Bond and Slip of Reinforcing Bars
The bond and slip of reinforcing bars embedded in concrete is an important and certainly difficult phenomenon. In the previous sections, the bond strength was considered a secondary “internal” problem which was treated by means of very simple models of perfectly plastic shear-slip behavior. The ACI Code, too, gives simple provisions for the so-called development length of reinforcing bars, representing tile length of embedment in concrete required for ensuring that the bar can develop its full yield capacity. These formulas are also based on plasticity. The development length is obtained by balancing the steel force at yield Fs — fyAs, againstthe bond strength TcpsLs, where ps is the peri meter of the reinforcement and Ls the slip (development) length. Taking then rc oc f[ oc ff’c leads to ACI empirical formulas in which the development length is Cj Asfyj s/Jc ог Сг/у/у/71< whcre Ct ai, d C-2 are empirical constants
defined in the code. The code also gives various modifications of these formulas to take into consideration the clear spacing of parallel bars, thickness of concrete cover, and corrections for the case of lightweight aggregate concretes and for epoxy coated reinforcement. The formulas, however, do not consider any sire effect associated with the brittleness of concrete.
In reality, the problem of bond and slip is a fracture problem; even more, it is a multiple fracture problem, and a three-dimensional one. Fig. 10.5.2 sketches the three kinds of cracks or crack systems involved in the pullout of a bar from a concrete cylinder or prism. The first system of cracks is transverse to the prism, as shown in Fig. 10.5.2a; the cracks are conical in shape (secondary cracks) or plane (principal cracks). The second system of cracks is longitudinal to the prism; the cracks are generated by hoop stresses and vary in number depending of the morphology of the cross-section (Fig. 10.5.2b). The third system of cracks consists of the shear cracks generated at the interface of steel and concrete; the jump in displacements at the interface is the crack sliding (Fig. 10.5.2c).
The role of transverse fracture of concrete in the bond between concrete and deformed reinforcing bars was first studied by Gylltoft (1984) and by Ingraffea et al. (1984). Gylltoft examined the role of fracture in axisymmetric pullout of bars from concrete blocks. He considered both monotonic and cyclic loading, earned out experiments, and was able to successfully predict the load-slip diagrams observed in the pullout tests. Special crack elements that involve linear strain softening of concrete in tension and a linear strain hardening in shear were used to model the interface. Ingraffea et al. (1984) used a discrete mixed-mode nonlinear fracture model in axisymmetric finite element analysis. They applied the cohesive crack model to characterize the tensile softening at each bond crack, and adopted the aggregate interlock model of Fenwick and Paulay (1968) to characterize the shear softening. The study of Ingraffea et al. (1984) indicated that secondary cracking around the primary cracks contributed to bond slip. Placing special interface elements at the primary crack locations, and comparing numerical results to test results for a center-cracked reinforced concrete plate under uniform tension, Ingraffea et al. (1984) calculated the degradation of stiffness and the crack opening profiles. Ingraffea et al. used in their finite element program (FRANC) a sophisticated technique for remeshing around the crack tip as the crack lip advances. Rots (1988, 1992) analyzed the problem of transverse cracking concomitant with longitudinal cracking. He used a smeared crack approach for the secondary cracks, and a discrete crack approach modeled by interface elements for the primary cracks.
A problem in the foregoing analysis is to correctly handle longitudinal cracking that occurs simultaneously with transverse cracking. In reality, a three-dimensional formulation ought to be used to analyze the problem in detail; however, this would require an enormous computational effort. Axisymmetric formulations have been used, with (he expedient of using a circumferential stress-strain relationship that is a smeared version of a cohesive crack. This is done as in Chapter 8 for the uniaxial case, except that now the stress is the circumferential stress ag and the cracking strain is Eg. These are related to each other and to the crack opening by
where w(r) is the opening of each crack at a distance r from the rotational axis, nc the number ot cracks, and f(w) the softening function (for a single crack). Note that the number of cracks nc must be assumed
before the calculation and cannot be inferred from the analysis. It is usually selected between 2 and 4, based on experience.
An entire family of simplified analyses of longitudinal splitting cracks taking into account the fracture behavior of concrete was developed based on modifications of the initial approach by Tepfers (1973, 1979). Tepfers assumes that the rise of interfacial shear stresses r is accompanied by a rise of a contact normal stress a. He further postulates that at splitting failure, a and r are related by a Coulomb-type law that he writes as
o – = rtan0 (10.5.7)
where ф is a constant complementary friction angle. Tepfers then reduces the analysis to an axisymmetric problem of a thick-walled concrete tube subjected to inner pressure a. Tepfers considers only elastic – brittle behavior and elastic-perfectly plastic behavior. Keeping this approach, several researchers extended Tepfers’ analysis to include softening and fracture. All these analyses use further simplifications, such as neglecting Poisson’s effect, and use the circumferential smeared cracking as given by (10.5.6). The main difference is in the kind of softening curve used by the various authors: van der Veen (1991) uses a power-law softening (Reinhardt 1984); Reinhardt and Van der Veen (1992) and Reinhardt (1992) use the CHR softening curve (Cornelissen, Hordijkand Reinhardt 1986b; see Section 7.2.1); Rosati and Schumm (1992) use a hyperbolic law; and Noghabai (1995a, b) uses a linear softening. As a further difference, Rosati and Schumm (1992) consider, instead of the Coulomb criterion (10.5.7), a Mohr-Coulomb condition given by a — (r – To) tan ф, in which to is a constant “cohesion”.
