designed before the advent of modern seismic regulations, the moment–curvature relationship has been evaluated and a wide comparison has been carried out in order to investigate the influence on flexural strength and ductility due to the FRP type, the number of FRP layers, the axial load and, in particular, the use of different constitutive laws which can be adopted to model the confined concrete behaviour. In addition, since the FRP wrapping is applied to a structural member not completely unloaded, also the influence of the loads acting on the structure before the strengthening intervention has been analysed. In particular, the initial strain occurring in concrete and steel before strengthening needs to be known in order to account for these pre-existing strains in the evaluation of the strengthened section. In addition, in order to evaluate the effectiveness of the strengthening intervention, it is also necessary to study the section in its initial configuration. In fact, the analysis of the pre-existing section, conducted by subdividing it in confined and unconfined concrete areas, provides the needed information to evaluate the increase of resistance and ductility that the strengthening intervention can provide. Moreover, the developed mechanical model also includes the possibility of buckling of reinforcing bars in compression, according to a model previously proposed [15]. The occurrence of buckling of longitudinal reinforcing bars can be important in evaluating the ultimate capacity of columns having few stirrups. In this work the proposed model is applied to typical cases of columns designed with outdated regulations in order to compare their flexural resistance and section ductility to those of the corresponding sections strengthened by means of FRP wrapping. 2. Concrete stress–strain constitutive laws Many researchers have developed and proposed different constitutive laws to model the behaviour of concrete confined by FRP wrapping. Some of them require an iterative procedure, some assume as collapse condition the attainment of the ultimate value of concrete axial strain while others refer to the ultimate strain of FRP; finally, some of them account only for the maximum value of the lateral confining pressure while others consider its whole development as a function of the concrete axial strain. In a previous work [1], thirteen constitutive laws have been extensively discussed pointing out their features and drawbacks, so that herein they are only recalled making reference to the original works (Table 1). The extreme variability deriving from the application of these constitutive laws is pointed out by the comparison provided in Fig. 1 with reference to one of the analysed sections (section 4B

Table 1 Stress–strain constitutive laws considered in the analysis. 1 = Mander, Priestley and Park for unconfined concrete [2] 2 = Mander, Priestley and Park [2] 3 = Lam and others [3]

4 = Saafi and others [4] 5 = Wang and Restrepo [5]

6 = Spoelstra and Monti modified [6] 7 = Lam and Teng [7,8] 8 = Harajli, Hantouche and Soudki [9] 9 = Realfonzo and Faella [10] 10 = CNR [11]

11 = Spoelstra and Monti [12] 12 = Iterative CNR [1,11] 13 = Spoelstra and Monti II [1,12]

80

3

70

Stresses (MPa)

86

9

2

11

60

10

6

12

50

4

40 30 8

20 7

10 0

5

13 1

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1

Strain Fig. 1. Constitutive laws compared for section 4B.

of Table 2). A more wide analysis and discussion is presented elsewhere [1]. It is useful to remember that the constitutive laws of concrete confined with FRP wrapping are influenced not only by number of layers and the type of material used, but also by the geometry of the section. Therefore, despite of the same number of layers and the same type of FRP used for the reinforcement of two different sections, two significantly different behaviours are obtained and, above all, the amount of the difference is strongly influenced by the adopted constitutive laws [1].

3. Adopted model for steel bars The current Italian code requires that the ratio fu/fy between the ultimate stress fu and the yield stress fy has to be greater than 1.25– 1.30 in case of steel bars with improved bond, depending on the type of steel while, in case of normal steel bars this ratio has to be greater than 1.50–1.55. Regulations of few decades ago provided acceptance limits of fu and fy leading to ratios fu/fy even greater than 1.80–2.00 [16]. This issue deserves to be considered in the analysis of existing reinforced concrete structures requiring the adoption of an appropriate model for the steel behaviour. A typical stress–strain curve of steel reinforcing bars is shown in Fig. 2, where esy is the yield strain, esh is the strain at the onset of strain-hardening (determined as esh = 10esy), esu is the ultimate strain (determined as esu = 100esy). The branch from the on-set of strain-hardening to the ultimate strain can be represented by a curve with the following expression:

rs

"    # fu fu esu  es 2 ¼ fy  1 fy fy esh  esu

ð1Þ

In addition, the possibility of buckling of longitudinal bars subjected to compression needs also to be modelled [13,14]. This issue can affect the results of the comparison between the flexural resistance of the original unstrenghtened section and the one of the section strengthened by means of FRP wrapping. In particular, with reference to the original unstrenghtened section, the possibility of buckling after the occurrence of concrete cover spalling has to be considered [15]. It means that the unconfined concrete constituting the concrete cover represents a lateral restraint against longitudinal bars and it is able to avoid buckling up to the achievement of the strain leading to the spalling of the unconfined concrete. This assumption provides the minimum effect in terms of resistance degradation due to buckling. In fact, if the unconfined concrete was assumed to be unable to laterally restrain the bars before the achievement of its spalling strain, the bars could prematurely buckle causing the spalling of the cover concrete before the occurrence of the corresponding strain limit.

