1 Introduction

With the development of the industrial community, structures are becoming larger and more complex, and safety and serviceability assessment requires the development of accurate and reliable methods for their analysis. Laboratory methods that may be subjected to static and dynamic loads are introduced to ensure the strength of structures against earthquakes. But it is important to note that laboratory methods require high costs and may provide little information. In concrete structures, due to the brittle behavior of concrete and its early rupture in loading, the analysis of its behavior is more complicated than steel. And it is usually a little difficult to analytically study the composite behavior of two completely different materials such as concrete and steel and the time-dependent changes of the two materials and the effects between the two materials, and researchers are still working on these topics. Due to the development of computers and numerical methods and a clear explanation of the properties of materials, numerical methods are widely used among researchers today. Reinforced concrete is one of the most widely used structural materials today. One of the suitable methods for the safety of structures or members of reinforced concrete against seismic loads is to increase their ductility. In addition to ductility, other very important parameters such as strength, stiffness, are obtained from the moment-curvature curve and the relationships between them. This curve examines the behavior of reinforced concrete structures under the effect of bending. In the moment-curvature curve, there are three important points related to the cracking of the reinforced concrete section, the yield of tensile rebars and finally the failure of the compressive concrete. This area of the curve indicates the ductile behavior of a flexural reinforced concrete beam, and the lower the cross-sectional tensile rebars leads to the greater the ductility in the beam behavior as well as the moment-curvature curve.

So far, various methods have been proposed to analyze the ductile behavior of reinforced concrete beams. It is important to consider the tensile effects of concrete and steel correctly. In this regard, a simple algorithm and formula to calculate the relationship between tensile reinforcement and flexibility of reinforced concrete beams Presented [1, 2].

Also, the amount of tensile reinforcement for a beam with the ratio of compressive to tensile reinforcement and the desired ductility is obtained. In an experimental study, 15 laboratory beams have been studied experimentally and theoretically [3] The experiments are divided into 5 groups, each group to study a factor in the behavior of the beam, including strength, maximum deformation and type of failure.

Another group of researchers presented experimental relationships to determine the capacity for elastic and final deformation. [4] These relationships are based on parameters such as final strength, ductility of steel and shear slenderness. In another numerical study, Mohemmi and Broujerdian [5], proposed an indirect method for considering the bond-slip interaction between rebars and concrete in analysis of reinforced concrete frames. In this study in order to accurately evaluate the nonlinear behavior of RC frames, a reduction factor has been considered and demonstrated well agreement between numerical analysis and experimental data.

In this research, after presenting Mander et al. [6] and Kent and Park [7] strain stress models, the relationships used to calculate the forces, which are generated from rebars and concrete in the tensile and compressive areas are presented and then an algorithm for calculating the moment-curvature graph is presented. Also, in the process of obtaining the moment-curvature graph, the cover of sections has been gradually removed due to cracking of unreinforced concrete. In this research 4 rectangular reinforced concrete sections are used to obtain moment-curvature graph, for this purpose MATLAB program has been used and the analysis of reinforced concrete sections has been done gradually by increasing the strain in concrete compressive fiber. Finally, for validation, the results of numerical analysis are compared with the KSU-RC program and there is a good compatibility between the results of this research and the output obtained from the program.

2 Stress-Strain Models for Confined Concrete

2.1 Mander, Priestly, Park 1988

In this section, the relations developed by for the stress strain curve of confined concrete are presented. Figure 1 shows the strain stress curve for confined and non-confined concrete [6], as can be seen the use of stirrup, hoop and spiral increase the compressive stress and ultimate compressive strain as well. Clearly it is important to have accurate information concerning the complete stress-strain curve of confined concrete in order to conduct reliable moment-curvature analysis to assess the ductility available from columns with various arrangements of transverse reinforcement.

