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This paper proposes an analysis procedure able to determine the flexural response of rectangular symmetrically reinforced concrete sections subjected to axial load and uniaxial bending. With respect to the usual numerical approaches, based on the fibre decomposition method, this procedure is based on the use of analytical expressions of the contributions to the equilibrium given by the longitudinal reinforcement and the concrete region in compression, which depend on the neutral axis depth and the curvature at each analysis step. The formulation is developed in dimensionless terms, after a preliminary definition of the geometrical and mechanical parameters involved, so that the results are valid for classes of RC sections. The constitutive laws of the materials include confinement effect on the concrete and postyielding behaviour of the steel reinforcement, which can be assumed to be softening behaviour for buckled reinforcing bars. The strength and curvature domains at the first yielding of the reinforcement in tension and at the ultimate state are derived in the form of analytical curves depending on the compression level; therefore, the role of a single parameter on the shape of these curves can easily be deduced. The procedure is validated by comparing some results with those numerically obtained by other authors.

The performance of reinforced concrete frames under severe earthquakes largely depends on the ability of the beam and column sections to undergo large inelastic deformations. Especially, this ability plays a decisive role in existing buildings, the safety level of which can be estimated by employing nonlinear analysis tools (like pushover), needing a careful input in terms of strength domains and moment-curvature relationships of the critical sections.

The evaluation of the moment-curvature response of critical sections of RC members is a complex issue mainly because of the interaction of various parameters: constitutive laws of materials in the elastic and plastic ranges, member geometry, buckling phenomena in reinforcing steel bars, and loading conditions.

In order to include in a computer software the cross-section strength domain and moment-curvature relationships of reinforced concrete members, two different approaches are usually followed: the use of strength domains [

The layered section approach gives a realistic and almost complete description of the behaviour of the critical regions of a RC framed structure, also allowing to include the bond slip effects in the moment-rotation response. However, the application of this approach to large structures with many degrees of freedom has some limitations, because numerical integrations and a prediction of the state of materials must be carried out considering the entire layers of each region where a plastic hinge can occur; thus, a high amount of information is required to characterize each section involved, and a large number of numerical operations is needed to reach an acceptable level of error. This results in heavy computational efforts and convergence problems for nonlinear structural analysis. Consequently, the use of moment-curvature relationships is still the more widespread and efficient approach.

The definition of moment-curvature relationships of RC sections has been a point of research interest for many years [

Historically, moment-curvature relationships with softening branch were first introduced by Wood [

Commercial structural analysis software programs now provide the analyst with the option of conducting moment-curvature analyses at critical sections. Although well-verified equations have been used in the software developments, some questions may remain: (i) for a given project, how can the computational analysis results be easily verified outside the software? (ii) how can general considerations on the role of different parameters characterizing the section and reinforcement geometry be derived?

To answer to these questions, a dimensionless formulation is required, able to fully characterize the nonlinear behaviour of an RC cross-section subjected to an assigned loading condition.

In a recent paper [

In this context the present paper proposes an analytical model for the evaluation of the moment-curvature relationship of rectangular RC sections subjected to axial load and uniaxial bending, by using an incremental strain technique, which could be an alternative to the classical fibre decomposition method. The model is based on a cross-sectional analysis satisfying strain compatibility and equilibrium conditions and can be utilized assuming any constitutive law for the confined concrete. The model is able to take into account several mechanical and geometrical parameters, such as section aspect ratio, longitudinal reinforcement amount and distribution, confinement effect on the concrete core, reinforcing steel hardening, and/or softening postbuckling behaviour. The formulation is carried out in dimensionless form, and it is able to stress the role of the aforementioned parameters in determining the shape of the moment-curvature relationship and the ultimate values of the bending moment and curvature.

The ability to analyze this role for classes of RC sections is the main contribution of the formulation proposed. On the other hand, the following limits are related to the basic assumptions: the model is applicable only to rectangular sections with symmetrical distribution of the longitudinal reinforcement; no cyclic behaviour is considered. Moreover, the use of the proposed model to define the moment-rotation response of the critical region to which the section considered ideally belongs requires a suitable modeling of the potential “plastic hinges” which is not discussed here.

The analysis model adopted is an updated version of that already utilized in [

The compressed concrete rectangular section is assumed to be reinforced by longitudinal steel bars symmetrically located with respect to the two principal axes of inertia of the section so that the geometrical and mechanical barycentres are coincident.