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96 Table 2 Considered study cases. Section

Rck (MPa)

Size (b  h) (cm)

Concrete cover (cm)

Corner radius (cm)

Axial load (kN)

Steel yield Strength (MPa)

Steel ultimate Strength (MPa)

Bars

Stirrups diameter (mm)/ spacing (cm)

FRP type

Layer thickness (mm)

Young modulus (Mpa)

Ultimate strain (%)

Number of FRP layers

1A 1B 2A 2B 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I

15 15 15 15 20 20 20 20 20 20 20 20 20 20 20

30  30 30  30 30  40 30  40 30  50 30  50 30  60 30  60 30  60 30  60 30  60 30  60 30  60 30  60 30  60

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

350 350 450 450 600 600 800 800 800 800 350 450 600 1100 800

220 220 220 220 380 380 380 380 380 380 380 380 380 380 380

330 330 330 330 570 570 570 570 570 570 570 570 570 570 570

4£12 4£12 4£14 4£14 6£16 6£16 6£18 6£18 6£18 6£18 6£18 6£18 6£18 6£18 6£18

£6/15 £6/15 £6/20 £6/20 £6/20 £6/20 £6/25 £6/25 £6/25 £6/25 £6/25 £6/25 £6/25 £6/25 £6/25

CFRP CFRP CFRP CFRP CFRP CFRP CFRP CFRP AFRP GFRP CFRP CFRP CFRP CFRP CFRP

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5

231,000 231,000 231,000 231,000 231,000 231,000 231,000 231,000 120,000 80,000 231,000 231,000 231,000 231,000 231,000

1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 3.7 4.5 1.4 1.4 1.4 1.4 1.4

1 3 1 3 1 3 1 3 1 1 1 1 1 1 6

N¼

2Mp w

ð2Þ

where Mp and w are, respectively, the yielding moment and the lateral displacement of the bar. In addition, the relation between the axial displacement d and the rigid rotation h is given by:

d ¼ s  2ðs=2Þ cos h ¼ sð1  cos hÞ

ð3Þ

where s is the stirrup spacing. By means of Eq. (3), the lateral displacement w can be expressed as:

ffi s s s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  cos2 h ¼ w ¼ sin h ¼ 2 2 2

Fig. 2. Diagram representing r–e relationship for steel.

Conversely, with reference to the strengthened section, if the bar is located in a zone of effectively confined concrete (Fig. 6)b, bar buckling is prevented due to the presence of the lateral restrain constituted by the FRP wraps. In Fig. 3, the assumed kinematic mechanism for a buckled longitudinal bar is depicted. The stirrups represent a restraint like a support, so that the whole bar can be regarded as a beam on multiple supports subjected to a compression axial load. Obviously, as far as the bar axial load increases, the part of the bar between two consecutive hoops can develop a kinematic mechanism characterised by three plastic hinges (Fig. 3). The equilibrium of the part of the bar located between two consecutive hoops provides:

Mp

δ

N

s

θ

s/2

θ w N

s/2 Mp N Mp Fig. 3. Kinematic mechanism of a buckled bar.

ð4Þ

By substituting such expression of w in Eq. (2), the N–d relationship between the axial load and the axial displacement of a compressed bar during the post-buckling behaviour is obtained. Therefore the average stress rs = N/Al (being Al the bar cross section area) and the axial strain of the buckled bar e = d/s can be obtained as follows:

rs ¼

4M p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sAl 2e  e2

ð5Þ

However, this relation does not account for the interaction between the axial force and the bending moment, i.e. the reduction of the plastic moment of the longitudinal bar due to the presence of the axial load. To take into account this effect, the relationship between N and Mp(N) has to be considered. By means of translational and rotational equilibrium the plastic stress distribution shown in Fig. 4 provides:

N ¼ R2 ðp  2/ þ sen2/Þfy Mp ðNÞ ¼

s/2

s

N

Mp

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d2 2  2 s s

4 3 R fy sen3 / 3

ð6Þ ð7Þ

where fy is yield strength of steel, R is bar radius and / is the parameter locating the plastic neutral axis. The use of relations (5), (6) and (7) requires an iterative procedure. In particular, for a given value of the axial strain e and a first attempt location of the neutral axis (/ = p/2), Eq. (5) provides the value of the average stress and, therefore, of the corresponding axial force N = rsAl. The obtained value has to be introduced in Eq. (6) to correct the location of the neutral axis (/). The obtained value of / allows, by means of Eq. (7), the evaluation of the reduced plastic moment which has to be adopted in Eq. (5) to correct the value of

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φ

example, in case of bars £20 and stirrup spacing equal to 10 cm, the difference between the two curves is about 45% for a strain equal to 1%. N

R

M p(N)

4. Theoretical model for the evaluation of moment curvature diagram

fy fy Fig. 4. Plastic stress distribution.

Without M-N interaction With M-N interaction

Fig. 5. Stress–strain curve of longitudinal bars including post-buckling.

