As recalled shortly here, the classical equations governing the bond-slip problem can be derived considering equilibrium and compatibility conditions of an infinitesimal r.c. tie.

A tie is considered, the length of which is dx (Figure 1), made up by one centered rebar and by the surrounding portion of concrete in tension, subject to a tensile force F.

The equation governing the global equilibrium of the tie is trivially

where F_s and F_c represent the quotas of the total force acting on steel and on concrete, respectively.

Let f the diameter of the steel bar and A_c, depending on x, the concrete area involved by the bond stress diffusion. Differentiating Equation (1) it results

and then

where s_s and s_c are the stresses in steel and concrete, respectively, and ρ(x) is the geometric reinforcement ratio.

Said τ(s) the bond stress, function of the slip s, the local equilibrium of the rebar over the length dx gives

and, finally, the compatibility condition,

yields

being e_s and e_c the strains in steel and concrete respectively.

Combining Equations (2), (4) and (6), the following non-linear differential equation is obtained

which can be solved, under appropriate boundary conditions, once input the stress-strain relationships for steel and for the tensile branch of concrete, the bond-slip law and the dependence of A_c on x.

To solve the non-linear Equation (7), the procedure described hereinafter can be implemented, based on the finite difference method.

It is worth to underline that the main difficulty in finding the solution concerns the assignment of the boundary conditions, since, in the present case, the boundary itself is unknown.

The most direct way to fix the boundary conditions is to assign the slip, s₀, and its first derivative in a cross section where s_c = 0, typically corresponding to a cracked section or to an end section of the tie. Moreover, since in this section e_c = 0, from Equation (6), it results, as soon as the steel strain e_s0 is assigned.

Since boundary conditions themselves depend on the strain field in the concrete tie, solution of Equation (7) requires an iteration process, like the one described hereinafter.

The proposed procedure is based on an iterative shooting solution of the system of ODEs

suitably derived from Equation (7).

The first phase (phase I), summarized in the flow chart shown in Figure 2, allows to define the boundary conditions up to the first crack opening in a reinforced concrete tie, the length of which is L^*, through the following steps:

1) set the origin of the x-axis in the starting cross section x = x₀ = 0, which coincides with an end section of the tie, where s_c = 0, and subdivide the half of the tie in N intervals of constant length Dx;

2) assign the s-e constitutive laws for steel and for concrete under tensile stresses; assign the bond-slip law, τ-s; assign the A_c(x) function;

3) set i = 1, assign tentatively the boundary conditions in x = 0, s₀ and or equivalently s₀ and e_s0, being, and, using the bond-slip law and the steel constitutive law, evaluate the bond stress τ₀(s₀) and s_s0(e_s0);

4) starting from integrate (8) using for instance the Runge-Kutta 4^th order method, and determine the slip s_i and the steel stress s_s,i at the end of the interval, x_i;

5) using the bond-slip law, evaluate the bond stress τ_i(s_i) in x_i;

6) using the equilibrium condition (3) of the tie segment, calculate, through the general equation

where A_c,i is the concrete area at section x_i, and ρ_i the geometric steel to concrete ratio at the i-th section;

7) stop the iteration when s_i = 0, and consequently τ_i = 0, or when i = N and check convergence (step 9);

8) set i = i + 1 and iterate the process from step 4;

9) check convergence: if condition e_s = e_c is satisfied at x_i, convergence is achieved; otherwise keep unchanged the value of e_s0 and assign a new value for s₀, iterating the process from step 6 till to convergence;

10) once the convergence is achieved, retain the actual value of s₀(e_s0) and the corresponding real curves e_s(x) and e_c(x) along the tie, together with the actual transfer length L_a(e_s0), as shown in Figure 3.

Of course, the searching of s₀ can be refined using one of the usual numerical root finding methods, like the secant method, the false position method or the Newton-Raphson method.

Considering that actual values of s₀(e_s0) and e_s0 represent the real boundary conditions at x = 0 when the stress in the naked bar is s_s0(e_s0), the actual boundary conditions can be derived, in terms of e_s0 and s₀, in the whole significant range of the loading process, simply increasing e_s0 and iterating again the procedure.

Clearly, in the loading process described by the increase of e_s0, the strains e_s and e_c at x = L_a and the transfer length L_a itself are increasing functions of e_s0.

As soon as the concrete ultimate tensile strain e_ct is attained, for, setting and and, provided that the condition for crack formation,

is fulfilled, a first crack opens at, satisfying the inequality

When the crack opens, the strain distributions e_s and e_c on the length abruptly change and a discontinuity occurs in the s₀-e_s0 diagram. The updated strain distributions as well as the updated slip value can be then derived iterating the procedure for the unique case, setting the tie length to, as described afterwards.

