# Alternative mathematical modeling for plastic hinge of reinforced concrete beam

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## Abstract

The presentation of the plastic zone for flexural members is basic as it administrates the load carrying and deformation capacities of the member. Therefore, plastic hinge study has been of great attention to structural researchers for decades. The behavior of plastic hinge is very tricky due to the high nonlinearity of materials, and relative movement between the component materials and strain localization. As a result, most researchers investigated the problem experimentally or by using programs which depend on the materials nonlinearity such as ANSYS, ABAQUS or ADINA programs. This paper tries to investigate the problem analytically by proposing a simple approximate equation to compute the plastic hinge rotation. A lot of parameters such as the tension and compression steel ratios, concrete strength, steel yield, section geometry and the load position were considered in the suggested equation. The analytical results of simply supported beams under middle concentrated load with different parameters were presented and compared with the conjectures of the various formulations, software programs (ANSYS and NONLACS2) and experimental results. The simplified suggested equation gave satisfying results. Moreover this paper presented a structural mathematical model based on the suggested equation to predict the deformations behavior of the beams. The plastic rotation and deflections can be predicted by the mathematical model after yielding till the failure moment, not only at the failure moment as in the common formulations. The mathematical model was verified and gave values of deflections matching experimental results.

## Keywords

Plastic hinge Mathematical model Plastic rotation R.C beams## 1 Introduction

The plastic hinge in which the plastic deformation localized in a small zone is critical for flexural members as it controls the load carrying and deformation capacities of the member. The important gain of including the inelastic behavior of the members in structural design is that the moment redistribution which occurs can increase appreciably the strength of the structure above the strengths calculated by the elastic theory. The inelastic behavior of Reinforced Concrete (RC) sections leads to a redistribution of moments and forces, resulting in an increased load carrying capacity of the members and the indeterminate structure. As the applied load is increased, hinges start forming in succession at locations where the hinge moment capacity is reached; with further increase in the applied load, these hinges continue to rotate until the last hinge forms converting the structure into a mechanism resulting in failure. The plastic rotation capacity is a complex issue, mainly because of interaction of the various parameters such as materials, member geometry, loading type. Due to the large variation of these factors experimental results from previous research show significant scatter of the measured values of rotation capacity. Various formulations given by Baker and Amarakone [4], Mattock (1966), Corley [11] and Riva and Cohn [21] were performed to predict the plastic rotation capacity. A parametric study is performed by Kheyroddin and Naderpour [2] to assess the influence of the tension reinforcement index, and the bending moment distribution (loading type) on the ultimate deformation characteristics of reinforced concrete (RC) beams, the analytical results for 15 simply supported beams with different amounts of tension reinforcement ratio under three different loading conditions are presented and compared with the predictions of the various formulations and the experimental data. Ko et al. [17] descript an experimental study on the plastic rotation capacity of reinforced high strength concrete beams, they tested thirty-six beams with various compressive strengths of concrete, tensile reinforcement ratios and loading type to evaluate the plastic rotation capacity and compared the experimental data with numerical analysis. A lot of research can be carried out in the effect parameters on the plastic rotation capacity such as steel ratios and size effects. It has for instance been suggested by Hilierbogr [14, 15], and Markeset [16]. that the rotation capacity of plastic hinges is member size dependent, approximately inversely proportional to the beam height. Some experimental work in this field has been done in the past, where such a tendency has noticed Mattock [18], Corley [11], Cederwall and Sobko [10] and Bosco et al. [6]. Bigaj and Walraven [1] proceeded test series on simply supported slender beams to study the size effect on the rotation capacity of plastic hinges; they descript a model for the rotation capacity. Many experimental and theoretical studies have been conducted to examine the ductile behavior and plastic rotation of reinforced high-strength concrete beams Arslan and Cihanli [3], Bernardo and Lopes [5] and Carmo and Lopes [7, 8]. The studies generally showed the high-strength concrete members have sufficient ductility to guarantee their structural safety provided that an appropriate choice of the amount and location of the reinforcement is made, and that the general rules adopted for normal-strength concrete structures can also be applied to high-strength concrete structures.

## 2 Various previous formulations to compute plastic rotation capacity

For the loading type of concentrated load at mid span, the most widely used \(\theta_{p}\) formulations in Europe and North American are presented as follows

**Baker and Amarakone**[4]

**Mattock**[19]

**Corley**[11]

**Riva and Cohn**[21]

## 3 Analytical solution

### 3.1 The materials modeling

- 1.
Concrete constitutive model

**Smith and Young**[23]

**Desayi and Krishnan**[12]

**Carrerira and Chu**[9],

**Ezeldin and Balaguru**[13],

**Nataraja et al**. [20]

- 2.
Reinforcement steel modeling

The reinforcement steel will be modeled as an elastic strain-hardening material as shown in Fig. 2.