the bar average stress. The procedure has to be repeated until convergence. The r–e curve describing the post-buckling behaviour of longitudinal bars is completely defined starting from the point corresponding to the occurrence of buckling, given by Eq. (5) for the corresponding buckling load. In particular, considering a buckling load equal to the critical Eulerian one, the curve is completely determined as shown in Fig. 5. In the same figure, the influence of the reduction of the plastic moment due to the interaction with the axial load is also represented. In particular, for large slenderness values (for example £12 bars with stirrup spacing equal to 30 cm) this difference becomes almost negligible, conversely, for small values of slenderness, the difference between the curves (i.e. the influence of M–N interaction) becomes significant. For

b

As shown in Fig. 6, within a reinforced concrete section strengthened by means of FRP wraps, four differently confined parts can be identified so that four different concrete stress–strain constitutive laws need to be considered: (1) unconfined concrete, (2) concrete effectively confined by stirrups, (3) concrete effectively confined by FRP, (4) concrete effectively confined by FRP and stirrups. Therefore, the first step to be made is the delimitation of confined and not confined parts of the original pre-existing section. To this aim, the longitudinal ‘‘confining bars’’ or ‘‘restraining bars’’, which are those located in the corners and the intermediate ones provided that they are restrained by steel ties, have to be identified. Starting from these restraining points, it is possible to determine the parabola arches dividing zones of effectively confined concrete from zones of unconfined concrete, as it is shown in Fig. 6a for the unstrengthened original section. It is well known that wrapping by means of FRP sheets needs preliminarily the rounding of the corners of the existing section. Therefore, in the model herein adopted, it is assumed that the confining effect is exerted along the whole rounded corner, so that the final points of the rounding radius can be considered as 00 restraining points00 (Fig. 6b). Similarly to the parabola between the longitudinal bars of the unstrengthened section, the span-to-depth ratio of the parabola between two corners, separating confined and unconfined concrete, has been set equal to 4. By means of such parabolas the four zones of differently confined concrete can be identified (Fig. 6b). As soon as the different stress–strain constitutive laws for all the different concrete zones constituting the cross section are known [1] and the steel stress–strain behaviour has been defined, a fibre model can be easily developed by assigning to each fibre the corresponding r–e law according to the fibre location. The proposed model has been implemented in a computer program, namely SCFRP (Strengthened Column with FRP), to calculate the moment–curvature diagram for fixed values of the axial load. In particular, for a given value of the curvature, by means of the translation equilibrium equation, the location of the neutral axis can be

b’=b-2r

r

Buckling not allowed

r

r Concrete confined by stirrups and FRP

Buckling allowed

h

h’=h-2r

Unconfined area

Stirrup

(h’-2r)/4

Tie

Unstrengthened section

(a)

Concrete confined only by FRP

r

Concrete confined only by stirrups

(b’-2r)/4 Strengthened section

(b)

Fig. 6. Different confined zones of rectangular sections.

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

In Appendix A (from Figs. A1–A15), the moment–curvature diagrams are given for all the cross sections and for all the stress–strain constitutive laws considered for modelling confinement effects. In partcular, each figure provides 13 moment–curvature curves evaluated according to the r–e laws of confined concrete proposed by different researchers as specified in Table 1. In these study cases, the influence of the load already acting on the pre-existing unstrengthened section has not been considered, but its influence will be discussed in next Section. By comparing figures from A1–A15, the influence of FRP type, of axial load and of FRP number of layers can be outlined.

c

c c

c h/2 h h/2

b

b

 Influence of FRP type

Fig. 7. Location of longitudinal bars.

evaluated. Successively, by means of rotation equilibrium equation, the bending moment is obtained. The procedure stops when the ultimate strain of at least one elementary area (i.e. a fibre) of effectively confined concrete is attained or when at least one reinforcing bar has buckled. As already mentioned, buckling of longitudinal bars is allowed only when the concrete cover is constituted by unconfined concrete. This is the case of longitudinal bars of unstrengthened sections (Fig. 6a) subjected to buckling after the occurrence of cover concrete spalling and the case of intermediate bars of strengthened sections (Fig. 6b). In fact, when the concrete cover belongs to a confined concrete zone, like in the case of corner bars (Fig. 6b), it can be assumed that concrete cover is always able to provide an effective lateral restraint preventing longitudinal bar buckling. Conversely, when the concrete cover belongs to an unconfined concrete zone, the bar can buckle when the concrete cover attains the spalling strain.

5. Analysis of study cases In order to investigate the influence of the different modelling options regarding the stress–strain constitutive laws of confined concrete zones, it is necessary to take into account all the geometrical and mechanical characteristics both of the pre-existing section and of the applied FRP wrapping. Therefore, 15 study cases have been properly selected (Table 2). In Fig. 7 the location of the steel longitudinal bars in case of 4 and 6 steel bars is depicted.

The influence of the FRP type is pointed out by comparing sections 4A, 4C and 4D of Table 2. These sections are characterised by the same geometrical and mechanical properties with the only differences regarding the FRP type and the corresponding layer thickness which has been properly modified to compensate the different values of the elastic modulus. In case of confinement with CFRP (Section 4A), the confined concrete strength exhibits an average increase equal to 114% and the concrete ultimate strain attains an average increase equal to 1050% (Table 3) when compared to the case of unconfined concrete (law 1). Conversely, the maximum flexural resistance which the strengthened section is able to exhibit corresponds to an average increase of only 5% (Table 4) with respect to the original section and the ultimate curvature shows an average increase equal to 50% (Table 5). In case of confinement with AFRP (Section 4C), confined concrete strength exhibits an average increase equal to 214% and the average increase of the ultimate strain is equal to 3396% (Table 3), compared to unconfined concrete. Conversely, the maximum bending resistance exhibits an average increase equal to 9% (Table 4) and the ultimate curvature shows an average increase equal to 472% (Table 5). Considering the confinement with GFRP (Section 4D), compared to unconfined concrete, confined concrete strength exhibits an average increase equal to 238% and the ultimate strain has an average increase equal to 4413% (Table 3), while the maximum bending moment shows an average increase equal to 10% (Table 4) and ultimate curvature exibits an average increase equal to 540% (Table 5). Starting from these observations, it is clear that by using different

Table 3 Confined concrete strength and ultimate strain for sections 4A, 4C and 4D.