As known, other cracks can form for e_s0 values slightly greater than the identified steel strain level until a steady situation is achieved in the whole tie. In this situation consecutive cracks are spaced less than 2, so that the condition s_c < f_ct is satisfied in each point of the tie and the first crack pattern is fully stabilized.

The actual crack locations are generally indeterminate, as the tensile strength of concrete, f_ct, is a random variable and the unique constraint for is the inequality (11). In any case, the distances along the tie, and consequently the first crack pattern through the Monte Carlo method can be obtained, assigning, for example, a random concrete tensile strength distribution along the tie itself.

Nevertheless, even if the first phase crack is stabilized, further cracking is not prevented for increasing values of the external tensile force F. In fact, since the slip s increases as F, and τ depends on s, it can happen that concrete tensile strength f_ct is reached again for F = F⁽²⁾ within a distance, , determining a new stabilized cracking pattern.

The subsequent phases can be investigated properly modifying the above illustrated procedure. Referring to the j-th step, the modified procedure is summarized in the flow chart in Figure 4:

1) set the origin of the x-axis in the starting cross section x = 0 of the considered portion of the tie, bounded by adjacent cracks spaced, and subdivide half of the interval in M parts of constant length Dx;

2) set i = 1, assign tentatively the boundary conditions s₀ and, , at x = 0 and evaluate the bond stress τ₀(s₀) and s_s0(e_s0), bearing in mind that, at the first step after the (j − 1)-th crack formation phase, it holds

the equality;

3) assume the function s(x) is antisymmetric in the interval: this implies that, disregarding friction forces, the convergence condition is met if the slip s vanishes at midpoint of the interval, where the concrete stress s_c reaches the local maximum s_cmax, then solve Equation (7) as above (steps 3 to 9);

4) as s_cmax increases with e_s0, a new crack opens at as soon as the strain e_s0 attains the value, corresponding to.

Set and iterate again the process from 1), until e_s0 attains the ultimate value, e_su.

Finally, crack formation ends and the crack pattern is completely defined when, along the whole tie, it results s_c < f_ct, whichever the value of e_s0. In the final stage, experimental evidence confirms that tension stiffening is negligible, so that tie behavior tends to fit the naked bar behavior.

Once completed the procedure, the evaluation of each crack width is trivial, being the sum of the quotas s₀ pertaining to the two parts of the tie delimited by the crack.

At the end of each iteration step, as soon as convergence conditions are satisfied, the couple of boundary conditions, s₀ and ε_s0, describing the evolution of the loading process, is finally obtained.

In Figure 5 a typical outcome of the procedure’s implementation is shown in terms of s₀-e_s0 curve for the end cross section of a reinforced concrete tie, which has been previously investigated adopting the CEB-MC90 [12] bond-slip relationship illustrated in §3.3 [8] . The example is referred to a r.c. cylindrical tie reinforced by an axially located 16 mm B450C steel bar. The diameter of the tie, 0.8 m long, was 112 mm.

Inspecting the diagram, it is possible to identify the different phases of the loading path, the formation of cracks and in particular the discontinuities corresponding to the cracking phases: for the given the slip s₀ at the crack location suddenly decreases, reflecting the abrupt redistribution of the strain fields in the concrete.

The reinforced concrete ties we tested in [9] (case study 1) were made up by a single bar, 16 or 20 mm in diameter, axially located in a concrete cylinder, whose diameter was about 132 mm.

More in detail, the results obtained from “short specimens”, 235 mm long, are considered here. These specimens were setup to prevent the crack formation, in such a way to have the opportunity to check the accuracy of the procedure till to the appearance of the first crack, since the successive phases (cracked specimen) are the reiteration with different boundary conditions of phase I itself.

The strains along the steel bar were measured using electrical strain gauges, spaced 25 mm one another and embedded in two diametrically opposite milled grooves (3.5 ´ 4 mm) (Figure 6).

Moreover, the main objective of tests was the definition of a suitably improved effective local bond-slip relation law, alternative to the CEB ones, able to take into account indirectly the effects of the splitting failure mode on the actual non-linear behavior of reinforced concrete members.

Bresler and Bertero [11] performed similar tests on r.c. ties, axially reinforced by one single rebar (case study 2), again instrumented with strain gauges. In this case the strain gauges were located in a milled groove inner to the steel bar, previously cut in two halves and subsequently reassembled, according to a technique previously

developed by Mains [14] .

The results, which have been considered for the comparison, refer to cylindrical specimens, 406 mm long (16”), whose diameter was 152 mm (6”). The diameter of the reinforcing bar, placed axially, was 28.6 mm (1”1/8) (Figure 7).

Since the concrete cylinder presented a notch at midspan to predetermine the location of the crack, the analyses reported here refer to the half-length L/2.