### 3.2 The curvature after yielding

*ϕ*is the ratio of segment height to the depth of neutral axis from compression fiber (c), in the current analysis

*ϕ*is assumed to equal to 0.1.

Hence, for any arbitrary value of concrete strain (\(\varepsilon_{c}\)) after yielding, the values of c and M can be found according to Eqs. (15)–(25).

### 3.3 Plastic rotation capacity

In which, \(l_{y}\) is the beam length over which the bending moment is larger than the yielding moment (My) or the distance between the critical section and the location where tension steel bars start yielding and \(\varphi (x)\) is the curvature at a distance x from the critical section at the ultimate load stage. The shaded area in Fig. 4c shows the plastic rotation (\(\theta_{pl}\)) that occurred in addition to the elastic rotation at the plastic hinge at the ultimate load stage.

### 3.4 Simplified equation for computing plastic rotation capacity

Five simply supported beams subjected to middle concentrated load will be analyzed by the analytical solution to compute the plastic hinge rotation, the steel reinforcement ratio was variable for these beams.

Details of the analyzed beams

Beam | \(\rho = {\raise0.7ex\hbox{${A_{s} }$} \!\mathord{\left/ {\vphantom {{A_{s} } {bd}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${bd}$}}\) | \(f_{c}^{{\prime }}\) (MPa) | \(f_{y}\) (MPa) | \(\rho^{{\prime }}\) | Type of loading |
---|---|---|---|---|---|

1 | 0.0294 | 23.4 | 328 | 0.0037 | Concentrated load |

2 | 0.022 | 23.4 | 328 | 0.0037 | Concentrated load |

3 | 0.0147 | 23.4 | 328 | 0.0037 | Concentrated load |

4 | 0.011 | 23.4 | 328 | 0.0037 | Concentrated load |

5 | 0.0074 | 23.4 | 328 | 0.0037 | Concentrated load |

### 3.5 Comparison of the plastic rotation capacity between the Suggested equation in the present study and various formulations

A paramedic study was performed by Kheyroddin and Naderpour [2] to study the influence of the tension reinforcement as one parameter on the ultimate deformations of (RC) beams. The details of these beams are as shown in Fig. 6. They analyzed these beams by finite element method by using NONLACS2 (NONLinear Analysis of Concrete and Steel Structures). Plastic rotations and curvature ductility factor of these beams were computed and compared with Baker and Amarakone [4], Mattock [18] and Corley [11] formulations.

The plastic rotation capacity will be computed for the same beams by the simplified Eq. (33), by using ANSYS program, and will be compared with the results of the various formulations. Beam modeling in ANSYS program.

### 3.6 Verified the suggested simplified equation experimentally

Characteristics of the test specimens

Code | Size | Concrete | Steel | ||||||
---|---|---|---|---|---|---|---|---|---|

d (mm) | B (mm) | L (mm) | \(f_{c}^{{\prime }}\) (MPa) | \(f_{ct}\) (MPa) | \(\rho_{s}\) (%) | \(f_{t}\) (MPa) | \(f_{y}\) (MPa) | \(\varepsilon_{t}\) (%) | |

B.01.1.4 | 90 | 50 | 1000 | 31.71 | 2.47 | 0.28 | 678 | 590 | 3.6 |

B.0.2.4 | 180 | 100 | 2000 | 34.4 | 2.37 | 0.279 | 641 | 562 | 9.17 |

B.0.2.16 | 180 | 100 | 2000 | 40.57 | 3.16 | 0.279 | 641 | 562 | 9.17 |

B.0.3.4 | 450 | 250 | 5200 | 33.52 | 2.31 | 0.279 | 641 | 568 | 9.36 |

B.0.3.16 | 450 | 250 | 5200 | 37.25 | 2.77 | 0.279 | 641 | 568 | 9.36 |

B. 1.1.4 | 90 | 50 | 1000 | 33.12 | 2.51 | 1.118 | 641 | 562 | 9.17 |

B. 1.2.4 | 180 | 100 | 2000 | 35.27 | 2.33 | 1.117 | 661 | 573 | 9.31 |

B. 1.2.16 | 180 | 100 | 2000 | 39.76 | 2.91 | 1.117 | 661 | 573 | 9.31 |

B. 1.3.4 | 450 | 250 | 5200 | 32.26 | 2.26 | 1.116 | 650 | 550 | 9.27 |

B. 1.3.16 | 450 | 250 | 5000 | 35.43 | 2.73 | 1.116 | 650 | 550 | 9.27 |

The comparison illustrated that the efficiency of the suggested equation as a simple formulation is acceptable. The suggested equation takes into account a lot of factors such as reinforcement ratios, member size and concrete and steel types. Based on this equation, the behavior of the beams by the load deflection curve can be predicted as shown later.