FRP young modulus (Mpa) FRP ultimate strain (%) Total layer thickness (mm) Adopted constitutive law Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Average of percentage increase with respect to unconfined concrete (Law 1)

Section 4A (CFRP)

Section 4C (AFRP)

Section 4D (GFRP)

231,000 1.4 0.5 fcc (Mpa) 16.6 38.0 36.3 29.7 39.8 42.5 26.1 27.8 45.6 36.2 40.7 34.1 28.9 35.5 114

120,000 3.7 1.0 fcc (Mpa) 16.6 54.2 69.9 47.0 42.2 60.0 42.8 43.3 68.1 54.9 64.7 38.2 40.1 52.1 214

80,000 4.5 1.5 fcc (Mpa) 16.6 57.5 81.4 52.4 38.0 60.4 48.4 49.1 74.0 60.3 71.0 38.9 42.7 56.2 238

ecc,u 0.002 0.014 0.013 0.018 0.028 0.044 0.011 0.013 0.053 0.012 0.040 0.016 0.014 0.023 1050

fcc/ fc 1.0 2.3 2.2 1.8 2.4 2.6 1.6 1.7 2.7 2.2 2.5 2.1 1.7 2.1 114

ecc,u/ ec,u 1.0 7.0 6.5 9.0 14.0 22.0 5.5 6.5 26.5 6.0 20.0 8.0 7.0 11.5 1050

ecc,u 0.002 0.025 0.028 0.084 0.048 0.174 0.037 0.062 0.143 0.017 0.160 0.024 0.037 0.070 3396

fcc/ fc 1.0 3.3 4.2 2.8 2.5 3.6 2.6 2.6 4.1 3.3 3.9 2.3 2.4 3.1 214

ecc,u/ ec,u 1.0 12.5 14.0 42.0 24.0 87.0 18.5 31.0 71.5 8.5 80.0 12.0 18.5 35.0 3396

ecc,u 0.002 0.027 0.033 0.118 0.090 0.203 0.048 0.088 0.173 0.019 0.213 0.026 0.045 0.090 4413

fcc/ fc 1.0 3.5 4.9 3.2 2.3 3.6 2.9 3.0 4.5 3.6 4.3 2.3 2.6 3.4 238

ecc,u/ ec,u 1.0 13.5 16.5 59.0 45.0 101.5 24.0 44.0 86.5 9.5 106.5 13.0 22.5 45.1 4413

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

Table 4 Increase in flexural resistance for section 4A, 4C and 4D. Adopted constitutive law

Section 4A (CFRP)

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Average of percentage increase with respect to unstrengthened section (Law 1)

Section 4C (AFRP)

Mmax (kNm)

M max;strengthened M max;unstrengthened

287.20 317.86 317.07 296.85 303.18 303.18 296.97 299.41 300.14 303.18 289.95 303.18 289.95 301.74 5

1.00 1.11 1.10 1.03 1.06 1.06 1.03 1.04 1.05 1.06 1.01 1.06 1.01 1.05 5

Section 4D (GFRP)

Mmax (kNm)

M max;strengthened M max;unstrengthened

Mmax (kNm)

M max;strengthened M max;unstrengthened

287.17 332.86 338.66 308.42 312.54 326.61 295.85 299.03 300.19 310.34 327.86 303.45 288.48 312.02 9

1.00 1.16 1.18 1.07 1.09 1.14 1.03 1.04 1.05 1.08 1.14 1.06 1.00 1.09 9

287.17 338.04 349.68 322.54 312.34 326.61 295.51 311.88 300.19 314.59 327.74 303.44 291.75 316.19 10

1.00 1.18 1.22 1.12 1.09 1.14 1.03 1.09 1.05 1.10 1.14 1.06 1.02 1.10 10

Table 5 Increase in ultimate curvature for section 4A, 4C and 4D. Adopted constitutive law

Section 4A (CFRP)

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Average of percentage increase with respect to unstrengthened section (Law 1)

Section 4C (AFRP)

Section 4D (GFRP)

vu (1/cm)

vu;strengthened vu;unstrengthened

vu (1/cm)

vu;strengthened vu;unstrengthened

vu (1/cm)

vu;strengthened vu;unstrengthened

0.00057 0.00083 0.00070 0.00070 0.00136 0.00202 0.00044 0.00057 0.00083 0.00057 0.00097 0.00070 0.00057 0.00086 50

1.00 1.46 1.23 1.23 2.39 3.54 0.77 1.00 1.46 1.00 1.70 1.23 1.00 1.50 50

0.00057 0.00216 0.00242 0.00374 0.00374 0.00474 0.00163 0.00269 0.00083 0.00970 0.00474 0.00110 0.00163 0.00326 472

1.00 3.79 4.25 6.56 6.56 8.32 2.86 4.72 1.46 17.02 8.32 1.93 2.86 5.72 472

0.00057 0.00242 0.00341 0.00507 0.00440 0.00474 0.00216 0.00407 0.00830 0.00123 0.00474 0.00123 0.00202 0.00365 540

1.00 4.25 5.98 8.89 7.72 8.32 3.79 7.14 14.56 2.16 8.32 2.16 3.54 6.40 540

Table 6 Confined concrete strength and ultimate strain increase for sections 4A, 4B and 4I. Adopted constitutive law

