Keywords

1 Introduction

Ductility is a vital parameter in seismic design. The ductility of reinforced concrete (RC) structure depends on the ductility of its elements such as columns. In addition to their great potential in restoring the load-bearing capacity, high ductile RC columns provide adequate ductility to the overall structure [1]. From here, it appears the great importance of RC columns ductility. In this field, the study of RC columns ductility requires a deep analysis at the limit states, where it is essential to know the behavior of materials such as concrete and steel [2]. Practically, the RC column cross section is subjected to lateral pressure resulted from the transverse reinforcement, named confinement [3, 4]. Under confinement, significant enhancements are occurred on concrete behavior, where higher strength and higher critical strains are achieved [5, 6]. These changes on concrete characteristics affect the overall behavior of RC elements, so it is necessary to take into account these new characteristics in structural elements studying [7].

Most of the research carried out on the flexural behavior of RC columns are experimental works, where the curvature ductility factor was obtained from the moment–curvature (M–φ) diagram. The yield curvature was measured when the tension reinforcement yielded in tension while the ultimate curvature was measured when the concrete strain reaches its ultimate value. In this field, experimental researches proved that the increase of lateral pressure enhances the curvature ductility of RC columns [8, 9].

On the other hand, a few theoretical researches were conducted on the curvature ductility of RC columns. The lack of theoretical researches in this field pushed the researchers to undertake the current study. A new hypothesis is introduced in order to calculate the yield curvature while the ultimate curvature is calculated according to the proposition of Fardis [10]. In this context, the model of confined concrete given by Eurocode 2 [5] is adopted for the present study. Finally, experimental data are collected in order to show the reliability of the developed model in predicting the curvature ductility factor of confined RC columns made with normal strength (NSC) and high strength concrete (HSC).

2 Stress–Strain Relationship for Confined Concrete Under Compression

Similar to unconfined concrete, Eurocode 2 [5] provides a parabola-rectangle diagram for confined concrete under compression as shown in Fig. 1. Compared to unconfined concrete, an increase in stress (σc) and strain (εc) can be noticed. This amelioration is due to the effective lateral confining pressure (σ2).

Fig. 1
A graph plots stress versus strain. It plots 2 concave-down increasing curves. Unconfined concrete has a maximum compressive strength of f prime c and a strain of epsilon c u. Confined concrete has a maximum compressive strength of f c c and a strain of epsilon c u c.

Stress–strain (σc–εc) model for confined concrete [5]

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The compressive strength of confined concrete (fcc) is given by the following expression:

$$ {\text{f}}_{{{\text{cc}}}} = \left\{ {\begin{array}{*{20}l} {{\text{f}}^{\prime}_{{\text{c}}} \left( {1.125 + 2.50\frac{{{\upsigma }_{2} }}{{{\text{f}}^{\prime }_{{\text{c}}} }}} \right)\;{\text{for}}\;{\upsigma }_{2} > 0.05{\text{f}}^{\prime }_{{\text{c}}} } \hfill \\ {{\text{f}}^{\prime }_{{\text{c}}} \left( {1.0 + 5.0\frac{{{\upsigma }_{2} }}{{{\text{f}}^{\prime }_{{\text{c}}} }}} \right)\;{\text{for}}\;{\upsigma }_{2} \le 0.05{\text{f}}^{\prime }_{{\text{c}}} } \hfill \\ \end{array} } \right. $$
(1)

where fʹc = compressive strength of unconfined concrete.

The strains at reaching the maximum strength (ε0c) and at ultimate (εcuc) are given as follows:

$$ {\upvarepsilon }_{{0{\text{c}}}} = {\upvarepsilon }_{0} \left( {\frac{{{\text{f}}_{{{\text{cc}}}} }}{{{\text{f}}^{\prime}_{{\text{c}}} }}} \right)^{2} $$
(2)
$$ {\upvarepsilon }_{{{\text{cuc}}}} = {\upvarepsilon }_{{{\text{cu}}}} + 0.2\frac{{{\upsigma }_{2} }}{{{\text{f}}^{\prime}_{{\text{c}}} }} $$
(3)

where ε0 and εcu = strains at reaching the maximum strength and at ultimate of unconfined concrete, respectively.

On the other hand, the effective lateral confining pressure (σ2) is calculated as follows:

$$ {\upsigma }_{2} = 0.5\,{\upalpha }\,{\upomega }_{{{\text{wd}}}} {\text{f}}^{\prime}_{{\text{c}}} $$
(4)

where ωwd = mechanical volumetric ratio of confining hoops within the critical region and α = confinement effectiveness factor. These two parameters can be calculated according to Sect. 5.4.3.2.2 of Eurocode 8 [11].

3 Curvature Ductility Factor

The curvature ductility factor (μφ) is calculated by dividing the ultimate curvature (φu) by the curvature at yielding (φy):

$$ {\upmu }_{\upvarphi } = \frac{{{\upvarphi }_{{\text{u}}} }}{{{\upvarphi }_{{\text{y}}} }} $$
(5)

3.1 Curvature at Yielding

The yield curvature (φy) is calculated when the tension reinforcement reaches their yield strength. As shown in Fig. 2b, the web reinforcement (Asw) creates a rigid region in the middle of the column cross section, where they prevent the increase of the tension region depth. In this context, the neutral axis depth (kd) is assumed to be greater than (h/2), where it is taken as follows:

$$ {\text{kd}} = \left( {\frac{{\text{h}}}{2} + \frac{{{\text{l}}_{{{\text{sw}}}} }}{2}} \right) $$
(6)
Fig. 2
3 diagrams. a and b. A rectangular-shaped yield curvature has a height and width of h and b. It has a diamond-shaped structure in the middle, with a rigid region and dots denoting reinforcement. c. An inverted triangle has a strain of epsilon s y with neutral axis at depth k d.