## 4 Mathematical model to compute the plastic deformations

The deformations of the beam after crack till yielding can be computed based on linear analysis, where the moment of inertia for the cross section in this stage can be replaced with cracking moment of inertia in virtual section, and using the same linear equations of the structural analysis. After yielding where clearly changing of modules of elasticity and the moment of inertia at the plastic zone as increasing the load, the deformations will be more difficult to determine. The deformations which occurred after yielding are computed by analytically using programs take into consideration materials nonlinearity or measuring experimentally. In this study, simple mathematical model will be suggested to compute the plastic deformations avoiding the complexity of materials nonlinearity.

Hence, it is easy to compute the plastic rotation at any stage of loading not only at the failure stage. This helps to use the linear analysis to determine the plastic deformations after the yield till failure. The RC beam can be modeled after the yield by representing the plastic zone as intermediate hinge with rotational spring. This spring has a rotational stiffness \(K_{\theta }\) which can be computed based on Eq. (40).

*δ*

_{p}is plastic deformation, Δ

*θ*

_{p}the relative plastic rotation is equal to 2

*θ*

_{p}due to the symmetry,

*M*′ is the bending moment at mid span due to the existence of unit force and equal to

*L*/4

## 5 Verified the mathematical model experimentally

Properties of tested beams

No. | Beams | Steel details | \(f_{c}^{\prime }\) | |
---|---|---|---|---|

Numbers of bars | \({\raise0.7ex\hbox{$\rho $} \!\mathord{\left/ {\vphantom {\rho {\rho_{b} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\rho_{b} }$}}\) | |||

1 | 6-30-1 | 2D13-1D10 | 0.33 | 66.6 |

2 | 6-50-1 | 2D16-1D13 | 0.54 | 66.6 |

3 | 6-65-1 | 2D19-1D10 | 0.66 | 66.6 |

4 | 6-75-1 | 2D19-1D16 | 0.79 | 66.6 |

The notations for beam designations are as follows:

6-30-1 where 6: compressive strength of concrete (60 MPa), (30): tension reinforcement ratio and (1): loading type “1-point loading”.

Properties of reinforcement steel

Bar size | Area (cm | Yield strength (MPa) | Tensile strength (MPa) | Elastic modulus (GPa) |
---|---|---|---|---|

D10 | 0.71 | 406 | 608 | 175 |

D13 | 1.27 | 413 | 579 | 200 |

D16 | 1.99 | 443 | 640 | 172 |

D19 | 2.87 | 419 | 615 | 174 |

D22 | 3.87 | 385 | 588 | 172 |

It was observed from the shown load deflection curves in Fig. 19a–d, that the mathematical model gives matching results comparing to the results of experimental tests. As a result, this mathematical model can be used in future work in computing moment redistribution of indeterminate R.C beams.

## 6 Conclusions

The following conclusions have been drawn out of the presented study:

A simplified equation was suggested in this research to compute plastic rotation capacity in R.C beams. The tension and compression reinforcement ratios, materials strengths, member geometry which are as affected parameters were included in this equation. By analytical programs and previous experimental studies, the simplified equation was verified and the results were satisfying.

To ease the way to compute the plastic rotation of R.C beams, that wasn’t the main destination of the research. A structural mathematical model based on the simplified equation was as a main aim of the research and suggested to compute the plastic deformations of R.C beams. This model can be used to compute the plastic deformations from yielding till the ultimate moment, wasn’t only at the ultimate moment as in various common formulations. In the suggested model, the yielding zone was represented as intermediate hinge with rotational spring. The rotational stiffness of the spring was computed according to the suggested simplified equation. The mathematical model was verified by analyzing tested simply supported beams with various parameters. A comparison of the load deflection curves between the results of the tested beams and the mathematical model, illustrated that the results were matching each other. The simplified mathematical analysis which was presented in this work can be used in future work in computing moment redistribution of indeterminate R.C beams.

## Notes

### Acknowledgements

We thank our colleagues from Delta University for Science and Technology who provided insight and expertise that greatly assisted the research, although they may not agree with all of the conclusions of this paper.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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