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Percentage of average increase with respect to law 1

Section 4A (1 layer)

Section 4B (3 layers)

Section 4I (6 layers)

Section 4B

Section 4A

Section 4I

Section 4A

ecc;u;3 ecc;u;1

f cc;6 f cc;1

ecc;u;6 ecc;u;1

1.00 1.00 2.31 2.33 0.71 1.66 2.45 1.38 1.74 1.50 1.65 1.56 1.00 1.59 59

1.00 1.65 3.66 2.53 1.09 1.56 2.82 2.69 2.41 2.25 2.31 2.13 1.66 2.20 120

1.00 1.00 4.31 4.06 0.43 1.14 4.64 2.00 2.45 2.00 2.28 2.13 1.00 2.08 108

fcc,1 (Mpa)

ecc,u,1

fcc,3 (Mpa)

ecc,u,3

fcc,6 (Mpa)

ecc,u,6

f cc;3 f cc;1

16.6 38.0 36.3 29.7 39.8 42.5 26.1 27.8 45.6 36.2 40.7 34.1 28.9 35.5 114

0.002 0.014 0.013 0.018 0.028 0.044 0.011 0.013 0.053 0.012 0.040 0.016 0.014 0.023 1050

16.6 54.5 74.8 49.2 44.2 64.6 45.2 45.6 76.0 57.2 67.7 52.6 39.5 55.9 237

0.002 0.014 0.030 0.042 0.020 0.073 0.027 0.018 0.092 0.018 0.066 0.025 0.014 0.037 1729

16.6 62.8 132.7 75.0 43.3 66.5 73.7 74.8 109.7 81.3 94.2 72.7 48.1 77.9 369

0.002 0.014 0.056 0.073 0.012 0.050 0.051 0.026 0.130 0.024 0.091 0.034 0.014 0.048 2296

1.00 1.43 2.06 1.66 1.11 1.52 1.73 1.64 1.67 1.58 1.66 1.54 1.37 1.58 58

FRP types the increase of flexural resistance is almost negligible, being no more than 10%, but a remarkable increase of ultimate cur-

vature, ranging from a minimum of 50% to a maximum of 540%, depending on the adopted constitutive law, is achieved.

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96 Table 7 Increase in flexural resistance and ultimate curvature for section 4A, 4B and 4I. Adopted constitutive law

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Percentage of average increase with respect to law 1

Section 4° (1 layer)

Section 4B (3 layers)

Section 4I (6 layers)

Section 4A vu;6 vu;1

vu,1

Mmax,3

vu,3

Mmax,6

(1/cm)

(kNm)

(1/cm)

(kNm)

(1/cm)

(kNm)

(1/cm)

(kNm)

(1/cm)

287.2 317.86 317.03 296.85 303.18 303.18 296.97 299.41 300.14 303.18 289.95 303.18 289.95 301.74 5

0.00057 0.00083 0.00070 0.00070 0.00136 0.00202 0.00044 0.00057 0.00083 0.00057 0.00097 0.00070 0.00057 0.00086 50

287.2 329.59 343.7 308.19 313.49 342.39 298.91 303.08 303.08 312.17 341.09 315.1 303.08 317.82 11

0.00057 0.00110 0.00282 0.00216 0.00163 0.00407 0.00136 0.00097 0.00083 0.00110 0.0044 0.00163 0.00070 0.0019 233

287.2 335.5 368.74 345.34 319.35 346.39 333.67 331.67 307.09 333.35 351.93 340.76 312.73 335.54 17

0.00057 0.00123 0.00374 0.0044 0.00149 0.00341 0.00341 0.00202 0.00097 0.00189 0.00407 0.00282 0.00083 0.00252 343

1 1.04 1.08 1.04 1.03 1.13 1.01 1.01 1.01 1.03 1.18 1.04 1.05 1.05 5

1 1.33 4.03 3.09 1.20 2.01 3.09 1.70 1.00 1.93 4.54 2.33 1.23 2.22 122

1 1.06 1.16 1.16 1.05 1.14 1.12 1.11 1.02 1.10 1.21 1.12 1.08 1.11 11

1 1.48 5.34 6.29 1.10 1.69 7.75 3.54 1.17 3.32 4.20 4.03 1.46 3.45 245

350

100 Section 4I (6 sheets)

80

Stresses (MPa)

Section 4I M max;6 M max;1

(kNm)

70

Section 4B (3 sheets)

60 50 Section 4A (1 sheet)

30 20 10 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Strains Fig. 8. Example of FRP thickness’s influence (Law 11).

300

Moment (kNm)

Regarding the influence of the number of FRP layers, it can be noted that by changing the CFRP layers from one (Section 4A) to three (Section 4B) the formulations for confined concrete modelling provide, an average increase of concrete strength equal to 58% and increase of concrete ultimate strain equal, on average, to 59% (Table 6). A smaller confinement effect is obtained dealing with the maximum bending moment; in fact, it exhibits an increase equal to 5% on average (Table 7). Conversely, dealing with the ultimate curvature an increase equal to 122% is achieved (Table 7) on average. By varying the number of FRP layers from one to six, the concrete strength increases of 120% and the concrete ultimate strain increases of 108% (Table 6), while the maximum bending moment and the ultimate curvature increase, respectively, of 11% and 245% (Table 7). It can be concluded that by increasing the number of FRP layers from one to six, a negligible increase of the maximum bending moment is obtained. On the contrary, a remarkable increase of the ultimate curvature is achieved. In order to clarify the above results, in Fig. 8 the stress–strain diagrams of the three considered sections (one, three and six FRP layers) are reported with reference to constitutive law number 11. In addition, in Fig. 9 the moment–curva-

