Mechanical shear strength model for reinforced concrete beams strengthened with FRP materials

Construction and Building Materials 124 (2016) 855–865

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Mechanical shear strength model for reinforced concrete beams strengthened with FRP materials Vincenzo Colotti Dept. of Civil Engineering, University of Calabria, Ponte P. Bucci, Cubo 39B, 87036 Rende (CS), Italy

h i g h l i g h t s
A theoretically-based shear strength model is proposed.
Some critical aspects of the shear behavior are taken into account.
An extensive numerical investigation shows the accuracy and reliability of the model.

a r t i c l e

i n f o

Article history: Received 18 April 2016 Received in revised form 30 June 2016 Accepted 1 July 2016

Keywords: Analytical modelling Beams Debonding FRP Plasticity theory Shear Strengthening

a b s t r a c t In this paper, a theoretically-based model for the prediction of the shear strength of reinforced concrete beams strengthened with FRP is presented. The model has been validated with the results available in literature of about 200 shear tests on beams strengthened with different FRP configuration schemes (fully wraps, U-wraps, side bonding). Numerical comparisons considering the values predicted by the ACI guidelines and a recent shear strength model existing in literature have been also carried out. These numerical investigations show that the proposed model appears as a very good predictor of shear strength and has a significantly smaller coefficient of variation compared to the other models. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The problem of accurately determining the shear capacity of structural concrete members appears rather difficult to solve, despite many studies carried out in recent decades. Contrary to the determination of the flexural capacity of reinforced concrete (RC) sections, for which a perfectly rational design approach it is possible, for shear capacity the semi-empirical shear design procedure proposed by the ACI-ASCE Committee in 1962 [1] still forms the basis of most current design shear code provisions [2–4]. With the advent of modern techniques for upgrading existing concrete structures by using externally bonded Fiber Reinforced Polymer (FRP) composites, the problem of a correct evaluation of the load-carrying capacity of shear-strengthened RC members has become even more stringent both for economic and safety aspects.

E-mail address: [email protected] http://dx.doi.org/10.1016/j.conbuildmat.2016.07.146 0950-0618/Ó 2016 Elsevier Ltd. All rights reserved.

One of the primary reasons for the poor quality of shear design provisions for RC was that the shear failure process is a very complex phenomenon influenced by many parameters, perhaps more of twenty parameters, as suggested by Leonhardt [5], and characterized by a multitude of nonlinearities that governs the behavior of diagonally cracked concrete. Typically, the shear strength of RC beams strengthened with composite materials is computed by adding the contribution of the FRP to that of concrete and internal shear reinforcement, as described by the following equation:

Vu ¼ Vc þ Vs þ Vf

ð1Þ

where Vu = shear resistance of the strengthened beam and Vc, Vs, and Vf = three components contributed by the concrete, internal steel stirrups, and external FRP shear reinforcement, respectively. In many studies [6–10], the experimental values of the FRP contribution (Vf) have been often obtained by subtracting the values acquired through the reference (unstrengthened) beam tests from the value obtained from the FRP strengthened beam tests. However, this approach can be misleading, since the influence of

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V. Colotti / Construction and Building Materials 124 (2016) 855–865

the different parameters that interact is not properly evaluated. Moreover, even if the contribution of the FRP was virtually estimated to be exact, from the structural safety point of view, what matters should be the total shear capacity and consequently the reliability of the theoretical predictions should be evaluated considering this capacity. Therefore, the reliability studies conducted on theoretical models calibrated considering only the contribution of the FRP [12] have reduced practical importance. In many cases, empirical equations were developed by fitting the results from a limited number of experiments, in which the parameters vary in a way not always representative of the actual design situations. For example, in laboratory shear experiments often the specimens were heavily reinforced in flexure to ensure that a flexural failure mode does not prematurely end the shear test, while in real design it is desirable to have the structural behavior governed by a ductile flexural failure [13]. A further major difficulty to develop adequate predictive shear strength models of FRP strengthening beams is that the interaction between the concrete, internal steel stirrups, and FRP contributions to shear strength has not been clearly verified in the experimental tests, due to complexity in independent measuring of such contributions. From the above observations, therefore, it is obvious that the current simple additive approach based on the implicit assumption that the three components of shear resistance reach their maximum values simultaneously in a real beam is unrealistic and unconservative. Recently, some studies have been carried out to consider the FRP-steel interaction effect in RC beams shear-strengthened with FRP U-strips or side strips [10,11,16,17]. This shear interaction effect is taken into account by introducing some modification factors determined throughout the loading process described in terms of crack opening [17], or by means a new formulation for the FRP and steel contributions determined on the basis of experimentally observed failure modes [10,11]. However, both these formulations present some drawbacks. The evaluation of the model proposed by Chen et al. [17], in fact, has been conducted using the FRP shear contribution as test parameter and, therefore, its reliability remains questionable, while the model proposed by Pellegrino et al. [10,11] is not clearly defined. The above considerations indicate that a possible and substantial way to improve shear provisions and to provide estimation with the least scatter, consists in the use of theoretically-based shear design procedure, in which the empirical assumptions are restricted to the level of constitutive relationships of materials and their adhesion property, and, moreover, in adopting mechanical schemes as much as possible adherent to the effective structural behavior, with simple but realistic simplified assumptions. In this paper, a new theoretical shear strength model modified from a previous model proposed by Colotti and Swamy [18] for FRP strengthened RC beams is first presented. The model is based on the Plasticity Truss Model approach, conveniently refined in order to capture the interaction effect between the internal and external shear reinforcement. For this purpose, the results in closed form solution obtained by a sophisticated analytical model describing the debonding process in a FRP shear-strengthened RC beam [19], are suitably implemented in the new shear strength model. Moreover, other effects on the shear performance, such as variable crack angle and shear span/depth ratio, are also considered. Predictions from the new proposed model are then compared with experimental results and values predicted by the ACI guidelines [20] and with the recent theoretical shear strength model proposed by Chen et al. [17]. These numerical comparisons, conducted using the results of about 200 shear test of beams collected from the literature, show that the proposed model provides clo-

ser prediction for the shear resistance of strengthened RC beams for all the configuration schemes considered (fully wraps, U-wraps, side bonding). As better indicated in the following sections, the proposed model appears as a very good predictor of shear strength and has a significantly smaller coefficient of variation than the other models considered in the numerical investigation. Since the proposed model is a variable angle truss model, the Vc term is not required, which instead would be necessary to correct the usual 45° truss, as in the ACI Code. In this way the problem of interaction between the concrete contribution and the other shear components (FRP and steel) is bypassed. 2. Modelling of the shear behavior or RC beams strengthened with bonded FRP reinforcement In order to develop a simplified mechanical model that adequately predicts the shear strength of a RC beam strengthened with bonded FRP reinforcement, the modelling scheme proposed in an earlier study [18] has been adopted. This scheme, based on the Plasticity Truss Model developed primarily by Nielsen et al. [21] and Thurlimann [22] for ordinary RC beams, idealizes a generic cracked RC beam as a plane truss consisting of an upper and a lower stringers (compression concrete and tensile reinforcement, respectively) and a web element (diagonal concrete struts and transverse reinforcement). On the basis of equilibrium (Fig. 1), kinematic (Fig. 2), and yield conditions, by using the mathematical theory of plasticity, the load-carrying capacity (Vu) of strengthened RC beams subjected to bending and shear is obtained in the following form [18]:

