Bond slip model in cylindrical reinforced concrete elements conﬁned with stirrups

An analytical model able to evaluate the bond-slip law of conﬁned reinforced concrete elements is developed and presented in this paper. The model is based on the studies developed by Tepfers and by den Uijl and Bigaj on the thick-walled cylinder model and extended to the case of the presence of transverse reinforcement. The bond strength and the considered failure modes (splitting or pull-out failure) are expressed as a function of the geometrical (concrete cover and transverse reinforcement) and mechanical (con-crete strength) parameters of the element. The application of the proposed methodology allows to forecast the failure mode, and equations for the bond-slip law are ﬁnally proposed for a range of steel strain lower than the yielding one.

List of symbols r r;r i Radial stress component on the cylinder at radius r i r t;r i Circumferential stress component on the cylinder at radius r i r r;st Radial stress on the stirrups r t;st Axial stress in the stirrups / st Stirrup diameter / eq Equivalent stirrup diameter / b Longitudinal rebar diameter c Rebar concrete cover c 1 Cylinder radius r s Rebar radius r cr Radius of the cracked part of the cylinder u r;r i Radial displacement at radius r i p st Stirrups spacing q cc Ratio between longitudinal reinforcement area and core section area q s Ratio of the volume of transverse confining steel to the volume of confined concrete core e r;r i Radial strain at radius r i m c Concrete Poisson coefficient f ct Concrete peak tensile strength f cm Mean compressive strength of the concrete w r Crack width at radius r s slip s b Bond stress s max Maximum bond stress

Introduction
The definition of the bond-slip law in reinforced concrete structures is very important in the context of a correct evaluation of deformability, ductility and crack evolution (opening and spacing). Rehm (1961) experimentally showed that the stress arising at the bar-concrete interface depends on the relative sliding, generated between the bar and the surrounding concrete. The bond failure of the adhesion may occur with the propagation of the splitting cracks through the concrete cover (splitting failure) or with the crisis of the concrete corbels (in compression) between the ribs (pull-out failure). The bond strength and the failure modes generally depend on the effectiveness of the confinement provided by the concrete cover and the distance between the bars (Ferguson et al. 1954;Morita and Kaku 1979) and by the transverse reinforcement (Orangun et al. 1977; Kemp and Wilhelm 1979;Skorogobatov and Edwards 1979;Kaku 1979, Morita andFujii 1982). The influence of the relative rib area and bar diameter on the local bond behaviour is pointed out in (Metelli and Plizzari 2014). Different authors (Rehm 1961;Eligehausen et al. 1983;Shima et al. 1987;Giuriani et al. 1991;Cairns and Jones 1996;Gambarova and Rosati 1997;Yasojima and Kanakubo 2008) proposed analytical formulations for the bond-slip behavior based on experimental results.
In particular, Yasojima and Kanakubo (2008) conducted pull-out bond tests for obtaining the local bond stress versus slip relationship in specimens with confinement of lateral reinforcement. The test results showed the increase of the maximum bond stress with the increment of the lateral confinement stress, and showed how the slippage at maximum bond stress was influenced by splitting crack width and shape of main reinforcement. Finally, based on the obtained results a relationship between bond stress and slippage in case of confinement of lateral reinforcement was proposed.
Finally, the Model Code 2010 suggests bond-slip relationships distinguishing between pull-out and splitting failure, but in this last case, only, allows accounting for the stirrups presence.
Aim of this paper is the definition of an analytical model that allows the evaluation of the bond-slip relationship in confined reinforced concrete elements for a range of steel strain lower than the yielding one. The main difference of the proposed procedure, with respect to the above cited ones, lies in the possibility of recognizing, through a physical, analytical model, the actual failure mode (splitting of the concrete cover or pull-out of the ribbed bar), accounting for the stirrup presence. In particular, the proposed model extends the studies by Tepfers (1979) and den Uijl and Bigaj (1996), developed for unconfined elements on a thick-walled cylinder and by Coccia et al. (2014) for the evaluation of the corrosion in reinforced concrete elements on the bond strength, introducing the effect of transverse reinforcement. Finally, a formulation of the bond-slip relationship, expressed as a function of the mechanical and geometrical parameters, is obtained through a regression with the ordinary least squares method, based on the analytical outcomes of a parametric survey. The results obtained with the proposed model are compared with the most recent formulations available in the literature and discussed above (Yasojima and Kanakubo 2008;Model Code 2010).