The longitudinal reinforcing bars are distributed as follows:

four bars of the same diameter are located at the corners, giving total sectional area

Figure

Section geometry and analysis model.

The symbols

These quantities had not been defined in the original model [

The mean distance of the barycentres of the sectional area of the upper and lower reinforcing bars from the nearest horizontal side of the core, of length

By assuming that the bending moment lies in the barycentric vertical plane of the section, the upper and lower reinforcing bars provide a primary contribution to the rotational equilibrium of the section, while the intermediate bars, distributed with constant pitch

With reference to the symbols in Figure

Since high curvature values are involved (ultimate limit state), unconfined concrete spalling reduces the resisting section to that of the confined concrete core; all the geometrical and mechanical parameters involved in the formulation proposed here are referred to this reduced section.

Therefore, the axial load

The strains of the confined concrete and the dimensionless curvature

For clarity’s sake, the geometrical parameters that must be considered assigned are denoted using Greek letters. Therefore, with reference to the symbols in Figure

Differently from what was necessary for the analysis at the ultimate state [

Figure

Constitutive laws of concrete.

The formulation proposed in the present work allows the use of any law; however, the applications were carried out using the expressions proposed in [

In normalized form these expressions are written as

The symbol

Equations (

All quantities referring to the unconfined concrete are here denoted by the subscript

For confined concrete, the characteristic quantities of which are denoted by the subscript

In the present formulation, the factors relating the cylindrical strength and the corresponding strain of the confined concrete to the ones of the originally unconfined concrete are denoted as

As a result, it can be shown that in the field

Figure

Constitutive law of longitudinal steel reinforcement.

With respect to the analyses made in [

Under tension the possible hardening behaviour is defined by the mean hardening modulus

The simple equations that analytically express the laws of the elastic and postyielding branches in Figure

The dimensionless parameters characterizing the slopes of the postyielding branches and the ultimate strain in an ideal normalized stress-strain diagram are denoted as

A further parameter relating the characteristic strain values of steel and concrete is

Denoting as

At the second member of these equations there are clearly indicated and ordered contributions offered by the principal reinforcement, secondary reinforcement, and concrete.

These contributions are analytically expressed in the following sections by assuming the classical hypothesis that the section remains plane and neglecting the tensile concrete strength.

In the analytical formulation shown in this section, the following parameters are involved: the geometrical parameter

Figure

Strain and stress states of principal reinforcement of section.

By using the symbols shown in the figure, the upper and lower reinforcement bars are subjected, respectively, to the strains:

The axial load and the bending moment that the principal reinforcement can bear with respect to the barycentre of the section are expressed by

By substituting (

On the basis of what was observed commenting on (

With respect to the formulation shown in the previous section, instead of

Figure

Strains and stresses of uniformly distributed secondary reinforcement.

These quantities are expressed, respectively, by

Figure

Strain state of concrete section: (a) uncracked section; (b) cracked section.

Considering that the strains involved in the functions

The flexural response of a class of RC sections, characterized by assigned values of the geometrical and mechanical parameters defined above, is obtained by imposing the equilibrium condition expressed by (

For each configuration the contributions to the equilibrium provided by the principal reinforcement ((

The field of validity of the procedure proposed here is limited by the fact that the constitutive law of the steel reinforcement does not include unloading branches starting from a point of the post yielding branches in compression or in tension. Therefore, for any loading step the upper principal reinforcement and the upper fibre of the distributed secondary reinforcement, both potentially subject to compression, cannot be in the postyielding field in tension; the lower principal reinforcement and the lower fibre of the distributed secondary reinforcement, both potentially subject to tension, cannot be in the postyielding field in compression.

Because of these assumptions, the field of values of

More precisely, in the absence of secondary reinforcement (

However, it must be observed that (

The more usual application of the analytical expressions derived in the previous sections is to derive by a step-by-step procedure the moment-curvature

To this purpose the procedure is applied by the following steps:

assigning the lowest value of curvature

for each value of the couple

when the assigned value of

considering an increased value of

The procedure stops when the maximum available ultimate confined concrete strain or the maximum available tensile strain of the lower reinforcement is exceeded. These two occurrences imply, respectively, that

If the stress-strain law of the steel reinforcement also exhibits a softening postyielding branch corresponding to a postbuckling behaviour up to the strain value

A typical

The analytical expressions shown in the previous sections make it possible to determine directly the coordinates of these points and to derive in closed form the

The constructions of the aforementioned curves can be made more easily than by using a classical discretized model of the section, because (

In order to construct the

Once

It must be observed that, if the principal reinforcement in compression is subject to buckling after the achievement of the yielding stress, the ultimate strain of this compressed reinforcement is the strain denoted as

The

Considering the field of validity of the procedure proposed, the minimum value of

Obviously, the

In relation to the field of applications (RC columns), the first values of

The reliability of the procedure proposed is shown here by using the results derived from two RC sections considered by other researchers.