V u ¼ minðV u;1 ; V u;2 Þ

ð2Þ

C=-M/dv+½Vcotθ

C’=-M/dv- ½Vcotθ

Ff V

V

dv=jd

Ff

θ

T=M/dv+½Vcotθ

hf

wf

sf

T’=M/dv+3/2Vcotθ

Δ=dvcotθ

(a) Rectangular free-body diagram -M/dv- ½Vcotθ

M

piy

V

pey

M/dv+ ½Vcotθ

Δ=dvcotθ

(b) Triangular free-body diagram Fig. 1. Equilibrium scheme for bending-shear [18].

dv=jd

V. Colotti / Construction and Building Materials 124 (2016) 855–865

Δ=dvcotθ

determined effectiveness factors on the concrete strength, the plastic solutions may provide good agreement between theoretical and experimental tests [23]. Therefore, since the theory of plasticity has been demonstrated to be a very effective way to tackle shear design in normal RC beams, it also might be the case for FRP shear-strengthened RC beams, despite the brittle behavior of the composite material. The basic idea of the present paper is the introduction of effectiveness factors not only for the concrete strength (mc), but also for the FRP bond strength, capable to take into account several main effects of the bond-slip behavior. The peculiar aspect of the present approach is that such effectiveness factors are determined on the basis of rational theoretical models, instead of fitting experimental results. The empirical assumptions are essentially limited at level of constitutive laws. For example, the bond-slip behavior between FRP and concrete substrate is expressed by means the fracture energy and correlated parameters, such as bond strength and slip at failure.

x O

M

ρ

dv δv(x) θ

857

V

δvo δho Fig. 2. Kinematic mechanism for bending-shear [18].

with

V u;1

8p a T d 2T d y v < 4y þ 2a for 0 < py 6 po ¼ 3ay 2 v qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ 2T d :p a2 þ py v  a for py > po y

ð3Þ

8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   > py < bdv f mc  bfpyc for 0 < bfpyc 6 m2c c bfc ¼ > mc p : bdv f c for bfyc > m2c 2

ð4Þ

3. Effectiveness factors for steel and FRP

y

V u;2

where

py ¼ piy þ pey

ð5Þ

T y ¼ ml Asl f ly þ Ap f pu

ð6Þ

In the above relations, py denotes the total equivalent stirrup/ link (transverse reinforcement) force; Ty is the tension force in the longitudinal steel reinforcement and/or external bonded plate/laminate; a is the shear span, and dv is the distance between the upper and bottom stringers (lever arm); Asl, fly and ml are the cross-sectional area, the yield strength and the effectiveness factor of the longitudinal reinforcement, respectively; Ap and fpu are the cross-sectional area and the yield (or rupture) stress of longitudinal bonded plate/laminate. The formulae in Eqs. (3), (4) represent the shear capacity for flexural-shear interaction failure mode and for shear webcrushing failure mode, respectively. They are both obtained as an upper bound and a lower bound solution and, therefore, they represent an exact solution in the context of the plasticity theory. In particular, Eq. (3) represents the plastic solution associated to a global failure mechanism characterized by diagonal tension failure combined with or without longitudinal reinforcement failure. Moreover, in order to account for the contribution of the external shear reinforcement, two mechanisms of local failure mode are considered, i.e. bond slip or tensile failure of the web link material. Similarly, Eq. (4) represents the plastic solution for the shear web-crushing failure mode, related to crushing of diagonal concrete struts and/or transverse reinforcement failure. More details can be found in Ref. [18]. As in any plastic solution, the problem to be overcome for the applicability and validity of Eqs. (3), (4) is the non-ductile nature of the materials (concrete and FRP) and their joining mechanism (bond-slip behavior), which renders the hypothesis of perfectly plastic material model a crude approximation. However, many applications of plastic theory in practical design of RC beams have shown that, by introducing empirically

The use of reduced strengths by means of effectiveness factors is a well-consolidated way in order to make less inappropriate the hypothesis of perfectly plastic material model and to make acceptable the application of plastic theory to reinforced concrete structures subjected to shear [23]. These factors should not be considered as simple empirical correction factors introduced to statistically calibrate the experimental results, but rather as factors that take into account some major effects that characterize all concrete structural problems (inhomogeneous strain or stress fields, as softening, cracking, and local damage of the materials). In this way, it is possible to provide a physical meaning of these effectiveness factors and, thus, they may be virtually evaluated with a theoretical approach. In the case of RC beams shear-strengthened with bonded FRP links, owing the debonding propagation process, the FRP stresses along a critical shear crack do not assume at every point the maximum value at the same time. Thus, instead of using the FRP debonding resistance, it is useful to consider a reduced strength that expresses the average stress maintained in the critical shear crack. Moreover, since the concomitant strains in steel stirrups intercepted by the critical shear crack may be below their yield strain, so that not all of them reach yielding at the shear failure of the beam, a similar reduced yield strength that expresses the average steel stirrups stress may be used. In this way, the shear interaction effect between internal and external transverse reinforcement is taken into account, too. Based on a theoretical study on the stress distribution in the FRP along a critical shear crack at rupture or debonding failure, the following expressions are obtained for the FRP effectiveness factor Df [19]:

Df ¼

  xd 1 p xd  þ  1 þ c1 1  D qpeak 4 D

ð7Þ

with

( c1 ¼

0  1 2

1þr

rr

f ;max



for side bonding andU  jacketing for complete wrapping

8 dv þdc bo Le > for side bonding > dv xd < for U  jacketing ¼ 1 D > > : 1þkc ð0:5hf Le Þ for complete wrapping wo

ð8Þ

ð9Þ

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V. Colotti / Construction and Building Materials 124 (2016) 855–865

qpeak ¼

8( D ½1  cosðp2 bo Þ > xd > >  > < D 1 þ p ðb  1Þ > > > > :

xd

o

2

1þkc ðdv þdt Le Þþco 2

if bo 6 1 if bo > 1

for side bonding

for U  jacketing for complete wrapping

wo

ð10Þ in which

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h 3 dv þ dc f A ; bo ¼ min @ pLe 2Le

rf ;max ¼ k2c

sf t fo kc

6 rr ;

s2f

1 ¼ ; 2Gf Ef t fo

ð11Þ

rr ¼ uR f fu

Le ¼

p 2kc

;