40

Section 4A vu;3 vu;1

Mmax,1

 Influence of the number of FRP layers

90

vu,6

Section 4B M max;3: M max;1

250 Section 4A (1 sheet)

200

Section 4B (3 sheets)

Section 4I (6 sheets)

150 100 50 0 0.000

0.001

0.002

0.003

Curvature (1/cm) Fig. 9. Example of FRP thickness’s influence (Law 12).

ture diagrams are depicted for the same sections with reference to the constitutive law number 12. The adoption of different constitutive laws leads to similar results concerning the influence of the number of FRP layers.  Influence of the axial load The axial load acting on the section obviously does not affect the constitutive laws of confined concrete. For this reason the data provided in columns (1) and (2) of Table 3 remain valid also for the study cases 4E (N = 350 kN), 4F (N = 450 kN), 4G (N = 600 kN), 4A (N = 800 kN) and 4H (N = 1100 kN), where, the only difference with respect to the study case 4A is the axial load. By considering an increase of the axial load from 350 kN (Section 4E) to 450 kN (Section 4F) the increase of the maximum bending moment is equal on average to 7% (Table 8) while the ultimate curvature is subjected to a reduction equal on average to 15% (Table 9). The increase of the axial load from 350 kN (Section 4E) to 600 kN (Section 4G) gives rise to an increase of the maximum bending moment equal on average to 17% (Table 8) while the ultimate curvature reduction is, on average, equal to 33% (Table 9). A similar trend is observed due to a further increase of the axial load (see study cases with N = 800 kN and N = 1100 kN). In conclusion, the increase of the axial load gives rise to the increase of the flexural resistance but reduces the curvature ductility. In Fig. 10 the whole moment–curvature diagrams as affected by

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

Table 8 Increase of the flexural resistance for section 4E (N = 350kN), 4F (N = 450kN), 4G (N = 600kN), 4A (N = 800kN) and 4H (N = 1100kN). Adopted constitutive law

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Percentage of average increase

Sec. 4E (N = 350 kN)

Sec. 4F (N = 450 kN)

Sec. 4G (N = 600 kN)

Sec. 4A (N = 800 kN)

Sec. 4H (N = 1100 kN)

Sec. 4F

Sec. 4G

Sec. 4A

Sec. 4H

Sec. 4E

Sec. 4E

Sec. 4E

Sec. 4E

Mmax,4E (kNm)

Mmax,4F (kNm)

Mmax,4G (kNm)

Mmax,4A (kNm)

Mmax,4H (kNm)

M max;4F M max;4E

M max;4G M max;4E

M max;4A M max;4E

M max;4H M max;4E

232.19 239.30 238.25 233.48 239.55 245.98 233.48 233.48 233.48 233.80 241.66 233.48 229.99 236.33 2

249.93 258.93 257.80 252.19 253.53 261.18 252.19 252.19 252.19 252.19 255.68 252.19 247.05 253.94 2

272.00 285.64 285.64 275.15 276.84 279.69 275.15 275.15 275.15 276.84 273.12 276.84 268.81 277.00 2

287.20 317.86 317.03 296.85 303.18 303.18 296.97 299.41 300.14 303.18 289.95 303.18 289.95 301.74 5

304.87 353.88 345.45 311.63 322.80 322.80 313.60 315.60 316.58 322.80 302.20 322.80 302.20 321.03 5

1.08 1.08 1.08 1.08 1.06 1.06 1.08 1.08 1.08 1.08 1.06 1.08 1.07 1.07 7

1.17 1.19 1.20 1.18 1.16 1.14 1.18 1.18 1.18 1.18 1.13 1.19 1.17 1.17 17

1.24 1.33 1.33 1.27 1.27 1.23 1.27 1.28 1.29 1.30 1.20 1.30 1.26 1.28 28

1.31 1.48 1.45 1.33 1.35 1.31 1.34 1.35 1.36 1.38 1.25 1.38 1.31 1.36 36

Sec. 4H

Table 9 Increase of the ultimate curvature for section 4E (N = 350kN), 4F (N = 450kN), 4G (N = 600kN), 4A (N = 800kN) and 4H (N = 1100kN). Adopted constitutive law

Law 1 Law 2 Law 3 Law 4 Law 5 Law 6 Law 7 Law 8 Law 9 Law 10 Law 11 Law 12 Law 13 Average (Law 2 – Law 13) Percentage of average increase

Sec. 4E (N = 350 kN)

Sec. 4F (N = 450 kN)

Sec. 4G (N = 600 kN)

Sec. 4A (N = 800 kN)

Sec. 4H (N = 1100 kN)

Sec. 4F

Sec. 4G

Sec. 4A

Sec. 4E

Sec. 4E

Sec. 4E

Sec. 4E

vmax,4E

vmax,4F

vmax,4G

vmax,4A

vmax,4H

(1/cm)

(1/cm)

(1/cm)

(1/cm)

(1/cm)

vmax;4F vmax;4E

vmax;4G vmax;4E

vmax;4A vmax;4E

vmax;4H vmax;4E

0.00070 0.00163 0.00136 0.00149 0.00268 0.00407 0.00097 0.00123 0.00136 0.00123 0.00341 0.00149 0.00123 0.00185 164