Confinement model
The effect of the confinement on the bond behavior is described with reference to the thick-walled cylinder model ( Fig. 1a) developed by Tepfers (1979) and den Uijl and Bigaj (1996) for elements without transverse reinforcement. The confinement effect introduced in these models is related to the cover concrete of the ribbed bar, only. In the present approach, the confinement due to transversal reinforcement is considered and it is modeled introducing an external radial compressive stress, hereinafter referred to as r r,st . This last variable is here defined as a function of the geometrical characteristics of the cylinder, of the mechanical properties of the stirrups and of the stress state in the cylinder wall.
The response of the thick-walled cylinder is examined in terms of radial stress (radial component of the bond action) and radial deformation at the interface between the ribbed bar and the concrete. A linear elastic distribution of the radial stresses is assumed, up to the reaching of the concrete tensile strength. At this stage, a certain number of radial cracks forms and the cracked part of the cylinder is characterized by a non-linear behaviour. Therefore, three stages can be considered: the uncracked, the partly cracked and the entirely cracked stages, pointed out, in the following, with the superscripts I, II and III, respectively.

Uncracked stage
Since in the first uncracked stage a linear elastic behavior of the cylinder is assumed, the equations given by Timoshenko (1976) can be adopted. In particular, the reference element is characterized by the stress boundary conditions related to the internal radial pressure and external stirrups: where the subscripts r,r i and r,r e correspond to the radial component at the interface and at the external radius of the cylinder, respectively, r s is the radius of the ribbed bar, r r;r s is the radial component of the bond stress and r r,st is the radial confinement caused by the transversal reinforcement. According to Mariotte's law (Fig. 2), the stress r r,st is evaluated as: where r t,st is the axial stress of the stirrups, / eq is the equivalent diameter of the transversal reinforcement, c 1 is the external radius of the cylinder (Fig. 1a). The equivalent diameter of stirrups is introduced in order to account for two aspects. Firstly, since the action of the transverse reinforcement is concentrated, it cannot be considered diffused on the entire length of the cylinder between individual stirrups; secondly not the whole concrete core is effectively confined by the stirrups. Thus, the equivalent diameter is expressed as: where / st is the diameter of the transverse reinforcement, p st is the transverse reinforcement spacing, k eq is the coefficient of efficiency proposed by Mander et al. (1988) that, in the case of the thick-walled cylinder, is equal to: being q cc the ratio between the area of longitudinal reinforcement and the area of core section. The axial stress of the transversal reinforcement is estimated by the circumferential stress on the cylinder at the external radius, neglecting the minor influence of Poisson effect and assuming an elastic behavior for steel: where r t;r e is the circumferential stress at the external radius of the cylinder, E s and E c the Young modulus of steel and concrete, respectively. Finally, in this stage, the equations given by Timoshenko with the boundary conditions (Eqs. (1) and (2)) in terms of stresses and displacement are: where k c ¼ / eq c 1 E s E c is the coefficient accounting for the confinement effect given by the stirrups.
The radial strain at the interface e r;r s is obtained by normalizing the radial displacement u r;r s to the bar radius.

Partly cracked stage
The uncracked stage ends when the circumferential stress at the interface reaches the tensile strength of the concrete. The fracture criterion, considered both by Tepfers (1979) and den Uijl and Bigaj (1996), is related to a uniaxial state of stress, even if the stress state is evidently biaxial. Talaat and Mosalam (2007) show that the solution accounting for a biaxial failure criterion is not so different from the one related to a uniaxial failure criterion. For this reason, in the proposed approach, the partly cracked stage starts when r t;r s in Eq. (8) is equal to the concrete uniaxial tensile strength f ct . Due to the cracks formation, the cylinder is divided in an internal cracked part and an external uncracked part (Fig. 1b). In the uncracked part the behavior can be considered linear elastic (superscript LE, in the following), while the behavior of the cracked part of the cylinder is non linear (superscript NL, in the following).
As already mentioned, at the crack front (r = r cr ) the circumferential stress is equal to the tensile strength f ct . Thus, substituting r t,r = f ct and r = r s = r cr into Eq. (8) the radial stress at the front crack can be evaluated: Hence, the contribution of the uncracked part of the cylinder at the interface is: If Eq. (10) is substituted into Eq. (9) and it is assumed f ct /E c = e cr , the radial displacement at the crack front is: Thus the radial strain at the interface is: The contribution of the cracked part of the cylinder is evaluated by considering the softening behavior of concrete in tension (Fig. 1d). In particular the model of bilinear softening, proposed by Roelfstra and Wittmann (1986) based on Hillerborg's fictitious crack model (1983), is adopted. According to this theory, the fictitious crack width at the generic radius r can be expressed as: where a i , b i and w 0 are the parameters of the softening model (a 1 ¼ À and w, f are the coordinates of the intersection point of the two softening lines (Fig. 1d).
Following the approach proposed by Van der Veen (1990) and neglecting the influence of the radial stress on the circumferential deformation, the total elongation D t,r of a circular fiber with radius r can be expressed as the sum of a rigid radial displacement, giving rise to a constant crack width (Fig. 3a), and an elastic elongation (Fig. 3b): where n is the number of radial cracks. At the crack front, where the circumferential stress is equal to f ct , the cracks are closed, therefore the total elongation depends only on the elastic deformation and, neglecting the Poisson effect, it is equal to 2pr cr e cr . Substituting into Eq. (15) and solving it: where C 2 ¼ 2pe cr nw 0 .
The radial stress granting the equilibrium of the circumferential state of stress in the cracked part of the cylinder is: In order to simplify the solution of Eq. (17), the strain e t,r is set equal to e cr . This assumption results in an overestimation of the circumferential deformation but, on the other hand, neglecting the Poisson effect results in an underestimation of this quantity. Talaat and Mosalam (2007) showed that these two assumptions balance each other, minimizing the committed error.
The solution of Eq. (17), with the substituting of Eq. (16), allows calculating the radial stress at the interface given by the cracked part of the cylinder: Thus, the total confinement radial stress in the stage II is given by the sum of Eqs. (11) and (18): In the evaluation of the radial deformation caused by the cracked part of the cylinder, the Poisson effect is negligible if compared to the effect of the radial cracks. The variation of the wall thickness can be expressed as: Solving Eq. (20) the two contributions are: Dr cr;2 ¼ e cr a i C 2 4 2r 2 cr ln r cr r s À 2r 2 cr À r 2 s þ 4r cr r s þ e cr b i r cr ln r cr r s À r cr þ r s ð22Þ The radial strain at the interface due to the cracked part of the cylinder is given by: Thus, the total radial strain at the interface in the stage II is given by the sum of Eqs. (13) and (23)