Fig. 1.
figure 1

Strain stress curve for confined and unconfined concrete by Mander, J., Priestley, M. & Park, R., 1988 [6]

Equations 1 to 7 show the relationships proposed by Mander et al. [6]. To calculate the strain stress curve for confined concrete. Equations 8 to 11 are used to calculate the confined stress for circular and 12 to 16 for rectangular sections. As well as Fig. 2 shows the confined core for circular and also Fig. 3 shows the confined core related to rectangular sections, these figures are taken from Mander et al., study [6].

$$ f_{c} = \frac{{f^{\prime}_{cc} \,x\,r}}{{r - 1 + x^{r} }} $$

Where f'cc is compressive strength of confined concrete

$$ x = \frac{{\varepsilon_{c} }}{{\varepsilon_{cc} }} $$
$$ \varepsilon_{cc} = \varepsilon_{c0} \left[ {1 + 5\left( {\frac{{f^{\prime}_{cc} }}{{f^{\prime}_{c0} }} - 1} \right)} \right] $$
$$ r = \frac{{E_{c} }}{{E_{c} - E_{\sec } }} $$
$$ E_{c} = 5000\sqrt {f^{\prime}_{c0} } $$
$$ E_{\sec } = \frac{{f^{\prime}_{cc} \,}}{{\varepsilon_{cc} }} $$

where εc, is longitudinal compressive concrete strain that is assumed, εc0 = 0.002

$$ f^{\prime}_{cc} = f^{\prime}_{c0} \left( { - 1.254 + 2.254\sqrt {1 + \frac{{7.94f^{\prime}_{1} }}{{f^{\prime}_{c0} }}} \, - 2\frac{{f^{\prime}_{1} }}{{f^{\prime}_{c0} }}} \right) $$

f'1 = lateral confining stresses

$$ f^{\prime}_{1} = \frac{1}{2}k_{e} \rho_{s} f_{yh} $$
$$ k_{e} = \frac{{\left( {1 - \frac{{s^{\prime}}}{{2d_{s} }}} \right)^{2} }}{{1 - \rho_{cc} }}\,\,for\,\,circular\,\,hoops $$
$$ k_{e} = \frac{{\left( {1 - \frac{{s^{\prime}}}{{2d_{s} }}} \right)}}{{1 - \rho_{cc} }}\,\,\,for\,\,circular\,\,spirals $$
$$ \rho_{s} = \frac{{A_{sp} \pi d_{s} }}{{\frac{\pi }{4}d_{s}^{2} s}} = \frac{{4A_{sp} }}{{d_{s} s}} $$

ps = ratio of the volume of transverse confining steel to the volume of confined concrete core.

Fig. 2.
figure 2

Confined core for circular sections by Mander, J., Priestley, M. & Park, R., 1988 [6]

$$ f^{\prime}_{1x} = k_{e} \rho_{sx} f_{yh} \, $$
$$ \rho_{sx} = \frac{{A_{sx} }}{{sd_{c} }} $$
$$ f^{\prime}_{1y} = k_{e} \rho_{sy} f_{yh} $$
$$ \rho_{sy} = \frac{{A_{sy} }}{{sb_{c} }} $$
$$ f^{\prime}_{1} = \sqrt {(f^{\prime}_{1x} )^{2} + (f^{\prime}_{1y} )^{2} } $$

where Asx and Asy = the total area of transverse bars running in the x and.

y directions and \(k_{e} = 0.75\,\)

Fig. 3.
figure 3

Confined core for rectangular sections by Mander, J., Priestley, M. & Park, R., 1988 [6].

2.2 Kent and Park Model

In this section, the Kent and Park model [7] is presented, as seen in Fig. 4 in this model confined concrete sustain more ultimate strain than unconfined concrete based on Kent, D. C, and Park, R. (1971) model [7], but according to the relationships of this compressive stress model, there is no change in its compressive stress.

Fig. 4.
figure 4

Stress strain curved for confined and unconfined concrete based on Kent, D. C, and Park, R. (1971) model [7]

In this model the ascending branch is represented by Eq. (17).

$$ \sigma_{c} = f_{c} \left[ {\frac{{2\varepsilon_{c} }}{0.002} - \left( {\frac{{\varepsilon_{c} }}{0.002}} \right)^{2} } \right]\, $$

The post-peak branch was assumed to be a straight line whose slope was defined primarily as a function of concrete strength, Eqs. (1820)

$$ \sigma_{c} = f_{c} \left[ {1 - Z\left( {\varepsilon_{c} - 0.002} \right)} \right] $$
$$ \varepsilon_{50u} = \frac{{3 + 0.0285f_{c} }}{{14.2f_{c} - 1000}} $$
$$ \varepsilon_{50h} = \frac{3}{4}\rho_{s} \sqrt {\frac{{b^{\prime\prime}}}{s}} $$

where; σ = Concrete stress; b″ = big size of the core concrete (area inside the stirrup), s = stirrup spacing, ρs = stirrup percent density.