0.00057 0.00136 0.00123 0.00123 0.00229 0.00341 0.00083 0.00097 0.00123 0.00097 0.00295 0.00136 0.00097 0.00157 175

0.00057 0.00110 0.00097 0.00097 0.00189 0.00282 0.00070 0.00070 0.00097 0.00083 0.00229 0.00097 0.00070 0.00124 118

0.00057 0.00083 0.00070 0.00070 0.00136 0.00202 0.00044 0.00057 0.00083 0.00057 0.00097 0.00070 0.00057 0.00086 50

0.00030 0.00057 0.00044 0.00057 0.00110 0.00149 0.00044 0.00044 0.00070 0.00044 0.00136 0.00057 0.00044 0.00071 138

0.81 0.83 0.90 0.83 0.85 0.84 0.86 0.79 0.90 0.79 0.87 0.91 0.79 0.85 15

0.81 0.67 0.71 0.65 0.71 0.69 0.72 0.57 0.71 0.67 0.67 0.65 0.57 0.67 33

0.81 0.51 0.51 0.47 0.51 0.50 0.45 0.46 0.61 0.46 0.28 0.47 0.46 0.46 54

0.43 0.35 0.32 0.38 0.41 0.37 0.45 0.36 0.51 0.36 0.40 0.38 0.36 0.39 61

6. Influence of initial internal actions 400 Section 4H (1100 kN)

350 Section 4A (800 kN)

Moment (kNm)

300 250 200 150

Section 4G (600 kN)

Section 4F (450 kN)

Section 4E (350 kN)

100 50 0 0.0000

0.0005

0.0010

0.0015

Curvature (1/cm) Fig. 10. Influence of axial load (Law 2).

the influence of the axial load are depicted with reference to constitutive law number 2.

FRP wrapping is a strengthening technology of reinforced concrete members whose main effectiveness is the improvement of the curvature ductility of the cross-section. It is applied to existing structures which are already loaded due to permanent loads and to the live loads still acting at the time of the strengthening intervention. Therefore, the influence of loads acting at the time of wrapping of the unstregthened structural member should be considered. In order to evaluate this influence, with reference to one of the analysed sections (Section 2A), an initial axial load Ni and an initial bending moment Mi acting on the unstrengthened original section have been considered. For sake of simplicity, reference is made to the stress–strain constitutive law number 12, for N = Ni the moment–curvature diagram, until the point corresponding to a value of bending moment equal to Mi. With reference to the initial internal actions Mi and Ni the strain of each fibre element constituting the section has been preliminarily evaluated with reference to the geometry of the original crosssection. The obtained strain distribution is the one due to the preexisting loads acting on the unstrengthened section and it has to be

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

60

90 80

50

Moment (kNm)

Moment (kNm)

70 60 N i = 200 kN; M i = 40 kNm

50

N i = 200 kN; M i = 30 kNm

40 30

N i = 200 kN; M i = 20 kNm

20

0

0.00002

0.00004

3 1

30

9

13

6

4

11

8

7

20 N = 350 kN

10 0

N i = 0 kN; M i = 0 kNm

0

12

2

N i = 200 kN; M i = 10 kNm

10

5 10

40

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Curvature (1/cm) 0.00006

0.00008

0.0001

Fig. A2. Section 1B.

Curvature (1/cm) Fig. 11. Influence of pre-existing loads (Section 2A – Law 12).

100

7. Conclusions and future developments In this paper the influence on the evaluation of the flexural resistance and curvature ductility of reinforced concrete columns strengthened by means of FRP wrapping, due to the use of different constitutive laws for confined concrete modelling, has been investigated with reference to 13 constitutive laws, deeply analysed in a previous work [1]. All the considered constitutive laws show a

90

Moment (kNm)

80 70

8

60

7

12

4

2

5

3

13

1

6

9

10

11

50 40 30

N = 450 kN

20 10 0

0

0.001

0.002

0.003

0.004

0.005

Curvature (1/cm) Fig. A3. Section 2A.

Moment (kNm)

accounted for in the evaluation of the moment–curvature diagram of the strengthened section. By varying Mi and Ni different moment–curvature diagrams have been obtained and compared for the same section (Section 2A). It can be observed (Fig. 11) that very significant differences occur with reference to the flexural stiffness of the cross-section, i.e. the initial slope of the moment–curvature curve. In particular, it can be noted that for a fixed value of the initial axial force Ni the slope of the curves increases as far as the initial bending moment Mi increases. Conversely, regarding the ultimate conditions both in terms of flexural capacity and in terms of ultimate curvature, the influence of the initial strain distribution due to the loads acting on the original pre-existing section is totally negligible. This result is very interesting from practical point of view, because it allows to evaluate the resistance and the curvature ductility of the strengthened section by a simplified approach where the initial strain distribution in the pre-existing original section is neglected. Conversely, of the above results the knee region of the moment– curvature relation is the most significantly affected by the loads acting on the pre-existing section. Similar results have been obtained for all the considered constitutive laws of confined concrete.

110 100 90 80 70 60 50 40 30 20 10 0

13

5

12

10

6

2 9

3

8

4 7

1

11

N = 450 kN

0

0.001

0.002

0.003

0.004

Curvature (1/cm) Fig. A4. Section 2B.