ð12Þ

tfo ¼ min ðtf ; t ref Þ

ð13Þ

co ¼ 1  cos ðkc dc Þ þ ½1  sin ðkc dc Þ tan ðkc dc Þ; wo ¼

rr hf ; 2Ef df

2Gf

df ¼

ð14Þ

sf

where hf = dc + dv + dt represents the total length of the FRP strips (see Fig. 3); tf is the thickness of the FRP links; tref is the reference thickness of concrete substrate or debonded concrete [24]; ffu and Ef are the tensile strength and modulus of elasticity of the FRP, respectively. In the above relations, rf,max represents the maximum stress in the FRP at debonding failure; rr is the effective tensile/rupture strength of FRP; /R is a reduction factor to account the detrimental effect of the corners of beams on the tensile strength of the FRP [25,26]; Le is the effective bond length; Gf, sf and df are the interfacial fracture energy, the local bond strength and the ultimate slip at FRP-concrete interface, respectively. For the validity of Eqs. (7)– (10), it is required that dc 6 Le and dv + dt > Le. The effectiveness factor for the steel stirrups Ds corresponding to the peak value of the average FRP stress can be expressed as follows [19]:

Ds ¼

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 qpeak wf

for qpeak 6 wy

:1 

for qpeak > wy

wy

3

1 3qpeak

wy wf

w

ð15Þ

f

w

piy ¼ Ds

Ast f ty s

ð17Þ

pey ¼ Df

2wf t f rf ;max sf

ð18Þ

where Ast, fty and s are the cross-sectional area, the tensile yield strength and the spacing of the stirrups, respectively; wf and sf are the width and spacing of the FRP strips, respectively. Based on the above factors, the total equivalent stirrup/link force py can be conveniently re-written in the form

py ¼ Dt ðpi;max þ pe;max Þ

ð19Þ

where

pi;max ¼

Ast f ty s

ð20Þ

pe;max ¼

2wf t f rf ;max sf

ð21Þ

Dt ¼ r¼

rDs þ Df 1þr

ð22Þ

pi;max pe;max

ð23Þ

In the above relations, pe,max and pi,max represent the maximum equivalent FRP and steel forces, respectively. The factors Ds, Df, and Dt can also be identified as interaction factors for internal, external and total shear reinforcement. Their expressions are applicable to beams strengthened with either vertical discrete strips or continuous sheets, arranged in various configuration schemes (complete wrapping, U-jacketing, or side bonding).

f

4. Comparison with other analytical proposals

where

e2y wy ¼ ; wf 2kdf

In the above relations, Es and ey are the modulus of elasticity and yield strain of the steel, respectively; so is the local bond strength at the steel-concrete interface; / is the diameter of the steel stirrups. The factors Df and Ds allow to describe the shear forces in the FRP links and steel stirrups at ultimate as a proportion of their maximum capacity. Therefore, the action of the internal steel stirrups and FRP at shear failure can be expressed as

k¼

4so Es u

ð16Þ

Most of current methods used for calculating the shear capacity of FRP strengthened beams are based on the format expressed by

dt

x O ρ h

y dv

hf

d

θ dc Δ=dvcotθ Fig. 3. Geometrical parameters defining the idealization of FRP strips crossing the critical shear crack.

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V. Colotti / Construction and Building Materials 124 (2016) 855–865

Eq. (1). Generally, the shear capacity of the concrete (Vc) and the internal steel stirrups (Vs) are determined according to the expressions provided by reference codes [2,3] for ordinary RC members, while the different models proposed in literature differ in the evaluation of the shear contribution of the external reinforcement (Vf). In the following, the above-cited basic codes, together with two models for the Vf contribution, are selected in order to compare and critically examine the results obtained by the proposed model with those of the selected models in the light of the observed experimental results.

rf ;max ¼ min ðf fu ; rdb;max Þ

ð35Þ

8 qffiffiffiffiffiffiffiffiffi 2Ef Gf > < for Lmax P Le tf ¼  qffiffiffiffiffiffiffiffiffi 2Ef Gf > : sin p Lmax for Lmax < Le 2 Le t

rdb;max

ð36Þ

f

8 < hfe þht þhb

for side strips

: hfe þht þhb

for U  strips

2 sin b

Lmax ¼

sin b

ð37Þ

For FRP side strips, Dfrp is given by 4.1. ACI 440 model Provisions of ACI Committee 440 [20] on shear strengthening of RC beams are based on the study of Khalifa et al. [27]. The total shear strength can be calculated as follows:

V u ¼ V c þ V s þ wf V f

ð24Þ

Af v Ef efe ðsin b þ cos bÞdf Vf ¼ sf

ð25Þ



efe ¼

for 2 and 3 sides bonded

0:004 6 0:75efu

for completely wrapped

k1 k2 Le 6 0:75 11900efu

kv ¼



fc 27

k1 ¼

for 2 side bonded

: df Le df

for 3 side bonded

23300 ðtf Ef Þ0:58

wf ¼

ð27Þ

ð28Þ

8 d 2Le < fd



ð26Þ

2=3

f

k2 ¼

Le ¼

kv efu 6 0:004

0:85 for 2 and 3 sides bonded 0:95 for completely wrapped

ð29Þ

f fe ¼ rf ;max Dfrp

Lm ¼ kh Le

ð41Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  p p p  p ðhfe þ hb Þ 2 þ1 1 1 þ 1 kh ¼ p 4 2 2 4 Le sin b p

ð42Þ

For FRP U-strips, Dfrp is given by

 p hdf Dfrp ¼ 1  1  4 hfe hdf ¼ 2df

wep ¼ df

ð43Þ

hfe wep sinðh þ bÞ   hfe 1 þ p2 Le sin 1 b

sinðh þ bÞ sffiffiffiffiffiffiffiffiffiffiffiffi df Ef t f qffiffiffiffi ft

0:55

ð48Þ f cu ¼ f c =0:8

2Gf

df ¼

ð45Þ

ð47Þ

sf ¼ 1:5bw f t f t ¼ 0:395f cu ;

ð44Þ

ð46Þ

sf

ð49Þ ð50Þ

sf

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  wf =ðsf sin bÞ bw ¼ 1 þ wf =ðsf sin bÞ

ð51Þ

The coefficient K, termed the shear interaction factor, is given by

(

K¼

l¼

1  lð1  K s Þ for U  strips B Bþl

Vs Vf

Ks ¼ ð33Þ ð34Þ

ð39Þ ð40Þ

ð31Þ

ð32Þ

ð38Þ

hdb ¼ Lm sin b  hb

Gf ¼ 0:308b2w

Chen et al. [17] proposed a shear strength model for the evaluation of the FRP shear contribution introducing a suitable shear interaction factor to take into account the interaction between internal and external reinforcement. It is implicitly assumed that the contribution of the concrete (Vc) to the shear capacity is the same as for the ordinary RC beam given the reference code. The model is valid only for strengthening beams with side bonding and U-jacketing schemes. The shear resistance of a RC externally strengthened beam can be expressed as

2wf t f f fe hfe ðcot h þ cot bÞ sin b sf

hfe  hdb 1 þ p2 ðkh  1Þ

Le ¼

4.2. Chen et al. model

Vf ¼

hdf ¼

ð30Þ

In the above formulae, b is the angle between principal fiber orientation and longitudinal axis of beam; df = hf  dc is the effective depth of the external reinforcement, dc is the concrete cover; efu is the ultimate tensile elongation of the FRP; Afv is the area of FRP shear reinforcement with spacing sf.