The first application refers to one of the sections examined by Zahn et al. [^{2}. The longitudinal reinforcement consisted of 12 steel bars of diameter 16 mm, having yielding strength ^{2}, uniformly distributed along the perimeter of the concrete core (constant pitch). The cover depth, measured from the external perimeter of the stirrups, was 13 mm. The transverse reinforcement, consisting of steel stirrups of diameter equal to 10 mm, according to the calculations made by the aforementioned authors, provided the effective confinement pressure

The flexural response was numerically derived in [

Mander’s model [

Calibration of constitutive law of confined concrete.

The procedure proposed in [^{2}. These values are also adopted for the dimensional stress-strain law expressed by the model used here [

Since for the stress-strain law of the unconfined concrete Zahn et al. [

On the basis of the above data, the square cross-section of the concrete core, measured inside the perimeter stirrups, has a side of

The principal reinforcement consists of four upper and four lower 16 mm steel bars while the secondary reinforcement consists of four bars (two on the right and two on the left) of the same diameter. The steel reinforcement constitutive law is defined by assuming

Therefore, the dimensionless parameters involved in the formulation proposed take on the following values:

Figure

Comparison of results:

Both curves in the figure clearly show the cusp produced by the yielding of the principal reinforcement in tension and the effects of the progressive cover spalling.

The second application refers to the middle-height section of an RC column, of height 1640 mm, experimentally tested by Saatcioglu et al. [

The loading condition was realized by imposing relative axial displacements so that the reactive compressive force acted with fixed eccentricity in a plane of principal inertia of the sections. Suitable devices were applied at the column ends so that the column itself behaved like a hinged vertical RC member under an eccentric compressive load.

As a consequence, each point of the moment-curvature curve characterizing the experimental response obtained by the aforementioned authors corresponds to a different value of axial load. Moreover, the actual bending moment acting on the middle-height section was influenced by the

The formulation proposed here is validated considering some points of the aforementioned curve for which the authors indicate the values of the axial load

The specimen considered had a square section of side ^{2} and average hardening modulus

The unconfined concrete had cylindrical strength ^{2}. The numerical model adopted in [

On the basis of these geometrical and mechanical data, and by evaluating the effective confinement pressure by the procedure proposed in [^{2},

Figure

Comparison of theoretical and experimental results.

It must be observed that the last point detected by using the present formulation, corresponding to the ultimate state of the section, had been already found in [

Figure

The dotted curve in Figure

Nevertheless, the axial load values that cannot be considered because of these limitations are well beyond the ones that can be assumed in the structural design of RC buildings in seismic areas or usually found in the columns of existing buildings.

A dimensionless formulation has been proposed, which provides the flexural response of classes of sections of RC columns having the same values of the geometric and mechanical parameters defined in this study.

At each loading stage, the resisting components of the section (cover and concrete core and principal and secondary longitudinal reinforcements) give a contribution to the equilibrium that can be expressed by analytical functions depending on the normalized neutral axis depth and the curvature of the section.

Special equilibrium conditions, like the ones corresponding to the first yielding of the principal reinforcement in tension and to the ultimate state, imply an analytical link between the neutral axis depth and the curvature, so that the sum of the aforementioned contributions becomes an analytical function of a single variable, which can express a strength or curvature domain.

For a generic equilibrium condition an iterative procedure to determine the neutral axis depth corresponding to the assigned values of curvature and compression level is required, but each iteration step implies the simple use of the same equilibrium equation for an updated value of the normalized neutral axis dept.

The dimensionless form of the formulation proposed makes also it easily possible to evaluate the influence of the parameters characterizing the class of sections examined on the strength and curvature corresponding to a given loading stage.

This work was carried out within the 2010–2013 Research Project “DPC-ReLUIS (Dipartimento Protezione Civile-Rete dei Laboratori Universitari di Ingegneria Sismica),” AT 1, Task 1.1.2. The related financial support was greatly appreciated.