60

Moment (kNm)

50 3

40 1

30

12

10

8 13

7

2

5

4 6

11

9

20 N = 350 kN

10 0

0

0.001

0.002

0.003

0.004

Curvature (1/cm) Fig. A1. Section 1A.

0.005

0.006

0.007

large increase both in terms of confined concrete strength and in terms of ultimate strain with respect to the unconfined concrete, but significant differences are obtained in the resulting stress– strain curve depending on the adopted constitutive law. Even though the confinement by FRP wrapping leads to a significant improvement of concrete behaviour in terms of ultimate strength, the influence on the whole section behaviour in terms of flexural strength is significantly attenuated, being the obtained increase of resistance not greater than 11%. Conversely, a more significant increase in curvature ductility has been obtained (up to 540%). However, it has been recognised that the estimated values of the curvature ductility are dependent on the constitutive law adopted for the stress–strain curve of confined concrete. Therefore, taking into account that the effects in terms of flexural strength are

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

Moment (kNm)

200

9

350

12

300 5

10

150 8

100

7

1

13 3

6 2

50 0

11

4

Moment (kNm)

250

0.001

0.002

12 3 4

250 9

200

8

6

100

N = 800 kN

50 0

0.003

Curvature (1/cm)

1 13

0

7

2

0.001

350

100

Moment (kNm)

Moment (kNm)

9

4

7

10

0.005

3 4 5

3

5

150

0.004

2

10

300 8

11 6

12

1 2

N = 600 kN

50

250

9

11 6

200

8

7 13

150

1

12

100

N = 800 kN

50 13

0

0.001

0.002

0.003

0.004

0.005

0

0.006

0

0.001

0.002

Curvature (1/cm)

0.004

0.005

Fig. A9. Section 4C.

400

350 9

350

11

250

4

200

7 12

150

6 5

10

100

3

8

N = 800 kN

50 1

0

3

2

10

300

5 8

250 9

200

11 6 4

7

150

1

12

13

100

N = 800 kN

50

2 13

0.001

Moment (kNm)

300

0

0.003

Curvature (1/cm)

Fig. A6. Section 3B.

Moment (kNm)

0.003

Curvature (1/cm) Fig. A8. Section 4B.

250

0

5

0.002

Fig. A5. Section 3A.

200

11

150

N = 600 kN

0

10

0.002

Curvature (1/cm)

0

0

0.001

0.002

0.003

0.004

0.005

Curvature (1/cm)

Fig. A7. Section 4A.

Fig. A10. Section 4D.

quite limited independently of the adopted confinement stress– strain model, it can be concluded that there is a limited interest in further developments of accurate confinement models aimed at the prediction of the whole stress–strain curve of confined concrete. Conversely, the attention and the efforts of researchers should be mainly devoted to the improvement of the accuracy of the formulations for predicting the ultimate strain of confined concrete which, obviously, plays a role of paramount importance for a reliable prediction of the curvature ductility of the cross-section and, as a consequence, of the plastic rotation supply of reinforced concrete columns confined by FRP wrapping. Also the influence of the load acting on the original pre-existing unstrengthened section has been considered. The obtained results show that significant effects occurs only in the first part of the mo-

ment–curvature diagram, in particular in the knee region of the curves. Conversely, regarding the ultimate resistance and curvature ductility the effects of such loads are negligible. Consequently, the choice of the confined concrete constitutive law has no significant importance on the resulting ultimate flexural strength of the strengthened section. Regarding the future developments, as this work has pointed out that the adopted stress–strain constitutive law of confined concrete has a significant influence on the prediction of the curvature ductility of cross-section, rather than on their flexural resistance, it is evident the need to provide a wide comparison with the experimental results available in the technical literature. In fact, the main drawback of the great number of proposals dealing with the stress–strain modelling of concrete confined by FRP wrapping is that they are often calibrated on the proposers

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R. Montuori et al. / Composites: Part B 54 (2013) 85–96

Moment (kNm)

200

10

3

400

12 5

2

4

350

6

11

Moment (kNm)

9

250

8

150 1 13

7

100

N = 350 kN

50 0

2 7

250 200

8 1

150

0 0

0.001

0.002

0.003

10 N = 1100 kN

0.004

0

0.001

9

0.002

Curvature (1/cm) Fig. A14. Section 4H.

400

2

2

350 5

7

200

4 8

150

6

11

Moment (kNm)

250

Moment (kNm)

6

50

300

12

3

1 13

10

100

N = 450 kN

50

300

12 10

250

13

9

8

7 6 3

5

200 150

11

4

1

100

N = 800 kN

50 0

0.001

0.002

0

0.003

Curvature (1/cm)

3

300 250 7

9

5

10

11

6

200 150

1 8

100

4 13

2 N = 600 kN

12

50 0

0

0.001

0.002

0.003

Curvature (1/cm)

0.004

0.005

Fig. A15. Section 4I.

Fig. A12. Section 4F.

Moment (kNm)

11

12

100

Fig. A11. Section 4E.

0

5

4

13

Curvature (1/cm)

0

9

3

300

0.001

0.002

0.003

Curvature (1/cm) Fig. A13. Section 4G.

own test results. Therefore, the need of a wide collection of experimental test results, aimed to compare the fibre model herein proposed with the experimental evidence, can be identified as the forthcoming development of this research activity with the aim to identify the most appropriate stress–strain confinement model, particularly in terms of ultimate strain, to be adopted to analyse the ultimate response of FRP wrapped reinforced concrete columns subjected to axial force and bending.

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