V u ¼ V c þ V s þ kV f

 p hdf hdb Dfrp ¼ 1  1   4 hfe hfe

ao ¼

for side strips

ð52Þ ð53Þ

w1:4 ep ao þ w1:4 ep A ðcos hÞ1:4

ð54Þ

ð55Þ

860

V. Colotti / Construction and Building Materials 124 (2016) 855–865

( A¼

104 x1:48½lnðhfe Þ  4:52ðf ty  173Þðu þ 0:935Þ for plain bar stirrups 105 x4:94½lnðhfe Þ  3:34ðf ty  245Þðu  0:767Þ for deformed bar stirrups

ð56Þ   8 5 hfe > þ 2:11 for plain bar stirrups < u1:01x10 0:854 f 1:88 Le ty   B¼ 5 hfe > : 2:05x10 þ 1:58 for deformed bar stirrups u1:13 f 1:71 Le

ð57Þ

ty

In the above equations, Dfrp is the stress distribution factor; rdb,max is the maximum stress in the FRP strips crossing the critical shear crack as governed by debonding failure; Lmax is the maximum bond length of the FRP strips intersected by the critical shear crack; hb is the thickness of concrete cover; ht is the vertical distance from the top end of FRP strips to the crack tip; hfe is the effective height of FRP; Lm is the maximum mobilized bond length. In the previous equations, units are in mm and MPa. Further details can be found in Ref. [17].

5. Numerical validation of the proposed shear strength model and comparison with other models To evaluate the performance of the proposed shear strength model, a large number of experimental results reported from a wide range of sources were used. An extensive database representing 201 RC beams strengthening in shear was collected from published literature. In detail the database contains 44 beams strengthened with FRP side-strips, 96 beams strengthened with FRP U-strips and 61 beams strengthened with FRP completely wrapped. The test data cover the following ranges of parameters: beam height h = 150–940 mm; web thickness b = 77–457 mm; shear to span ratio a = 0.9–6; cylinder compressive strength of concrete fc = 15–56 MPa. Due to space limitations, only the experimental shear capacities of the collected test data are reported in Appendix. Further details on the geometric and material properties can be found in the original sources. For purposes of comparison, the predictions of the proposed model have been compared with those of the ACI 440 model, Chen et al. model combined with the basic codes ACI 318 and Eurocode 2. It is emphasized that in the Chen et al. model the governing parameters are not applicable to beams with FRP fully wrapped. In the numerical investigation, the following assumption have been made: dv = jd, with j = 0.9; mc = 0.7; ml = 1 for nl = 1 and ml = 0.85 for nl > 1, where nl = number of longitudinal steel layers; /R = 0.8 for FRP fully wrapped or U-jacketed and /R = 1 for side bonded strips; tref = 0.25 mm. Furthermore, for the fracture energy Gf, and the correlated bondslip parameters sf, df, bw, the same expressions proposed by Lu et al. [28] and adopted by Chen et al. [17] as given by Eqs. (47)–(51) have been assumed. Moreover, for the steel bond strength so, the expression provided by the Eurocode 2 [3] has been adopted. Since the Chen et al. model concerns only the FRP shear contribution (Vf), the predictions of overall shear strength have been obtained by combining both basic model codes selected in order to estimate the contributions of concrete and internal steel stirrups. The comparative study between the measured and predicted shear capacities was realized through the ratio between experimental and theoretical values Vexp/Vtheor. The main statistical measures of this ratio include its average (AVG), the coefficient of variation (COV), and the maximum (MAX) and minimum (MIN) values.

Table 1 Results of comparison: statistical analysis. Statistical parameter

Proposed model

ACI model

Chen-ACI model

Chen-EC2 model

N. AVG COV (%) Max Min

201 1.10 20.30 1.59 0.62

201 1.56 40.36 4.15 0.42

140 1.23 32.33 2.83 0.46

140 0.90 37.76 2.47 0.32

Note: N. = number of beam tests; AVG = average; COV = coefficient of variation.

The results of the comparative study are presented in Appendix, while a synthesis of the statistical characteristics is shown in Table 1. It is seen that the proposed model provides significantly good predictions, with a mean Vexp/Vtheor value of 1.10 and a coefficient of variation of 20.3%. At the contrary, the overall predictions of the other three models (ACI 440, Chen-ACI, Chen-EC2) do not appear to be equally satisfactory, with a mean Vexp/Vtheor values ranging between 0.90 and 1.56 and the coefficients of variation between 32.33% and 40.36%. The Chen et al. model also show a discrete agreement with the experimental data, while the ACI model provides very conservative results as indicated by the mean Vexp/Vtheor value of 1.56. At the same time, the ACI model appears also as the less accuracy as reflected by the high value of coefficient of variation equals to 40.36%. It is important to note that among the models considered, the ACI model is the only one that does not take into account the interaction effect between the internal and external shear reinforcement. To better evaluate the reliability and the performance of different shear strength models, taking into account not only the safety and precision but also the economy, the methodology proposed originally by Collins [29] could be very useful [30]. In this methodology, called Demerit Point Scale Methodology (DPSM), a score is attributed for each range of Vexp/Vtheor ratio, as shown in Table 2. The total demerit point score of each predictive equation is obtained by summing the products of the percentage of Vexp/Vtheor falling in each range times the demerit points attributed to that range. As the demerit point scale is structured, it is clear that the larger is the total demerit point score and worse is the performance of the model. The procedure proposed by Collins [29] allows us to overcome the limitation of a common classification system based only on the main statistics measures (average, range, COV, etc.) regarding the behavior of the Vexp/Vtheor parameter. In fact, from a structural safety point of view, having Vexp/Vtheor lower than 0.5 is worse than Vexp/Vtheor > 2.0, which is not taken into account on current statistics analysis. With the procedure suggested by Collins [29], a penalty is assigned to each range of Vexp/Vtheor parameter and the total of penalties determines the performance of each predictive model. The Demerit Point Scale (DPS) assumes that the predictions within the range of Vexp/Vtheor = 0.85–1.30 are considered appropriate and so the

Table 2 Demerit point classification. Classification

Vexp/Vtheor

Score

Extremely dangerous Dangerous Low safety Appropriate safety Conservative Extremely conservative

<0.50 0.50–0.65 0.65–0.85 0.85–1.30 1.30–2.00 >2.00

10 5 2 0 1 2

V. Colotti / Construction and Building Materials 124 (2016) 855–865 Table 3 Results of comparison: demerit point classification.

a b

861

6. Conclusions

Vexp/Vtheor

Score

Proposed model

ACI model

Chen-ACI model

Chen-EC2 model

<0.50 0.50–0.65 0.65–0.85 0.85–1.30 1.30–2.00 >2.00 Total demerit score

10 5 2 0 1 2 point

0a 1 14 65 19 0 52b

1 4 7 24 43 20 127

1 9 4 44 38 4 109

7 15 29 37 11 1 216

Percentage values of Vexp/Vtheor results. 52 = (0  10) + (1  5) + (14  2) + (65  0) + (19  1) + (0  2).