Entirely cracked stage
The entirely cracked stage starts when the crack front reaches the external radius of the cylinder r cr = c 1 (Fig. 1c). The cracks become wider, the confining action of the concrete diminishes due to the softening behavior and the confining action of the transverse reinforcement increases due to its stiffness which contrasts the crack opening. The circumferential tensile stress is evaluated assuming a constant elongation of the fiber D t,r = D tot and the circumferential strain e t,r is set equal to e cr . Rewriting Eq. (15) the circumferential stress is found: where C 3 ¼ D tot nw 0 and C 2 is defined in Eq. (16). The radial stress is given by two contributing factors: the radial stress in the concrete wall which makes equilibrium with the circumferential stress and the radial stress due to the transversal reinforcement: where r t;st ¼ D tot 2pc 1 E s Substituting Eq. (25) into Eq. (26) and solving the radial stress at interface is found: The radial strain at the interface consists in three contributions: the radial strain due to the confining action of the transversal reinforcement at the end of the second stage, the radial deformation due to rigid motion and the radial deformation due to the variation of the wall thickness caused by the radial stress: Finally, the solution of Eq. (28) gives the radial strain e III r;r s : Bond model The bond model is formulated, in agreement with den Uijl and Bigaj (1996), by assuming a boundary layer with a conical surface, in order to skip the problem of local high non-linear stress and strains. The bond mechanism is based on dry friction, thus the bond stress is directly proportional to the radial stress r r,rs : where cot h is the friction coefficient. Splitting failure occurs when the crack front reaches the external face of the cylinder, while when pull-out failure takes place, a cylindrical sliding plain surrounding the ribbed bar forms.
As proposed by (den Uijl and Bigaj 1996) for the splitting failure the relationship between the radial displacement and the slip s is: where u is the angle between cone surface and bar axis, assumed equal to 0.1 f cm (den Uijl and Bigaj 1996) with f cm the mean compressive strength of the concrete.
For the pull-out failure it is considered the relationship proposed by (den Uijl and Bigaj 1996) for steel strains lower than the yielding one. In this case, the radial deformation-slip law simplifies in a piecewise function defined by the points a, b, c and d (Fig. 4) whose coordinates are given in the Table 1. Branches A and B are cubic parabola, branch C is an exponential function; the point b is characterized by a horizontal tangent and the point c is a point of inflection.
The transition from splitting to pull-out failure is identified by the maximum bond stress which must be limited to 5 f ct (den Uijl and Bigaj 1996). When this value is exceeded a pull-out failure occurs.