The problem of discrete longitudinal splitting cracks was directly addressed by Choi, Darwin and McCabe (1990). They used a three-dimensional finite element method to analyze test results and design code provisions on the bond failure of epoxy-coated or uncoated steel bars as a function of the bar size, variations in interface characteristics, and specimen geometry. .Splitting fractures observed in the tests were well reproduced by the computational model —based on a cohesive crack model— and the results provided support to some empirical code provisions. The computational results described well the increase of pullout strength with the cover thickness.
Recently, Noghabai (1995a, b) considered the numerical analysis of longitudinal splitting cracks based on the boundary conditions in Tepfers’ approach (concrete thick-walled tube with internal stress a) and analyzed localized cracking using three numerical approaches with the same underlying material mode! (a cohesive crack with linear or nonlinear softening). The first numerical procedure —the so-called discrete crack approach— was carried out by placing 28 radial layers of interface elements incorporating the stress – crack opening relationship. The strength of the layers was randomly assigned to promote localization into a small number of cracks. The second numerical procedure was the classical smeared crack approach, with the crack opening distributed within the elements. The third procedure used enriched shape functions to describe the displacement jump within each element —the so-called inner softening band finite elements (Klisinski, Runesson and Sture 1991; Klisinski, Olofsson and Tano 1995). The three procedures gave similar results for the curves of pressure vs. radial deformation, although none were able to continue into the structural softening branch. The inner softening band method seems very promising for capturing the cracking pattern. Still, the weakest link in the model is the relationship between the normal and shear stresses. A realistic relationship between the normal and shear stresses at the interface must somehow be related to the slip between steel and concrete.
The third type of crack involved in the pullout process is the shear-slip (mode II) crack occurring at the steel-concrete interface (see Fig. 10.5.2c, where the separation between the bar and the concrete is grossly exaggerated). A straightforward approach is to treat this shear-slip crack as a cohesive crack, i. e., to postulate that a certain relationship exists between the transferred shear stress т and the relative slip s:
T = t(s) (10.5.8)
where t(s) is the softening function for shear-slip. Introduced by Bazant and Dcsmorat (1994), this is a very simplified model which does not take into account friction and dilatancy occurring at the interface. More sophisticated models involving the crack opening due to dilatancy and the influence of the normal stress may be formulated, but will not be further described (for an overview of models and a thorough discussion of the coupling between normal and shear stresses and displacements, see Cox 1994). The simple model defined by (10.5.8) can, however, suffice to get a rough picture of the influence of the bond degradation on the overall response in bar pullout.
A simple and crude mathematical model which, nevertheless, realistically captures some aspects of fracture and the size effect was used by Bazant and Desmorat (1994), who considered a uniaxially stressed bar (or fiber) embedded in a concrete bar also behaving in an uniaxial manner, each with the cross section remaining plane and orthogonal to the bar axis. The interface between the bar and the concrete tube is characterized by the r — s relation (10.5,8). For the sake of simplicity, this relation was assumed linear (triangular stress-displacement diagram). The solution can then be obtained analytically, by integration of the differential equation of equilibrium in the axial coordinate x. The solution yields simple formulas. It is found that, during failure, zones of slip initiate at the beginning of embedment of the bar or at the bar end, or both, and spread along the bar as the end displacement of the bar is increased, as sketched (for a more general case) in Fig. 10.5.2c. For geometrically similar situations, a strong size effect is observed. The size effect is caused by the fact that the ratio of the length of the slipped zone to the bar diameter decreases with increasing diameter, i. e., the slip zone localizes. For a sufficiently small size, the slip zone at the maximum load extends over the entire length of the bar embedment, and for a size approaching infinity, the relative length of the slipped zone tends to zero. The calculated size effect curve turned out to be very close to the generalized size effect law proposed by Bazant [Eq. (9.1.34)], with the exponent r = 1.25 for a concave nonlinear softening law. The one-dimensional solution may, of course, be expected to be good only when the slip zone is very long or very short compared to the bar diameter and the concrete cover around the bar is not too thick. In general, three-dimensional fracture analysis is, of course, required. Nevertheless, despite the one-dimensional simplification, it seems that the generalized size effect law indicated by this analysis may be applied as a simple approach to practical problems.
Further tests of bar pullout from normal and high strength concrete cubes were conducted by Bazant, Li and Thonta (1995). In these tests, it was tried to separate the effect of radial fractures emanating from the bar and the bond crack along the bar. The tests were designed so that no radial fractures would form and the bar would fail only due to bond fracture and slip. This was achieved by using a relatively short embedment of the bar in the concrete cube. The results again revealed a strong size effect. Because of the absence of radial cracks, it was admissible to compare the results to the aforementioned one-dimensional solution of Bazant and Desmorat (1994). The comparison was satisfactory although large scatter prevents considering this as a validation of the Bazant and Dcsmorat’s equation.
A similar degree of brittleness as in pullout occur in the failure of splices of reinforcing bars in which the lapped bars are not connected and the tensile force in the bars is transmitted through the concrete in which the bars are embedded. The codes provide empirical provisions for the length of overlap and for the so-called development length over which the yield force of the bar can be transmitted from concrete to the bar. These formulas are of the strength theory type which exhibit no size effect. §ener (1992) reports experiments which confirm that splices indeed exhibit a strong size effect which may be well described by Bazant’s size effect law and is rather close to LEFM (in more detail, §ener, Bazant and Becq-Giraudon 1997). The aforementioned type of correction of the existing formula —Eq. (10.1.13)— is also needed in this case.