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Refined non-linear static or dynamic analyses of reinforced concrete structures require the knowledge of the actual force-displacement or bending moment-rotation curves of each structural member, which depend on the crack widths and on the crack pattern, and after all on the slip between concrete and reinforcing steel. For this reason the definition of improved local models taking into account all these local aspects is a fundamental prerequisite for advanced assessment of r.c. structures. A numerical procedure which allows to predict the relative displacement between steel reinforcement and the surrounding concrete in a reinforced concrete element, once assigned the stress in the naked steel bar and the bond-slip law is discussed. The method provides as final outcomes the sequence of crack openings and the individual crack widths, regardless of the particular bond-slip correlation adopted. The proposed procedure is implemented referring to two relevant experimental case studies, demonstrating that it is able to predict satisfactorily actual strain fields and slips along the investigated reinforced concrete elements.

Non-linear static or dynamic analyses are increasingly used to study the behavior of reinforced concrete structures, especially for seismic design [

These curves represent also the starting information for the evaluation of the available ductility of structural elements and for an appropriate estimation of their relevant parameters, required to perform equivalent elastic analyses, as defined in seismic codes. Evidently, the curves can be obtained experimentally, through sophisticated and typically expensive ad hoc tests, or numerically, resorting to refined theoretical models, suitably validated. Since the precision of the results depends on the type of non-linear analysis which is carried out, the degree of sophistication of the theoretical model can even vary from case to case.

As the non-linear force-displacement or bending moment-rotation relationships of r.c. members depend on the crack widths and on the crack pattern, and after all on the slip between concrete and reinforcing steel, advanced models should consider appropriately all these local aspects, usually ignored in classical simplified approaches, which represent, on the contrary, key issues in refined structural analysis [

Moreover, experimental studies [

A numerical procedure for the solution of the differential equation governing the interaction between steel bars and concrete is proposed, which allows to predict the relative displacement between steel reinforcement and the surrounding concrete in a reinforced concrete tie, once assigned the stress in the naked steel bar and the bond-slip law.

The proposed approach, which is obviously independent of the particular bond-slip correlation adopted to perform calculations, provides as final results the sequence of crack openings and the individual crack widths, or in other words the evolution of the crack pattern in the tie, from which it can be derived, regarding the tie as a basic sub-element of a r.c. element in bending, the actual non-linear moment-rotation diagram of the element itself.

To validate it, the procedure has been applied to some relevant case studies, aiming at replicating experimental results on r.c. ties, obtained by Bresler and Bertero [

In the examples, beside the bond-slip models provided in CEB-FIP Model Codes [

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 (

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, s_{0}, and its first derivative _{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_{s}_{0} 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 ^{*}, through the following steps:

1) set the origin of the x-axis in the starting cross section x = x_{0} = 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, s_{0} and _{0} and e_{s0}, being_{0}(s_{0}) and s_{s}_{0}(e_{s}_{0});

4) starting from ^{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

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_{s}_{0} and assign a new value for s_{0}, iterating the process from step 6 till to convergence;

10) once the convergence is achieved, retain the actual value of s_{0}(e_{s}_{0}) and the corresponding real curves e_{s}(x) and e_{c}(x) along the tie, together with the actual transfer length L_{a}(e_{s}_{0}), as shown in

Of course, the searching of s_{0} 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 s_{0}(e_{s}_{0}) and e_{s}_{0} represent the real boundary conditions at x = 0 when the stress in the naked bar is s_{s}_{0}(e_{s}_{0}), the actual boundary conditions can be derived, in terms of e_{s}_{0} and s_{0}, in the whole significant range of the loading process, simply increasing e_{s}_{0} and iterating again the procedure.

Clearly, in the loading process described by the increase of e_{s}_{0}, the strains e_{s} and e_{c} at x = L_{a} and the transfer length L_{a} itself are increasing functions of e_{s}_{0}.

As soon as the concrete ultimate tensile strain e_{ct} is attained, for

is fulfilled, a first crack opens at

When the crack opens, the strain distributions e_{s} and e_{c} on the length _{0}-e_{s}_{0} diagram. The updated strain distributions as well as the updated slip value

As known, other cracks can form for e_{s}_{0} values slightly greater than the identified steel strain level _{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

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

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

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

2) set i = 1, assign tentatively the boundary conditions s_{0} and_{0}(s_{0}) and s_{s}_{0}(e_{s}_{0}), 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_{c} reaches the local maximum s_{c}_{max}, then solve Equation (7) as above (steps 3 to 9);

4) as s_{c}_{max} increases with e_{s}_{0}, a new crack opens at _{s}_{0} attains the value

Set _{s}_{0} 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_{s}_{0}. 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 s_{0} 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, s_{0} and ε_{s}_{0}, describing the evolution of the loading process, is finally obtained.

In _{0}-e_{s}_{0} curve for the end cross section of a reinforced concrete tie, which has been previously investigated adopting the CEB-MC90 [

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 _{0} at the crack location suddenly decreases, reflecting the abrupt redistribution of the strain fields in the concrete.

To validate its effectiveness, the procedure has been to simulate the outcomes of two experimental test campaigns carried out by the authors [

Essentially, the actual (measured) strains on the steel reinforcing bars are compared with those obtained numerically by means of the proposed method.

It must be stressed that the classical CEB correlation is valid only when the directly loaded rebar is pulled out from a fully confined member, where the slip attains values of several millimeters, while in usual structures, like bending beams, the slip rarely exceeds 0.0 - 0.2 mm. This behavior can be explained considering that, due to tangential tensile stresses around the rebar, the concrete cover splits before full activation of bond strength (splitting failure mode).

Obviously, the splitting failure mode cannot be taken into account directly in one-dimensional bond-slip models, but its effects can be indirectly modeled by using effective bond-slip relationship, suitably determined. On the basis of this consideration, to widen the significance of the analysis and further corroborate the results, beside the classical CEB bond-slip relationship, an alternative effective bond-slip relationship, illustrated in §3.3, has been adopted.

The reinforced concrete ties we tested in [

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) (

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 [

developed by Mains [

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) (

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 [

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 [

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 [_{c} » 30 MPa (

As discussed in [

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 [_{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

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 (

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

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 [

· 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.

A numerical method for the study of the evolution of the crack pattern in r.c. ties, under monotonic loading, simulating the tensile part of a more general r.c. member in bending, has been illustrated.

As soon as the stress-strain relationships for steel and concrete in tension are given and a suitable effective bond slip model is adopted, the method allows to calculate the actual distributions of steel and concrete strains and stresses along the tie and, consequently, the actual distributions of bond stresses and slips between the steel reinforcement and the surrounding concrete.

A comparative implementation of the procedure is discussed with reference to the CEB bond-slip model and to an alternative effective bond-slip model, derived from experimental results obtained in a previous experimental study on concrete ties reinforced with instrumented rebars.

The numerical simulations of the ad hoc tests as well as of other similar experimental data available in Literature, confirmed that the proposed algorithm is satisfactory and can be suitably adopted for refined non-linear analyses of r.c. ties, whichever the bond-slip model taken into account.

The numerical procedure can be easily extended, with slight modifications, to more general cases, like beams in bending or beam to column joints, in particular when refined and realistic evaluations of the actual behavior are required.