Analytical results and comparison with existing formulations
The main results of the proposed model are here reported in the hypothesis of a crack number equal to three, in agreement with den Uijl and Bigaj (1996).  Table 1 Parameters of radial displacement-slip relationship (Fig. 6) for pull-out failure (den Uijl and Bigaj 1996)

Point Deformation
Slip a e ra radial deformation corresponding to r r = s b1 = 5f ct L key length of the concrete between two subsequent ribs Figure 5 shows the influence of the transverse reinforcement on the bond behavior in case of splitting failure. The presence of stirrups leads to an increase of the maximum bond stress and an improvement of the softening branch. It is interesting to notice that in the first branch of the curve, which is followed in the case of pull-out failure, there isn't any effect of the transverse reinforcement. This observation is confirmed by the experimental and numerical results of pull-out tests of Torre-Casanova et al. (2013).
The most recent bond-slip relationship is proposed in Model Code 2010(2012 and allows distinguishing between pull-out and splitting failure, and in this last case, only, allows accounting for the stirrups presence. The Code suggests applying the pull-out curve when there is a good confinement of concrete (concrete cover bigger than 5[ b and distance between the bars bigger than 10[ b , with [ b the longitudinal bar diameter) or there is a ''suitable confining reinforcement'' (MC2010). In Fig. 6a the proposed bond model is compared to the one suggested by Model Code 2010Code (2012 firstly for pull-out failure, case in which the transverse reinforcement does not play any role in both the models. It appears that the maximum bond strength is comparable, while the slip associated to this value, evaluated with the Model Code prescription, is higher than that estimated with the proposed model. This result was already noted by den Uijl and Bigaj (1996) in comparing some test results with the formulation proposed by Model Code 1990. More significant differences are found for splitting failure (Fig. 6b). In this case, the MC2010 provides two models related to the unconfined and confined behaviour, independently of the stirrups amount. Furthermore, the maximum bond strength depends on the concrete compressive strength only, without any influence of the concrete cover to bar diameter ratio. The MC2010 unconfined model appears to be more conservative, with respect to the proposed one. The maximum bond strength of the MC2010 confined model is similar to the one obtained with the proposed model with a stirrup spacing higher than 100 mm but the code relationship presents a lower stiffness in the ascending branch. Yasojima and Kanakubo (2008) proposed a formulation for bond stresses, in case of splitting-failure only, depending on the confining action of the stirrups. In particular, the authors, in order to account for the stirrup presence, add a bond increment (s bc ), to the local bond behavior without lateral reinforcement, calibrated on the basis of experimental results from pull-out tests. The bond term s bc is a function of the mechanical and geometrical properties of the stirrups, of the slip, of the concrete mechanical properties.
In Fig. 7 their results are compared with the one obtained with the proposed model, for compressive concrete strength f cm equal to 30 MPa, bar diameter

Parametric analysis and analytical formulation of the bond-slip model
Finally, a parametric survey is carried out, in order to highlight the main parameters affecting the bond behaviour. In particular, three variables have been considered and varied, i.e. the concrete compressive strength (f cm ), the ratio between the concrete cover and the bar diameter (c/ [ b ) and the ratio of the volume of transverse confining steel to the volume of confined concrete core (q s ). The first parameter f cm has been assumed equal to 30, 35 and 40 MPa; the (c/[ b ) ratio has been set equal to 1.5, 2, 2.5 and 3 and the q s ratio is supposed equal to 0, 1, 1.5, 2, 2.5 and 3 %. All the obtained bond-slip curves can be approximated with three functions defined by the points of coordinates (s 0 ; s max ) and (s 1 ; s r ): Figure 8 shows the influence of the concrete compressive strength (f cm ) and non-dimensional cover (c/[ b ) on the values of bond strength (s max , s r ) and slip (s 0 , s 1 ) in absence of transverse reinforcement (q s = 0). In this case, it can be noted an almost linear variation of the bond strength (s max , s r ) and slips (s 0 , s 1 ) with f cm and c/[ b . In particular s max and s 1 are highly affected by concrete strength (the first one increases, the last one decreases) with respect to s r and s 0 . In particular, this last parameter is almost constant with f cm . The non-dimensional cover ratio influences both strengths and slips that increase with it.
The effect of the stirrups on the bond-slip law is shown in Fig. 9, where the analyzed parameters are plotted as a function of the coefficient q s , assuming f cm = 35 MPa and c/[ b = 2.5. It can be noted the bond stress is influenced by the stirrups only when q s exceeds a minimum percentage of transverse reinforcement (q s,lim ). A formulation of q s,lim , calibrated on the results of the parametric survey is here proposed as a function of the non dimensional cover (c/[ b ): Moreover, Fig. 9 shows the slips are independent of the stirrups confinement while bond stresses increase with it.
The final result of the parametric analysis is the bondslip law proposed in this paper, whose governing parameters are reported in Table 2 for the splitting or the pull-out failure. The first parabolic branch (Eq. 32) is the same for both the failure modes, but is limited to a maximum bond equal to 5f ct in the case of pull-out failure (Fig. 10b).