Demerit Point (DP) is zero. Moreover, the predictions within the unsafe region are penalized more intensively than those on the conservative side (refer to Table 2). The results of comparative study based on the DPSM are presented in Table 3. The analysis of these results reveals that the proposed model provides the better predicting results in terms of safety, precision and economy, with a total demerit point score equal to 52, compared with the values of 127, 109 and 216 obtained for the ACI model, Chen-ACI and Chen-EC2 model, respectively. A careful analysis of results in Table 3 reveals that ACI model has 20% of Vexp/Vtheor values in the extremely conservative range, much larger in relation to the other models, thus confirming the very conservative nature of such model. On the other hand, the proposed model presents a large percentage (65%) of results in appropriate safety range, and only 1% of results in dangerous and extremely dangerous ranges, which confirms the smallest value of the total demerit point score and the very good performance of this model. Again, for the models under appraisal, the sum of the percentage of results in the conservative and appropriate safety ranges are 84, 82, 48 and 67 for the proposed model, Chen-ACI model, Chen-EC2 model, and ACI model, respectively. This findings reveal that the models that explicitly consider the shear interaction effect have the better performance among the appraised models. From these models, the largest percentage of results in appropriate safety range provided by the proposed model (65 versus 44 and 37) justifies the smallest value of the demerit point score and, therefore, the very satisfactory performance of the proposed model. A separate comment deserve the different performances obtained with the Chen et al. model when combined with the two basic codes taken into account. There are two fundamental features in which the methods of the American and European codes differ. The first is the choice of a value for the angle h of the truss model. The second is whether the truss has to bring the entire shear force or if a part of the shear is being carried by the concrete. ACI 318 code recommends to use the sum of concrete and steel contributions with the angle of the shear crack h = 45°, while Eurocode 2 recommends to use only the steel contribution with an angle h variable between 21.8° and 45°. The results of the present analysis show that when the Chen et al. model is combined with the ACI 318 code, it provides a discrete agreement with the test data, and it is generally conservative. At the contrary, when the Chen et al. model is combined with the Eurocode 2, it provides results that overestimate the experimental data and it is strongly unconservative. This aspect is confirmed by the very high total demerit point score equal to 216, resulting of the largest percentage of results (22%) in dangerous and extremely dangerous range, as reported in Table 3.

In this paper a new theoretically-based model is presented to predict the shear strength of RC beams strengthened with externally bonded FRP reinforcement. The model is derived by a previous authors’ work and is based on the PlasticityTruss Model approach, opportunely refined in order to capture the interaction effect between the internal and external shear reinforcement. For this purpose, the results in closed form solution obtained by a sophisticated analytical model describing the debonding process in a FRP shear-strengthened RC beam [19] are suitably implemented in the new shear strength model. Significant features of the model are that it takes into account other critical aspects of the shear behavior, such as variable crack angle and shear span/depth ratio and, moreover, empirical assumptions are essentially restricted to the level of constitutive relationships of materials and their adhesion property. The model has been validated against 201 RC beam tests from a large number of different sources reported in published literature and representing a wide range of test geometries, structural variables, and FRP configuration schemes (fully wraps, U-wraps, side bonding). The predicted results compare very well with experimental data, with a mean experimental/theoretical failure load ratio Vexp/Vtheor of 1.10 and an acceptable coefficient of correlation of about 20%. For comparison purposes, the predicted results of the model have been compared with those of the ACI 440 model and with those of a recent shear strength model proposed by Chen et al. [17]. It has been shown that when the Chen et al. model is combined with the ACI 318 basic code, there is a discrete agreement with the experimental data, with a mean Vexp/Vtheor values of 1.23 and a coefficient of correlation of 32%, while when it is combined with the Eurocode 2, it tends to overestimate the experimental results and it is strongly unconservative, with a mean Vexp/Vtheor value of 0.90 and a coefficient of correlation of 37%. The ACI 440 model, instead, provides very conservative results, with a mean Vexp/Vtheor value of 1.56 and, at the same time, it appears also as the less accurate, with a coefficient of variation of about 40%. It should be noted that among the models considered, the ACI 440 model is the only one that does not take into account the shear interaction effect. To better evaluate the reliability and the performance of the shear strength models examined, taking into account not only the safety and precision but also the economy, the Demerit Point Scale Methodology proposed by Collins [29] has been utilized. The comparative analysis based on this methodology reveals that the proposed model provides the better performance, with a total demerit point score equal to 52, compared with the values of 127, 109 and 216 for the ACI 440 model, Chen-ACI model and Chen-EC2 model, respectively. The obtained results have shown that the proposed model appears as a very good predictor of shear strength, with sufficient accuracy and reliability and, therefore, can be usefully utilized as an effective tool for developing safe and economic design procedures for FRP shear-strengthened RC beams. The possibility to extend the applicability of the presented shear strength model to other FRP strengthening technique, such as Near Surface Mounted, will be the subject of future research.

Compliance with ethical standards This study was funded in part by the Ministry of University and Scientific Research (MIUR), Italy. The author declares that he has no conflict of interest.

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V. Colotti / Construction and Building Materials 124 (2016) 855–865

Appendix A. Appendix Test database: comparison between experimental and analytical results obtained with different models. Reference

Beam N.