3 Calculation of Moment and Curvature

To calculate the moment-curvature curve, we need to calculate the forces in the section that these forces must be calculated step by step with increasing strain and after establishing the tensile and compressive balance in the section, the total moment and curvature are obtained in each step.

It should be noted that the concrete used in the cover is crushed due to not being reinforced at a strain of 0.003 and must be removed from the calculation stage. The subject is included in the analysis. Equations 21 to 29 show how to calculate the forces related to concrete and reinforcements in section and finally the moment and the corresponding curvature as well. Figure 5 shows the forces for concrete and reinforcements for a section with tensile and compressive rebars.

Fig. 5.
figure 5

Concrete and reinforcements forces for a section with compressive and tensile reinforcement

$$ \varepsilon_{c} = \frac{{\varepsilon_{ca} }}{{k_{d} }} \times y $$
$$ A_{c} = \int_{0}^{{k_{d} }} {f_{c} (y)} \,dy $$
$$ C = \frac{{\int_{0}^{{k_{d} }} {f_{c} (y) \times y} \,dy}}{{\int_{0}^{{k_{d} }} {f_{c} (y)} \,dy}} $$
$$ CC = A_{c} \times b $$
$$ TS = As \times E \times \varepsilon_{s} $$
$$ CS = As^{\prime} \times E \times \varepsilon^{\prime}_{s} $$
$$ P = CC + CS - TS $$
$$ M = CC \times \left( {d - k_{d} + C} \right) + CS \times (d - dp) $$
$$ \phi = \frac{{\varepsilon_{ca} }}{{k_{d} }} $$
$$ \begin{gathered} \varepsilon_{ca} = Concrete\,\,Strain\,\,that\,\,is\,\,assumed \hfill \\ k_{d} = Depth\,\,of\,Narural\,\,Axis\,\,must\,\,be\,\,assumed \hfill \\ b = width\,\,of\,\,\sec tion \hfill \\ A_{c} = Stress\,\,Area\,\,of\,\,Concrete \hfill \\ C = Centroid\,\,of\,\,concrete\,\,force \hfill \\ CC = Concrete\,\,force \hfill \\ CS = Compressive\,\,rebars\,force \hfill \\ TS = Tensile\,\,rebars\,force \hfill \\ P = Equilibrium\,\,of\,\,forces\,\,(must\,be\,\,zero) \hfill \\ M = Moment \hfill \\ \phi = Curvature \hfill \\ \end{gathered} $$

Figure 1 illustrates the calculation of moment-curvature algorithm. In this diagram, first the input data including the compressive strength of unconfined concrete, section depth, section width, area of reinforcement, etc. are given. Then fc and the final strain of the confined concrete are calculated according to the Kent Park and Mander models. And then the balance of forces in the cross section must be checked. If there is balance between tensile and compressive forces then the moment and curvature are calculated. This process should be continued with a gradual increase in compressive strain until the section fails. It should be noted that due to the unconfined concrete cover, this part of the section should be gradually removed from the section after reaching the fracture strain of concrete. These steps are given in full in the algorithm.

Fig. 6.
figure 6

Proposed algorithm for moment-curvature calculation

As it is noted in previous sections and shown in Fig. 6 algorithm, due to non-confinement of concrete cover it must be eliminated during the analysis. Concrete cover is eliminated when strain in concrete compressive fiber reaches approximately to 0.003. Figure 7 depicts the elimination of concrete cover that leads to depth of section decrease due to experiencing strain upper than 0.003.