Column cross section at yielding

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Hence, the yield curvature is written as follows:

$$ {\upvarphi }_{{\text{y}}} = \frac{{{\upvarepsilon }_{{{\text{sy}}}} }}{{{\text{d}} - \left( {\frac{{\text{h}}}{2} + \frac{{{\text{l}}_{{{\text{sw}}}} }}{2}} \right)}} $$
(7)

where h = height of column cross section; lsv = length of the rigid region created by web reinforcement; εsy = fy/Es = strain corresponding to yield strength of steel reinforcement; d = distance measured from the extreme compression fiber to the centroid of the tension reinforcement; fy and Es = yield strength and modulus of elasticity of tension steel reinforcement; b = width of column cross section; d1 = distance measured from the extreme compression fiber to the centroid of the compression reinforcement; As, Aʹs and Asw = areas of tension, compression, and web reinforcement.

3.2 Curvature at Ultimate

The ultimate curvature is calculated when the strain in the extreme compression fiber of the confined zone reaches the ultimate value (εcuc). Figure 3 shows the strain distribution at this stage. Based on this figure, the ultimate curvature is written as follows:

$$ {\upvarphi }_{{\text{u}}} = \frac{{{\upvarepsilon }_{{{\text{cuc}}}} }}{{{\text{c}} - {\text{e}}}} $$
(8)
Fig. 3
2 diagrams. a. A rectangular-shaped yield curvature with height h and width b. A diamond-shaped structure of the confined area is in the middle, with a rigid region and dots denoting reinforcement. b. An inverted triangle has a strain of epsilon c u c with a neutral axis at depth c and distance e.

Column cross section at ultimate

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where c = neutral axis depth at ultimate and e = distance measured from extreme compression fiber of concrete to confined zone extremity.

In this case, the neutral axis depth (c) was given by Fardis [10] as follows:

$$ \frac{{\text{c}}}{{\text{d}}} = \frac{{\left( {1 - {\updelta }_{1} } \right)\left( {{\upnu } - {\upomega }_{{\text{s}}} + {\upomega^{\prime}}_{{\text{s}}} } \right) + \left( {1 + {\updelta }_{1} } \right){\upomega }_{{{\text{sw}}}} }}{{\left( {1 - {\updelta }_{1} } \right)\left( {1 - \frac{{{\upvarepsilon }_{{0{\text{c}}}} }}{{3{\upvarepsilon }_{{{\text{cuc}}}} }}} \right) + 2{\upomega }_{{{\text{sw}}}} }} $$
(9)

where

  • δ1 = d1/d

  • ν = N/b0 d0 fcc: ratio of the axial load (N)

  • ωs = Asfy/b0 d0 fcc, ωʹs = Aʹsfy/b0 d0 fcc, ωsw = Aswfy/b0 d0 fcc: mechanical reinforcement ratios of tension, compression, and web reinforcement, respectively

  • h0 = h − 2(e0 + dbh/2), b0 = b − 2(e0 + dbh/2), d0 = d − 2(e0 + dbh/2)

  • dbh: diameter of transverse reinforcement

  • e0: concrete cover.

4 Method Verification

In this section, the performance of the presented method is tested against large experimental results collected from the literature. In this field, 30 experimentally tested columns are selected from previous researches. The details and properties of the selected specimens are shown in Table 1.

Table 1 Properties and details of the selected beams
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The predicted curvature ductility factors (μφ,pre) of the selected specimens are presented in Table 2 with those obtained during experiments (μφ,exp). The comparison shows a good agreement between the predicted and the experimental values, where the mean value (MV) and the standard deviation (SD) of the errors (μφ,pre/μφ,exp) are equal to 1.02 and 0.17, respectively. In the same context, Fig. 4 shows the predicted and the experimental results head-on. The dispersion of the points affirms the good agreement between the predicted and the experimental results. Moreover, the value of the coefficient of determination (R2 = 0.90) confirms this deduction, where it is very close to 1. Consequently, these statistical results indicate the good performance of the proposed method in predicting the curvature ductility factor of NSC and HSC columns.

Table 2 Comparison between predicted and experimental results
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Fig. 4
A scatter plot compares the predicted and experimental values of mu phi. It plots the data in an increasing trend, with a linearly increasing regression line of R squared = 0.90.

Comparison between predicted and experimental results

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5 Conclusion

The current research presented a detailed method in order to predict the curvature ductility factor of confined RC columns made with NSC and HSC. Eurocode 2 [5] and Eurocode 8 [11] were used to calculate the strength and strains of concrete under confinement, i.e., under lateral pressure due to transverse reinforcement. In this field, the results obtained by using the proposed method were compared with those extracted from experimental researches. The statistical comparison showed that the proposed method has a good performance and accuracy in predicting the curvature ductility factor of confined RC columns with concrete strength up to 100 MPa.

On the one hand, the present study provided a simple method that can be used with high accuracy. On the other hand, it can be a starting point for studying the ductility of RC columns confined with other types of materials or under active confinement. Moreover, it motivates researchers to undertake other studies on different confined RC elements such as beams.