Beam Specimen

Type S-U-W

Vexp (kN)

Vmod (kN)

VACI (kN)

VC-A (kN)

VC-E (kN)

Vexp/ Vmod

Vexp/ VACI

Vexp/ VC-A

Vexp/ VC-E

Araki et al. [31]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

CF-045 CF-064 CF-097 CF-131 CF-243 AF-060 AF-090 AF-120 No.2 No.3 No.7 No.8 A2 A3 A5 Bb Bc Be Bf SB-F1 SB-F2 MB-F1 MB-F2 LB-F1 LB-F2 PC1 PC2 A290W A290WR W7 BDF-00 BDF-R6 BDF-R8 S5-CS S5-CA RS4Wa RS3Wa RS2Wa ST1b ST2b ST3b C1 C2 G1 G2 G3 G4 GS1 GS1a GS3 GS5 RR2 RR4 RR5 RR6 BS7 V12-A V18-A V20-A V18-B V16-A BT2 BT4 S3 S5 A10_M A12_M B10_M B12_M

W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W U U U U U U U U

236,0 262,0 307,0 358,0 407,0 237,0 259,0 312,0 285,2 236,0 568,8 529,6 185,0 187,0 227,0 136,0 121,0 178,0 161,0 66,1 66,7 236,3 250,3 871,5 881,1 177,5 155,0 367,0 388,0 1110,5 132,1 180,9 213,6 725,0 888,0 250,0 330,0 300,0 242,0 270,0 279,0 603,1 716,0 469,0 551,5 581,2 689,5 319,5 356,2 427,5 515,0 60,7 56,0 63,2 61,6 235,5 116,4 127,3 140,1 202,4 184,0 155,0 162,0 202,1 198,2 61,0 89,8 55,6 71,5

262,0 287,5 304,8 320,4 358,2 282,8 298,1 312,5 258,8 258,4 549,7 550,8 126,9 119,8 159,6 131,4 111,9 154,7 117,0 60,6 60,6 171,8 171,8 733,0 733,0 122,8 118,9 268,8 279,3 814,6 136,0 153,0 165,4 589,4 622,5 208,8 210,0 212,1 243,7 262,1 272,1 506,1 550,4 459,3 501,6 529,1 548,6 426,0 438,9 500,4 535,1 42,4 40,0 40,0 40,0 202,8 137,6 137,6 137,6 156,0 156,0 140,6 127,9 134,9 146,2 47,6 85,0 44,3 77,0

115,5 127,2 147,0 166,6 232,8 114,4 124,0 133,8 128,1 113,7 315,0 280,0 68,1 61,1 68,1 49,8 38,8 49,8 38,8 33,3 33,3 130,9 130,9 566,7 566,7 106,6 100,8 181,3 175,6 385,8 66,5 97,1 134,5 560,5 768,7 369,4 386,2 419,6 124,1 146,2 168,2 303,3 385,9 246,3 271,8 297,3 322,8 229,3 233,5 259,0 284,5 72,5 72,4 72,3 73,1 131,5 62,4 62,4 62,4 87,7 87,7 119,8 79,8 78,3 99,7 72,5 95,6 49,4 73,7

NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA 142,68 107,33 94,09 108,56 80,00 105,10 51,35 67,52

NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA 223,6 135,2 96,0 134,9 77,3 140,4 67,5 108,4

0,90 0,91 1,01 1,12 1,14 0,84 0,87 1,00 1,10 0,91 1,03 0,96 1,46 1,56 1,42 1,04 1,08 1,15 1,38 1,09 1,10 1,38 1,46 1,19 1,20 1,45 1,30 1,37 1,39 1,36 0,97 1,18 1,29 1,23 1,43 1,20 1,57 1,41 0,99 1,03 1,03 1,19 1,30 1,02 1,10 1,10 1,26 0,75 0,81 0,85 0,96 1,43 1,40 1,58 1,54 1,16 0,85 0,92 1,02 1,30 1,18 1,10 1,27 1,50 1,36 1,28 1,06 1,26 0,93

2,04 2,06 2,09 2,15 1,75 2,07 2,09 2,33 2,23 2,08 1,81 1,89 2,72 3,06 3,33 2,73 3,12 3,58 4,15 1,99 2,00 1,80 1,91 1,54 1,55 1,66 1,54 2,02 2,21 2,88 1,99 1,86 1,59 1,29 1,16 0,68 0,85 0,71 1,95 1,85 1,66 1,99 1,86 1,90 2,03 1,95 2,14 1,39 1,53 1,65 1,81 0,84 0,77 0,87 0,84 1,79 1,86 2,04 2,24 2,31 2,10 1,29 2,03 2,58 1,99 0,84 0,94 1,13 0,97

NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA 1,09 1,51 2,15 1,83 0,76 0,85 1,08 1,06

NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA 0,69 1,20 2,11 1,47 0,79 0,64 0,82 0,66

Kamiharako et al. [32]

Cao et al. [33]

Leung et al. [8]

Diagana et al. [34] Carolin et al. [35] Godat et al. [36] Teng et al. [14]

Colalillo et al. [37] Grande et al. [15]

Ianniruberto et al. [38]

Lee et al. [39]

Al-Amery et al. [40]

Matthys [41] Beber et al. [42]

Khalifa et al. [43] Sato et al. [44] Barros et al. [45]

863

V. Colotti / Construction and Building Materials 124 (2016) 855–865 Appendix (continued) Reference

Beam N.

Beam Specimen

Type S-U-W

Vexp (kN)

Vmod (kN)

VACI (kN)

VC-A (kN)

VC-E (kN)

Vexp/ Vmod

Vexp/ VACI

Vexp/ VC-A

Vexp/ VC-E

Adhikary et al. [46]

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142

B-7 B-8 U4 U5 U6 SO3-2 SO3-3 SO3-4 SO3-5 SO4-2 SO4-3 SW3-2 SW4-2 BS2 BS5 BS6 B1 B3 B5 B19 UF90 LB-U1 LB-U2 MB-U1 MB-U2 PU1 PU2 B2S B3S B3FS RC8NA RC12NA RS4Ub RS3Ua RS2Ua ED1_S0_0.5L ED1_S0_1L ED1_S0_2L ED1_S1-0.5L ED1_S1-1L ED1_S1_2L ED1_S2_1L ED1_S2_2L ED2_S0_1L ED2_S0_2L ED2_S1_1L ED2_S1_2L JO JP BT1-1 BT1-1I BT1-2I BT2-1 B-U1-C-14 B-U2-C-14 B-U1-C-17 B-U2-C-17 V10-A V10-B V17-A V11-A V11-B V17-B V15-B V16-B S0-0.12R S0-0.17R1 S0-0.20R2 S0-0.20R1 S0-0.23R S0-0.33R S0-0.66R S1-0.17R2

U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U

68,5 85,8 101,5 404,5 1009,0 131,0 133,5 144,5 169,5 127,5 155,0 177,0 180,5 247,5 170,0 166,7 701,0 60,7 66,6 55,5 125,0 563,0 559,8 154,6 159,7 142,5 130,0 112,5 75,0 77,5 850,9 765,1 225,0 270,0 280,0 102,0 120,0 122,0 282,0 255,0 267,0 309,0 297,0 59,0 68,0 96,0 105,0 50,1 62,3 67,4 87,3 67,4 67,4 252,9 264,8 238,9 243,3 107,5 106,0 102,8 98,4 124,8 92,9 138,4 112,4 120,9 134,5 131,1 135,7 150,6 120,0 121,7 246,7