Fig. 7.
figure 7

Gradual elimination of non-confined concrete cover

4 KSU-RC Software

KSU_RC is a program for analyzing the behavior of reinforced concrete columns, the first version of which has been created and updated by Esmaeily as USC_RC (2002) [8] and then by Esmaeily as KSU_RC (2007) [9]. The current version has the ability of performance analysis under changes in lateral and vertical load patterns but can only be used for rectangular, circular and hollow sections. Shir Mohammadi [10] updated the ksu version in 2015, which was part of his doctoral dissertation.

Rohleder [11] had a research also led to the current version of KSU-RC, which underwent many changes compared to the original version. The program is based on analytical models of material behavior under monotonic and cyclic loading, all sections can be triangulated by each sectional element following the behavioral model of strain stress based on its deformation. Elements can be defined as both confined and unconfined.

5 Result and Discussion

Figure 8 demonstrates rectangular reinforced concrete sections used in this research in order to calculate moment-curvature diagram. It should be noted that compressive strength of unconfined concrete and the reinforcement yielding strength are assumed; Fc = 22 N/mm2 assumed; Fy = 400 N/mm2 respectively.

Fig. 8.
figure 8

Reinforced concrete sections used in this study

In this section the result of Moment-Curvature for Mander and Kent-Park models which were created by MATLAB are compared with KSU-RC software.

As can be seen in Fig. 1, 2, 3 and 4, the moment-curvature diagram for the 4 sections are shown. First, the relationship between moment and curvature is linear, and with increasing curvature, the moment also increases. And then after cracking the concrete in the tensile zone, the moment-curvature relationship gradually becomes non-linear to reach the peak point and maximum moment. Then, by increasing the amount of curvature and strain in the concrete compressive fiber, the concrete cover that is not confined is gradually cracked and removed from the section. And we will have a gradual decrease in strength and bending moment. Then, with increasing strain in the compressive farthest wire, the curvature also increases and the tensile rebars reach the yielding point and experience a large deformation without increasing the strength (due to the elastoplastic behavior used in this research for reinforcement) and this causes a large increase in curvature and Keep the amount of moment in section constant. The increase in curvature continues until the concrete reaches its ultimate strain and the section fails.

Figure 9, 10, 11 and 12 show the moment-curvature diagrams for the Kent-Park and Mander models developed in the MATLAB program and the Mander model obtained from the KSU-RC program. This comparison shows that the algorithm proposed in this research to calculate the moment-curvature diagram in confined reinforced concrete sections has a suitable agreement with the KSU-RC reinforced concrete sections analysis program and the part related to concrete cover is removal during analysis with increasing strain well considered in concrete.

Using this algorithm can be used to analyze larger reinforced concrete structures, while similar KSU-RC programs are usually used to analyze a section.

Fig. 9.
figure 9

Comparison of Moment-Curvature diagrams for Sect. 1

Fig. 10.
figure 10

Comparison of Moment-Curvature diagrams for Sect. 2

Fig. 11.
figure 11

Comparison of Moment-Curvature diagrams for Sect. 3

Fig. 12.
figure 12

Comparison of Moment-Curvature diagrams for Sect. 4

6 Conclusion

Experimental tests are the most accurate methods for analyzing reinforced concrete structures in linear and non-linear zones, and reinforced concrete structures have a complex behavior due to nonlinear behavior and cracking of concrete in tension, and researchers still study on the methods of analysis of these structures. But doing laboratory research requires a lot of time and money, and also it is not possible to test a 3D full-scale structure in a model laboratory. For this reason, the use of numerical methods has been developed due to high accuracy and reasonable cost reduction. In this research, an algorithm is proposed to analyze and obtain the moment-curvature diagram in reinforced concrete sections. In this algorithm, Mander and Kent-Park models are used to calculate the bending moment, and step by step, by increasing the strain in the section compressive fiber, the moment and the corresponding curvature are obtained. The important point about this algorithm is that due to the lack of confinement in concrete cover, this part is gradually eliminated from the section at a strain above 0.003, which is very important in gradually reducing the RC sectional strength. Finally, the moment-curvature diagram obtained for four reinforced concrete sections numerically, is compared with the moment-curvature diagram obtained from non-linear analysis software of reinforced concrete structures KSU-RC and shows high accuracy. It is also possible to extend this algorithm to the structure and analyze larger frames and structures.