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; Morita and Kaku 1979, Morita and Fujii 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.
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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.
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:
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 \(\sigma_t,r_s \) 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 σ t,r = f ct and r = r s = r cr into Eq. (8) the radial stress at the front crack can be evaluated:
Following the approach proposed by Van der Veen (1990) and neglecting the influence of the radial stress on the circumferential deformation, the total elongation Δ 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):
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:
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.
where Tenv is the toughness and χenv is the dimensionless factor in environmental conditions. The effects of the environmental conditions on the geometries of subsurface cracks and stress fields can be quantified by combining Eqs. (8) and (9) to yield [40] 2ff7e9595c
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