69,6 70,4 82,2 358,0 766,1 141,1 150,3 158,7 158,7 120,5 132,7 185,9 148,6 207,7 186,4 186,4 477,3 58,3 59,4 57,7 145,4 612,8 612,8 143,9 143,9 118,2 114,6 101,6 58,9 71,3 791,1 763,4 225,2 227,2 230,7 163,4 174,2 185,8 200,7 201,9 203,9 213,7 213,3 57,9 61,2 65,3 66,2 37,1 50,9 89,5 89,5 85,5 65,4 291,5 303,9 285,7 299,8 130,2 130,2 130,2 130,2 130,2 130,2 141,4 141,4 158,0 167,2 172,0 172,3 175,7 181,6 193,8 206,1

47,6 65,2 38,4 162,8 366,0 56,3 67,7 90,6 90,6 56,3 90,6 185,3 185,3 152,2 122,5 122,3 535,6 71,2 93,9 74,4 169,7 456,5 456,5 112,6 112,6 99,9 94,9 68,0 68,0 68,0 522,7 521,3 242,0 258,1 290,4 65,7 80,0 98,5 195,4 209,6 228,1 339,3 357,8 18,9 27,5 53,3 62,0 57,9 57,9 55,7 55,7 49,9 45,2 202,8 253,4 185,9 236,5 59,8 59,8 59,8 59,8 59,8 59,8 82,4 82,4 62,4 68,1 71,9 72,1 75,6 86,9 109,6 176,7

65,34 65,91 49,11 174,11 356,07 69,61 80,91 93,25 93,25 69,61 93,25 187,97 187,97 160,97 132,26 132,02 468,13 99,76 126,72 107,08 234,51 498,40 498,40 127,80 127,80 114,24 107,24 80,67 80,67 80,67 583,12 588,61 249,02 263,17 291,47 95,31 111,75 138,08 215,47 227,46 246,01 343,18 353,95 39,71 49,37 71,92 80,17 57,52 57,52 73,27 73,27 65,91 63,17 190,04 207,62 175,30 194,08 74,25 74,25 74,25 74,25 74,25 74,25 89,59 89,59 87,74 99,19 105,29 105,61 110,36 119,82 147,49 202,26

101,0 101,6 72,1 220,6 408,2 90,5 118,7 149,6 149,6 90,5 149,6 193,5 193,5 192,0 120,5 120,5 738,9 168,2 190,4 175,7 311,2 777,6 777,6 210,5 210,5 140,5 123,0 118,4 118,4 118,4 698,8 594,8 408,9 422,0 445,6 127,4 168,5 234,4 301,4 309,9 321,6 350,5 353,4 64,7 81,9 97,1 101,7 101,2 101,2 102,1 102,1 83,7 94,5 339,9 358,9 325,8 347,0 92,6 92,6 92,6 92,6 92,6 92,6 130,9 130,9 95,9 124,6 139,8 140,6 152,5 176,2 245,3 329,4

0,98 1,22 1,24 1,13 1,32 0,93 0,89 0,91 1,07 1,06 1,17 0,95 1,21 1,19 0,91 0,89 1,47 1,04 1,12 0,96 0,86 0,92 0,91 1,07 1,11 1,21 1,13 1,11 1,27 1,09 1,08 1,00 1,00 1,19 1,21 0,62 0,69 0,66 1,40 1,26 1,31 1,45 1,39 1,02 1,11 1,47 1,59 1,35 1,22 0,75 0,98 0,79 1,03 0,87 0,87 0,84 0,81 0,83 0,81 0,79 0,76 0,96 0,71 0,98 0,80 0,77 0,80 0,76 0,79 0,86 0,66 0,63 1,20

1,44 1,32 2,64 2,48 2,76 2,33 1,97 1,59 1,87 2,26 1,71 0,96 0,97 1,63 1,39 1,36 1,31 0,85 0,71 0,75 0,74 1,23 1,23 1,37 1,42 1,43 1,37 1,65 1,10 1,14 1,63 1,47 0,93 1,05 0,96 1,55 1,50 1,24 1,44 1,22 1,17 0,91 0,83 3,13 2,47 1,80 1,69 0,87 1,08 1,21 1,57 1,35 1,49 1,25 1,04 1,28 1,03 1,80 1,77 1,72 1,65 2,09 1,55 1,68 1,36 1,94 1,97 1,82 1,88 1,99 1,38 1,11 1,40

1,05 1,30 2,07 2,32 2,83 1,88 1,65 1,55 1,82 1,83 1,66 0,94 0,96 1,54 1,29 1,26 1,50 0,61 0,53 0,52 0,53 1,13 1,12 1,21 1,25 1,25 1,21 1,39 0,93 0,96 1,46 1,30 0,90 1,03 0,96 1,07 1,07 0,88 1,31 1,12 1,09 0,90 0,84 1,49 1,38 1,33 1,31 0,87 1,08 0,92 1,19 1,02 1,07 1,33 1,28 1,36 1,25 1,45 1,43 1,38 1,33 1,68 1,25 1,54 1,25 1,38 1,36 1,25 1,28 1,36 1,00 0,83 1,22

0,68 0,84 1,41 1,83 2,47 1,45 1,12 0,97 1,13 1,41 1,04 0,91 0,93 1,29 1,41 1,38 0,95 0,36 0,35 0,32 0,40 0,72 0,72 0,73 0,76 1,01 1,06 0,95 0,63 0,65 1,22 1,29 0,55 0,64 0,63 0,80 0,71 0,52 0,94 0,82 0,83 0,88 0,84 0,91 0,83 0,99 1,03 0,49 0,62 0,66 0,86 0,81 0,71 0,74 0,74 0,73 0,70 1,16 1,14 1,11 1,06 1,35 1,00 1,06 0,86 1,26 1,08 0,94 0,97 0,99 0,68 0,50 0,75

Godat et al. [36]

Khalifa et al. [47]

Matthys [41]

Islam et al. [48] Mosallam et al. [49]

Monti et al. [26] Leung et al. [8]

Diagana et al. [34] El-Ghandour [50]

Belarbi et al. [51] Grande et al. [15]

Bousshelam et al. [7]

Al-Sulaimani et al. [52] Jayaprakash et al. [53]

Pellegrino et al. [10]

Beber et al. [42]

Mofidi et al. [54]

(continued on next page)

864

V. Colotti / Construction and Building Materials 124 (2016) 855–865

Appendix (continued) Reference

Baggio et al. [55]

Godat et al. [56]

Monti et al. [26] Khalifa et al. [43] Pellegrino et al. [9]

Al-Sulaimani et al. [52]

Sato et al. [44] Adhikary et al. [46] Carolin et al. [35]

Zhang et al. [57]

Beber et al. [42]

Grande et al. [15]

Kim et al. [58]

Beam N.