Among the available data for case study 2, they have been selected those referring to the specimen “H” for the increasing branch of the load cycle nr. 2, which can be assimilated to a monotonic loading phase extending from 17.8 kN (4 kips) up to 133.4 kN (30 kips).

In numerical simulations, constitutive laws for steel and concrete have been modeled according to the Ramberg- Osgood formulation [15] , in the form

and, (12)

being f_y the yield stress of the steel, f_ct the tensile strength of concrete, and a_s, n, a_c and m the curve parameters.

The mechanical properties and the curve parameters pertaining to case study 1 were obtained suitably fitting the experimental data derived from ad hoc tests [9] , while in case study 2, due to lack of information, the limited available experimental data were used, appropriately adapting the parameters evaluated in case 1.

The actual yield stress of steel rebars was f_y » 545 MPa in case 1 and f_y » 690 MPa in case 2, while the compressive strength of concrete was f_c = 29.6 MPa in case 1 and f_c = 40.8 MPa, in case 2.

Furthermore, it has been assumed that the effective concrete area A_c(x) varied steeply from zero to A_c within a distance approximately twice the rebar’s diameter, according to an exponential law. Anyhow, sensitivity analyses showed that results do not vary significantly from those obtained assuming a constant value for A_c.

As recalled above, a bond-slip correlation currently adopted in Literature is that proposed by the CEB in Model Code 90 [12] , slightly modified in the final draft of fib Model Code 2010 [13] . Inspecting the bond-slip laws proposed in Model Codes 90 and in Model Code 2010 for concrete class f_c » 30 MPa (Figure 8), it results that, within the scopes of the present work, the differences between the two curves are not critical.

As discussed in [15] , the bond-slip curves proposed by CEB/fib, in particular in the field where bond deteriorates, are associated with slip values well beyond values actually measured in r.c. members, where rebars are efficiently anchored. Moreover, as said before, these values would be incompatible with effective crack widths currently measured in real structures, which are directly correlated with actual slip values.

Consequently, as clearly demonstrated by the implementation of the procedure too, the adoption of CEB τ-s laws implies that the maximum level of bond stress cannot be attained in real structures, so that the decreasing right branch of τ-s curve, associated with bond deterioration, cannot be covered.

On the basis of the refined measurements of steel stress, carried out on the above mentioned r.c. ties during monotonic tensile tests [9] , an alternative effective bond-slip relationship can be proposed. The discussion of criteria adopted to derive this law, which is illustrated in Figure 9, again for a f_c » 30 MPa concrete, is out of the

scope of the present work; nevertheless, however it must be highlighted that the maximum bond stress (which is the same adopted in CEB-FIP models) is associated to a slip value which is significantly lower than the corresponding ones in MC90 and in MC2010 models.

Using the proposed procedure, the behavior of the two ties tested in case studies 1 and 2 have been simulated, for increasing value of the external force applied to the rebar. For selected load steps, chosen to appropriately cover those for which experimental data were available, the results are summarized in Figure 10 and Figure 11, pertaining to case study 1 and case study 2, respectively.

In the diagrams, for each load step, the curves representing the measured experimentally strain distributions e-x along the steel bar are compared with those obtained by the implementation, in turn, of the bond-slip correlation suggested by the CEB Model Code 90 (Figure 8) and of the alternative effective bond-slip law suggested in Figure 9. In order to facilitate the comparisons, in the figures the curves obtained adopting CEB model, designated as “CEB”, are plotted in grey, the curves obtained adopting the alternative model, designated as “A”, are plotted in black and finally the experimental ones are plotted with piecewise linear curves.

Aiming to supplement the information, the three families of τ-x experimental and numerical curves, describing the bond stress variation along the tie in case 1, are reported in Figure 12. The diagram clearly shows that, as expected, CEB model excites very low bond stresses.

Inspecting the diagrams it can be remarked that, in both cases, the proposed procedure allows to perform refined local investigations and that, even if outside the scope of the present paper,

· the use of CEB bond-slip model leads to underestimate the bond stress transfer from steel to concrete; in fact, the calculated strains in the rebar are markedly higher than those actually measured and the calculated bond stress are sensibly smaller than the actual ones;

· on the contrary, the use of the bond-slip model proposed in [9] leads to a good fit of bond stresses and of measured steel strains;

· moreover, the slip values calculated adopting the two alternative τ-s models at end sections of the tie do not fully reflect the significant differences registered above; in fact in both cases maximum slips are limited to 0.2 mm; but, in case of CEB correlation, these values are one order of magnitude lower than the slip interval associated with the plastic range of the bond-slip CEB curve. This confirms that, as just explained, in r.c. members actual slips are well below those corresponding to the horizontal plateau or to the descending branch of CEB curves.