Beam Specimen

Type S-U-W

Vexp (kN)

Vmod (kN)

VACI (kN)

VC-A (kN)

VC-E (kN)

Vexp/ Vmod

Vexp/ VACI

Vexp/ VC-A

Vexp/ VC-E

143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201

S1-0.23R S1-0.33R Beam 2 Beam 6 Beam 7 LSB-140-1L LSB-140-2L LSB-203-1L LSB-203-2L LSB-203-3L LSB-406-1L LSB-406-2L LSB-610-1L LSB-610-2L LSB-610-3L SF90 BT3 BT5 TR30C2 TR30C3 TR30C4 TR30D10 TR30D2 TR30D20 TR30D3 TR30D4 TR30D40 SO SP WO WP S2 S4 B-4 A-290b B-290 B-390 Z11-S90 Z22-S90 Z31-F90 Z31-FD Z42-F90 Z42-FD V9-A V9-B V21-A V13-A V13-B V22-B V20-B RS4Sb RS3Sb RS2Sb CP2-VW CP3-VW CS2-VW CS3-VW CP2-1VS CP3-1VS

U U U U U U U U U U U U U U U S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S

253,9 250,6 186,5 167,0 152,5 268,8 288,1 250,3 276,4 295,7 231,8 268,8 216,7 213,4 216,7 112,5 157,5 121,5 120,0 112,8 140,2 193,0 213,3 247,5 161,4 208,8 212,0 41,5 41,2 42,0 45,2 160,5 156,3 58,6 298,0 298,0 298,0 84,6 104,6 76,5 87,7 128,6 133,5 98,1 104,3 115,2 122,0 125,8 112,5 142,9 180,0 205,0 240,0 173,0 154,0 170,0 108,0 163,0 94,5

208,2 209,4 165,9 158,7 158,7 194,5 198,5 194,2 198,4 200,0 193,8 198,3 193,6 198,3 199,9 139,9 135,2 121,6 112,1 129,3 129,3 165,8 173,2 173,2 158,9 165,8 165,8 35,9 48,7 37,3 51,2 129,8 137,9 64,5 233,1 235,9 241,0 84,5 111,0 67,9 67,9 86,6 86,6 123,9 123,9 123,9 133,2 133,2 177,6 177,6 213,5 216,1 220,5 140,5 115,9 142,6 117,2 132,7 111,2

184,2 195,4 131,4 106,1 103,9 472,8 566,9 422,2 516,3 585,3 366,0 460,2 347,3 441,4 510,3 161,8 105,0 73,9 70,6 114,4 114,4 173,1 193,1 193,1 145,2 173,1 173,1 35,3 35,3 31,7 31,7 80,0 91,4 29,4 172,2 178,2 210,6 84,3 84,3 44,1 44,1 44,1 44,1 54,8 54,8 54,8 72,3 72,3 122,0 122,0 231,1 247,2 279,5 170,3 170,3 197,8 197,8 112,2 112,2

212,37 219,76 150,65 126,26 126,26 435,78 493,73 403,89 469,29 501,61 368,47 442,13 356,62 433,05 472,35 190,86 130,23 99,80 83,91 106,16 106,16 148,93 154,86 154,86 139,29 148,93 148,93 42,37 42,37 51,31 51,31 92,77 99,89 54,88 186,31 181,37 193,70 67,80 67,80 56,27 56,27 56,27 56,27 69,34 69,34 69,34 82,65 82,65 105,46 105,46 214,08 226,67 252,65 121,69 121,69 126,28 126,28 101,86 101,86

339,0 346,2 231,7 170,0 170,0 462,7 469,6 456,6 467,5 470,0 446,7 464,2 442,4 462,9 468,3 276,2 192,5 116,4 127,8 183,5 183,5 239,3 244,5 244,5 230,2 239,3 239,3 59,7 59,7 82,2 82,2 86,3 117,0 77,6 227,8 269,6 300,7 113,3 113,3 84,4 84,4 84,4 84,4 80,3 80,3 80,3 113,6 113,6 170,6 170,6 332,6 360,7 405,2 169,2 169,2 180,7 180,7 119,7 119,7 Mean COV (%)

1,22 1,20 1,12 1,05 0,96 1,38 1,45 1,29 1,39 1,48 1,20 1,36 1,12 1,08 1,08 0,80 1,16 1,00 1,07 0,87 1,08 1,16 1,23 1,43 1,02 1,26 1,28 1,16 0,85 1,13 0,88 1,24 1,13 0,91 1,28 1,26 1,24 1,00 0,94 1,13 1,29 1,49 1,54 0,79 0,84 0,93 0,92 0,94 0,63 0,80 0,84 0,95 1,09 1,23 1,33 1,19 0,92 1,23 0,85 1,10 20,30

1,38 1,28 1,42 1,57 1,47 0,57 0,51 0,59 0,54 0,51 0,63 0,58 0,62 0,48 0,42 0,70 1,50 1,64 1,70 0,99 1,23 1,11 1,10 1,28 1,11 1,21 1,22 1,18 1,17 1,32 1,43 2,01 1,71 1,99 1,73 1,67 1,41 1,00 1,24 1,74 1,99 2,92 3,03 1,79 1,90 2,10 1,69 1,74 0,92 1,17 0,78 0,83 0,86 1,02 0,90 0,86 0,55 1,45 0,84 1,56 40,36

1,20 1,14 1,24 1,32 1,21 0,62 0,58 0,62 0,59 0,59 0,63 0,61 0,61 0,49 0,46 0,59 1,21 1,22 1,43 1,06 1,32 1,30 1,38 1,60 1,16 1,40 1,42 0,98 0,97 0,82 0,88 1,73 1,56 1,07 1,60 1,64 1,54 1,25 1,54 1,36 1,56 2,29 2,37 1,41 1,50 1,66 1,48 1,52 1,07 1,36 0,84 0,90 0,95 1,42 1,27 1,35 0,86 1,60 0,93 1,23 32,33

0,75 0,72 0,80 0,98 0,90 0,58 0,61 0,55 0,59 0,63 0,52 0,58 0,49 0,46 0,46 0,41 0,82 1,04 0,94 0,61 0,76 0,81 0,87 1,01 0,70 0,87 0,89 0,70 0,69 0,51 0,55 1,86 1,34 0,76 1,31 1,11 0,99 0,75 0,92 0,91 1,04 1,52 1,58 1,22 1,30 1,43 1,07 1,11 0,66 0,84 0,54 0,57 0,59 1,02 0,91 0,94 0,60 1,36 0,79 0,90 37,76

Note: S = side bonded; U = U wrapped; W = fully wrapped; NA = not applicable. VC-A = Chen-ACI model; VC-E = Chen-Eurocode model.

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